The Broken-Born program

The cosmological constant Λ

Mon, 01 Jan 0001 00:00:00 +0000

The cosmological constant Λ is a constant term in Einstein’s equations of gravity that acts as an energy density of empty space: the same at every point and at every time, with a pressure equal to minus that density, so that its equation of state w, the ratio of pressure to energy density, is exactly −1. Matter and radiation thin out as the universe expands while this density does not, so a positive Λ comes to dominate, makes the expansion accelerate, and leaves an empty universe expanding exponentially forever, called de Sitter space; with a negative Λ, the expansion of a universe containing matter halts and reverses. Λ is the dark energy of the standard cosmological model, ΛCDM, a cosmological constant plus cold dark matter, and it is the standard expectation in two rows of the program’s prediction table: a constant w = −1 in row 4, and eternal expansion toward de Sitter space in row 5. The program assigns the observed dark energy to a field, τ, in its place, and data that settle on a constant w = −1 would kill that sector. The fits the program cites report evidence that dark energy is weakening, which a constant cannot do; the second data release of the Dark Energy Spectroscopic Instrument also favours an earlier phase with w below −1, which τ cannot have (DESI DR2’s phantom crossing). The vacuum energy of quantum fields acts exactly as a Λ does, and why it is not far larger than observed is a separate, unsolved problem.

De Sitter space

Mon, 01 Jan 0001 00:00:00 +0000

De Sitter space is the spacetime of a universe that contains nothing but a positive cosmological constant Λ, an energy density of empty space that stays the same as space expands. Its expansion rate, the Hubble rate H, never changes, so the distances between distant galaxies grow exponentially, and every observer is surrounded by a horizon beyond which events can never be seen. Its curvature is the same at every point and in every direction; anti-de Sitter space is the counterpart for a negative cosmological constant. With a positive cosmological constant as its dark energy, standard cosmology expects our universe to approach de Sitter space as matter thins: “Eternal de Sitter” is the standard expectation of the program’s row 5, where the program requires instead a far-future turnaround whose premise is unresolved. De Sitter space is also the background on which a massive spin-2 field, a graviton with mass, must satisfy the Higuchi bound, the setting of partially massless gravity, and, in two dimensions, the space of values of the field K in the program’s first toy model, where its metric, positive in one direction and negative in the other, would give a ghost: a mode with negative energy.

The velocity dispersion σ

Mon, 01 Jan 0001 00:00:00 +0000

The velocity dispersion σ of a galaxy, a dark-matter halo or a galaxy cluster is the spread of the velocities of its stars or particles about their mean: the standard deviation of their velocity distribution along one direction, quoted in km/s. It measures how hot a system is in the dynamical sense. For a system held together by its own gravity and in equilibrium, the dispersion is set by the mass and the size, which is how the dark matter of a dwarf galaxy is weighed from the speeds of its stars along the line of sight. The values used in the Broken-Born program’s notes run from 3 km/s, for an ultra-faint dwarf galaxy, to 1000 km/s, for a galaxy cluster, and none of them is cited. In the program σ is the scale against which the velocity gate of conjugate matter is measured: where σ is below the gate velocity v*, as in dwarf galaxies, most pairs of dark-matter particles interact, and where it is above, as in the Milky Way and in clusters, only particles moving together do. σ also enters the Debt-3 calculation that killed velocity clumping, the occupation number that divides particle from wave dark matter, and the power law (v*/σ)⁴ of row 3, the island discriminant. The quantity is standard physics; this page explains it and lists where the program uses it.

Acceptance criterion 1: the toy Lagrangian (OP11)

Mon, 01 Jan 0001 00:00:00 +0000

Acceptance criterion 1 is the referees’ demand for one explicit action, a toy model being enough, in which Born reciprocity ρ, the exchange (x, p) → (p, −x), is a genuine symmetry that the vacuum breaks, with the neutral metric η as the order parameter. In ordinary quantum mechanics ρ is only a change of description, carried out by the Fourier transform; the program claims more, and the referees asked to see it in a model. The referees of review rounds 1 and 2 named it independently, and the program made it the first acceptance criterion for anything it proposes. Checkpoint #11 records it as met in the referees’ wording by model I, a sigma model of a phase-space metric and a real structure, with a bounded ρ-preserving potential. A simulation with all fields dynamical, a derivation of the potential and forced locality of matter remain open. One clause of an older wording, that η also select the electric sublattice, is not addressed.

Born's question

Mon, 01 Jan 0001 00:00:00 +0000

Born’s question asks why physics singles out position when quantum kinematics treats position and momentum as equals. A quantum particle can be described by a wavefunction of its position or by a wavefunction of its momentum. The Fourier transform, the operation that turns a sound’s waveform into its spectrum, converts either description into the other exactly, and exchanging the two variables, with one change of sign, leaves the rules of quantum kinematics unchanged. The laws of motion do not share this evenness: atoms are small in space rather than in speed, forces fall off with distance rather than with difference in velocity, and general relativity curves space-time while leaving the space of momenta flat. Max Born raised the question in 1938. The Broken-Born program is one proposed answer: the symmetry is exact in the laws and broken by the vacuum.

CLAUDE.md and the README

Mon, 01 Jan 0001 00:00:00 +0000

The root CLAUDE.md is the repository’s file of standing instructions for Claude Code sessions. It says what the program is, maps the documents to their folders, records what has changed since the program moved from claude.ai, defines the gate that must pass before work is called finished, and sets the method and the working arrangement with Paul. Its section “Since the handoff” is the most current statement of the program’s status, and it ranks first when the documents disagree. The README is the repository’s one-line description from its first commit, and what it says about the project is out of date.

Coset and hyperbolic sigma models

Mon, 01 Jan 0001 00:00:00 +0000

A sigma model is a field theory whose field takes values, at each point of spacetime, in a curved space called the target, such as a circle, a sphere or a hyperbolic plane; the action charges a cost whenever neighbouring points take distant values in the target. In a coset sigma model the target is a group divided by a subgroup, so the group acts on the target as a symmetry and every constant field is a lowest-energy state. Two results from this literature shape model I, the program’s coset model of phase-space structures. The target’s metric must be positive, or the kinetic term carries a ghost, a mode of negative energy. And in two dimensions a compact continuous symmetry cannot break spontaneously, while the non-compact symmetry of a hyperbolic target can.

Dirac quantization and the DSZ pairing

Mon, 01 Jan 0001 00:00:00 +0000

Dirac quantization is the rule, found by Paul Dirac in 1931, that a magnetic monopole can exist in quantum mechanics only if the product of its magnetic charge g with every electric charge e is a whole-number multiple of a fixed constant: e·g ∈ 2πℤ in the normalization the checkpoint uses. Without the rule, the line along which the monopole’s field enters its vector potential would shift the phase of a charge carried around it, and that line would be observable. Schwinger and Zwanziger extended the rule to dyons, particles that carry both kinds of charge, as the Dirac–Schwinger–Zwanziger (DSZ) pairing of two charges, an antisymmetric integer form. Because the product of the two charges is fixed, a small electric charge forces a large magnetic one, so a weakly coupled force has strongly coupled monopoles. The program builds its charge-lattice argument on the pairing, and the weak-to-strong consequence is one of the results that survived the retired soliton candidate.

Friedmann cosmology and minisuperspace

Mon, 01 Jan 0001 00:00:00 +0000

Friedmann cosmology is the description of a universe that looks the same at every place and in every direction, so that its entire geometry reduces to one function of time, the scale factor a(t), which multiplies the distance between any two distant galaxies. The Friedmann equation ties the expansion rate to the energy density, and each kind of content, whether matter, radiation or a cosmological constant, thins out at its own rate as a grows. Minisuperspace is the name for gravity reduced to this one function, with the freedom to choose the time coordinate kept as a single function called the lapse. The program writes its cosmology in this reduced form and adds a second scale factor for the conjugate universe, the momentum-space description of the same state.

General relativity and the graviton

Mon, 01 Jan 0001 00:00:00 +0000

General relativity is Einstein’s theory of gravity. A single field, the spacetime metric g_μν, fixes distances and durations; its curvature is gravity; and a freely falling body follows the straightest path the metric allows, feeling no force. Small ripples of the metric about flat spacetime are the graviton, a massless spin-2 field whose gauge symmetry is the linearized form of a change of coordinates (a diffeomorphism), and a single such field that couples consistently to itself and to matter rebuilds general relativity. The program keeps general relativity as the long-range gravity of ordinary matter; its second, massive spin-2 field reaches ordinary matter only at gravitational strength and over a short range. It uses the graviton as the template for that second field, the momentum graviton X, whose local symmetry is a second copy of the same linearized diffeomorphisms, and its conjugate special relativity reads the equivalence principle as the relativity principle of the momentum polarization.

History of the predictions table

Mon, 01 Jan 0001 00:00:00 +0000

The predictions table is the program’s list of observations that could kill it: for each observable it gives what standard physics expects, what the Broken-Born program requires, and the result that would end the claim. The table has appeared in three documents: ten rows in the lab notebook (#1 §9, v0.1), ten in the first consolidated paper (#9 §9, v1) and eleven in the current checkpoint (#11 §13). Between them, two rows were withdrawn when the Debt-3 calculation killed velocity clumping, a diagnostic row took their place, the lead row softened from an exponential cliff to a knee, a graviton row and a dark-sector row were added, and the retired soliton candidate joined the list of withdrawn predictions. This page records those changes so that a row number in an older document can be read correctly.

Lakatos's methodology of research programmes

Mon, 01 Jan 0001 00:00:00 +0000

Lakatos’s methodology of scientific research programmes is an account, from the philosophy of science, of how a body of scientific work is judged over time rather than one hypothesis at a time. A programme keeps a hard core of assumptions fixed and surrounds it with auxiliary hypotheses that are revised when predictions fail. It counts as progressive while its revisions lead to new predictions, some of which are then confirmed, and as degenerating while its revisions only absorb failures after the fact. The Broken-Born program describes its own status in these terms, “a Lakatosian research program, not a theory”, and checkpoint #11 closes with an assessment sorted into hard core, progressive signs, degeneration risks and falsification exposure.

Noncommutative field theory and the Langmann–Szabo duality

Mon, 01 Jan 0001 00:00:00 +0000

Noncommutative field theory places fields on a plane whose two coordinates do not commute, [x₁, x₂] = iθ: the plane behaves like the phase space of a quantum particle, with θ in the role of Planck’s constant. Fields on it are multiplied with the Moyal star product, which builds that relation in. On this plane Langmann and Szabo found a duality that exchanges a field’s description in position with its description in momentum. It is a Fourier transform of the field, relating the two descriptions as a waveform is related to its spectrum. Grosse and Wulkenhaar’s renormalizable model of a self-interacting scalar field is carried into itself by the duality at one value of its oscillator parameter, Ω = 1, and there its β-function vanishes to all orders in four dimensions. This literature supplied model II of the program’s toy models: an interacting field theory with an exact exchange of position and momentum, used to test whether a vacuum alone could break that exchange.

Obituary: superheavy conjugate solitons

Mon, 01 Jan 0001 00:00:00 +0000

Superheavy conjugate solitons were the third dark-matter candidate of version 2.0 of the checkpoint: magnetic monopoles of a hidden gauge force, with a benchmark mass of 3×10¹³ GeV in the final version, stable because their field configuration cannot unwind and dark because no gauge force of ordinary matter acts on them. The proposal was written up as a standalone manuscript, #14, and reviewed blind three times. The third review re-ran every number in code and found that the count of monopoles formed in the early universe had been normalized far too low. Counted correctly, the benchmark mass is overproduced by a factor of 2.1×10³ to 1.2×10⁴, and the only version that survives is a much lighter one, which is an existing published scenario that does not depend on the program. The candidate was retired on 2026-10-19 and stays dead.

Obituary: velocity clumping, the founding image

Mon, 01 Jan 0001 00:00:00 +0000

Velocity clumping was the program’s founding picture of dark matter: a smooth dark-matter halo would break up into a few streams, each made of particles moving with nearly the same velocity, so that dark matter clumps in velocity instead of in position. The lab notebook described it as the Jeans problem, the gravitational instability that makes lumps in space, with the roles of position and velocity exchanged. The Debt-3 calculation tested it and found it impossible for particle dark matter, fatal where it would be mandatory for wave dark matter, and merely optional elsewhere. The claim is dead as dynamics and stays dead.

Occupation number: particles or waves

Mon, 01 Jan 0001 00:00:00 +0000

The occupation number is the average number of particles that share one quantum state, or mode, of a gas. When it is much smaller than one, the particles behave as separate particles; when it is much larger than one, so many quanta share each mode that the gas behaves as a classical wave, in the way that the enormous number of photons per mode in a radio broadcast behaves as an electromagnetic wave. For dark matter it is set by the particle mass, the halo’s density and the spread of its velocities, and it equals the number of particles inside a cube whose side is the de Broglie wavelength. In the Broken-Born program it is the quantity that divides the dark-matter question into a particle island and a wave island.

P1: the phase-space arena

Mon, 01 Jan 0001 00:00:00 +0000

P1 is the program’s first axiom: physics is formulated on phase space, the space whose points are a position together with a momentum, rather than on spacetime alone. The axiom gives phase space two structures. The symplectic form ω is the antisymmetric pairing of position with momentum that quantum mechanics fixes through Planck’s constant; the generalized metric H is a positive measure of length that treats position and momentum directions together. Much of the program is built from these two: the quarter-turn that exchanges position and momentum is generated by the combination I = ω⁻¹H, and the space of possible metrics H becomes the program’s modular field once only ω is held fixed. The lab notebook’s first version of P1 also listed a third structure, the neutral metric η; the current version lists only ω and H, and P2′ makes η a property of the vacuum.

Quintessence and the equation of state

Mon, 01 Jan 0001 00:00:00 +0000

Quintessence is dark energy made of a scalar field, a quantity with a single value at each point of space, whose energy density changes slowly as the universe expands instead of staying fixed as a cosmological constant’s does. Its behaviour is summarized by the equation of state w, the ratio of pressure to energy density: a cosmological constant has w = −1 exactly, and a field whose energy is almost all potential energy comes close to that value. In a thawing model the expansion holds the field nearly still at early times and it starts to change late, so w begins near −1 and rises. The program’s dark-energy field τ is a model of this kind, with the periodic potential of an axion-like field, sitting near the potential’s maximum.

Relic abundance mechanisms

Mon, 01 Jan 0001 00:00:00 +0000

A relic abundance is the amount of a particle species left over from the hot early universe, and a relic-abundance mechanism is the process that sets that amount. In thermal freeze-out, particles annihilate in equilibrium with the hot plasma until the expansion of the universe outruns their annihilation, after which their number per unit volume only dilutes; the weaker the annihilation, the more survive. Other mechanisms set the amount differently: by an excess of particles over antiparticles, as for the baryons of ordinary matter; by slow production that never reaches equilibrium (freeze-in); or, in a sector whose particles can only annihilate among themselves, by number-changing self-interactions that turn rest mass into heat (cannibal dark matter). The program uses this standard physics to argue that conjugate matter χ cannot get its abundance from freeze-out, and the round-3 review used it to compute the relics of the retired soliton candidate’s hidden sector.

Round-3 pre-registration

Mon, 01 Jan 0001 00:00:00 +0000

The round-3 pre-registration is the forecast of the third blind review of the soliton dark-matter manuscript #14, written into review/round3-preregistration.md and committed on 2026-10-19 before the referee was dispatched, so that the forecast could not be fitted to the outcome. It predicted that the referee would grade version 3 B (“flawed but salvageable”) with medium confidence, that the weakest link it named would be “the framework does no work”, that no numerical indictment would survive, and that all 23 citations would resolve. The referee graded the manuscript C. The adjudication scored the verdict a miss, the first after two hits; the weakest link a miss on the referee’s first choice and a hit on its second; the arithmetic a miss; and the citations a hit. The commit that recorded the forecast also stated it in its subject line, which the referee’s environment displayed, so the round was less blind than intended, and forecasts have been kept out of commit subjects since.

Self-interacting dark matter

Mon, 01 Jan 0001 00:00:00 +0000

Self-interacting dark matter (SIDM) is dark matter whose particles scatter off one another through a force of their own, in addition to gravity. The scattering carries heat from the outer parts of a halo into its colder center, which turns the steep central density peak predicted for collisionless cold dark matter (CDM) into a flat core. How strongly the particles scatter, and how that strength changes with their relative speed, is the observable that the Broken-Born program’s velocity gate modifies. This page is the established background; the program’s own law is the gated scattering law σ_T(v).

Shift symmetry: X is never auxiliary

Mon, 01 Jan 0001 00:00:00 +0000

The shift symmetry is the statement that adding a constant symmetric matrix B to the momentum graviton X leaves every geometric action of the program unchanged. X is the real part of the modular field Z = X + iY, and the shift Z → Z + B is one of the maps of the group Sp(2d, ℝ) that preserves the rigid symplectic form ω. Any geometric action built from ω and the phase-space metric H therefore takes the same value before and after the shift. Such an action cannot give X a potential energy, a mass, or any coupling without derivatives: X is axion-like, a field that enters only through how it varies from place to place. That rules out one of the three outcomes the program had defined for X, an auxiliary field with no motion of its own: X either propagates or does not exist. The calc note that found the result rates it rigorous; it is the first step of the argument that X must be a massive spin-2 field, and no script checks it.

Special relativity: the template

Mon, 01 Jan 0001 00:00:00 +0000

Special relativity is Einstein’s 1905 theory of how measurements of length and time compare between observers moving at constant velocity relative to one another, built on one speed that all of them measure alike, the speed of light c. Its machinery (boosts, the light cone, the relativity of simultaneity, time dilation, length contraction and the mass shell) is textbook physics. The program uses it as a template: the paper #13 copies each piece into the momentum polarization, the program’s mirror description in which momentum plays the role of position, with force in place of velocity and a maximum force c̃ in place of c. This page states the template and lists the copies; #13 tags most of its copies as derived, but no review round or script has checked them, and the checkpoint #11 does not mention them.

Spontaneous symmetry breaking

Mon, 01 Jan 0001 00:00:00 +0000

Spontaneous symmetry breaking is the situation in which the equations of a system are unchanged by some transformation but its lowest-energy state is not. A cooled magnet is the standard example: the laws governing its atoms treat every direction alike, yet the magnet ends up magnetized along one direction, and any other direction would have been an equally good ground state. The symmetry is not lost. It now relates the possible ground states to one another, and the system occupies one of them. The Broken-Born program uses this pattern for its hard core: it proposes that the exchange of position and momentum, an exact symmetry of quantum kinematics, is broken by the vacuum rather than by the laws. This page is the textbook background and the program’s operational version of it; the program’s own claims follow on the next pages of this section.

The blind-review protocol and verdict scale

Mon, 01 Jan 0001 00:00:00 +0000

The blind-review protocol is the fixed procedure the program uses to test a manuscript: a fresh AI subagent receives a copy of the manuscript and the protocol and nothing else, checks every citation against its source, re-runs every number in its own code, re-derives the mathematics independently, audits the argument for consistency and omitted constraints, and returns a grade of A, B or C with the single weakest link named. The author then rules on each finding, re-running disputed numbers in separate scripts, and scores the forecast that was committed before the referee was dispatched. The grade scale comes from the referee prompt of the skeptic review console, the tool of rounds 1 and 2; rounds 3 and 4 ran the full five-step form with a referee that could execute code. The protocol’s own stated limit is that the referee is the same model family as the author.

The canonical commutator and the uncertainty principle

Mon, 01 Jan 0001 00:00:00 +0000

The canonical commutator [x, p] = iℏ is the rule of quantum mechanics that position and momentum do not commute: measuring one and then the other is not the same as the reverse. Its best-known consequence is the uncertainty principle: no state has sharp values of both, and the product of the two spreads cannot fall below a bound set by Planck’s constant ℏ. The rule is unchanged when position and momentum are exchanged by the quarter-turn Born reciprocity, so on its own it treats the two variables alike. The program leans on it twice. It argues that because nothing can be sharp in both variables, any world with local physics has to favor one of them; and it reads the fixed value of ℏ as the one rigid structure of phase space.

The gated condensation rate Γ_self

Mon, 01 Jan 0001 00:00:00 +0000

Γ_self is the rate at which the gated attraction of conjugate matter makes a wave-like dark-matter halo condense: the rate at which the halo’s waves, scattering off one another, move their power into a narrow range of low relative momenta. It applies on the wave island, the range of very light χ masses in which each quantum state of the halo holds so many quanta that the halo is a classical wave field rather than a gas of particles. There every scattering rate is multiplied by that number of quanta. Levkov, Panin and Tkachev (LPT) worked out this kind of condensation for gravity; the gated attraction adds a second channel to it. When the halo’s velocity spread is wider than the gate, the rate falls as the fourth power of their ratio, a power law rather than the exponential cutoff that particles show, and that difference became a test of whether dark matter is made of particles or waves. Debt 3 compared the rate with the gravitational channel and with the stirring of the halo by its own orbits. Its form is derived; its numerical coefficient is a heuristic estimate that has not been calibrated.

The graveyard: dead claims

Mon, 01 Jan 0001 00:00:00 +0000

The graveyard is the register of the Broken-Born program’s dead claims: ideas the program put forward that its calculations, a blind review or an observation then killed. Each entry says what was claimed and what killed it, and links the claim’s obituary page and the calculation, review ruling or decision that settled it; dead items told inside other pages, and the predictions withdrawn from the program’s table, are listed with them. A claim recorded here is not revived.

The iron rule: no arithmetic by eyeball

Mon, 01 Jan 0001 00:00:00 +0000

The iron rule is the program’s requirement that no number be accepted on inspection: every number a document asserts, and every number a reviewer disputes, is computed by code that is actually run and prints its result, and the code is kept in the repository’s verify/ folder. It was adopted after the first two blind reviews, in which AI referees working without code execution made three confident accusations of order-of-magnitude errors, all of which came from their own unit mistakes. The rule makes the arithmetic checkable by anyone who runs the scripts, and the gate runs them before any work counts as finished. It does not make the physics right, which round 3 showed.

The misalignment potential

Mon, 01 Jan 0001 00:00:00 +0000

The misalignment potential is the energy cost that the program assigns to a mismatch between the polarization J, the split of phase space into position and momentum directions that the vacuum has chosen, and the local phase-space geometry, which the modular field Z carries. The momentum graviton X, the real part of Z, gets no potential from the geometry alone, because constant shifts of X are a symmetry (the shift symmetry), so its mass has to come from this term. The calc notes call it the η-selection, or Higgs, sector. The natural measure of the mismatch is the anticommutator {J, I[Z]} of the polarization with the complex structure built from Z, which is zero when the two are compatible; its block form is derived exactly and checked by machine. Calc note #7 reads off the blocks that the mismatch vanishes only at X = 0 with the metric pinned to a reference metric (see flag 3 in review/wiki-build-flags.md), so that the potential involves two tensors, as the potentials of massive gravity do. Whether such a potential avoids the Boulware–Deser ghost is the question of the tuning test.

The palindrome βₙ = β₄₋ₙ

Mon, 01 Jan 0001 00:00:00 +0000

The palindrome is the constraint βₙ = β₄₋ₙ on the five coefficients β₀ to β₄ of the potential that gives the program’s second spin-2 field its mass: β₀ = β₄ and β₁ = β₃, with β₂ free, so the list of coefficients reads the same forwards and backwards. The potential couples two frames, one built from the dynamical geometry and one determined by the polarization that the vacuum selects, and exchanging the two frames reverses the order of its five terms. The program’s claim is that Born reciprocity, on the broken vacuum, acts as exactly this exchange, so a potential invariant under reciprocity must be a palindrome. The constraint places the program’s gravity sector in the interchange-symmetric branch of bimetric gravity and reads that branch’s two-element exchange symmetry as Born reciprocity. It is a derivation of the program [D] with no machine check, and the branch itself is known in the literature, where it contains the partially massless point.

The radiation era: thermal history and N_eff

Mon, 01 Jan 0001 00:00:00 +0000

The radiation era is the early period in which the universe’s energy was dominated by radiation and by particles moving at nearly the speed of light, so that the temperature alone set how fast the universe expanded. Three standard quantities describe it. The first is the expansion rate as a function of temperature, H = 1.66√g* T²/M_Pl. The second is the entropy density: it falls as the volume grows, so the ratio of a relic’s number density to it, the yield, stays fixed once the relic is no longer made or destroyed. The third is N_eff, the effective number of light neutrino-like species, which nucleosynthesis and the cosmic microwave background both measure. The program uses these quantities twice: in its optional prediction of fewer effective species from the conjugate universe, and in the review that retired its soliton dark-matter candidate, whose central error was a dropped factor of the radiation-era expansion rate.

The Siegel domain Sp(2d, ℝ)/U(d)

Mon, 01 Jan 0001 00:00:00 +0000

The Siegel domain is the space of complex symmetric d × d matrices Z = X + iY whose imaginary part Y is positive definite. For a single coordinate (d = 1) it is the upper half of the complex plane; for larger d it is the matrix version of that half-plane. Each point Z names exactly one positive phase-space metric H_Z that is compatible with a fixed symplectic form ω, and every such metric has a point, so the domain is the space of all of them. The symplectic group moves its points by fractional-linear (Möbius) maps, the matrix versions of z → (αz + β)/(γz + δ). The program uses the domain as the arena of the modular field once only ω is held rigid. The space and its parametrization are standard mathematics, and battery test T2 checks the parametrization’s identities at one sampled point.

The structures ω, η, H and the split-quaternion algebra

Mon, 01 Jan 0001 00:00:00 +0000

Born geometry equips phase space, the space of positions and momenta taken together, with three structures: the symplectic form ω, the antisymmetric pairing behind the quantum commutator; a neutral metric η, a symmetric pairing with as many positive as negative directions, whose null directions mark out which coordinates count as position and which as momentum; and a positive generalized metric H, a ruler that measures lengths in phase space. Combined in pairs, the three give operators I, J and K that multiply like the split quaternions: I squares to −1, J and K square to +1, and IJ = K. Freidel, Leigh and Minic named phase spaces carrying such a compatible triple Born geometries. The quarter-turn ρ, which the Fourier transform implements, preserves ω and H in their standard forms and reverses the sign of η. Under the program’s corrected postulate η is the order parameter of the breaking, so ρ changes the order parameter’s sign.

The velocity gate

Mon, 01 Jan 0001 00:00:00 +0000

The velocity gate is a factor in the self-interaction of conjugate matter χ that turns the interaction off between two particles that approach each other with a large relative velocity. For an ordinary force carried through position space, the scattering amplitude depends on how much momentum the two particles exchange; the gate makes it depend also on the momentum with which they approach, a variable that, according to the checkpoint, no carrier local in position produces. In signal terms, an ordinary contact force acts where two waveforms overlap, and the gate acts only where their spectra overlap as well. The program first postulated the gate as a Gaussian, then derived its mechanism from exchange of the momentum graviton X between conjugate particles. The mechanism stands as a structural derivation; the exact shape of the cutoff is open. The gate is what lets conjugate matter scatter off itself in dwarf galaxies, where dark matter moves slowly, and not in galaxy clusters, where it moves fast.

Timeline

Mon, 01 Jan 0001 00:00:00 +0000

The timeline lists the program’s events in date order, from Max Born’s paper of 1938 to the third data release of the Dark Energy Spectroscopic Instrument (DESI), which the handoff names the program’s external clock, since its data can kill the dark-energy sector or the whole framework. The work done on claude.ai in September and October 2026 is almost entirely undated, so those events follow the creation order of the artifacts in HANDOFF.md §3. From the move to this repository on 2026-10-19, the commit history dates each step to the minute: review round 3 and the retirement of the soliton candidate, the toy models and review round 4, the checkpoint’s versions 2.1 and 2.2, and the building of this wiki, with the file of suspected source errors its verifiers kept and the audit that re-checked every item; then, on the next day, a rule on how the wiki states negative results, new pages of established physics, two further flags, a worksheet for ruling on the flags, a second round of new pages, and a static site built from the wiki.

A hidden sector at its own temperature

Mon, 01 Jan 0001 00:00:00 +0000

A hidden sector at its own temperature is a collection of particles and forces that takes no part in the strong, weak or electromagnetic interactions of ordinary matter and, because nothing lets it exchange energy with our plasma, keeps a temperature T_h of its own, which can differ from ours. The ratio of the two temperatures, written ξ, is set when the universe is reheated after inflation, or when the two sectors last exchanged energy; afterwards it changes only when the heavy particles of one sector annihilate and their entropy heats the light particles of that sector alone. The ratio fixes how much extra radiation the hidden sector adds in the radiation era, an excess that grows as ξ⁴ and that nucleosynthesis and the cosmic microwave background measure. It also fixes when, by our clock, the hidden particles freeze out, and it lasts only while no coupling between the sectors, such as a Higgs portal or kinetic mixing, brings the two plasmas into contact. The program met this physics in the review of its soliton dark-matter candidate, whose hidden sector contained a massless hidden photon: the withdrawn manuscript #14 required ξ < 0.77, and round 3 corrected the bound to ξ < 0.760. The candidate is dead, and these numbers are kept as its record.

Planck units and the two Planck masses

Mon, 01 Jan 0001 00:00:00 +0000

Planck units are the units of length, time, mass and energy built from three constants of nature: Planck’s constant ℏ, the speed of light c and Newton’s constant G. The Planck length √(ℏG/c³) and the Planck energy √(ℏc⁵/G) mark the scale at which the quantum behaviour of gravity is expected to matter, and their ratio, c⁴/G, is a force that contains no ℏ. The Planck mass √(ℏc/G) also measures the strength of gravity: a spin-2 field such as the graviton couples to matter with a strength inversely proportional to it, so the Planck mass appears wherever the coupling of gravity or the expansion rate of the universe enters a formula. It is written in two conventions, M_Pl = G^{−1/2} in units with ℏ = c = 1 and the reduced M̄_Pl = M_Pl/√(8π), which describe the same physics provided that each formula keeps one of them throughout. The program’s conjugate special relativity sets its Born length equal to the Planck length, which makes its maximum force c⁴/G. The review that retired the program’s soliton dark-matter candidate found that the candidate’s reported factor of about 3 between the two conventions came from mixing them.

Thawing and freezing quintessence

Mon, 01 Jan 0001 00:00:00 +0000

Thawing and freezing are the two classes into which quintessence, dark energy made of a slowly changing scalar field, divides according to how its equation of state w, the ratio of its pressure to its energy density, changes with time. A thawing field is held almost still by the expansion of the universe at early times, so w stays close to −1, the value of a cosmological constant; the field starts to change only late, and w rises above −1 toward the present. A freezing field is already changing at early times and slows down, so w falls toward −1. A field that starts near the maximum of its potential, a hilltop model, thaws. The program’s dark-energy field τ is taken to be such a field, and row 4 of its predictions requires a thawing drift that never takes w below −1. The second data release of the Dark Energy Spectroscopic Instrument (DESI) favours a w that was below −1 earlier in cosmic history, a phase that a field of this kind, with the standard kinetic energy, cannot produce, so the requirement is under real pressure.

Big Bang nucleosynthesis

Mon, 01 Jan 0001 00:00:00 +0000

Big Bang nucleosynthesis (BBN) is the formation of the lightest atomic nuclei, mostly helium-4 with small amounts of deuterium, helium-3 and lithium-7, in the first minutes of the hot early universe. Its outcome depends on the expansion rate at the time. Neutrons and protons stop converting into each other once the weak interactions that convert them become slower than the expansion, and nearly every neutron left when nuclei can survive ends up in helium, so the primordial helium abundance measures how fast the universe was expanding in its first minutes. That makes nucleosynthesis a test of two things at that time, the radiation content, counted as the effective number of light species N_eff, and the strength of gravity; it also limits particles that decay or release energy during or after it. The program uses this timing in two places. The lab notebook dates an optional step in the effective strength of gravity, G_eff, “before BBN” (row 10), and the round-3 review found that the light hidden fermion ψ of the retired soliton candidate would, at MeV masses, still have been freezing out during nucleosynthesis, so the lower edge of its relic window was wrong.

Axion-like fields and pseudo-Goldstone bosons

Mon, 01 Jan 0001 00:00:00 +0000

An axion-like field is a scalar field whose value is an angle. Its interactions are unchanged when the field is shifted by a constant, φ → φ + c, except for a small explicit breaking of that symmetry, which leaves a potential that repeats as the angle goes around its circle, in its standard form V₀[1 + cos(φ/f)]. Such a field is a pseudo-Goldstone boson: the massless mode that a spontaneously broken continuous symmetry leaves behind, given a small mass by a slight explicit breaking. The shift symmetry keeps the field light and lets it couple to other matter only through how it changes in space and time. The axion of the strong interaction is the original example; it and lighter fields of the same kind are studied as dark matter, and the lightest as dark energy. The Broken-Born program’s dark-energy field τ is one: an angle with a cosine potential, starting near the potential’s maximum. The program also calls its momentum graviton X “axion-like” for its exact shift symmetry, and the worklog of its toy models describes the vacuum’s choice of position and momentum as “a pseudo-Goldstone polarization angle”.

Redshift

Mon, 01 Jan 0001 00:00:00 +0000

Redshift is the stretching of the wavelength of light between its emission and its observation. In cosmology it is written z and measures how much the universe expanded while the light was travelling: 1 + z = a₀/a_e, the scale factor when the light arrives divided by the scale factor when it left, the scale factor being the number that multiplies the distances between distant galaxies. Every wavelength in a spectrum stretches by the same factor, the momentum of every freely moving particle falls in inverse proportion to the scale factor, and the frequency of gravitational waves and the temperature of freely travelling radiation fall with it; in wave terms, stretching a waveform compresses its spectrum by the same factor. Because the scale factor has grown throughout the history of the universe, a measured redshift dates the emission, and surveys report the expansion history as a function of z, including the equation of state w of dark energy, its pressure divided by its energy density. The program meets redshift there. The second data release of the Dark Energy Spectroscopic Instrument (DESI) favours a w that lies below −1 at redshifts above about 0.4 and above −1 since; the program requires that w never fall below −1, so its dark-energy sector is under real pressure. The popular article #12 reads the redshift as the momentum spectrum of the universe contracting, which the wiki takes as a plain-language gloss on the program’s constraint a·ã = 1; the technical documents do not discuss the connection.

The CPL parametrization w(a) = w₀ + wₐ(1 − a)

Mon, 01 Jan 0001 00:00:00 +0000

The Chevallier–Polarski–Linder (CPL) parametrization is the standard way to describe, with two numbers, how the equation of state w of dark energy, its pressure divided by its energy density, changes as the universe expands: w(a) = w₀ + wₐ(1 − a), where a is the scale factor, the size of the universe relative to its size today. w₀ is the value of w today and wₐ measures how it has drifted; a cosmological constant is the point w₀ = −1, wₐ = 0. Cosmological surveys commonly report their dark-energy results in this form, and the program states its dark-energy requirement in it: a thawing field, with w₀ > −1 and wₐ < 0, that is never phantom, meaning that w never falls below −1. The form is a straight line in the scale factor, an approximation to the true history rather than a physical model. The second data release of the Dark Energy Spectroscopic Instrument (DESI) favours a w that crosses −1, lying below it earlier in cosmic history, at redshifts above about 0.4, and above it since, and calls that preference “rather independent of parametrization choices”. The crossing stands against the program’s requirement and puts its dark-energy sector under real pressure. In these terms the favoured shape has w₀ > −1 with w₀ + wₐ < −1, although a CPL fit in that region does not by itself show a crossing, since a line fitted to a field that never falls below −1 can lie there too (flag 43 in review/wiki-build-flags.md).

The equivalence principle: free fall and local inertial frames

Mon, 01 Jan 0001 00:00:00 +0000

The equivalence principle is the statement that gravity acts on every body in the same way. In its weak form, the universality of free fall, a body released at a given place with a given velocity follows the same path whatever it is made of. In Einstein’s form, a small enough freely falling laboratory is an inertial frame, one in which a body on which no force acts moves in a straight line at constant speed, and the laws of special relativity hold in it. Equivalently, a laboratory accelerating uniformly through empty space cannot be told apart from one held at rest in a uniform gravitational field. General relativity builds the principle in by coupling all matter and energy to one spacetime metric. The program postulates it for both of its kinds of matter: postulate P4 couples ordinary matter and conjugate matter χ to the same geometry, so χ falls, orbits and bends light like any other dark matter. Separately, the paper on conjugate special relativity (#13) reads the principle as the relativity principle of the momentum polarization, the program’s mirror description in which momentum plays the part of position. That reading is a claim of the program, alive but unreviewed, with its own page: the equivalence principle as dual relativity.

The modular group SL(2, ℤ)

Mon, 01 Jan 0001 00:00:00 +0000

The modular group SL(2, ℤ) is the group of 2 × 2 matrices with whole-number entries and determinant 1. Each such matrix [[a, b], [c, d]] moves the points z of the upper half of the complex plane by the fractional-linear map z → (az + b)/(cz + d), and two of the matrices generate all the others: S, which sends z to −1/z, and T, which sends z to z + 1. Because the matrices and their inverses have whole-number entries, they carry a two-dimensional lattice, a regular grid of points, onto itself, and in gauge theory the group acts on the lattice of electric and magnetic charges and on the complexified coupling τ_c, which combines the coupling g with the vacuum angle θ. T, the shift of θ by 2π, relabels the charges and leaves the set of particles unchanged. S exchanges electric and magnetic charges and, at θ = 0, turns g into 4π/g; as a symmetry of a quantum theory it is S-duality, which the checkpoint records as exact in maximally supersymmetric Yang–Mills theory and which is not expected to be exact without supersymmetry. The program meets the group through the matrix of S, which the checkpoint notes is also the matrix of Born reciprocity and of the program’s modular inversion Z → −Z⁻¹; on phase space, though, the symmetry group that contains that matrix is continuous, not this discrete one. Whether the two settings are parts of one enlarged “modular” structure is an open conjecture, and an earlier use, in which reciprocity, acting as the modular S transformation, was conjectured to fix a hidden sector’s coupling, was superseded.

The transfer cross-section σ_T

Mon, 01 Jan 0001 00:00:00 +0000

The transfer cross-section σ_T measures how strongly particles scatter off one another by counting each collision according to how much forward momentum it removes: σ_T = ∫(1 − cos θ) dσ, where θ is the angle through which the collision turns the particles’ relative motion. A grazing collision, which barely bends their paths, counts for almost nothing, and a head-on rebound counts most. For dark matter that scatters off itself, σ_T, rather than the plain count of collisions, is the standard measure of how fast scattering moves momentum and heat through a halo. Because the motions of stars and gas reveal the mass density of dark matter but not the mass of its particles, observations constrain σ_T per unit mass, σ_T/m, quoted in cm²/g. How σ_T changes with the speed of a collision depends on the force: it stays constant for a short-range contact force, and it falls steeply with speed for a long-range force, whose scattering is concentrated at small angles. This page covers the established quantity. The Broken-Born program writes its law for dark matter’s self-scattering in it: the gated scattering law σ_T(v) bends down at a knee near a gate velocity v*, with isotropic scattering below it, and the first three rows of the predictions table concern that scattering.

Thermal freeze-out

Mon, 01 Jan 0001 00:00:00 +0000

Thermal freeze-out is the standard account of how many particles of a stable species are left over from the hot early universe when the species was once in equilibrium with the plasma. While the plasma is hot, the particles are made and annihilated in pairs at equal rates. Once the temperature falls below their mass, their equilibrium number falls exponentially; when annihilation becomes too slow to keep pace with the expansion of the universe, particles stop meeting partners, their number per unit entropy stops changing, and the species has frozen out. Typically this happens when the temperature has fallen to about a twentieth of the mass. The weaker the annihilation, the more particles survive: the relic density is roughly inversely proportional to the annihilation cross-section. The program meets freeze-out in three places. The lab notebook argues that particle χ, connected to ordinary matter only by gravity, cannot get its abundance this way, which leaves the ratio of dark to ordinary matter as an open problem, OP7. The standard expectation against which row 7 predicts a permanent null in direct detection is a freeze-out relic coupled to ordinary matter. And the relics of the hidden sector of the retired soliton candidate are freeze-out relics, computed with the standard formula and recomputed in round 3 of blind review: in the sector’s minimal version the heavy W′± would outweigh the observed dark matter 10¹⁶–10¹⁷ times, a result that survives the candidate, and the mass window of its light fermion ψ was wrong at both edges.

Topological defects and the vacuum manifold

Mon, 01 Jan 0001 00:00:00 +0000

A topological defect is a wall, line or point where a field that has broken a symmetry is held out of its lowest-energy states, because the vacuum values it takes around that place cannot be joined smoothly across it: a domain wall between two regions that settled into different vacua, a string around which the field’s vacuum value runs once round a loop, or a point-like monopole around which it covers a sphere. The set of lowest-energy states is the vacuum manifold. When a symmetry group G is spontaneously broken, the vacuum manifold is G/G₀, the group taken modulo the subgroup G₀ that leaves one vacuum in place. Its dimension counts the massless Goldstone modes of a relativistic theory, and its shape, measured by its homotopy groups, decides which defects a cooling universe can leave and why, once formed, they cannot decay on their own. In the Broken-Born program the vacuum manifold of the polarization, the Lagrangian Grassmannian Λ(d) of all possible choices of which phase-space directions count as position, contains a loop that cannot be shrunk, and the program takes the periodicity of its dark-energy field τ from that loop. The retired dark-matter candidate was a defect of the monopole kind in a hidden gauge sector; it died on the count of how many of them a transition makes, not on their stability. One open question, without a candidate, asks whether topology could keep a single object localized in momentum.

The cosmic microwave background

Mon, 01 Jan 0001 00:00:00 +0000

The cosmic microwave background (CMB) is the thermal radiation left over from the hot early universe, released when the universe became transparent. Until then electrons and nuclei formed a plasma in which light scattered constantly; once the plasma had cooled enough for them to combine into neutral atoms, an event called recombination, the light travelled freely, and the expansion has since stretched it to microwave wavelengths. Small differences in its temperature across the sky preserve the sound waves that ran through the plasma until that moment. Their pattern measures the densities of ordinary and dark matter, the length of the ruler that galaxy surveys use, and the amount of radiation in the early universe, counted as the effective number of light species N_eff. The program has used the CMB in the round-3 review of its retired soliton dark-matter candidate: the Planck satellite’s 2018 measurement of N_eff bounded the candidate’s extra radiation, and Planck’s dark-matter density is the target of the review’s abundance calculations. Row 10, the program’s optional early-universe prediction, names a proposed CMB experiment, CMB-S4, as its probe.

The relic yield Y = n/s and entropy dilution

Mon, 01 Jan 0001 00:00:00 +0000

The yield of a relic is its number of particles per unit of entropy, Y = n/s: the number density n divided by the entropy density s of the hot plasma that filled the early universe. While nothing creates or destroys the particles, both quantities fall in inverse proportion to the volume as the universe expands, so the yield stays fixed, and multiplying it by today’s entropy density gives the relic’s density now. Entropy dilution is the standard way a yield can still fall after it has been fixed: a large release of entropy into the plasma, such as the decay of heavy particles that dominated the universe for a time (a period called early matter domination), divides every yield fixed before the release by the factor by which the entropy grew. Both entered the review that retired the program’s soliton dark-matter candidate. Its count of monopoles per unit entropy had dropped a factor of about 4.6×10³ that comes from the expansion rate (Obituary: the Kibble normalization). Counted correctly, monopoles at the candidate’s benchmark mass could make up the dark matter only with a dilution factor of at least 10³–10⁴ left free, and the program rejected that route because a free factor removes the mass prediction.

Yukawa forces and their range

Mon, 01 Jan 0001 00:00:00 +0000

A Yukawa force is the force carried by a field whose quanta have a mass M. Between two sources its potential falls as e^(−Mr)/r: at short distances it is the inverse-square force of a massless field, but beyond a distance of about ℏ/(Mc), the mediator’s reduced Compton wavelength, it dies away exponentially, so the mass of the mediator sets the range of the force. In momentum the same law is the factor 1/(q² + M²), and that is the form in which such a force enters scattering. The program uses it in several places. Its momentum graviton X is massive, so the force X exerts on ordinary matter is “Yukawa-cut at 1/M_X”, which the program gives as the reason experiments see no sign of it (safety by range). The standard self-interacting dark-matter models that the program’s scattering predictions are measured against scatter through a light Yukawa mediator, and the program’s own amplitude for conjugate matter keeps the Yukawa factor in the momentum transfer and multiplies it by a velocity gate. Conjugate special relativity reads X’s short range as a range in momentum, which the checkpoint it cites does not state. The physics is textbook. The same name also covers a different object, the Yukawa coupling that gives particles their masses, which the program’s toy models use as an analogy.

Ghosts and perturbative instabilities

Mon, 01 Jan 0001 00:00:00 +0000

A ghost is a mode of a field whose kinetic term, the part of its energy that comes from the field’s rate of change, has the wrong sign, so that its excitations carry negative energy or, in the quantum theory, negative norm. On its own a ghost oscillates like any other field. Coupled to ordinary fields it leaves the energy with no lower bound, and the vacuum can decay into ghosts and ordinary particles without violating energy conservation, so a theory in which a ghost interacts with ordinary matter is not considered physical. The other instabilities of small perturbations about a solution are different: a tachyon, a mode of negative mass squared, grows at long wavelengths, and a gradient instability, a negative squared propagation speed, grows fastest at short wavelengths. Under strong coupling, which is not an instability, a mode’s kinetic term is so small that linear perturbation theory cannot decide its fate. Whether a mode is a ghost is read from the signs of the kinetic terms that remain once constraints have removed the components that do not propagate. Ghost freedom decides what the program’s momentum graviton X can be: it excludes an X without its gauge symmetry, and the tuning test asks whether the potential that gives X its mass carries the Boulware–Deser ghost of generic massive gravity. It also fixes the kinetic term of the real structure in model I, the program’s coset toy model, and a ghost field is one route to the phantom side of the dark-energy divide, which the program’s dark-energy field, an ordinary scalar, cannot reach.

Claims that stand

Mon, 01 Jan 0001 00:00:00 +0000

This register lists the Broken-Born program’s claims that currently stand: postulates, derived results and predictions that no calculation, review or observation has killed. Each line gives the claim, the tags its sources attach and what verifies it, grouped by the sections of the wiki. Standing is not the same as confirmed: most of these claims are untested by experiment, a machine check confirms the algebra but not whether the algebra describes nature, and the dark-energy sector is under pressure from current data.

Double field theory and the generalized metric

Mon, 01 Jan 0001 00:00:00 +0000

Double field theory is a formulation of the low-energy fields of closed string theory (the metric, the antisymmetric Kalb–Ramond B-field and the dilaton) on a doubled space, in which each ordinary coordinate has a partner “dual” coordinate. Its central object, the generalized metric, packages the metric and the B-field into one matrix that preserves a neutral metric η, and its symmetry group O(d, d) is the group of transformations that preserve η. For the theory to be consistent, fields may depend on only half of the doubled coordinates: the section, or strong, condition. The known derivations of that condition leave open which half, and no mechanism that selects one is known. For the program, double field theory is the contrast case. Doubling geometry with η gives the metric an antisymmetric partner, the B-field; doubling it with the symplectic form ω gives it a symmetric partner, a second graviton; and holding ω rigid as well as η removes the B-field altogether.

Dual diffeomorphisms: a second graviton

Mon, 01 Jan 0001 00:00:00 +0000

Dual diffeomorphisms are the local form of the momentum graviton’s shift symmetry: instead of adding the same constant matrix to X everywhere, X may be shifted by the symmetrized derivative of a vector field λ_μ(x) that varies from point to point, δX_μν = ∂₍μλ_ν₎. In linearized general relativity, the small ripple of the metric has exactly this kind of redundancy under small changes of coordinates, and that redundancy is what makes it a massless spin-2 field, the graviton. X is a symmetric tensor with the same redundancy, so it is a second graviton-like gauge field. A theorem excludes a second massless graviton that interacts with the first, and this identification, together with the ghost that a symmetric tensor without the gauge symmetry carries, is what forces X to become massive. The step is derived from the structure of the doubled geometry and rests on standard linearized gravity [D + T]; no document writes an action for X in which the symmetry can be checked, and no script checks it.

Dual stress T̃

Mon, 01 Jan 0001 00:00:00 +0000

The dual stress, written T̃, is the quantity through which matter couples to the momentum graviton X, in the way the stress-energy tensor is the quantity through which matter couples to the ordinary graviton. Stress-energy is built from derivatives of a matter field along spacetime and describes how energy and momentum are distributed in position; the dual stress is built from derivatives along the momentum directions and so reflects how the field is structured across momentum, the spectrum side of the same field. Exchange of X between the dual stresses of two particles of conjugate matter produces the velocity gate. The calc notes define the dual stress in words and give no formula for it. Why it is strong for conjugate matter and weak for ordinary matter is an open problem.

Dwarf galaxies: cores and ultra-faint dwarfs

Mon, 01 Jan 0001 00:00:00 +0000

Dwarf galaxies are small, faint galaxies, many of them satellites of the Milky Way, whose mass is mostly dark matter. In many of the brighter, classical dwarfs the dark matter’s central density is flatter than collisionless cold dark matter (CDM) predicts — the cusp–core problem — which a self-interaction of the dark matter, the wave nature of very light dark matter, or the gravitational stirring caused by gas that supernovae blow out of the center can explain. The ultra-faint dwarfs (UFDs) are fainter still, and they are the densest, oldest and most dark-matter-dominated systems known. In the Broken-Born program, the classical dwarfs fix the strength of conjugate matter’s self-scattering, and the ultra-faint dwarfs, Segue 1 among them, falsified Scenario A, one branch of the momentum-clumping test.

Goldstone modes and the Higgs mechanism

Mon, 01 Jan 0001 00:00:00 +0000

A Goldstone mode is a massless excitation that appears when a continuous symmetry of the laws is broken by the ground state. Moving along the family of equally good ground states costs no energy, so a slow, long-wavelength wave of that motion costs arbitrarily little, and the particle it describes has no mass. If a small explicit term tilts the family, the mode acquires a small mass and is called a pseudo-Goldstone mode. When the broken symmetry is a gauge symmetry, the Higgs mechanism applies instead: the gauge field absorbs the would-be Goldstone modes as extra polarizations and becomes massive. The program uses all three ideas. In its coset toy model the polarization angle is a Goldstone direction, which an added potential turns into a pseudo-Goldstone mode, and its momentum graviton X gets its mass from a gravitational version of the Higgs mechanism.

Inflation, reheating and entropy dilution

Mon, 01 Jan 0001 00:00:00 +0000

Inflation is a proposed early period of extremely rapid, accelerating expansion, which thins any relics produced before it until essentially none remain in the observable universe; reheating is its end, when the energy that drove inflation turns into hot particles and sets the starting temperature of the radiation era. Relics made after reheating can still be thinned later if something releases a large amount of entropy into the plasma, for example the decay of heavy particles that dominated the universe for a time, a period called early matter domination. The program uses this physics in two places. Inflation is the standard reason no magnetic monopoles of our own sector are seen, which is one of the two reasons the early claim that the conjugate sector explains their absence is dead. And late dilution was the only way the retired soliton dark-matter candidate could have kept its heavy mass, a route the program rejected because it removes the mass prediction.

Maximum force and maximal acceleration in the literature

Mon, 01 Jan 0001 00:00:00 +0000

The maximum-force conjecture is a proposal within general relativity that no force can exceed c⁴/4G, a combination of the speed of light c and Newton’s constant G; Gibbons named it the maximum tension principle, and Schiller took a principle of maximum force as the starting point for deriving general relativity. An older and separate proposal, Caianiello’s, is a maximal acceleration obtained from a metric on phase space. Both are proposals rather than settled results, and #13, the program’s paper on conjugate special relativity, notes that the maximum-force conjecture is contested. They matter to the program because #13’s maximum force, c̃ = c⁴/G, has the same form up to a factor of 4, and #13 offers a reason for such a limit. This page covers the literature; the Planck units it is written in are on their own page, and #13’s own claim is on the maximum force page.

Model I: action and symmetries

Mon, 01 Jan 0001 00:00:00 +0000

Model I’s action is the integral, over a flat two-dimensional base, of a Lagrangian with three terms: two kinetic terms, which cost action when the fields vary from point to point, and a potential that vanishes exactly when the two fields are compatible. The fields are geometric structures on the phase plane of one particle moving in one dimension: a metric H, parametrized by a point of the upper half-plane, and a real structure K, whose two eigenlines are the candidate position and momentum axes. The action is unchanged by every symplectic (area-preserving) linear map of the phase plane, which includes Born reciprocity ρ, and by the separate flip K → −K. That invariance is criterion S1, checked symbolically, and it supplies the “genuine Noether symmetry” that acceptance criterion 1 asks for.

Model II: the matrix base and the action

Mon, 01 Jan 0001 00:00:00 +0000

Model II is the self-dual Grosse–Wulkenhaar model of a self-interacting scalar field on the noncommutative plane, rewritten as a model of matrices. The field is expanded in functions f_mn built from harmonic-oscillator states; in this basis the star product of two fields becomes the product of their matrices, and the integral over the plane becomes a trace. At the self-dual point Ω = 1 the quadratic part of the action is diagonal, 8π(m + n + 1)|φ_mn|², so each matrix entry carries an oscillator energy of its own, and the interaction is the trace of a fourth power of the matrix. A complex field is an arbitrary matrix and a real field a Hermitian one. Every formula was checked against an independent anchor before it was used, and the first check of the kinetic form, made with finite differences, failed until exact derivatives replaced them. The duality lemma, the vacua and the restoration argument of model II are all computed on this set-up.

Obituary: Scenario A and the ultra-faint-dwarf endpoint

Mon, 01 Jan 0001 00:00:00 +0000

Scenario A was the assumption that, on the wave island, the gated attraction that condenses wave-like dark matter is also what gives ordinary dwarf galaxies their smooth, constant-density cores. Fitting those cores fixes the strength of the coupling, and with the coupling fixed, condensation is forced to happen, across the whole halo, in the ultra-faint dwarf galaxies, the faintest, densest and oldest galaxies known. Because the attraction makes a condensate collapse whenever it passes a critical mass, throwing off fast waves that escape the galaxy, these galaxies would by now have lost much of their dark matter. Instead they are among the most dark-matter-dominated systems observed: Segue 1 holds a dense, intact halo around stars about 13 billion years old. A much lighter mass, which would avoid the collapse, is excluded by other observations, and no other change of parameters rescues the scenario. Scenario A is dead.

Original P2: both structures rigid (superseded)

Mon, 01 Jan 0001 00:00:00 +0000

The original P2 was the lab notebook’s second axiom: of the three structures on phase space, the symplectic form ω and the neutral metric η are fixed background, and only the generalized metric H carries degrees of freedom. It was meant to express the constancy of Planck’s constant, and the notebook gave it three further jobs: removing a would-be second graviton, fixing the relation between the two cosmological scale factors, and supplying the dark-energy field’s potential. The program’s geometry showed that it fixed too much. With both structures fixed, H is forced into a rigid block form, the polarizations (position–momentum splittings) compatible with both include no circle for the dark-energy field to live on, and the theory reduces to general relativity plus one bespoke model of self-interacting dark matter. It was replaced by P2′, which fixes ω alone. Of its three jobs, one was reversed, one survives from ω alone, and one became an open problem.

Round-3 verdict, rulings and citation audit

Mon, 01 Jan 0001 00:00:00 +0000

The round-3 verdict and rulings are the outcome of the third blind review of #14, the manuscript proposing that dark matter consists of superheavy magnetic monopoles of a hidden gauge sector: the referee’s grade and findings, and the adjudicator’s ruling on each finding. The referee graded version 3 C, fundamentally flawed, with medium confidence. The predicted monopole mass rested on a count of monopoles formed in the early universe that was normalized too low, by a factor of about 4.6×10³, and every number derived from that mass carried the error. The adjudicator agreed for v3 as written. Of the thirteen numbered findings it upheld most, accepted others, some without re-checking them, and overturned one, the claim that a cited paper lacked the operator it was cited for. The audit also found every one of #14’s 23 references real, identified the manuscript’s one theorem as a known result of Born geometry, and listed constraints the manuscript had not evaluated, which the adjudication accepted as a to-do.

Row 1: the knee in σ_T(v)

Mon, 01 Jan 0001 00:00:00 +0000

Row 1 of the predictions table tests how strongly dark matter scatters off itself as a function of the speed at which two particles collide. If dark matter is the program’s conjugate matter χ in particle form, its self-interaction passes through the velocity gate: it is strong in slow collisions, as in dwarf galaxies, and absent in fast ones, as in galaxy clusters, with a knee near the gate velocity v* ~ 10² km/s between the two. Standard cold dark matter does not scatter at all, and the usual self-interacting models, in which a light particle carries the force, weaken gradually with speed, as a power law, rather than at a knee. Observations of halos across the range of masses, and so of internal speeds, can find the knee or exclude it. Row 1 is one of the three rows that the checkpoint names as the program’s near-term falsification exposure.

S-duality and Montonen–Olive

Mon, 01 Jan 0001 00:00:00 +0000

S-duality is the proposed equivalence between a gauge theory at coupling g and a theory at a coupling of order 1/g in which electrically charged particles and magnetic monopoles exchange roles, so that a strongly coupled theory has a weakly coupled description. It grows out of a symmetry of Maxwell’s equations in empty space, which are unchanged when the electric and magnetic fields are exchanged. Montonen and Olive proposed the quantum version in 1977. The checkpoint records it as exact in maximally supersymmetric Yang–Mills theory, one of the few theories in which it has passed non-trivial tests, and Seiberg’s related dualities supply the “magnetic quarks” of a dual description. In string theory S-duality sits with T-duality inside a larger U-duality. The program uses S-duality as a structural comparison, because it acts on the coupling through the same matrix as Born reciprocity, and it carries a standing caution: outside special supersymmetric theories the duality is an expectation, not an established symmetry.

Success criteria S0–S5 and the pre-registered predictions

Mon, 01 Jan 0001 00:00:00 +0000

The success criteria are six pass-or-fail tests, S0 to S5, written down on 19 October 2026 before either toy model was built, together with a forecast of how each would come out. Four of them encode the referees’ literal demand: prior work is cited, the action is symmetric, the symmetry permutes the vacua, and an order parameter distinguishes them. Two encode the deeper objection that the framework does no work: the breaking must survive quantum fluctuations, and it must make matter’s interactions local in position. Fixing the tests first means the results are judged by standards chosen before the results were known, as the program’s pre-registered forecasts do for its review rounds. The worklog scored all four forecasts as hits, before the round-4 review narrowed several of the results that the scoring relied on.

The 't Hooft–Polyakov monopole

Mon, 01 Jan 0001 00:00:00 +0000

The ’t Hooft–Polyakov monopole is a stable, particle-like configuration of fields in a gauge theory whose SU(2) symmetry is broken to U(1) by a Higgs field in the adjoint representation; far from its core it looks like a magnetic monopole of the surviving U(1). It cannot decay, because its Higgs field winds once around the space of possible vacua, a sphere, and no smooth change of the fields can undo the winding. Its mass is the symmetry-breaking scale v times 4π/g_d, with g_d the gauge coupling, times a factor f between 1 and about 1.79 set by the strength of the Higgs field’s self-interaction relative to the gauge coupling. In the program it was the object of the withdrawn manuscript #14, which proposed superheavy monopoles of this kind as dark matter; that proposal is retired, and the physics on this page is textbook.

The collaboration: roles, attribution and conventions

Mon, 01 Jan 0001 00:00:00 +0000

The Broken-Born program is the work of two parties: Paul (P.R.), an independent developer, and Claude, an AI model made by Anthropic. Paul supplies conjectures and direction and holds a veto; Claude supplies derivations, counterexamples, verification and drafting, and the papers carry an attribution line that records this split. Paul cannot independently verify most of the technical physics, so the provenance tags, the machine checks and the review forecasts written down in advance are his means of telling sound results from confabulation. The same conventions fix how decisions are taken and delegated, how work is divided into chunks that a new session can resume, and how the science is written.

The cosmological constraint a·ã = 1

Mon, 01 Jan 0001 00:00:00 +0000

The cosmological constraint is the relation a·ã = 1 between the scale factor a of our universe and the scale factor ã of its momentum-space description, the conjugate universe: as our universe expands, its momentum-space description contracts by exactly the same factor. It follows from the program’s postulate that the symplectic form ω, the structure that fixes the size of a cell of phase space and with it the value of Planck’s constant, is rigid. A comoving cell therefore keeps its phase-space volume while position-space distances grow, which forces momentum-space distances to shrink; in signal terms, stretching a waveform compresses its spectrum. The relation is derived [D], and it turned one of the program’s founding suppositions, that the conjugate universe shrinks as ours expands, into a theorem.

The Fierz–Pauli mass term and the Boulware–Deser ghost

Mon, 01 Jan 0001 00:00:00 +0000

The Fierz–Pauli mass term is the only mass term that gives a free spin-2 field a mass without a ghost, a mode whose energy has the wrong sign. In matrix notation it is proportional to Tr(h²) − (Tr h)², where h is the field; any other relative weight of the two terms lets an extra mode propagate, and that mode is a ghost. When the theory is made nonlinear, a generic completion brings the extra mode back, the Boulware–Deser ghost, and keeping it out takes the special de Rham–Gabadadze–Tolley (dRGT) and Hassan–Rosen potentials. A related standard fact, the signature obstruction, says that a ghost-free two-derivative kinetic term for a symmetric tensor field must have the gauge symmetry of the massless graviton; the Fierz–Pauli theory keeps that kinetic term and breaks the symmetry only through its mass term. The program uses all three facts to decide what its second field, the momentum graviton X, can be. Its tuning test checks X’s mass term against the Fierz–Pauli form, and the version built from metrics fails that check.

The handoff document

Mon, 01 Jan 0001 00:00:00 +0000

HANDOFF.md is the note that carried the program from the claude.ai conversation where it was developed to Claude Code and this repository. It was prepared on 2026-10-19 so that a fresh session could continue without the original chat: it gives the scientific state on one screen, numbers the artifacts #1–#15, lists the results that exist only in the chat, and records the method, the review history, the working conventions and a ranked list of next actions. Much of it still stands, including its list of dead claims, which stays binding. Its soliton dark-matter candidate and its work queue were superseded on 2026-10-19, the queue that replaced it was revised again on 2026-10-19 and 2026-10-20, several of its statuses have moved on, and the root CLAUDE.md and checkpoint #11 now rank above it.

The hard core: spontaneously broken Born reciprocity

Mon, 01 Jan 0001 00:00:00 +0000

The hard core is the assumption the Broken-Born program keeps fixed while its other claims are tested and revised: Born reciprocity, the quarter-turn (x, p) → (p, −x) of quantum phase space (the space of positions and momenta taken together), is an exact symmetry of the laws, and the vacuum, the universe’s lowest-energy state, breaks it spontaneously. Only one structure of phase space is fixed in advance, the symplectic form ω, which encodes the constancy of Planck’s constant. The neutral metric η, the structure that marks which directions count as position, belongs to the vacuum state rather than to the laws, and it is the order parameter that distinguishes our vacuum from its mirror image. A cooled magnet is the standard picture: the laws governing its atoms single out no direction, but its magnetized state points in one. Checkpoint #11 v2.2 records the hard core as intact and sharpened twice by corrections that later derivations forced. The toy models, two small field theories built to test the idea, added two qualifications: a definite polarization, that is, a definite choice of which directions count as position, needs an extra potential, and the breaking has physical effect only once matter couples to the sign of the order parameter.

The metric tier fails twice

Mon, 01 Jan 0001 00:00:00 +0000

The metric tier is the naive way, in calc note #7’s word, of writing the mass term of the momentum graviton X: as a function of metric-level, second-order quantities, the blocks of the misalignment potential. It fails the ghost-freedom test in two independent ways. At quadratic order the natural potential gives the metric fluctuation h a mass term proportional to Tr(h²) without the −(Tr h)² term that the Fierz–Pauli form requires. Beyond quadratic order it contains Tr(g²), which is not a linear combination of the elementary symmetric polynomials from which every ghost-free Hassan–Rosen potential is built. Either failure lets the Boulware–Deser ghost, an extra mode with negative kinetic energy, propagate [T]. Calc note #7 calls the result a “theorem-grade negative”, and tests T4 and T7 of the verification battery check both failures by machine. The way out was a change of variables, to the frame tier.

The modular field Z = X + iY

Mon, 01 Jan 0001 00:00:00 +0000

The modular field Z = X + iY is a complex symmetric matrix field that, in the program, carries the whole geometry of phase space at each point of spacetime once only the symplectic form is held rigid. Each value of Z is a point of the Siegel domain and names one phase-space metric compatible with that form. The program reads the imaginary part Y as the spacetime metric g, and the real part X as a new symmetric tensor field, the momentum graviton, which records how far the split between position and momentum tilts from place to place. In the vacuum X = 0 and Z = i·g. These readings are the program’s derived claims, tagged structural in the note that introduced them; the algebra of the parametrization is machine-checked, and later notes show that X must be a massive spin-2 field.

The phantom divide and the null energy condition

Mon, 01 Jan 0001 00:00:00 +0000

The phantom divide is the value w = −1 of the dark-energy equation of state, the value a cosmological constant has, and it separates two kinds of behaviour: above it, dark energy thins out as space expands; below it, in the regime called phantom, dark energy becomes denser as space expands. An ordinary scalar field cannot reach the phantom side, because the quantity that would have to become negative is the square of the field’s rate of change. That fact is a case of the null energy condition, a standard requirement on physically reasonable matter. A measured w below −1 would therefore point to something unusual, such as a field with negative kinetic energy, several interacting fields or a change to gravity. The program’s dark-energy field is an ordinary scalar, so a robust intrinsic w < −1 would kill the program’s framework outright.

The skeptic review console

Mon, 01 Jan 0001 00:00:00 +0000

The skeptic review console is a single HTML page, artifact #15 (review/skeptic-review-console.html), that sends a manuscript to a fresh Claude model with no conversation context, an adversarial referee prompt and web search turned on, and prints the review it gets back. It was the tool of review rounds 1 and 2, both on #14, the standalone manuscript proposing superheavy hidden-sector solitons as dark matter. Its prompt defines the A/B/C verdict scale that the later rounds kept. The referee it calls can search the web but cannot execute code, the limitation behind the referees’ arithmetic errors in those rounds. The page embeds #14 version 2 verbatim and is kept as the record of rounds 1 and 2; the proposal in it has since been retired.

The symplectic form and Sp(2d, ℝ)

Mon, 01 Jan 0001 00:00:00 +0000

The symplectic form ω is the antisymmetric pairing of positions with momenta that underlies Hamiltonian mechanics and the quantum commutator; in the plane of one position and one momentum it measures signed area. The symplectic group Sp(2d, ℝ) is the set of linear maps of phase space (d positions, d momenta) that preserve it. The group contains rotations of phase space — the quarter-turn Born reciprocity is one — together with stretches that lengthen one direction and shorten another, called boosts and squeezes. In the program ω is the one structure held fixed: its corrected postulate reads the rigidity of ω as the constancy of Planck’s constant, and every other piece of phase-space geometry is allowed to vary.

The three debts

Mon, 01 Jan 0001 00:00:00 +0000

The three debts are the three open problems that the program’s first write-up, the lab notebook #1, ranked most severe and called its “three named debts”; the calc notes #2 to #8, written to close them, call them Debt 1, Debt 2 and Debt 3. Debt 1 asked for the field equation of momentum-space geometry, the question left open since Born’s paper of 1938. Debt 2 asked where the velocity gate comes from, the factor that confines the self-interaction of conjugate matter to particles of nearly equal velocity. Debt 3 asked whether that interaction makes dark matter gather into streams of shared velocity, the program’s founding image. Debt 3 was closed first, against the program: such clumping is nowhere required. Debt 1 was closed in structure: momentum-space geometry is carried by a massive spin-2 field of a known class of bimetric gravity, though the class was selected rather than derived from the program’s action. Debt 2’s mechanism, exchange of that field, came out of the first Debt-1 note; the gate’s exact shape is still open.

The vacuum-energy identity V(1) = 2T

Mon, 01 Jan 0001 00:00:00 +0000

The vacuum-energy identity is a short algebraic result about the program’s spin-2 interaction potential once its coefficients obey the palindrome βₙ = β₄₋ₙ. At the aligned point, where the potential’s two frames coincide, the value of the potential, V(1), is exactly twice its tadpole T, the coefficient of the term linear in a small departure from that point. A flat aligned vacuum separately requires V(1) = T, and the two equations together force T = 0 and V(1) = 0, so the spin-2 interaction sector adds no vacuum energy there. Once the palindrome holds, that single flatness condition does the work of the two tunings that generic bimetric gravity needs. The integer arithmetic is machine-checked. The identity says nothing about the vacuum energy of matter loops, the “old” cosmological-constant problem, and whether it is new is the subject of an open literature comparison.

The χ-particle candidate

Mon, 01 Jan 0001 00:00:00 +0000

The χ-particle candidate is the proposal that dark matter consists of particles of conjugate matter χ: matter in the momentum-polarized representation, the mirror image of ordinary, position-polarized matter. χ gravitates exactly like ordinary matter, so it falls, orbits and bends light like standard cold dark matter, but no gauge force of ordinary matter acts on it. Its own self-interaction works only between particles with nearly equal velocities — the velocity gate — which predicts strong self-scattering in slow dwarf galaxies, none in fast galaxy clusters, and a knee in between. It is one of the program’s two remaining dark-matter candidates; its stability is assumed rather than derived, and its abundance is unexplained.

Topological dark matter in the literature

Mon, 01 Jan 0001 00:00:00 +0000

Topological dark matter is the established proposal that dark matter consists of topological defects, such as magnetic monopoles, left behind when a symmetry of a hidden sector breaks in the early universe. The defects are stable because their field configuration cannot unwind, and how many survive is set by how many form at the transition and how many then annihilate in pairs. Murayama and Shu found that such defects could be the dark matter at masses of 1–10 PeV, noting that Kibble’s simple count underestimates how many form; Liveoak, Maharana and Wells showed in 2026 that hidden monopoles overclose the universe if the symmetry breaks above about 100 PeV, unless a later epoch dilutes them. This low-scale scenario is what remains of the program’s retired soliton candidate once its abundance is counted correctly. Nothing in it depends on Born reciprocity, so the program claims compatibility with it and nothing more.

Verification checks arithmetic, not the model

Mon, 01 Jan 0001 00:00:00 +0000

A verification script confirms that a calculation was carried out correctly for the inputs and formulas written into it; it cannot tell whether those inputs and formulas are the right physics. The program learned this from its soliton dark-matter manuscript: every number in it was computed and checked by a gated script, yet one formula, the normalization of how many monopoles the early universe forms, was wrong, and only an independent review caught it. Checkpoint #11 has carried the caution since version 2.1, the rule on independent anchors is the repair, and every script page in this section states what its checks leave to the derivation.

Vlasov stability and Landau damping

Mon, 01 Jan 0001 00:00:00 +0000

The Vlasov equation describes a gas whose particles interact only through a smooth, averaged force field, so that the gas is fully specified by how densely it fills each region of position and velocity. Its stability theory asks whether a small ripple in that density grows or dies away. In a plasma a ripple usually decays even without collisions (Landau damping), while in a self-gravitating gas a ripple longer than a critical length grows into a lump (the Jeans instability). Where collisions do act, they drive any distribution toward the smooth thermal shape (Boltzmann’s H-theorem). The program used this textbook machinery in Debt 3 to test whether dark matter could clump in velocity.

Wave kinetics and Bose-star formation

Mon, 01 Jan 0001 00:00:00 +0000

Wave kinetics is the theory of how a field made of many waves with random phases relaxes when the waves scatter off one another. In a dark-matter halo made of a very light boson, the waves interact through gravity, and each scattering moves power toward the lowest-energy modes at a rate that grows with how many quanta those modes already hold. The result is a dense, coherent ground-state core, a Bose star, which forms at the halo’s center and then grows. Levkov, Panin and Tkachev (LPT) worked out this kinetic condensation and followed it in simulations. The Broken-Born program’s Debt-3 calculation used it as the anchor for a gated version driven by conjugate matter’s self-attraction.

Stückelberg fields

Mon, 01 Jan 0001 00:00:00 +0000

Stückelberg fields are extra fields written into a theory of a massive gauge field so that the gauge symmetry its mass term breaks holds again. For a vector field A_μ the mass term m²A_μA^μ changes when A_μ is shifted by the gradient of a function; written in terms of the combination A_μ + ∂_μπ, where π is a new scalar field that shifts back by the same function, it does not. Setting π to zero, the choice called the unitary gauge, returns the original theory, so the construction changes the description and not the physics. The new fields are the would-be Goldstone modes of the Higgs mechanism written out as fields: the extra polarizations that a gauge field absorbs, or “eats”, when it becomes massive. A massive spin-2 field needs a vector of them, and the vector’s gradient part carries the polarization of helicity zero, the one with no spin along the direction of motion. In the Broken-Born program the momentum graviton X becomes massive by absorbing the would-be Goldstone modes of its broken dual diffeomorphisms (the local shifts of X), and checkpoint #11 calls these modes “the Stückelberg sector”. The dark-energy field τ lies outside that sector at leading order; how strongly the two mix has not been computed and is open problem 6 (OP6).

Inverse-Higgs constraints

Mon, 01 Jan 0001 00:00:00 +0000

An inverse-Higgs constraint is a relation that fixes one would-be Goldstone field as a derivative of another, so that the first drops out of the theory as an independent field. Such relations arise when the symmetries broken by the ground state act on space and time themselves, as translations, rotations and boosts do. A flat membrane is the standard example. It breaks the translation perpendicular to itself and the rotations that would tilt it, but its local tilt is set by the slope of its local displacement, so the displacement is its only massless field. In a relativistic theory each broken generator of an internal symmetry brings its own massless Goldstone mode; when spacetime symmetries break, there can be fewer modes than broken generators. The program’s record mentions the idea in a single document: calc note #5 expects such constraints to remove most of the nine directions, besides that of the dark-energy field τ, in which the vacuum’s polarization can vary. No document works this out, and the question is open.

The Bullet Cluster

Mon, 01 Jan 0001 00:00:00 +0000

The Bullet Cluster is a pair of galaxy clusters seen shortly after the smaller one passed through the larger. A cluster has three components: its galaxies; hot, thin gas between them, which emits X-rays and outweighs the stars; and dark matter, which outweighs both. In the collision the two gas clouds struck each other and were slowed, while the galaxies passed through. The total mass, mapped by the way the clusters’ gravity bends light from more distant galaxies, moved on with the galaxies and not with the gas. The observation is widely cited as direct evidence that most of a cluster’s mass is a separate, nearly collisionless dark matter, and it limits how strongly dark matter can scatter off itself at the speeds of cluster collisions. The Broken-Born program calls its conjugate matter “Bullet-Cluster-safe”: it gravitates and bends light as ordinary matter does, and its self-interaction is switched off at cluster speeds by the velocity gate. The program’s documents quote no number for the limit and do not compare the gate with it.

The Kalb–Ramond B-field

Mon, 01 Jan 0001 00:00:00 +0000

The Kalb–Ramond B-field is an antisymmetric tensor field, B_μν = −B_νμ, that does for a string what the electromagnetic potential does for a charged particle: a particle’s quantum phase changes by the potential summed along its path, and a string’s by the B-field summed over the surface the string sweeps out. Like the electromagnetic potential it is a gauge field, so it can be changed by a gauge transformation without changing anything measurable, and what is measurable locally is its field strength, built from its derivatives. It is one of the massless fields of closed string theory, with the metric and a scalar, the dilaton. T-duality, the symmetry of strings on a circle that exchanges momentum around the circle with winding around it, turns parts of the metric into parts of the B-field, and for that reason double field theory puts the two in one matrix, the generalized metric, which preserves a neutral metric η. The program uses the B-field as established physics in three ways. Holding the symplectic form ω rigid as well as η removes it (corollary 2 of the compatibility theorem). It is the antisymmetric counterpart of the program’s own partner of the metric, the symmetric momentum graviton X: X is a second graviton, and the B-field is not. And its healthy propagation alongside the metric and the dilaton is the program’s precedent for a doubled geometry whose kinetic terms have no ghosts, modes that carry negative energy.

The signature obstruction

Mon, 01 Jan 0001 00:00:00 +0000

The signature obstruction is the standard result that a symmetric tensor field in spacetime, such as the perturbation h_μν of the metric, has no ghost-free kinetic term with two derivatives unless that kinetic term has the gauge symmetry of the massless graviton. A ghost is a mode whose energy has the wrong sign. The obstruction comes from the signature of spacetime, in which time enters with the opposite sign to space: in a contraction such as h_μν h^μν the components h₀ᵢ, with one time index and one space index, enter with the opposite sign to the others, and in the simplest kinetic term built from such contractions that sign becomes the sign of their kinetic energy. The gauge symmetry avoids the problem by keeping those components from propagating: the kinetic term of linearized general relativity contains no time derivatives of h₀₀ and h₀ᵢ, which are constrained rather than propagated. The Fierz–Pauli theory of a massive spin-2 field keeps that kinetic term and breaks the symmetry only through its mass term. The program uses the result to close one option for its second symmetric tensor, the momentum graviton X: without its gauge symmetry X would carry a ghost, and with the symmetry unbroken X would be a second massless graviton interacting with the metric, which the multi-graviton no-go theorem excludes, so X must have a mass. Calc note #6 tags the statement as standard literature without citing a source, and no script checks it.

The Witten effect

Mon, 01 Jan 0001 00:00:00 +0000

The Witten effect is the electric charge that a magnetic monopole carries in a gauge theory with a nonzero vacuum angle θ, the coefficient of a term in the Lagrangian proportional to the product E·B of the electric and magnetic fields. In such a theory a monopole’s electric charge is not a whole multiple of the unit of charge e: it is shifted by θ/2π of e for each unit of magnetic charge, so the monopole is a dyon, a particle carrying both kinds of charge. Edward Witten derived the result in 1979. Increasing θ by 2π gives back the same set of charges with the dyons relabelled, which is the shift T of the duality group SL(2, ℤ). The effect entered the program once, through the retired soliton candidate. The round-3 review accepted, without re-checking it, that a hidden vacuum angle together with kinetic mixing between the hidden photon and ours would give the candidate’s hidden monopoles a small electric charge under our photon, contradicting the manuscript’s statement that the decisive constraints act on its hidden fermion, not on the dark matter itself. Strict sector separation, the program’s reading of postulate P4, closes that channel.

Attractive condensates: critical mass and the bosenova

Mon, 01 Jan 0001 00:00:00 +0000

An attractive condensate is a Bose star — a self-gravitating clump of a light boson field, all in one coherent state — whose particles also attract one another through a self-interaction. Gravity and the self-attraction both pull inward against the outward pressure that comes from the field’s wave nature, and the self-attraction grows faster with density, so such a condensate has a maximum stable mass. Above it the condensate collapses, throws off part of its mass as fast waves in a burst called a bosenova, and the remnant grows again. In the Broken-Born program this cycle is the endpoint that, in Scenario A of the wave island, would have drained the dark matter out of the ultra-faint dwarf galaxies; Scenario A is dead.

Bigravity dark matter

Mon, 01 Jan 0001 00:00:00 +0000

Bigravity dark matter is the proposal, from the literature on bimetric gravity, that dark matter is the massive spin-2 particle of a theory with two interacting metrics. In ghost-free bimetric gravity the two metrics combine into a massless graviton, which carries ordinary gravity, and a massive spin-2 field; Babichev and collaborators proposed in 2016 that this heavy field makes up the dark matter. The program’s momentum graviton X is a massive spin-2 field of the same bimetric kind, so this literature is the lineage of the program’s X-condensate candidate. The program cites it only for that lineage, from model memory, and has not yet checked the source.

DESI baryon acoustic oscillations

Mon, 01 Jan 0001 00:00:00 +0000

The Dark Energy Spectroscopic Instrument (DESI) is a survey that measures the redshifts of galaxies and quasars in order to map how the universe has expanded over a long span of its history. Its main tool is baryon acoustic oscillations (BAO): sound waves in the hot plasma of the early universe left a preferred separation between concentrations of matter, a fixed length that serves as a ruler at every later epoch. In Fourier terms the imprint is a single bump in the correlation between galaxy positions and a series of wiggles in its transform, the power spectrum. Measuring the ruler’s apparent size at different redshifts gives distances, and the dark-energy equation of state is inferred from those distances by fitting a model; it is not observed directly. DESI’s data releases are the external clock for the program’s dark-energy sector, and its second release already leans against the program’s requirement.

Gravitational waves from phase transitions

Mon, 01 Jan 0001 00:00:00 +0000

A first-order phase transition in the early universe proceeds by bubbles of the new phase forming, growing and colliding, much as water boils, and the collisions and the sound waves they leave in the plasma produce gravitational waves that arrive today from every direction as a random background. The waves’ characteristic frequency is set at the transition by its temperature and duration and has been stretched by the expansion ever since, so a transition at a higher temperature leaves a higher frequency today. A related background, the primordial one, comes from inflation, and its spectrum records how fast the universe expanded at each early epoch. The program met this physics in two places: the hidden-sector transition of its retired soliton dark-matter candidate, whose signal would have lain far above the frequencies of current detectors, and an early idea, since dropped, that its conjugate gravity sector would leave a step in the primordial gravitational-wave background.

Halo-substructure probes: streams and dynamical friction

Mon, 01 Jan 0001 00:00:00 +0000

Halo substructure is the small-scale lumpiness of a galaxy’s dark-matter halo, and two kinds of observation probe it. Cold stellar streams — long, thin trails of stars pulled off globular clusters or small galaxies by the Milky Way’s tides — are disturbed by any lump of dark matter that passes through them, which leaves gaps and kinks along the stream. Dynamical friction — the drag a massive body feels from the gravitational wake it raises in a sea of lighter particles — makes globular clusters sink toward the center of a dwarf galaxy, at a rate set by the dark matter’s distribution. The Broken-Born program’s lab notebook used both as tests of its founding image of dark matter clumping by velocity; both tests are withdrawn, because the Debt-3 calculations showed that velocity clumping is nowhere a required consequence of the program.

Independent anchors

Mon, 01 Jan 0001 00:00:00 +0000

An anchor is a check that a verification script must pass against something known independently of the paper being checked, such as a textbook identity or a published value, before its own numbers are trusted. A check that only reproduces the paper’s own figure is circular: if the paper’s formula is wrong, the check repeats the error and passes. Review round 3 found such a case in the soliton manuscript, whose script anchored the monopole count to the previous draft’s own figure and computed a normalization too small by a factor of about 4.6×10³ without failing. The rule has been in the program’s method since, and three of the current scripts open with anchor checks taken from textbooks or from the cited literature.

Model I vacua, the order parameter and the Goldstone spectrum

Mon, 01 Jan 0001 00:00:00 +0000

The vacua of model I are its lowest-energy configurations: constant fields H and K that are compatible, so that the potential vanishes. They form a three-dimensional family, the upper half-plane of metrics times a circle of real structures, and Born reciprocity ρ carries each vacuum to a different one, so no vacuum is symmetric (criterion S2). The entry K₁₁ of the real structure takes opposite signs in a vacuum and in its image, which makes it an order parameter (criterion S3). Three of the four directions in which the fields can vary are flat. One of them is the polarization angle: moving along it turns the line of the phase plane that counts as position, at no cost in energy. Checkpoint #11 identifies that circle with the one on which its dark-energy field τ lives, as a matter of consistency, not as a test.

Obituary: the hidden SU(2) → U(1) proposal

Mon, 01 Jan 0001 00:00:00 +0000

The hidden SU(2) → U(1) proposal was the particle content the soliton manuscript #14 assumed for its dark sector: a hidden copy of a simple gauge force, SU(2)_d, whose Higgs field breaks it down to a single unbroken U(1)_d. The breaking leaves heavy charged W′± bosons, a massless hidden photon, and ’t Hooft–Polyakov monopoles that are stable for topological reasons, which were the dark-matter candidate. A light hidden fermion ψ was added because without it the W′± would overclose the universe. #14 identified this sector with the momentum-polarized half of the program’s matter, but nothing numerical in the proposal depended on that identification. As a model it is a standard hidden-sector monopole scenario already in the literature; as the program’s own proposal it died with the soliton candidate on 2026-10-19.

Open problems and the queue

Mon, 01 Jan 0001 00:00:00 +0000

This register lists the problems the Broken-Born program has not yet solved and the order in which work on them is queued. Its core is the eleven numbered open problems of checkpoint #11, the program’s current reference, which says that none of them is any longer about whether the structure can exist. Around that list it collects the current work queue, the open ends of the toy models, the open problems raised in the notes, the reviews and the other papers, and the open sub-questions recorded on pages about other subjects. #11 says the word “theory” still waits on three of the numbered problems: deriving the interaction potential of the program’s second graviton from its geometric action, comparing a vacuum-energy identity with the literature on partially massless gravity, and extending proofs made for positive-definite metrics to the signature of spacetime. The register ends with the 44 suspected errors in the source documents recorded while this wiki was built. Independent agents, each told to refute a flag, then re-checked the first 43. Of the first 42, 32 were confirmed, 8 depend on a convention or a reading the documents support, 2 are unclear because they rest on model memory, and none was refuted; flag 43, added later, was confirmed by both of its two checks. Flag 44, a minor point of wording raised after the audit, has not yet been audited. Each still awaits Paul’s ruling.

Open: the partially-massless comparison (OP2)

Mon, 01 Jan 0001 00:00:00 +0000

The partially-massless comparison is a pending literature check on the two results of this section, the palindrome and the vacuum-energy identity. A live literature search found that the palindromic branch contains the partially massless point of bimetric gravity, a parameter point with an extra gauge symmetry, and that Hossenfelder’s exchange-symmetric bimetric theory already cancels vacuum energy by a different mechanism. The comparison asks whether the identity is new, whether it holds up against that literature once conventions are aligned, and whether reciprocity is the origin of the partially massless symmetry. Until it is done, the identity is presented as adjacent to known results rather than as new. It is open problem 2 of the checkpoint and part of the first item of the current work queue, and no calculation for it exists yet.

Pre-registration and its calibration record

Mon, 01 Jan 0001 00:00:00 +0000

Pre-registration is the practice of writing down, and committing to the repository, the verdict a blind review is expected to return and the weakest point the referee is expected to find, before the referee is dispatched. Because the forecast is fixed in advance it cannot be adjusted to the outcome, and scoring it afterwards measures how well the author judges the author’s own work. The record so far: the verdict was forecast correctly in rounds 1 and 2, missed in round 3 and forecast correctly in round 4; the location of the weakest link is now forecast better, and its severity is still underestimated. Since round 3 the forecast is kept out of the commit subject, because a referee subagent sees recent commit subjects.

Provenance tags and citation marks

Mon, 01 Jan 0001 00:00:00 +0000

Provenance tags are the bracketed labels that the program’s papers attach to every claim to say where it comes from: [D] derived in the program, [T] taken from the established literature, [P] postulated, [O] open, or [V] checked by machine. Citations carry a second mark: [S] if the reference was retrieved by a live search during the project, † if it was recalled from the model’s memory and still has to be checked against the source. The tags let a reader who cannot check the physics see what kind of support each statement has, and they change only when the support changes. The wiki copies them exactly as the sources attach them and never upgrades one.

Reciprocity as modular inversion Z → −Z⁻¹

Mon, 01 Jan 0001 00:00:00 +0000

Modular inversion is the map Z → −Z⁻¹, “take minus the reciprocal”, applied to the modular field Z. The program shows that Born reciprocity, the quarter-turn of the position–momentum plane that the Fourier transform performs on a wavefunction, acts on Z in exactly this way. The map leaves the point Z = i unchanged, and that point is the program’s self-dual vacuum. On the vacuum slice the map sends the metric g to its inverse, which for a uniformly scaled metric turns the scale factor a into 1/a; the circle |Z| = 1 is identified with the program’s relation a·ã = 1 between our scale factor and that of the conjugate description, an identification queried in flag 23 of review/wiki-build-flags.md. The same quarter-turn matrix generates the S-duality of electric and magnetic charges. The claim is derived; for one coordinate (d = 1) a script checks it symbolically, and no script checks the general case.

Round-3 decisions and lessons

Mon, 01 Jan 0001 00:00:00 +0000

The round-3 decisions are the four choices that set the Broken-Born program’s course after its third blind review failed the soliton dark-matter manuscript, recorded at the end of review/round3-adjudication.md together with the round’s two lessons. Paul delegated the decisions on 2026-10-19, Claude made them, and Paul keeps the veto. Decision 1 took at face value the postulate that gauge forces do not cross between the two polarizations of matter, so that the hidden and visible photons mix no more than gravity makes them. Decision 2 retired the soliton candidate rather than keep it alive with a free dilution factor or claim a published model as the program’s own. Decision 3 dropped a queued calculation whose premise the review had inverted, and Decision 4 made the toy Lagrangian the next task, after a cleanup of the checkpoint. The lessons were to keep a review’s forecast out of the commit subject, and to check numbers against anchors that do not come from the paper being checked.

Row 2: isotropic scattering below the gate

Mon, 01 Jan 0001 00:00:00 +0000

Row 2 of the predictions table concerns the directions into which colliding dark-matter particles scatter. In the usual self-interacting models, where a light particle carries the force, most collisions deflect the particles only slightly, so the scattering is forward-peaked. In the program’s gated model, collisions slower than the gate velocity scatter equally in all directions, because the gate depends on the speed of the collision and not on its angle. The two patterns move energy and momentum through a halo with different efficiencies, so in principle they leave different imprints on the shapes of relaxed halos. The row stands with one stated assumption, that the particle carrying the gated force is heavy. It holds for dark matter made of particles; for a wave field it has not been restated.

Scenario B: tunable streams (quarantined)

Mon, 01 Jan 0001 00:00:00 +0000

Scenario B is the alternative to Scenario A on the wave island. In it, the smooth cores of dwarf galaxies are made by something other than conjugate matter’s gated attraction, such as the quantum pressure of extremely light wave dark matter or the feedback of ordinary matter, so nothing fixes the strength of that attraction. With the coupling free, it can be set anywhere in a window about ten orders of magnitude wide in which velocity streams form, or outside the window, where they do not. Streams are then possible but never required. That makes them an option the program can accommodate rather than a prediction it is committed to, which in Lakatos’s terms is a sign of a degenerating research programme. The checkpoint keeps the scenario alive but quarantined, among the program’s listed degeneration risks.

The Born connection

Mon, 01 Jan 0001 00:00:00 +0000

The Born connection is the connection of a Born geometry, phase space with the three compatible structures ω, η and H of Born geometry: the rule for comparing vectors at neighbouring points that respects all three structures. A connection defines how fields are differentiated, and its curvature is what the field equations of a geometry are built from. Freidel, Rudolph and Svoboda showed that a Born geometry has exactly one connection compatible with all three under the conditions of their theorem, so its curvature R_B[H] is well defined. The program’s first action for the phase-space metric was built on that curvature. In the current reference the connection appears in a citation and in the argument that frames, not metrics, are the natural variables for the mass term of the momentum graviton. Under the corrected postulate only ω and H are fundamental, and whether a unique compatible connection still exists then has not been shown.

The dual superconductor

Mon, 01 Jan 0001 00:00:00 +0000

The dual superconductor is the leading picture of quark confinement, the observed fact that quarks are never found in isolation. In an ordinary superconductor a condensate of paired electrons expels magnetic field and squeezes whatever magnetic flux remains into thin tubes. In the dual picture, proposed by ’t Hooft and Mandelstam, the vacuum of the strong interaction holds a condensate of magnetic monopoles, which squeezes the color-electric field between quarks into a string whose energy grows with its length, so that pulling two quarks apart never frees either of them. It is one of the three pieces of established physics that absorbed the program’s founding supposition 11, a magnetic counterpart for every electric particle; the program adds no claim of its own here.

The duality dictionary w_eff = −w̃

Mon, 01 Jan 0001 00:00:00 +0000

The duality dictionary is the rule that translates energy in the conjugate, momentum-space description of the universe (the conjugate universe) into the form in which it acts on our own expansion. Each kind of energy is labelled by its equation of state w, the ratio of its pressure to its energy density, and the rule is that conjugate-side energy with equation of state w̃ acts on us like energy with w = −w̃. Pure kinetic energy on the conjugate side therefore acts here like a cosmological constant, a conjugate cosmological constant acts like the stiffest possible fluid, and dust acts like dust. The rule is derived [D] from the cosmological constraint a·ã = 1, and it replaced an earlier heuristic that gave a different answer; #11 lists the change as one of the two forced corrections of the program’s hard core.

The duality on the matrix base

Mon, 01 Jan 0001 00:00:00 +0000

The lemma states how the Langmann–Szabo duality acts on the matrix-base functions f_mn of model II: it multiplies each one by (−1)^m, a sign fixed by its left index alone. The duality is a Fourier transform of the field, which exchanges its position profile with its momentum profile, and on this basis it reduces to parity, the reflection that leaves the even oscillator levels unchanged and reverses the sign of the odd ones, applied to the left index of the field’s matrix. Because Grosse and Wulkenhaar alternate the sign of the Fourier exponent from one factor of the interaction to the next, parity acts on the left index of one factor and on the right index of the next. The result came from a direct numerical Fourier integration, which settled a clash between conventions recalled from memory. The round-4 referee confirmed it and judged it correct but not new, and the note’s second version presents it as a lemma credited to the work it follows from. The rest of model II turns on it.

The Fourier transform as a phase-space rotation

Mon, 01 Jan 0001 00:00:00 +0000

The Fourier transform is the exact, reversible map that turns a quantum state’s description in position into its description in momentum — the same operation that turns a sound’s waveform into its spectrum. Neither description approximates the other; each contains the whole state. Applied once, the transform carries out Born reciprocity, the quarter-turn ρ: (x, p) → (p, −x) of phase space; applied twice, it reflects x → −x; applied four times, it returns every state unchanged. The program takes the analogy literally: position and momentum are two descriptions of one state, and its toy models use the transform to relate a vacuum in which a matter coupling is local in position to its mirror image, in which the coupling is local in momentum.

The frame tier: the symplectic vielbein

Mon, 01 Jan 0001 00:00:00 +0000

The frame tier is the reformulation that rescued the tuning test: the mass term of the momentum graviton X is built from frames, or vielbeins, instead of from metrics. A frame is a set of local reference axes from which a metric is assembled quadratically, as H = EᵀE, so the program calls frames first-order variables and metrics second-order ones. In frame variables the ghost-free potentials for two spin-2 fields are plain polynomials, wedge products of the two frames in the form found by Hinterbichler and Rosen (the vielbein formulation), with no matrix square roots and no ratio of coefficients to get wrong. The program has a frame of this kind, the symplectic vielbein E, which sits on the same first-order footing as the polarization J. Calc note #7 argues that the minimal coupling of the polarization to the geometry is therefore a wedge polynomial of E and a polarization frame E₀[J], and such a polynomial lies in the ghost-free class by construction, with the Fierz–Pauli structure automatic. The argument is structural and rests on the literature. The class was selected by minimality and consistency rather than derived from the program’s geometric action, and no machine check covers it.

The gate mechanism: X exchange over fiber overlap

Mon, 01 Jan 0001 00:00:00 +0000

The gate mechanism is the program’s derivation of the velocity gate: two particles of conjugate matter interact by exchanging the massive momentum graviton X, and they can do so only where their wavepackets overlap in momentum space. A conjugate quantum has a finite width μ along the momentum directions, as an atomic nucleus has a finite radius in space. A nucleus’s size cuts off its scattering at large momentum transfer; the χ quantum’s momentum width cuts off its interaction at large relative momentum, which is the same construction with position and momentum exchanged. Because X is heavy, its exchange acts as a contact interaction, and the overlap of the two wavepackets supplies the gate. The derivation is structural: it fixes what the interaction depends on, but the exact shape of the cutoff depends on a propagator and a profile that no document has computed.

The gated Vlasov model

Mon, 01 Jan 0001 00:00:00 +0000

The gated Vlasov model is the simplified description of dark matter that Debt 3 used to test velocity clumping. Dark matter is treated as a smooth gas in position and momentum. Besides gravity, each particle feels an attraction only from neighbours whose momentum lies within a narrow window around its own, the velocity gate, and the attraction’s range in space is shrunk to a point. Whether a small ripple in this gas grows then becomes an eigenvalue problem on momentum space, which the calculation brackets between a broad-gate limit and a narrow-gate limit instead of solving in general. The model also retired the earlier plan to race the growth of streams against orbital stirring, because below the threshold there is no growth to race.

The Kibble mechanism

Mon, 01 Jan 0001 00:00:00 +0000

The Kibble mechanism, proposed by Tom Kibble in 1976, is the standard account of how topological defects such as monopoles, strings and domain walls form when the early universe cools through a symmetry-breaking transition. Regions farther apart than the horizon, the distance light has travelled since the Big Bang, have had no contact, so the field settles into unrelated vacuum states in different regions, and defects are left where those regions meet. This gives at least about one defect per horizon volume, and that count is a minimum: slow transitions (Kibble–Zurek dynamics) and first-order transitions, which proceed by bubbles, make more. In the program the count decided the fate of the soliton dark-matter candidate, whose manuscript had normalized it too low and treated it as an upper end; that error and its correction are recorded with the candidate.

The lab notebook (#1) and its addenda

Mon, 01 Jan 0001 00:00:00 +0000

The lab notebook, artifact #1, is the program’s first formal write-up: version 0.1, dated September 2026, with five postulates, an action, checks against established physics, a predictions table with kill conditions, and a ranked list of open problems. As the first calculations came in, their results were appended as addenda v0.2 to v0.6 rather than edited into the text, so the notebook preserves the order in which the program changed. Because the body was never revised, claims that later died, momentum-space clumping among them, still read there as live, and the original form of postulate P2, which the first Debt-1 calculation overturned, still stands among its postulates. The notebook was consolidated into the program v1.0 (#9), superseded as a reference by checkpoint #11, and is kept as the audit trail.

The Mermin–Wagner–Coleman theorem

Mon, 01 Jan 0001 00:00:00 +0000

The Mermin–Wagner–Coleman theorem says that in two dimensions a continuous symmetry of the compact kind, such as rotation through an angle, cannot be spontaneously broken. The two dimensions can be two of space at nonzero temperature, or one of space and one of time in a quantum field theory. Slow, long-wavelength wobbles of the would-be ordered direction cost so little energy that they grow without limit as the system gets larger, and they wash the order out. Discrete symmetries escape the theorem, as the two-dimensional Ising magnet shows, and so do some non-compact symmetries. With only a few degrees of freedom, as in ordinary quantum mechanics (0+1 dimensions), not even a discrete symmetry breaks, because tunnelling mixes the would-be ground states. These limits determined how the program’s toy models were built: they start in 1+1 dimensions, and their compact polarization angle can order only after a potential reduces its symmetry to a discrete one.

The multi-graviton no-go theorem

Mon, 01 Jan 0001 00:00:00 +0000

The multi-graviton no-go theorem, proved by Boulanger, Damour, Gualtieri and Henneaux, states that several massless spin-2 fields cannot interact with one another consistently. Under its assumptions (local couplings, at most two derivatives, no ghosts) no coupling can mix them. Each massless graviton can be made consistent on its own, as general relativity is, but a theory in which two of them couple to each other cannot. Massive spin-2 fields fall outside the theorem, which is how bimetric gravity, with one massless and one massive spin-2 field, avoids it. In the program the theorem closes one route for the momentum graviton X: X has the gauge symmetry of a second massless graviton and must couple to the metric and to conjugate matter, so it cannot stay massless. Together with the signature obstruction, this leaves a mass as the only consistent option for X.

The nontriviality theorem

Mon, 01 Jan 0001 00:00:00 +0000

The nontriviality theorem states that the Broken-Born program predicts anything beyond general relativity plus one model of self-interacting dark matter only if the neutral metric η, the structure that marks which directions count as position, is a property of the vacuum rather than a fixed part of the background. It is the argument that forced the program to change its second axiom. With η fixed, the position–momentum splittings compatible with both fixed structures include no Maslov circle, the loop of splittings on which the program’s dark-energy field lives, and the geometric action depends only on the ordinary spacetime metric, which leaves no second, momentum-space graviton. With η a property of the vacuum, all of these return. The result is a structural argument of the program, tagged derived [D]; the compatibility theorem it starts from is a known result of Born geometry and is machine-checked.

The symplectic partner of the graviton is another graviton

Mon, 01 Jan 0001 00:00:00 +0000

The observation that the symplectic partner of the graviton is another graviton compares two ways of extending the geometry of spacetime to a doubled space, in which each coordinate has a partner. Each way holds one pairing of the doubled coordinates fixed, and the field that partners the metric is whatever that fixed pairing allows. Double field theory, the doubled geometry of string theory, fixes a symmetric pairing η, and the metric’s partner is the antisymmetric B-field, a 2-form gauge field that is harmless as far as spin goes. The program’s phase space fixes the antisymmetric symplectic form ω, and the metric’s partner is the momentum graviton X, which is symmetric and has its own linearized diffeomorphisms: a second graviton, and two interacting massless gravitons are inconsistent. Checkpoint #11 offers this as the structural reason string theory’s doubling works while Born’s stalled, and its assessment lists it as the one observation that stands regardless of the program’s fate. It is derived and rests on literature [D + T]; it is a structural argument, and no script checks it.

Two causal cones

Mon, 01 Jan 0001 00:00:00 +0000

The two causal cones are the light cones of position space and of momentum space. In the program’s broken vacuum the metric on phase space has one block for position and the inverse of that block for momentum, and each block defines its own cone of allowed motion: the familiar cone limits speed, |dx/dt| ≤ c, and its twin limits the rate of change of momentum, |dp/dt| ≤ c̃. A rate of change of momentum is a force, so the twin of the speed of light is a maximum force. This is the starting point of #13, the program’s paper on the special relativity of the momentum polarization. It is alive but unreviewed: #13 tags its starting point as derived, no script checks the cones, and the checkpoint #11 does not mention them. #13 also uses two different versions of the momentum cone in different sections, an inconsistency that is open.

Born reciprocity ρ: (x, p) → (p, −x)

Mon, 01 Jan 0001 00:00:00 +0000

Born reciprocity is the map ρ: (x, p) → (p, −x), which exchanges position and momentum, with one sign change, as a quarter-turn of phase space. The rules of quantum kinematics are unchanged by it, and the Fourier transform carries it out on wavefunctions. Max Born proposed in 1938 and 1949 that a fundamental theory should respect it, and hoped for a physics with a fundamental mass and a fundamental length on equal footing. Observed dynamics does not respect it — interactions are local in position — and the program’s starting assumption, its hard core, is that ρ is an exact symmetry of the laws that the vacuum breaks.

Claim tiers and selection bias

Mon, 01 Jan 0001 00:00:00 +0000

The claim tiers are checkpoint #11’s ranking of its own statements by how much trust each deserves. Tier 1 is explicit algebra checked by machine; Tier 2 is established literature, correct at the level of well-worn results; Tier 3 is the program’s new structural physics, which #11 rates as plausible, at a prior of ~50–80% per claim pending expert review. Within Tier 3, #11 counts its negative results as the most reliable items. Alongside the tiers it names one specific risk, the selection-versus-derivation bias, which the wiki reads as presenting a structure that was chosen from the known consistent options as though the program’s premises had forced it. The tiers and the bias check are part of #11’s methodological preface, §2.

Complex versus real fields: symmetry versus covariance

Mon, 01 Jan 0001 00:00:00 +0000

Whether the Langmann–Szabo duality is a symmetry of model II depends on whether its field is complex or real. On the matrix base the duality multiplies the rows of the field’s matrix by alternating signs. A complex field is an arbitrary matrix, so the duality takes every allowed field to another allowed field with the same action: a genuine symmetry, and one element, the half-turn, of a continuous family of phase rotations that also leave the action unchanged. A real field is a Hermitian matrix, and the duality’s image of a Hermitian matrix is in general not Hermitian, so the duality leads out of the space of real fields. The real action keeps its value only when it is rewritten in the new variable, which makes the duality a covariance of the real model, not a symmetry. The distinction decides what each version can show: a symmetry can be broken spontaneously and keeps the vacua it relates exactly degenerate, while without one a preference between vacua can arise explicitly, as it does in the real model at one loop.

Conjectures: chat-only and popsci-only claims

Mon, 01 Jan 0001 00:00:00 +0000

This register collects the claims that are not part of the program’s derived record. Some exist only in the original claude.ai conversation and are known from a list in the handoff document; some appear only in the two popular articles; and a few sit inside the record but are called conjectures by their own sources. No document in the repository derives them and no script checks them, so they are kept apart from the program’s results. The popular article “The Universe’s Other Half” presents the framework as if it were confirmed and never decides a claim’s status. Each line says where the claim is recorded and how it stands against the record.

dRGT massive gravity

Mon, 01 Jan 0001 00:00:00 +0000

dRGT massive gravity, after de Rham, Gabadadze and Tolley, is a theory of a single massive graviton whose nonlinear potential is built so that the Boulware–Deser ghost never appears; it completes the Fierz–Pauli mass term to all orders. Its potential is a linear combination, with constant coefficients, of the elementary symmetric polynomials of the matrix square root √(g⁻¹f), where g is the dynamical metric and f a fixed reference metric. With its two-metric extension by Hassan and Rosen, it is the one known consistent structure for a massive spin-2 field coupled to gravity. The program adopted this class as the health test for its momentum graviton X: X’s sector is healthy if and only if the potential that gives X its mass lands in it. Built from metrics, X’s potential misses the class; built from frames, it lands in it.

Mandatory breaking: the uncertainty argument

Mon, 01 Jan 0001 00:00:00 +0000

Mandatory breaking is the program’s argument that Born reciprocity, the exchange of position and momentum, cannot remain unbroken in any world that contains local physics. The uncertainty principle rules out anything being sharp in both position and momentum at once, much as a waveform and its spectrum cannot both be narrow. A world whose interactions are local must therefore be local in one variable at the cost of the other; in #11’s phrase, “the magnet must point”. The argument is why the program treats the breaking as forced rather than optional, while leaving open which of the two variables ends up as position. Since v2.2 it carries qualifications from the toy models: the polarization is a continuous angle, a definite polarization needs an added potential, and the breaking has physical effect only once matter couples to the sign of the order parameter.

Obituary: conjugate collapse powers our expansion

Mon, 01 Jan 0001 00:00:00 +0000

This page records a dead claim: that the gravitational collapse of the conjugate universe, the momentum-space description of our own, is what drives the accelerating expansion of our universe. It was one of the program’s founding suppositions. Written down with the heuristic translation rule of the early discussion, it required dark energy with equation of state w = −2, a value the data exclude. With the translation rule derived properly, the duality dictionary, collapsing conjugate matter acts on our expansion as ordinary matter does, slowing it rather than accelerating it, so the mechanism disappeared. The claim died twice and stays dead.

Obituary: the coupling α_d ≈ 34

Mon, 01 Jan 0001 00:00:00 +0000

α_d ≈ 34 was the strength the soliton candidate assigned to the interaction between its hidden monopoles: 1/(4α), the reciprocal of four times the fine-structure constant α ≈ 1/137 of ordinary electromagnetism. The program tried three ways to fix that value. The first applied Dirac’s charge-quantization condition using our photon’s charge, which contradicted the claim that the monopoles do not couple to our photon. The second had Born reciprocity act on gauge couplings as the modular S transformation of electric–magnetic duality; a referee called it numerology, and it was set aside. The third assumed that the smallest hidden charge equals the electron’s; it was motivated but not derived, and it fell with the candidate. The number itself shifts by a factor of four with the convention used. What survives is a general fact: a weakly coupled force always has strongly coupled monopoles.

Obituary: the naive ρ-preserving potential

Mon, 01 Jan 0001 00:00:00 +0000

The naive ρ-preserving potential was the term −μ[Tr(K K_ref)]², added to model I so that its circle of vacua would reduce to two and the breaking of Born reciprocity could survive in two dimensions. It is symmetric under ρ and gives the two intended vacua in the frozen-H reduction, where the metric H is held at 1 and K stays on its circle of angles. Once H is allowed to vary it is unbounded below: along a family of boosted configurations the compatibility term stays at zero while the potential falls without limit, and the intended vacuum is a saddle. The worklog’s first pass of criterion S4 and the first version of the toy-model note rested on it. The round-4 referee found the defect, the adjudication upheld it, and the potential survives only as the counterexample that the bounded potential V_b answers.

P2′: only the symplectic form is rigid

Mon, 01 Jan 0001 00:00:00 +0000

P2′ is the program’s corrected second axiom: of the structures on phase space, only the symplectic form ω is fixed in advance, and holding it fixed is the program’s way of saying that Planck’s constant does not change. The neutral metric η, which marks which directions count as position, is not part of the background. It is a property of the vacuum: the order parameter of the broken symmetry, the quantity whose value tells a vacuum from its mirror image, as the direction of magnetization tells a magnet pointing one way from one pointing the other. The original P2 had fixed η as well, and the nontriviality theorem showed that this left the program with no content of its own. Checkpoint v2.2 added a refinement from the toy models: with only ω fixed, the polarization is a continuous angle. Two consequences follow. Pinning the angle to a definite value needs an additional potential, which brings back a little fixed structure, and the choice has physical consequences only once matter couples to it.

Relic abundance by misalignment on the wave island

Mon, 01 Jan 0001 00:00:00 +0000

Misalignment is a way for a very light field to acquire its relic density in the early universe: the field starts away from the minimum of its potential, stays there while the universe expands quickly, and later oscillates coherently about the minimum; the oscillation then behaves as cold dark matter. It is the standard production route for axions. Debt 3 noted that conjugate matter χ on the wave island, a field this light, coupled only through gravity and its own self-interaction, would be produced the same way. On the wave island only, that removes a difficulty the lab notebook had recorded: a light, strongly self-coupled sector connected to ordinary matter only through gravity cannot get its density by ordinary thermal freeze-out. It does not explain why dark matter outweighs ordinary matter by 5.4 to 1, a question that stays open for both remaining dark-matter candidates.

Round-3 arbitration and the referee's constants

Mon, 01 Jan 0001 00:00:00 +0000

The round-3 arbitration is the adjudicator’s own recomputation of the central numbers the referee disputed, run before ruling on them: a short script, review/round3/adjudicator/round3_arb.py, that rebuilds the Kibble normalization from the expansion rate of the radiation-dominated universe, finds the monopole mass at which the abundance matches the dark matter, and recounts the relic density of the light hidden fermion ψ. It confirmed the referee’s central finding: one monopole per Hubble volume gives p = 104.31, where v3 had used 0.0228, so the mass that closes the abundance falls to 2.9–4.4×10¹² GeV and v3’s benchmark of 3×10¹³ GeV is overproduced. The referee had done the same work independently, taking every physical constant from outside sources, chiefly the Particle Data Group’s 2024 review and Planck 2018, rather than from the manuscript. The two calculations agree on the Kibble numbers and, on ψ, “to within ×1.6”, and the adjudicator’s numbers were carried into the gate.

Row 3: the island discriminant

Mon, 01 Jan 0001 00:00:00 +0000

The island discriminant is a proposed way to read from astronomical data whether dark matter is made of particles or of a classical wave field. The program’s gated self-interaction weakens above the gate velocity v* in both cases, but differently: for particles the cutoff is sharp, while for a wave field it falls off only as a power law, (v*/σ)⁴, where σ is the velocity dispersion of the halo. The program calls the two regimes the particle island and the wave island, and the shape of the knee in the self-interaction strength, the subject of row 1, would say which one dark matter occupies. The diagnostic came out of the calculation that killed velocity clumping (Debt 3). #11 counts it among its predictions not previously posed, and it selects a branch rather than killing the program. Its particle side assumes a sharp tail, which the program now treats as open.

The compatibility theorem

Mon, 01 Jan 0001 00:00:00 +0000

The compatibility theorem is a statement of linear algebra about metrics on phase space: if the symplectic form ω and the neutral metric η are both held fixed in their standard forms, the only positive metrics H compatible with both are block-diagonal, H = diag(g, g⁻¹), with the momentum block equal to the inverse of the position block. “Compatible” means that H fits each structure in the way Born geometry requires, (η⁻¹H)² = 1 and (ω⁻¹H)² = −1. The statement is Freidel, Leigh and Minic’s; the short block proof and its machine check are the program’s, and an independent reviewer re-derived it in round 3 and found that one step of the proof is not needed. Its use in the program is that it forced a change of postulates: with both structures fixed, the program reduces to general relativity plus one dark-matter model, so the program replaced its postulate P2 with P2′ and let η vary.

The conjugate universe: one state, two descriptions

Mon, 01 Jan 0001 00:00:00 +0000

In the current program the conjugate universe is a second description of our universe rather than a second world: the description of the one physical state in momentum variables, related to the ordinary description as a sound’s spectrum is related to its waveform. It began as one of the program’s founding suppositions, a second universe organized around momentum and existing alongside ours. Two derived relations turned it into a second description. The constancy of Planck’s constant fixes the size of a cell of phase space, so as our universe expands its momentum-space description contracts by the same factor; and a derived translation rule says how energy described on the momentum side acts on our expansion. The two descriptions are synchronized rather than causally linked, so an event in one is the same event read in the other, which is how the program reads its predicted far-future halt and reversal of our expansion as a bounce of the conjugate universe. The idea that observers might live in the momentum description appears only in the popular articles and has no status in the program.

The Debt-3 calc notes (#2–#4)

Mon, 01 Jan 0001 00:00:00 +0000

The Debt-3 calc notes are three short working notes, artifacts #2 to #4, that tested the program’s founding image: dark matter clumping by velocity rather than in space. The first showed that dark matter made of particles cannot clump this way. The second found that dark matter light enough to behave as a wave can, but only as an option, except in one scenario where clumping becomes compulsory in the smallest galaxies. The third showed that this scenario would have drained those galaxies of their dark matter, which observation rules out. Together they closed Debt 3 with a negative result, which checkpoint #11 now reports as an obituary. The notes are final for their own scope; no script in the gate checks their numbers.

The gated scattering law σ_T(v)

Mon, 01 Jan 0001 00:00:00 +0000

The gated scattering law gives the momentum-transfer cross-section σ_T of conjugate matter, the measure of how strongly its particles scatter off one another, as a function of the relative velocity v of a colliding pair. Because the velocity gate turns the interaction off at large relative velocity, self-scattering is strong in dwarf galaxies, where dark matter moves at tens of kilometres per second, and absent in galaxy clusters, where it moves at thousands; between the two, σ_T bends down at a knee near the gate velocity v*. Below the knee the scattering goes equally in all directions, unlike the forward-peaked scattering of the usual models with a light force carrier. The lab notebook first derived a sharp exponential cliff at v*; once the gate’s mechanism was found, the program kept only the knee, and the shape of its tail became an open question that observations could answer.

The Lagrangian Grassmannian Λ(d)

Mon, 01 Jan 0001 00:00:00 +0000

The Lagrangian Grassmannian Λ(d) is the space of all ways to split a 2d-dimensional phase space into a “position” half and a “momentum” half. Each Lagrangian subspace, a d-dimensional plane on which the symplectic form vanishes, is one point of it. For d = 1 it is the set of lines through the origin of the (x, p) plane, which is a circle; in general it is the compact manifold U(d)/O(d), of dimension 10 when d = 4. It contains a loop that cannot be shrunk to a point, the Maslov circle, so any angle measured around that loop is periodic. It also forms the distinguished (Shilov) boundary of the Siegel domain. In the program it is the space of possible vacuum polarizations, and its circle direction carries the dark-energy field. The mathematics is standard.

The magnetic mirror-family

Mon, 01 Jan 0001 00:00:00 +0000

The magnetic mirror-family is the idea that every electrically charged particle, quarks included, has a magnetic counterpart. It entered the program as P.R.’s founding supposition 11, and the checkpoint records it as established physics: Montonen–Olive duality, Seiberg’s magnetic quarks and the dual-superconductor picture of quark confinement already contain it. Dirac’s condition accounts for why the counterparts are not seen: a weak electric coupling forces a strong magnetic one, so magnetic particles are heavy and tightly bound. The checkpoint’s figure for the magnetic coupling of our electromagnetism, about 34, is not the hidden-sector coupling α_d ≈ 34 that died with the soliton candidate. The program’s own addition is the observation that choosing which particles count as electric is a choice of polarization, which has its own page.

The maximum force c̃ = c⁴/G

Mon, 01 Jan 0001 00:00:00 +0000

The maximum force is the limit that the momentum-space light cone places on how fast momentum can change, as the ordinary light cone limits speed to c: since a rate of change of momentum is a force, the limit is a largest possible force, c̃ = ℏc/λ², where λ is a length that the paper #13 calls the Born length. Setting the Born length equal to the Planck length gives c̃ = c⁴/G, about 1.2 × 10⁴⁴ newtons, which #13 describes as “the scale black holes reach when they collide”. The general-relativity literature already contains a conjectured maximum force, c⁴/4G; #13’s value is larger by a factor of 4, and the factor is open. The result is alive but unreviewed: no review round has examined it, no script checks the number, and the checkpoint #11 does not mention it.

The mean-field criterion and its 10²⁸–10³¹ shortfall

Mon, 01 Jan 0001 00:00:00 +0000

The mean-field criterion is the condition under which a gas of dark-matter particles with the gated attraction would break up into velocity streams on its own: the attraction energy among particles that share a velocity has to match the kinetic energy of their random motion. Debt 3 derived it in two limits of the gated Vlasov model and evaluated it at the scattering benchmark, where the coupling is already fixed by fitting the cores of dwarf galaxies. There the attraction falls short by a factor of about 10²⁸ in dwarf-galaxy cores and 10³¹ in the Milky Way’s halo near the Sun. Collisions cannot make up the difference, because they only drive the gas toward a smooth thermal distribution. The result rules out velocity clumping for particle dark matter, and the result itself stands.

The measured referee failure mode

Mon, 01 Jan 0001 00:00:00 +0000

The measured referee failure mode is the program’s record of where its blind reviews have gone wrong, and on which side. In rounds 1 and 2 the referees, which could search the web but not run code, were reliable on structure, literature and logic and unreliable on arithmetic: their three confident order-of-magnitude indictments were all their own unit errors, and none survived. That finding made the iron rule load-bearing and was the main reason the project moved to Claude Code, where a referee can execute code. In round 3 the failure moved to the author’s side: the author’s scripts had checked the arithmetic of a model with a wrong physics input, so a check that tests only arithmetic passed it. Round 4’s decisive defect was again on the author’s side, in a check run on a reduced model.

The misalignment mechanism

Mon, 01 Jan 0001 00:00:00 +0000

The misalignment mechanism is a way of producing dark matter as a coherent field rather than as a thermal gas of particles. In the early universe a very light scalar field starts at some value away from the minimum of its potential and stays frozen there while the universe expands faster than the field can oscillate; once the expansion rate falls below the field’s mass, the field oscillates about its minimum, and the energy of the oscillation then thins out with the expansion as cold matter does. It is the standard production route for axions. The Broken-Born program borrows it as the production channel for the wave branch of conjugate matter, a channel the particle branch still lacks; the ratio of dark to ordinary matter stays unexplained on both.

The monopole problem

Mon, 01 Jan 0001 00:00:00 +0000

The monopole problem, in the form the program cites (John Preskill, 1979), is the finding that the hot early universe should have produced superheavy magnetic monopoles in unacceptably large numbers, while none has ever been detected. Every grand unified theory contains such monopoles, the cooling universe makes at least about one per horizon volume when the unified symmetry breaks, and monopoles and antimonopoles annihilate one another too slowly to remove enough of them. The standard resolution is inflation, a period of rapid expansion after the monopoles form that dilutes them to an unobservable density. The program meets the problem twice: inflation is one of the two reasons its founding claim that conjugate monopoles explain the null result is dead, and round 3 ruled that the strong-coupling annihilation question of its retired soliton candidate is Preskill’s calculation, already done for hidden sectors.

The reduced-model stability rule

Mon, 01 Jan 0001 00:00:00 +0000

The reduced-model stability rule says that when a calculation simplifies a model by holding some of its fields fixed, the full model’s stability must still be checked with every field free to move, and the text must say that the calculation used a reduction. Review round 4 supplied the reason: the potential first used in the program’s coset toy model had been checked only with the phase-space metric H held fixed, and it is unbounded below once H is allowed to vary, so the model’s intended vacuum was a saddle. The repaired potential was checked with all fields dynamical, and the toy-model note and its lattice script now name the simulation as a frozen-H reduction.

The verdict: a conditional pass

Mon, 01 Jan 0001 00:00:00 +0000

The verdict of the tuning test is a conditional pass. The potential that gives the momentum graviton X its mass carries the Boulware–Deser ghost when it is written in metric, or second-order, variables, and it is ghost-free when it is written in frame, or first-order, variables. The condition is the choice of variables, and the calc notes argue that the first-order formulation is the natural one anyway, because the polarization J and the symplectic vielbein E are both first-order objects. Before the test, calc note #6 had spelled out what a failure would leave: only a phenomenological shell of gated self-interacting dark matter and a dark-energy field on the Maslov circle, with the geometric story demoted to heuristics. Checkpoint #11 tags the tuning test [D][V]. The machine checks cover the metric-tier failures; the frame-tier pass is a structural argument with no test, and its potential was selected rather than derived.

The X-condensate candidate

Mon, 01 Jan 0001 00:00:00 +0000

The X-condensate candidate is the proposal that dark matter is a coherent, spatially uniform oscillation of X, the program’s momentum graviton: the massive spin-2 field that the broken reciprocity adds to gravity. If it holds, dark matter and the hidden half of the geometry are one field rather than two separate things. The idea was logged during the Debt-1 work as a direction deliberately left unopened, became one of three dark-matter candidates in version 2.0 of the checkpoint, and has been one of two since the third was retired. The record holds its profile and its history, not a calculation; like the χ-particle candidate, its abundance is unexplained.

Why X must be massive

Mon, 01 Jan 0001 00:00:00 +0000

The forcing chain is the sequence of arguments in checkpoint #11 §7.1 that leaves the momentum graviton X one consistent option: to be a massive spin-2 field. Shift symmetry rules out an auxiliary X, a field with no motion of its own. The dual diffeomorphisms make X a second graviton-like gauge field. If that gauge symmetry stays unbroken, X is a second massless graviton, and a theorem excludes two massless gravitons that interact; X has to interact, with the metric and with dark matter. If the gauge symmetry is dropped from the kinetic term, the time–space components of a symmetric tensor carry kinetic energy of the wrong sign in spacetime’s signature, a ghost. Mass is the exit that remains, and it comes from the vacuum’s choice of polarization. The chain replaced an earlier three-way question about X (healthy, auxiliary, or a ghost that would have killed the program’s geometric sector) with the answer “massive spin-2, conditionally healthy”, the condition being the tuning test. It is derived and rests on literature theorems [D + T]; no script checks it.

τ: hilltop quintessence on the Maslov circle

Mon, 01 Jan 0001 00:00:00 +0000

The program identifies dark energy with τ, the angle that sets how the vacuum divides phase space into position and momentum. Because τ is an angle, its potential energy must repeat when τ goes once around its circle, and the program writes it as a cosine, V = V₀[1 + cos(τ/f)]. Today τ is taken to sit near the top of that potential, where its energy is almost all potential energy and it acts almost like a cosmological constant, while drifting slowly away from the top, so that dark energy weakens over time and its equation of state w never falls below −1. Thawing hilltop quintessence of this kind is an established class of models; the program’s contribution is the identification of the field, which the popular article calls “the angle of the mirror”. The claim stands under real pressure: the second data release of the Dark Energy Spectroscopic Instrument (DESI) favours a phase with w below −1, which τ cannot produce, and the next release is the decisive test.

Bounds on fuzzy dark matter

Mon, 01 Jan 0001 00:00:00 +0000

The bounds on fuzzy dark matter are observational lower limits on the mass of wave dark matter at its light extreme, where the particles are so light that their de Broglie wavelength is comparable to the core of a small galaxy. Two kinds of observation set them. In a fuzzy halo the interfering dark-matter waves form density fluctuations that heat stars over time, so the compact, cold stellar populations of ultra-faint dwarf galaxies exclude the lightest masses; and fuzzy dark matter suppresses small-scale structure, which the Lyman-α forest — the absorption lines of intergalactic hydrogen in quasar spectra — would reveal. In the Broken-Born program these bounds set the lower edge of the wave island and close the light-mass escape from the falsification of Scenario A.

Complex-model vacua and the polarization Δ(β)

Mon, 01 Jan 0001 00:00:00 +0000

The vacua of the complex model II are its configurations of lowest action. At the parameters the toy-model note uses, μ² = 4 and λ = 1, the two lowest oscillator levels condense: the diagonal matrix entries φ₀₀ and φ₁₁ take fixed nonzero magnitudes, and their relative phase β is free, so the vacua form a circle. The duality shifts β by π, half-way round the circle, so it maps every vacuum to a different one and leaves none unchanged: classically, the vacuum breaks the exchange of position and momentum. The polarization Δ(β) measures how much broader the vacuum’s profile is in position than in momentum. It varies as cos β, so the β = 0 vacuum is broad in position and narrow in momentum and the β = π vacuum is the reverse, as a narrow pulse has a broad spectrum and a broad pulse a narrow one. The sign was written reversed in the first draft and corrected the same day. The breaking is classical only: quantization restores the symmetry.

Constrained bimetric gravity

Mon, 01 Jan 0001 00:00:00 +0000

Constrained bimetric gravity is the picture of the program’s gravity that the tuning test leaves. Hassan–Rosen bimetric gravity is a ghost-free theory of two interacting metrics, each with its own dynamics. In the program’s version the second frame is not a free field: it is the polarization frame, built from the polarization J, and J ranges only over the compact space Λ(d) of splittings of phase space into position and momentum, which is ten-dimensional at d = 4. Calc note #7 calls the result a predictive truncation of bimetric theory, with fewer free functions. In it the massless combination of the two spin-2 fields is the ordinary graviton, the massive one is the momentum graviton X, and the light direction of the compact space is the dark-energy field τ. Reciprocity then restricts the potential to its palindromic branch, and with the condition for a flat vacuum two free parameters remain. The picture is a structural derivation; the standard bimetric facts it uses come from the literature, and nothing in it is machine-checked.

Corollaries: unimodularity, no B-field, slaved momentum metric

Mon, 01 Jan 0001 00:00:00 +0000

Three consequences follow from the compatibility theorem’s answer H = diag(g, g⁻¹). The determinant of H is 1, so the phase-space volume that H measures cannot change, which the program reads as the constancy of Planck’s constant built into the geometry. The off-diagonal block, where double field theory keeps its antisymmetric B-field, must vanish. And the momentum metric is fixed as the inverse of the position metric, g̃ = g⁻¹, so momentum space has no geometry of its own. That last corollary contradicted the program as it then stood: the dark-energy field and the geometric origin of the dark-matter gate both disappeared. It forced the postulate change to P2′, under which the block form, and with it the slaved momentum metric, holds only on the vacuum slice of the modular field.

Fifth-force safety of τ

Mon, 01 Jan 0001 00:00:00 +0000

Fifth-force safety is the requirement that a new light field not produce a detectable extra force between ordinary bodies. The program’s dark-energy field τ is extremely light, so if it coupled to ordinary matter through its value, as the gravitational potential does, it would add a long-range force of the kind that torsion-balance experiments search for. The lab notebook argues that τ is an approximate Goldstone field, one whose couplings are fixed by a symmetry under shifting its value, so it couples to matter only through its rate of change, which produces no such static force between ordinary bodies. The argument appears only in the lab notebook, in its first version; the current checkpoint does not restate it, and no script checks it.

Force boosts and force composition

Mon, 01 Jan 0001 00:00:00 +0000

A force boost is the momentum-space counterpart of a Lorentz boost: where Einstein’s boost relates observers moving at constant velocity, the force boost of #13 relates an observer under no force to one under a constant force, with the maximum force c̃ in the role of the speed of light. The same algebra then gives forces that combine like relativistic velocities, so that no composition of forces passes c̃, and observers under different forces who disagree about which events are simultaneous when the events differ in momentum. At what #13 calls the strongest sustained macroscopic force in nature, on a kilogram at a neutron-star surface, the correction is of order 10⁻⁶⁵, which #13 gives as the reason force looks exactly additive; only merging black holes approach the limit. The results are alive but unreviewed: #13 tags them as derived, but no review round or script has checked them, the checkpoint #11 does not mention them, and no observable signature in merger data has been derived.

Founding suppositions and their fates

Mon, 01 Jan 0001 00:00:00 +0000

The founding suppositions are the eleven ideas P.R. brought to the program at its start. Appendix A of the checkpoint (#11) lists each with the fate the later work gave it. One became the program’s hard core. Others survived in changed form: the conjugate universe became a second description of one state, conjugate matter became a dark-matter candidate that clusters in space like any other, and the idea that momentum space has its own gravity became a heavy, hidden field. One was promoted to a theorem, one split into a dead half and a living half, one was already established physics, and two died, including the founding image of dark matter clumping in velocity rather than in space. The audit is the record of which parts of the starting picture the calculations kept.

Hassan–Rosen bimetric gravity

Mon, 01 Jan 0001 00:00:00 +0000

Hassan–Rosen bimetric gravity is a theory of two dynamical metrics, g and f, each with its own Einstein–Hilbert term, coupled through a potential that keeps the Boulware–Deser ghost out. The potential is a sum of five terms with constant coefficients β₀ to β₄, the n-th term built from the elementary symmetric polynomial eₙ of the matrix √(g⁻¹f). An identity of these polynomials makes the exchange of the two metrics reverse the order of the terms, so the list of coefficients is read backwards. About proportional backgrounds the theory has one massless and one massive spin-2 field. The program’s geometric sector, the home of its momentum graviton X, lands in this theory with the second frame confined to a compact space, and Born reciprocity, acting as the exchange of the two frames, then forces the list of coefficients to read the same in both directions, the palindrome βₙ = β₄₋ₙ.

Monopole flux bounds: Parker and MACRO

Mon, 01 Jan 0001 00:00:00 +0000

The monopole flux bounds are upper limits on how many magnetic monopoles pass through the Galaxy, stated as a flux per square centimetre per second per steradian. The Parker bound comes from the Galaxy’s magnetic field: monopoles moving through it would gain energy from the field and drain it faster than the galactic dynamo regenerates it, so the field’s survival caps the flux; the extended Parker bound applies the same argument to the weaker seed field of the young Galaxy. Direct searches, among them the Monopole, Astrophysics and Cosmic Ray Observatory (MACRO) whose limit the withdrawn #14 quotes, have found none. The program used these bounds only counterfactually, to show that the retired soliton candidate could not have been magnetically charged under our electromagnetism.

Obituary: conjugate monopoles explain the null result

Mon, 01 Jan 0001 00:00:00 +0000

This was the claim that magnetic monopoles exist but belong to the conjugate, momentum-polarized half of physics, which would explain why no experiment has found one. It was the program’s founding supposition 5. The checkpoint’s audit of its monopole claims split the supposition in two and killed this explanatory half on two counts: the symmetry that exchanges electric and magnetic charge is electric–magnetic duality, not the position–momentum reciprocity the program is built on, and inflation already accounts for the absence of monopoles without new physics. The claim is dead and stays dead. The other half, that such monopoles may exist and would gravitate, survives on a separate page, and so does the prediction that no monopole of our sector will be found.

Obituary: the Kibble normalization

Mon, 01 Jan 0001 00:00:00 +0000

The Kibble normalization is the number p in the estimate of how many monopoles a symmetry-breaking phase transition in the early universe leaves behind: the yield n/s = p (T_c/M_Pl)³, the number of monopoles per unit entropy, where T_c is the transition temperature and M_Pl the Planck mass. Version 3 of the soliton manuscript #14 used p ≈ 0.023 and treated p = 1 as an upper end. The round-3 referee showed that 0.023 is what the formula gives when the expansion rate of the universe is written without its numerical prefactor; with the prefactor, one monopole per Hubble volume gives p ≈ 104. Because causality requires at least about that many, the value is a minimum, not a maximum. The correction moved the mass at which the monopoles would make up all the dark matter from v3’s (0.9–3.7)×10¹³ GeV down to 2.9–4.4×10¹² GeV, and the candidate was retired.

Open: polarization-genesis and the 5.4 : 1 ratio (OP7)

Mon, 01 Jan 0001 00:00:00 +0000

Polarization-genesis is the program’s name for a mechanism it does not yet have: the process that would set how much dark matter the early universe made compared with ordinary matter, whose densities stand in the observed ratio Ω_χ : Ω_b = 5.4 : 1. For particle χ the lab notebook argues that the standard route is closed: a light sector that interacts strongly with itself and touches ordinary matter only through gravity cannot get its abundance from thermal freeze-out, so the abundance has to come from an asymmetry, as the abundance of ordinary baryons does. The notebook sketched a mechanism in which heavy fields decay into both kinds of matter, with the split set by an angle of the vacuum’s polarization, and called it “a shape, not a mechanism”. The problem is open and applies to both remaining dark-matter candidates.

Open: the † literature battery (OP10)

Mon, 01 Jan 0001 00:00:00 +0000

The literature battery is the planned check of every reference that the program’s papers cite from model memory, marked †, against the published source. A † citation was recalled by Claude rather than retrieved, so its authors, title, journal reference, or the statement it is cited for may be wrong. Checkpoint #11 lists the battery as open problem 10 of its §14 and calls itself “not for submission until the † audit (open item 10) completes”. Some of the works involved have been checked in other contexts, and one suspected error in #11’s list has been flagged, but no complete battery is on record. It is part of the first item of the current work queue.

P3: a reciprocity-invariant action

Mon, 01 Jan 0001 00:00:00 +0000

P3 is the program’s symmetry axiom: the action, the single expression from which the field equations follow, is unchanged by the Born rotation, the quarter-turn of phase space that sends position to momentum and momentum to minus position. The rotation is generated by the complex structure I = ω⁻¹H built from the two structures of P1. In ordinary quantum mechanics this exchange is a change of description, like switching from a waveform to its spectrum: a unitary map between equivalent ways of writing the same physics. P3 claims more, a symmetry of a specific action, which is the kind of symmetry a vacuum can break spontaneously. The referees asked for an explicit action with that property, and the coset toy model supplied one in a simplified setting.

Review rounds 1 and 2

Mon, 01 Jan 0001 00:00:00 +0000

Review rounds 1 and 2 were the program’s first two blind reviews, both of #14, the standalone manuscript proposing superheavy hidden-sector solitons as dark matter. Both ran through the skeptic review console: a fresh referee with web search and no code execution. Round 1 graded version 1 B, flawed but salvageable, with medium confidence, and every valid finding was folded into version 2; round 2 graded version 2 “B near B/C” and called the derivation of the hidden coupling numerology. Round 2’s findings became the pending list R1–R7, from which version 3 was cut for round 3. The rounds are known only from the handoff’s summary: the referees’ reports and the full adjudications were never brought into the repository.

Row 4: thawing dark energy, never phantom

Mon, 01 Jan 0001 00:00:00 +0000

Row 4 is the program’s prediction for how dark energy, the cause of the accelerating cosmic expansion, changes over time. The program identifies dark energy with τ, the angle that locates the vacuum’s choice of polarization on a circle; τ has a periodic potential and starts near its maximum, a setup called hilltop quintessence. Such a field makes dark energy weaken slowly ("thaw") and keeps its equation of state w at or above −1, the value of a cosmological constant, so it can never enter the “phantom” regime w < −1. The test is the Dark Energy Spectroscopic Instrument (DESI). Its Data Release 3 (DR3), expected in late 2026, is the decisive one. Its Data Release 2 (DR2) already favours a phantom phase in the past, so the sector is under real pressure. Row 4 is one of the three rows the checkpoint names as the program’s near-term falsification exposure.

Superseded statements

Mon, 01 Jan 0001 00:00:00 +0000

This register lists the statements the program once made and later replaced with corrected versions. A superseded statement differs from a dead claim: the subject survives, but its earlier form is no longer the program’s position, often because a calculation or a review forced a repair. Each line names the old statement, what replaced it and where the replacement is recorded. The older documents are kept as an audit trail and still contain the old forms: the lab notebook, for example, keeps its version-0.1 body with the corrections appended as addenda, so the old forms read there as if they were current.

The bounded potential V_b

Mon, 01 Jan 0001 00:00:00 +0000

V_b is the repaired ρ-preserving potential of model I: the naive potential divided by a normalizing factor, 2 Tr KᵀK, which keeps it between −μ and 0 for every real structure K. Like the naive form it is unchanged by Born reciprocity ρ and by the flip K → −K, and it reduces the circle of real structures to the pair ±K_ref. Unlike the naive form, it leaves the vacuum a stable minimum when the metric H is free to vary: the boosted configurations along which the naive form fell without limit now cost energy. The round-4 referee proposed it, version 2 of the toy-model note adopted it, and checkpoint #11 records acceptance criterion 1 as met with it, in the referees’ wording. One flat direction remains that no symmetry protects, and where such a potential comes from is not derived.

The classicality theorem and the particle island

Mon, 01 Jan 0001 00:00:00 +0000

The classicality theorem says that dark matter coupled strongly enough to clump in velocity must be so light that it behaves as a wave, not as a collection of separate particles. It follows from the mean-field criterion: at the scattering strength that fits dwarf-galaxy cores, the criterion can be met only for masses below about 0.1 meV, and at such masses about 10²⁰ quanta share each quantum state, which is the regime of a classical wave. The program therefore splits dark matter into two regimes: a particle island, with χ masses of MeV to GeV, where dark matter scatters but cannot clump, and a wave island of far lighter masses. The theorem is derived at mean-field level and stands.

The Debt-1 calc notes (#5–#8)

Mon, 01 Jan 0001 00:00:00 +0000

The Debt-1 calc notes are four working notes, artifacts #5 to #8, on the most severe of the lab notebook’s open problems: what equations govern the geometry of momentum space. The first, written overnight, showed that the original postulates left the program with no content beyond general relativity and one model of self-interacting dark matter, repaired them, and gathered the geometry into a single modular field; on the way it found the mechanism of the velocity gate. The second showed that the field carrying momentum-space geometry must be a second graviton, and a massive one. The third tested whether that field avoids the Boulware–Deser ghost, the instability that generic massive-graviton theories carry, and found that it does when the field’s potential is written in frame variables, sets of local reference axes from which a metric is assembled; written in metric variables it fails, and the note argues that the frame formulation is the natural one. The fourth used Born reciprocity to constrain the remaining coefficients, leaving two free parameters, and to cancel the sector’s vacuum energy. Together they closed Debt 1 in structure; their results are now stated in checkpoint #11 §5–§8, and the battery checks their explicit algebra.

The gravitational Higgs mechanism

Mon, 01 Jan 0001 00:00:00 +0000

The gravitational Higgs mechanism is how the momentum graviton X acquires its mass in this program. The vacuum value of the modular field, ⟨Z⟩ = i·g, sits at X = 0, and the local shifts of X, the dual diffeomorphisms, move it away from there. So when the vacuum selects a polarization, a split of phase space into position and momentum, those gauge symmetries are broken spontaneously. In the ordinary Higgs mechanism, a gauge field absorbs the would-be massless modes of a broken symmetry and becomes massive; here X absorbs the would-be Goldstone modes of the broken dual diffeomorphisms and becomes a massive spin-2 field with five polarizations. Its mass M_X is the scale of the polarization breaking, and no document gives it a value. The mechanism is a structural derivation [D — structural]; whether the mass term it produces is free of ghosts was a separate question, the tuning test.

The Monte Carlo error-bar rule

Mon, 01 Jan 0001 00:00:00 +0000

The Monte Carlo error-bar rule says that a number produced by a random simulation must come with an error bar estimated from several independent runs. A simulation started from a fixed random seed gives the same output every time it is run, which makes the number reproducible but says nothing about how far it lies from the average the simulation is estimating. Review round 4 found the case that prompted the rule: the high-temperature control values in the toy-model note came from single fixed-seed runs without error bars, and the smallest lattice’s value was a short-run fluctuation. The lattice script now reports means and standard errors over four independent seeds, and its checks use those error bars.

The order parameter: J, η and K

Mon, 01 Jan 0001 00:00:00 +0000

An order parameter is the quantity that tells a broken-symmetry state from a symmetric one, as a magnet’s magnetization does: it is zero while the symmetry is intact, and once the symmetry is broken it takes a definite value that the symmetry would change. In the Broken-Born program that role belongs to the neutral metric η, the structure on phase space that marks which directions count as position and which as momentum. Born reciprocity reverses the sign of η, so a vacuum that carries a definite η is not reciprocity-symmetric, and its mirror image carries −η. The program has written this order parameter in three notations over its history, using J, η and K, and the documents do not agree on which of J and K to call “the real structure”. An early description of the breaking as a discrete pattern has been replaced by the Goldstone-angle picture.

The scattering benchmark

Mon, 01 Jan 0001 00:00:00 +0000

The scattering benchmark is the set of numbers obtained by requiring the self-scattering of conjugate matter to produce the flat central cores observed in dwarf galaxies. Fitting the cores fixes the cross-section per unit mass at low velocity; with an assumed mass for the carrier of the momentum transfer, that fixes the mass of a χ particle at about 70 MeV for a coupling of order one. The checkpoint keeps only this mass relation. The other benchmark numbers, among them the prediction that galaxy clusters show no self-interaction at any sensitivity, appear only in the lab notebook, rest on a Gaussian gate tail that is now an open question, and have never been checked by machine. The benchmark also fixes the coupling that the program’s Debt-3 calculation used to rule out collective clumping of χ particles.

The turnaround as the conjugate bounce

Mon, 01 Jan 0001 00:00:00 +0000

The turnaround is a far-future event the program predicts: our universe stops expanding and begins to contract, which happens if the potential energy of the dark-energy field τ becomes negative. Because our scale factor and that of the conjugate universe are reciprocals (the cosmological constraint a·ã = 1), the moment our universe is largest is the moment its momentum-space description is smallest, so the same event, read in momentum variables, is a contracting universe reaching its minimum size and rebounding. The program states this as one event with two descriptions, not as two linked events. It is a long-horizon prediction whose near-term proxy is the dark-energy drift that survey data are testing now, and whether τ’s potential, as the documents write it, can become negative at all is an unresolved conflict in the sources.

Unitary equivalence versus Noether symmetry

Mon, 01 Jan 0001 00:00:00 +0000

Unitary equivalence and Noether symmetry are the two different ways a map like Born reciprocity can relate to a physical theory. It can be a change of description, a unitary equivalence — the same physics written in other variables, as a Fourier transform rewrites a waveform as a spectrum — or it can be a symmetry of the dynamics: an operation that leaves a specific action unchanged, and therefore constrains what the theory can do (a Noether symmetry). In ordinary quantum mechanics ρ is the first kind, because the Stone–von Neumann theorem makes all the standard ways of representing the position–momentum commutation rules equivalent. The program needs the second kind, since only a symmetry of an action can be spontaneously broken. Its referees made exhibiting such an action the first acceptance criterion for anything it proposes, and #11 records the criterion as met, in the referees’ wording, by the toy models.

τ as the Maslov phase of the boundary

Mon, 01 Jan 0001 00:00:00 +0000

The program’s dark-energy field τ is an angle: the angle around the Maslov circle of the Lagrangian Grassmannian, the space of all position/momentum splittings, which forms the distinguished (Shilov) boundary of the Siegel domain where the modular field takes its values. Because that circle cannot be shrunk to a point, τ is periodic, with period 2πf, as a matter of topology rather than of modeling. Of the ten directions of the Grassmannian at d = 4, τ is assumed to be the only light one. The identification is derived [D], the topology is literature [T], and the lightness is a postulate [P]. How τ behaves as dark energy, and how current data bear on it, is worked out in the cosmology section.

Conjugate monopoles as ontology

Mon, 01 Jan 0001 00:00:00 +0000

Conjugate monopoles are hypothetical magnetic monopoles, or other topological solitons, belonging to the program’s momentum-polarized matter sector rather than to ours. The checkpoint keeps one statement about them: if they exist, they gravitate, because every kind of matter in the program couples to the same geometry, so their existence could not be without consequence. Their use as the dark matter itself, the superheavy soliton candidate, was retired on 2026-10-19 after its abundance arithmetic failed review. What remains is a statement about what such objects would do, not a claim that they exist.

Elementary symmetric polynomials

Mon, 01 Jan 0001 00:00:00 +0000

The elementary symmetric polynomials of a list of numbers are the sums of all products of distinct entries taken n at a time: e₁ is the sum, e₂ the sum of all pairwise products, and so on up to the product of all the entries, with e₀ = 1. Applied to the eigenvalues of the matrix √(g⁻¹f), built from the two metrics of bimetric gravity, they are the building blocks of the ghost-free bimetric potential, which is a linear combination of them with constant coefficients. Power sums beyond the first, such as p₄, the sum of fourth powers, can be written in terms of the eₙ by Newton’s identities, but only as nonlinear polynomials, so they lie outside that family. The program’s verification battery checks two facts of this algebra: the linear coefficients of the eₙ near the identity matrix, which give the tadpole of the program’s potential (test T6), and that p₄ is not a linear combination of the eₙ (test T7).

Force dilation

Mon, 01 Jan 0001 00:00:00 +0000

Force dilation is the momentum-space counterpart of time dilation. In special relativity a fast-moving clock runs slow; in the dual kinematics of #13, a system under a large sustained force evolves slowly, by the same square-root formula with force in place of velocity and the maximum force c̃ in place of the speed of light. Its mirror is a contraction of the range of momenta a forced system spans, in place of the contraction of its length. #13 restricts the effect to systems of the conjugate (momentum) polarization, leaves open which physical clocks respond, and states that both effects are unmeasurably small below the forces of black-hole mergers. The result is alive but unreviewed: #13 tags it as derived, but no review round or script has checked it, and the checkpoint #11 does not mention it.

Lagrangian subspaces and polarizations

Mon, 01 Jan 0001 00:00:00 +0000

A Lagrangian subspace of phase space is a set of directions, half the dimension of phase space, on which the symplectic form vanishes: no two of its directions form a position–momentum pair. Choosing one, a polarization, is choosing which coordinates count as “position”. The ordinary position axis is one Lagrangian subspace, the momentum axis is another, and a continuous family of tilted ones lies in between. A quantum state can be written as a wavefunction over any of them, and the different descriptions are related by unitary transforms such as the Fourier transform. The program’s postulates say that the vacuum carries one such choice, and that ours is tilted toward position.

Negative results and dead claims

Mon, 01 Jan 0001 00:00:00 +0000

The program’s rule on negative results is that a claim killed by calculation or by review is reported in an obituary of its own and is not revived. Checkpoint #11 ranks such negative results as the most reliable of its new structural results. The founding image of the program, dark matter clumping in velocity rather than in space, is one of the dead claims, and so is a later dark-matter candidate, superheavy solitons of a hidden gauge sector. The wiki gives every dead claim its own obituary page and lists them all in the graveyard.

Open: dual kinetic energy versus τ's potential

Mon, 01 Jan 0001 00:00:00 +0000

This open problem is a gap between two explanations of dark energy that stand side by side in the program’s lab notebook. The first comes from the duality dictionary: kinetic energy in the conjugate, momentum-space description of the universe (the conjugate universe) acts on our expansion like a cosmological constant, so the notebook names it the natural source of a near-constant dark energy. The second is the explanation the program uses: dark energy is the potential energy of the field τ (hilltop quintessence), held near the top of its potential. No document derives the second from the first, so the step from the motivation to the identification of dark energy with τ is asserted rather than derived.

Open: field content of the broken vacuum

Mon, 01 Jan 0001 00:00:00 +0000

The field content of the broken vacuum is the list of fields that the program’s geometry leaves in play once the vacuum has chosen a polarization: the ordinary metric, the momentum graviton X, the ten moduli of the Lagrangian Grassmannian (the dark-energy field τ plus nine more), and matter. Calc note #5 wrote the list down and marked it partial. The later Debt-1 notes settled what X is: a massive spin-2 field. Whether the nine extra moduli survive as fields, or are removed by inverse-Higgs constraints (relations that turn some Goldstone-type fields into derivatives of others), was never worked out. The question is open, and it is not on the checkpoint’s numbered list of open problems.

Open: Lorentzian dressing of the Euclidean skeleton (OP3)

Mon, 01 Jan 0001 00:00:00 +0000

The Lorentzian dressing of the Euclidean skeleton is the open problem of carrying the program’s main geometric results over from a positive-definite phase-space metric, the Euclidean case, to the Lorentzian signature of physical spacetime, in which time enters with the opposite sign to space. Those results, the compatibility theorem, the parametrization of the Siegel domain and the chain of arguments that forces the momentum graviton X to be massive, were proven in the Euclidean case. The extension is open problem 3 of the checkpoint #11. The change is not a formality: with a Lorentzian metric the group that leaves the vacuum fixed becomes non-compact, and the metric on the space of field values becomes indefinite, the situation in which some field components would carry kinetic energy of the wrong sign. The working notes expect the gauge symmetry and mass mechanism already in the chain to keep the theory healthy there, and list the extension as a check. Nothing has been computed, and the checkpoint counts it among the three items on which the word “theory” still waits.

Open: the gate shape (OP4)

Mon, 01 Jan 0001 00:00:00 +0000

The gate shape is the exact form of the function F that turns off the self-interaction of conjugate matter at large relative velocity. A Gaussian F falls off faster than any power of the velocity; a rational F, a ratio of polynomials, falls off as a power law. Which one holds depends on how the momentum graviton X propagates along the momentum directions of phase space, and on the profile of a single conjugate quantum along those directions; neither has been computed. The question is open, and it is observable: the tail of the knee in the self-interaction cross-section is the same function. The checkpoint counts the tail, not yet forced by any derivation, among the program’s risks of degeneration.

Overproduction at the benchmark

Mon, 01 Jan 0001 00:00:00 +0000

Overproduction at the benchmark is the corrected abundance result that retired the soliton candidate. If one monopole forms per Hubble volume when the hidden symmetry breaks, the least that causality allows, the monopoles match the observed dark-matter density only for a mass M* of 2.9–4.4×10¹² GeV. Heavier monopoles are overproduced in proportion to the fourth power of their mass, because a heavier monopole forms at a higher temperature, where more of them form per unit entropy. At the benchmark of #14 v3, 3×10¹³ GeV, they would exceed the observed density 2.1×10³ to 1.2×10⁴ times, and realistic production at the transition makes the excess larger still. The result stands and is checked by the program’s verification gate. It is why a superheavy monopole could survive only with an arbitrary dilution factor, which would remove its mass prediction.

P4: polarized matter and strict sector separation

Mon, 01 Jan 0001 00:00:00 +0000

P4 is the program’s axiom about matter. Ordinary matter, the Standard Model, is x-polarized: its fields are organized around position, the variable in which its interactions are local. A second kind, conjugate matter χ, is p-polarized, organized around momentum. Both kinds respond to the same gravity, so the equivalence principle holds across them, but no gauge force (the electromagnetic, weak or strong interaction) acts between them. Since checkpoint v2.1 the program reads the last clause strictly: the only mixing allowed between the two sectors is what gravity itself induces, which is zero or extremely small. P4 is the basis of the program’s prediction that dark-matter direct-detection experiments will see nothing, and in the toy models it is also what gives the vacuum’s choice of polarization a physical effect.

Quantum restoration in the complex model

Mon, 01 Jan 0001 00:00:00 +0000

Quantum restoration is the result that the classical breaking found in the complex model II does not survive quantization. The vacua form a circle labelled by the relative phase β of two condensed oscillator levels, and an exact symmetry turns the circle at no cost in action, so the quantum theory weights every value of β equally. A spontaneous choice can still survive in a system large enough that a vanishingly small bias, acting over its whole volume, pins it. This condensate occupies two oscillator levels and is a localized lump, so it supplies no such volume factor. The quantum average of the polarization is therefore zero, and criterion S4, survival of the breaking under quantum fluctuations (success criteria), fails for this route. Round 4 confirmed the conclusion, asked for the argument to be restated on these two grounds, and added that the bare mass must be renormalized as the truncation grows. The comparison with an exactly solved related model, the natural independent check, has not been made.

Row 5: the fate of the expansion

Mon, 01 Jan 0001 00:00:00 +0000

Row 5 is the program’s prediction for the far future of the cosmic expansion. With a constant dark energy, standard cosmology expands forever toward an empty de Sitter universe. In the program, the expansion halts and reverses once the potential energy V(τ) of the dark-energy field τ becomes negative. Through the cosmological constraint a·ã = 1, which ties our scale factor a to the scale factor ã of the conjugate description, the same event is the conjugate universe’s bounce: “our maximum is their minimum”. The turnaround lies beyond any near-term observation, so #11 names row 4 as its near-term test. Whether the potential as the documents write it can go negative at all is unresolved: as written, it never does.

The 'framework does no work' critique

Mon, 01 Jan 0001 00:00:00 +0000

“The framework does no work” is the referees’ objection that the program’s phase-space framework, Born reciprocity included, contributed nothing to the results of its dark-matter proposal. It was stated most fully in the third review round, against the withdrawn manuscript #14, which proposed superheavy monopoles of a hidden gauge sector. The referee found that every number in it came from standard hidden-sector physics plus one choice of coupling, which the framework motivated but did not fix. The objection shaped what came next. The program retired the candidate rather than claim a version of it that did not depend on the framework, and it made its next task an explicit model: an action in which reciprocity is a genuine symmetry that the vacuum breaks. The toy models meet the referees’ wording of that demand, but by the program’s assessment they answer the objection only in part, because the locality of ordinary matter in position still enters through an assumed coupling rather than following from the vacuum.

The checkpoint: #9 to #11 v2.2

Mon, 01 Jan 0001 00:00:00 +0000

The checkpoint is the program’s paper-length statement of everything it has derived, predicted, retracted and left open, with every claim tagged by where it came from. Its first version, #9 (“The Broken-Born Program” v1.0), consolidated the lab notebook and the calc notes into one document with a plain-language passage opening every section. Its successor, #11, is the current reference for the full program. Version 2.0 added a methodological preface with a machine-verification battery, sections on charge lattices and on the dark-sector candidates, a reference list and an audit of the founding ideas; version 2.1 recorded the outcome of review round 3, retiring the soliton candidate and putting the dark-energy prediction under pressure; version 2.2 recorded the toy-model results. #11 is a working checkpoint, not to be submitted until its citations from memory have been checked against their sources.

The frozen-H reduction

Mon, 01 Jan 0001 00:00:00 +0000

The frozen-H reduction is model I with its metric field held fixed at H = 1, so that the only remaining field is the polarization angle φ of the real structure K(φ). In that form the model is a two-dimensional spin model of one angle, with a stiffness term and a ρ-preserving potential, and it can be simulated on a lattice. The ordered ℤ₂ phase the simulation finds is a result about this reduction; with H dynamical, ordering is not established. Round 4 required the text to say so, after showing that the first potential tried was stable only in this reduction. It also replaced the claim of “Ising-class” breaking with “an ordered ℤ₂ phase”, because data at one temperature establish order but not a universality class.

The gate: scripts/check.sh

Mon, 01 Jan 0001 00:00:00 +0000

The gate is the shell script scripts/check.sh, which runs the repository’s six verification scripts in verify/ and then the wiki’s own consistency check, and exits with a failure if any check fails. The project calls it “the definition of done”: work is not finished until it passes, and a check may not be weakened to make it pass. Each verification script prints one line per check, a pass or a fail together with the measured value where there is one, so the record of a run shows what was tested as well as the verdict. What the gate tests is the arithmetic of the models the scripts encode and the form of the wiki; which claims it covers, and which it leaves to derivation, review or nothing, is listed below.

The geometric action S_H and the v0.1 action (superseded)

Mon, 01 Jan 0001 00:00:00 +0000

The v0.1 action is the complete action written in the program’s lab notebook in September 2026: a sum of five terms, for the phase-space geometry, the dark-energy angle τ, the Standard Model, conjugate matter, and an unspecified mixing term. Its geometric term S_H is an Einstein-like action for the phase-space metric H, built from the curvature of the Born connection and written with both the symplectic form ω and the neutral metric η frozen, as the original P2 required. From that rigidity the notebook argued that no second independent metric, and so no Boulware–Deser ghost, could arise. Both points were later overturned: postulate P2′ made η emergent, the momentum graviton X was found to be a second graviton, and ghost freedom now comes from the frame tier. This form of the action is therefore superseded, and checkpoint #11 writes no full action. The name S_H survives as the action from which open problem 1 asks for the ghost-free potential to be derived.

The no-phantom bound w ≥ −1

Mon, 01 Jan 0001 00:00:00 +0000

The no-phantom bound is the program’s requirement that the dark-energy equation of state w never fall below −1, the value of a cosmological constant. It holds because the program’s dark-energy field τ is an ordinary (canonical) scalar field, whose energy density can fall or hold steady as space expands but never rise, while w < −1, the phantom regime, would make it rise. The bound makes the dark-energy sector a sharp test: a robust measurement of an intrinsic w < −1 would kill the program’s framework outright. The second data release of the Dark Energy Spectroscopic Instrument (DESI) favours a phase with w < −1 at redshifts above about 0.4, so the bound stands against the direct reading of current data and is under real pressure; it survives only because fits without a phantom phase remain compatible and because w is inferred rather than observed.

The polarization as a Goldstone angle

Mon, 01 Jan 0001 00:00:00 +0000

The polarization as a Goldstone angle is the toy models’ finding that the vacuum’s polarization, which fixes the direction in the position–momentum plane that counts as position, is an angle that can turn smoothly rather than a switch with two settings. Born reciprocity is a quarter turn of that plane, the operation the Fourier transform performs when it turns a waveform into its spectrum, and in the toy models it is one member of a continuous family of symmetries. Turning the angle costs no energy, which makes it a Goldstone direction: the position-sharp and momentum-sharp vacua lie on one continuous family. This finding, recorded in #11 v2.2, is why a definite polarization needs an added potential on the angle. In the coset model the angle is the same circle that the program identifies with its dark-energy field τ, which shows that the structures fit together but does not test the cosmology. It is evidence from two toy models, not a theorem.

τ is not eaten

Mon, 01 Jan 0001 00:00:00 +0000

This result says that the dark-energy field τ is not one of the modes the momentum graviton X absorbs when it becomes massive. τ is an angle on the Lagrangian Grassmannian Λ(d), the compact space of all the ways of splitting phase space into position and momentum: it is the Maslov phase of the vacuum’s polarization. The modes X absorbs belong to a different sector, the gauge directions of the broken dual diffeomorphisms, which shift X; constant shifts of X do not generate the compact directions in which τ lies. So τ survives the gravitational Higgs mechanism as the light field the program’s dark-energy sector needs. Calc note #6 flagged the point as open; calc note #8 derived it at leading order [D — leading order]. How strongly τ still mixes with the absorbed modes through the potential has not been computed and is an open problem.

Assessment: a live research program

Mon, 01 Jan 0001 00:00:00 +0000

The assessment is the checkpoint’s own judgement of where the program stands, written in the terms of Imre Lakatos’s account of research programmes: a hard core that is kept fixed, auxiliary claims that may be revised, and the question of whether the program keeps producing new predictions (progressive) or only accommodates its failures (degenerating). Checkpoint #11 v2.2 finds the hard core intact and sharpened twice by corrections that later derivations forced. It lists the signs of progress and the risks of degeneration side by side, it names three predictions that observations could falsify soon, and it notes that the current dark-energy data already lean against one of them. Its verdict is that this is a live research program, and that the word “theory” waits on three open problems.

DESI DR2's phantom crossing

Mon, 01 Jan 0001 00:00:00 +0000

DESI DR2’s phantom crossing is the finding, reported in 2025 from the second data release (Data Release 2, DR2) of the Dark Energy Spectroscopic Instrument (DESI), that the dark-energy equation of state favoured by the data lies above −1 at low redshift and crosses below −1, into the phantom regime, at redshifts above about 0.4, in every combination of datasets analysed. The program requires dark energy never to be phantom, so its requirement stands against the direct reading of these data, and its dark-energy sector is under real pressure. The sector survives on two stated points: fits without a phantom phase remain compatible at about two standard deviations, and the equation of state is inferred from distance measurements rather than observed. The third data release is the decisive test.

Monte Carlo: an ordered ℤ₂ phase

Mon, 01 Jan 0001 00:00:00 +0000

The Monte Carlo is a lattice simulation of the frozen-H reduction of model I: a square grid of angles φ, one per site, with an energy that is lowest when neighbouring angles agree and, when the ρ-preserving potential is switched on, lowest at the two angles 0 and π. Random local changes are accepted or rejected by the Metropolis rule at a fixed temperature, and the lattice average m of cos φ, which changes sign under ρ, shows whether one of the two vacua has been singled out. With the potential the mean of |m| stays at 0.954 on every lattice size, an ordered ℤ₂ phase. Without it the order decays slowly with size, as a continuous symmetry in two dimensions requires, and at high temperature it falls as the inverse of the lattice size. This is criterion S4 for the reduction, with error bars from four independent seeds, adopted after round 4 found the first control numbers under-sampled.

Obituary: the √N topological floor (chat-only)

Mon, 01 Jan 0001 00:00:00 +0000

The √N topological floor was an argument, made only in the claude.ai chat that preceded this repository, about how many of the soliton candidate’s monopoles could survive annihilation in the early universe. A monopole and an antimonopole that meet can annihilate, and at the hidden sector’s strong magnetic coupling, about 34, that annihilation might have been fast enough to deplete them. The floor argument held that the Kibble mechanism leaves a net topological charge that annihilation cannot remove, so the surviving abundance would lie between a Kibble “ceiling” and that floor. It was never written into any document. In round 3 the referee showed that Kibble’s count is a minimum, not a ceiling, and that annihilation at this coupling has already been worked out for hidden sectors; with the soliton candidate retired, the planned calculation was dropped.

Open: deriving the wedge sector from S_H (OP1)

Mon, 01 Jan 0001 00:00:00 +0000

This open problem asks for the ghost-free potential of the momentum graviton X to be derived from the program’s geometric action instead of being chosen. The tuning test found that wedge polynomials of the symplectic vielbein E and the polarization frame E₀[J] form a ghost-free class (the frame tier), and argued that this is the minimal way to couple the polarization to the geometry. That argument selects the class; it does not obtain it line by line from an action. The only geometric action written out, the lab notebook’s S_H, predates the change of postulate that brought X into the theory. The problem is item 1 of the open problems in checkpoint #11 §14 [O], and one of the three items, with the partially-massless comparison and the Lorentzian dressing, on which #11 makes the word “theory” wait. Nothing has been computed toward it, and it is not on the current work queue.

Open: the dual-stress enhancement (OP5)

Mon, 01 Jan 0001 00:00:00 +0000

The dual-stress enhancement is the unexplained gap between how strongly conjugate matter scatters off itself and how weakly ordinary matter couples to the same carrier. In the program both kinds of matter couple to the momentum graviton X: conjugate matter through its dual stress, ordinary matter through the spacetime metric. Fitting the flat cores of dwarf galaxies needs a self-scattering cross-section that the checkpoint calls nuclear-scale, while ordinary matter must feel X only at gravitational strength. Something must make conjugate matter’s coupling far more effective than ordinary matter’s; what it is, and by how much, has not been worked out. The checkpoint calls it the main open quantitative item of the conjugate sector.

Open: the origin of the ρ-preserving potential

Mon, 01 Jan 0001 00:00:00 +0000

This open problem asks where the potential that fixes the polarization angle comes from. In the program’s coset toy model the vacuum’s polarization, the choice of which direction in the position–momentum plane counts as position, is an angle that can take any value at no cost in energy. In wave terms the angle runs continuously from describing a state by its waveform to describing it by its spectrum, through the intermediate descriptions that fractional Fourier transforms give. A definite polarization appears only after a potential is added that keeps Born reciprocity as a symmetry but pins the angle to one of two opposite settings. The program gives its dark-energy field τ a potential of the same kind. Such a potential brings in a fixed, unordered pair of axes, which is more fixed structure than the program’s postulate P2′ (“only ω is rigid”) provides, and nothing in the program derives it. Its first proposed origin, explicit breaking by the rigid background of the original postulate P2, did not survive the replacement of P2.

Open: the real model's one-loop preference (a lead)

Mon, 01 Jan 0001 00:00:00 +0000

The real model’s one-loop preference is the finding that, in the real-field version of model II, the first quantum correction (the one-loop term) gives the vacuum that is sharper in position a lower free energy than its mirror image, and so favours it, although the two have exactly the same classical action, so that before quantum effects are included neither is preferred. That version’s vacua are discrete: the field’s vacuum value occupies the two lowest oscillator levels, with either the same sign on both or opposite signs, and the duality, the model’s exchange of position and momentum, maps one choice to the other. The quantum fluctuations around the two vacua differ, and the lower free energy belongs to the opposite-sign vacuum, the one sharper in position than in momentum. That is a selection of the position-sharp polarization, but an explicit one: the real model is not symmetric under the duality, because its field is real in position, so nothing is broken spontaneously. The round-4 referee found the split, the adjudicator identified which vacuum it favours, and the toy-model note’s second version showed the value unchanged at three truncations, that is, with three different numbers of oscillator levels kept. It is recorded as a lead, not a result: it holds at a fixed bare mass, the mass parameter as written in the action, which itself has to be adjusted as more levels are kept, and what happens to the preference as the number of levels grows without limit, with the mass so adjusted, is not known.

Open: τ–Stückelberg mixing (OP6)

Mon, 01 Jan 0001 00:00:00 +0000

This open problem asks how strongly the dark-energy field τ mixes with the Stückelberg modes, the would-be Goldstone fields that the momentum graviton X absorbs when it becomes massive. Two fields mix when the quadratic part of a theory contains a term coupling them, so that neither has a definite mass on its own. The program’s leading-order result is that τ is not among the absorbed modes and stays light (τ is not eaten). Calc note #8 logged a residual mixing through the potential and expected it to be small “by compactness”; no document computes it. It is item 6 of the open problems in checkpoint #11 §14 [O].

P5: the vacuum selects a polarization

Mon, 01 Jan 0001 00:00:00 +0000

P5 is the program’s axiom of spontaneous breaking: the vacuum of the universe selects a polarization, a choice of which half of the phase-space directions count as position, and ours is the one organized around position in the ordinary sense (“x-tilted”). The mirror choice, a vacuum organized around momentum, satisfies the same laws equally well, because the reciprocity symmetry of P3 carries one onto the other. Which one occurs is a property of the state, not of the laws, as the axes of an ice crystal belong to the crystal rather than to the laws governing water. Freidel, Leigh and Minic had already described a preferred polarization as breaking Born’s symmetry; the program’s addition is to treat the choice as spontaneous symmetry breaking with an order parameter. The toy models qualify the axiom: with only the symplectic form fixed, the choice is a continuous angle, like an ice axis that could point anywhere, and a definite polarization needs an additional potential on that angle.

Review round 4

Mon, 01 Jan 0001 00:00:00 +0000

Review round 4 was the blind review of the toy-model note, papers/reciprocity-toy-models.md, which builds two explicit field theories in which Born reciprocity is a symmetry of the action and asks whether the lowest-energy state breaks it. A fresh referee that could execute code reproduced the note’s algebra and most of its numerics, but found that the note’s one claim of quantum symmetry breaking rested on a potential that is unbounded below once the field H is allowed to vary. It graded the note B, flawed but salvageable, and the author agreed; the pre-registered forecast had also been B, though it had placed the weakest link correctly and underestimated how serious it was. Every revision was accepted; version 2 of the note applied them on 2026-10-20, with three departures, and the round added two rules to the project’s method.

Row 6: the exactly massless graviton

Mon, 01 Jan 0001 00:00:00 +0000

Row 6 is the program’s prediction that the graviton, the quantum of gravitational waves, has exactly zero mass. The program’s geometry carries two spin-2 fields, the ordinary metric and a partner describing momentum space, and Born reciprocity exchanges them. The combinations of definite mass are their sum, which the exchange leaves unchanged (reciprocity-even), and their difference, which changes sign (reciprocity-odd). The even combination is the ordinary graviton and is exactly massless; the odd one is the momentum graviton X, which is massive, so the force it carries has a finite range. Observations of gravitational waves set upper bounds on the graviton’s mass, and in the program those bounds apply to the even combination, which meets them identically. A nonzero mass attributed to the even combination would kill the row; #11 does not count it among the program’s near-term exposures.

The invariant-length shell and Planck closure

Mon, 01 Jan 0001 00:00:00 +0000

The invariant-length shell is the momentum polarization’s counterpart of the mass shell. Every observer assigns an ordinary particle the same mass; #13, the paper on conjugate special relativity, proposes that every observer of the mirror polarization assigns a conjugate particle the same size, an irreducible length λ_χ, and obtains this by applying Born’s exchange of position and momentum to the mass shell. Two consequences are drawn. A length times the maximum force is an energy, and at the Planck length it equals the Planck energy exactly, a result #13 calls closure at the self-dual point. And a fixed size in position fixes a width in momentum, as a short pulse fixes a broad spectrum; #13 identifies that width with the momentum scale of the dark-matter velocity gate, which makes the knee in dark-matter self-interaction a dual-relativistic invariant. The shell is alive but unreviewed: #13 tags it as derived, but no review round or script has checked it, and the checkpoint #11 does not mention it. Its description of dark-matter quanta by a length has not been reconciled with the checkpoint’s ordinary treatment of them, which governs by the wiki’s order of precedence.

The net theorem: clumping is nowhere mandatory

Mon, 01 Jan 0001 00:00:00 +0000

The net theorem is the summary result of Debt 3: in every part of the program’s parameter space that survives, dark matter clumping by velocity is never a required phenomenon. For particle dark matter it cannot happen. For wave dark matter it was required only under one assumption about the coupling, Scenario A, which observation rules out; with the coupling left free, Scenario B, it is merely optional. The theorem closes the test of the program’s founding image with a negative result. It covers collective dynamics only, so whether a single topological object could be localized in momentum is a separate, untested question.

The phase-space diameter relation (chat-only)

Mon, 01 Jan 0001 00:00:00 +0000

The phase-space diameter relation is a conjecture from the program’s original chat sessions: the diameter of the universe in position space, multiplied by its diameter in momentum space, equals Planck’s constant times a large number fixed by how many photons the universe contains. It is a cosmological counterpart of the time–bandwidth product of signal analysis, which says that a pulse cannot be both very short and very narrow in frequency. The relation exists only in the chat, as summarized by the handoff document: no paper or note derives it and no script checks its numbers. A short note on it is in the current work queue.

The popular articles (#10, #12)

Mon, 01 Jan 0001 00:00:00 +0000

The popular articles are two plain-language essays about the program, written on claude.ai for general readers. “The Angle of the Mirror” (#10) is a field report on the first week: the bet that Born’s symmetry between position and momentum is exact but hidden, the two founding ideas the calculations killed, and the structure that survived. “The Universe’s Other Half” (#12) is a tour of the whole framework written, in its own opening words, “as if it were confirmed. It is not.” Neither has been revised since the import into this repository, so both still present the superheavy soliton candidate as alive and read the dark-energy data more favorably than the checkpoint now does, and #12 adds claims that no other document makes. They are kept for their exposition, and neither is ever a source for a claim’s status.

The symplectic charge lattice

Mon, 01 Jan 0001 00:00:00 +0000

The symplectic charge lattice is the program’s observation that the set of allowed electric and magnetic charges has the same geometry as phase space. The Dirac–Schwinger–Zwanziger (DSZ) pairing of two charges is an antisymmetric whole-number form, like the pairing of position with momentum; charges that can all be described together by one Lagrangian form a “Lagrangian” half of the lattice; and deciding which particles count as electric is therefore a choice of polarization, as the program’s vacuum chooses which coordinates count as position. One quarter-turn matrix generates Born reciprocity, the program’s modular inversion and S-duality. The checkpoint counts this as a structural unification [D — structural; math T]: the mathematics is established, the identification of the two choices is the program’s, and no calculation depends on it. Whether the two polarizations are one physical structure is open problem 8.

The verification battery T1–T7

Mon, 01 Jan 0001 00:00:00 +0000

The verification battery is a set of seven numerical and exact-integer tests, T1 to T7, of the matrix algebra at the core of the program: the compatibility theorem, the parametrization of the modular field, and the algebra behind the mass of the momentum graviton and the zero vacuum energy of its potential. It was written and run on claude.ai, carried into the repository as Appendix A of the handoff document, and now runs as verify/battery.py in the gate, with the same computations and random seeds. Checkpoint #11 calls the results it confirms Tier 1 and tags them [V]. Six of the tests evaluate an identity at randomly drawn sample points and one is exact integer arithmetic, so the battery confirms the algebra at sample points rather than proving it, and none of the tests asks whether the algebra describes nature.

The Wigner function and Weyl symbols

Mon, 01 Jan 0001 00:00:00 +0000

The Wigner function is a way of drawing a quantum state as a distribution over phase space, with position on one axis and momentum on the other. Adding it up over all momenta gives the probability of each position, and adding it up over all positions gives the probability of each momentum, so one picture holds both descriptions of the state; unlike a true probability, it can be negative. Weyl symbols extend the same idea from states to operators, turning each operator into a function on phase space. The program’s toy-model note uses the Wigner function to build a matter coupling that reads one axis of phase space as “position”, and to check that a coupling blind to that choice treats both axes alike.

Vielbein formulation and wedge potentials

Mon, 01 Jan 0001 00:00:00 +0000

In the vielbein, or frame, formulation of gravity the basic variable is a frame field e: at each point of spacetime, a set of reference axes that are orthonormal by definition, from which the metric is built as a quadratic expression, g = eᵀηe. The program calls frames first-order variables and metrics, being quadratic in frames, second-order. Hinterbichler and Rosen showed that the ghost-free potentials coupling two spin-2 fields can be written as wedge products of the two frames: polynomials, with none of the matrix square roots that the same potentials need when written with metrics. The program’s tuning test passed only in this form: candidate mass potentials for the momentum graviton X built from metrics fail the ghost test, while the frame-built one lands in the ghost-free class. That the program’s potential takes the frame form was selected for minimality and consistency rather than derived, which is an open problem.

Momentum fibers and the cotangent bundle

Mon, 01 Jan 0001 00:00:00 +0000

A momentum fiber is the set of all momenta a particle can have at one point of space-time. Phase space, the space of positions and momenta taken together, is made of these fibers, one attached to each point, much as a spectrogram attaches a whole frequency axis to each instant of a recording; geometers call the structure the cotangent bundle of space-time. In general relativity every fiber is flat. Momenta at a point are measured with the inverse of the space-time metric, which is the same everywhere along the fiber, so the curvature of space-time changes how neighbouring fibers are joined but not the geometry inside any one of them. The program’s checkpoint lists this flatness among the places where the laws treat position and momentum differently and calls it “an architectural assumption, not a measurement” (Born’s question). In the program’s own geometry the compatibility theorem fixes the momentum metric as the inverse metric on the vacuum, any independent momentum-space geometry is assigned to the momentum graviton X, and the dark-matter gate acts through the width of conjugate-matter quanta along the fibers. How X propagates along the fibers, which would fix the shape of that gate, has not been computed.

Open: a Monte Carlo with H dynamical

Mon, 01 Jan 0001 00:00:00 +0000

A Monte Carlo with H dynamical is a lattice simulation of model I in which the phase-space metric H varies from point to point and fluctuates together with the real structure K, instead of being held at the constant value H = 1. The program’s simulation is of the simpler model with H held fixed, the frozen-H reduction, which keeps only the polarization angle, the angle that sets which line of the phase plane counts as position. With the bounded potential V_b added, that angle stays near one of the two values that Born reciprocity ρ exchanges: an ordered ℤ₂ phase. Whether the order survives when H fluctuates as well has not been computed. Round 4 required the lattice result to be stated for the reduction only, after finding that the first potential tried, stable in the reduction, fell without limit once H was allowed to move. For the full model there is a classical check instead: the vacuum is a minimum of the potential with all four field directions free, though along one of them the potential is flat and no symmetry keeps it so. Checkpoint #11 lists the simulation as open, and no source says how it would be set up.

Conflicts between sources

Mon, 01 Jan 0001 00:00:00 +0000

The program’s documents were written over several weeks, and the later ones correct, re-tag or retire what the earlier ones said, so the same question can get different answers in different files. This page collects those disagreements between documents, and between two statements of the same document. Most are settled by a fixed order of precedence in which the most current document wins. The rest are cases where documents of equal standing disagree, or where no document addresses the point; for those the wiki reports both readings and does not choose. Suspected errors inside a single statement, such as a slip in a formula or a wrong section pointer, are listed separately for Paul’s ruling.

Curved momentum space

Mon, 01 Jan 0001 00:00:00 +0000

Curved momentum space is the idea, developed in a line of quantum-gravity research, that the space of momenta can carry a geometry of its own instead of being flat. Snyder used it in 1947 to build a quantized space-time; in quantum gravity with two space dimensions, which can be solved exactly, the momenta of a particle live on a curved space; and “relative locality” made the geometry of momentum space a principle. General relativity curves space-time but leaves momentum space flat, which the program counts as an assumption rather than a measurement; its own version of momentum-space geometry is the momentum graviton X. One known difficulty of the idea, the soccer-ball problem, was listed as open in the lab notebook and never resolved.

Dual inertia: free fall as rest

Mon, 01 Jan 0001 00:00:00 +0000

Dual inertia is #13’s account of rest and uniform motion in the momentum polarization, the mirror description in which momentum plays the part that position plays for ordinary matter. An ordinary inertial body keeps its velocity constant. In #13’s dual kinematics the rate of motion through momentum space, dp/dt, is a force, and it takes the place of velocity, so the dual-inertial motions are those of constant force: zero force, which in general relativity is free fall, and a constant nonzero force, which is a uniformly accelerated (Rindler) frame. In #13’s words, “An astronaut coasting in orbit is, to the momentum polarization, at rest.” The identification of dual-inertial frames with free fall is on #13’s list of fixed results; the dual second law built on it is marked conjecture-tier, the dual first law carries no tag, and their formalization waits on dynamics that have not been written. Like the rest of #13, the material is unreviewed and unchecked by script, and the checkpoint #11 does not mention it.

Obituary: velocity-space granularity in halos (old row 3)

Mon, 01 Jan 0001 00:00:00 +0000

Velocity-space granularity was the lab notebook’s prediction that the dark matter of a relaxed galaxy halo, such as the Milky Way’s, is split into of order ten broad streams, each with its own velocity and coherent over kiloparsec distances, and that their gravity leaves correlated ripples along thin streams of stars. It was row 3 of the notebook’s prediction table and one of its three flagship discriminants. It rested on the founding image of velocity clumping, so when Debt 3 showed that clumping is impossible for particle dark matter and never mandatory for wave dark matter, the row was withdrawn permanently, on both islands. The current table’s row 3 is a different prediction, the island discriminant.

Open: does ρ's SO(2) break in the coupled model?

Mon, 01 Jan 0001 00:00:00 +0000

The open question is whether model I, with its metric field H free to vary and no added potential, spontaneously breaks the rotation group SO(2) of the phase plane, the circle of symmetries that contains Born reciprocity ρ. Breaking it would mean that the vacuum settles on one polarization angle despite quantum fluctuations. In two dimensions a compact continuous symmetry cannot break on its own (the Mermin–Wagner–Coleman theorem), so the polarization angle by itself cannot order. But model I’s metric sector is a hyperbolic sigma model, a known case in which a non-compact symmetry does break in two dimensions, and the two sectors are coupled. Round 4 required the toy-model note to restrict the theorem to the compact SO(2) and to state the coupled case as open; the note lists it as not shown.

Open: one modular structure for both polarizations (OP8)

Mon, 01 Jan 0001 00:00:00 +0000

This open problem asks whether two choices the program treats in parallel are parts of one structure: the vacuum’s choice of which phase-space directions count as position (postulate P5), and the choice of which charges count as electric, which is a polarization of the charge lattice. Both choices are made with the same quarter-turn matrix, and the checkpoint conjectures that they are components of one larger “modular” structure, on the model of string theory, where T-duality and S-duality are both contained in a larger U-duality. It is open problem 8 of the checkpoint, tagged [O]. The checkpoint does not say what the larger structure would be, no calculation toward it exists in the repository, and it is not in the current work queue. The toy models that met the program’s first acceptance criterion did not address the related demand that the order parameter also select the electric charges.

Partially massless gravity

Mon, 01 Jan 0001 00:00:00 +0000

Partially massless gravity is a spin-2 field on de Sitter space, the spacetime of a universe expanding at a constant rate under a positive cosmological constant, whose mass takes the one value at which an extra gauge symmetry appears and removes the field’s helicity-0 polarization [T]. That value is the edge of the Higuchi bound, below which a massive spin-2 field on de Sitter space carries a ghost. Bimetric gravity has a parameter point, studied as partially massless bimetric gravity, at which the theory has de Sitter solutions in which the two metrics are proportional, with a ratio that the equations leave undetermined, and its potential is symmetric under exchange of the two metrics. The point matters to the program because at α = 1, which in the literature’s usual notation means equal Planck masses for the two metrics, it lies inside the palindromic branch that Born reciprocity selects. That raises the open question whether reciprocity is the origin of the partially massless symmetry, and it means the program’s vacuum-energy identity must be compared with this literature before it can be called new.

Row 7: a permanent null in direct detection

Mon, 01 Jan 0001 00:00:00 +0000

Row 7 predicts that experiments searching for dark matter through any interaction other than gravity will never see it. Direct-detection experiments look for dark-matter particles striking atomic nuclei in shielded underground detectors, and collider experiments look for dark matter produced in particle collisions; both need dark matter to couple to ordinary matter, the particles of the Standard Model, through some interaction besides gravity [T]. In the program, the two remaining dark-matter candidates, χ particles and an X-condensate, have no such coupling: postulate P4 lets no gauge force act across the two sectors of matter, and since 2026-10-19 it is read strictly, excluding even the small kinetic mixing that could otherwise link them. The expected signal is zero, permanently. One confirmed non-gravitational coupling would kill the row, which is one of the three that #11 lists as the program’s near-term falsification exposure.

Spectrum, parameters and reciprocity parity

Mon, 01 Jan 0001 00:00:00 +0000

The potential that gives the momentum graviton X its mass is a bimetric potential with five coefficients, and this page covers what is left of it once reciprocity has acted, and how the program’s two spin-2 fields sort under reciprocity. The palindrome removes two coefficients and the requirement of a flat vacuum removes a third, leaving two numbers: one sets X’s mass M_X, the other shapes its cubic interactions. Reciprocity exchanges the two spin-2 frames of the theory, so the fields of definite mass are their sum, which the exchange leaves unchanged and which is the ordinary massless graviton, and their difference, which changes sign and is the massive X. The same symmetry forces the two frames to have equal Planck masses, a statement of calc note #8 §3 that #11 does not repeat. The count and the parities are derivations [D] in checkpoint #11 §7.5, not machine-checked, and the masslessness of the even combination is prediction row 6.

The dark-energy over-constraint

Mon, 01 Jan 0001 00:00:00 +0000

The dark-energy over-constraint is a counting argument: the program’s dark-energy model has fewer free numbers than the quantities it predicts, so the data can test the model against itself. The model has three free numbers, the height V₀ and scale f of the potential of the dark-energy field τ and τ’s initial value τᵢ, and the height is fixed by the measured amount of dark energy, which leaves two. Those two must account for three observables: today’s equation of state w₀, its rate of change wₐ, and the time of the future turnaround, t_turn. One relation among the three is therefore a prediction, which the checkpoint counts among the program’s progressive signs. The third observable lies in the far future, and the turnaround it times needs the potential to go negative, which the potential as written never does.

The soliton-obituary script

Mon, 01 Jan 0001 00:00:00 +0000

The soliton-obituary script, verify/soliton_obituary.py, recomputes the numbers in the record of why the program’s superheavy-soliton dark-matter candidate failed: the corrected count of monopoles formed in the early universe, the mass at which their abundance would match the observed dark matter, the overproduction at the withdrawn manuscript’s benchmark mass, the corrected relic window of the hidden sector’s light fermion, the dark-radiation bound, and the exclusion of the minimal hidden sector. Before computing any of these it reproduces standard cosmology results that do not depend on the manuscript. It replaced an earlier script, knot_numbers.py, which had checked the manuscript’s own claims and had passed a wrong normalization. The candidate it documents is dead; the script keeps the arithmetic of that death reproducible.

The twisted element ρ∘(K → −K)

Mon, 01 Jan 0001 00:00:00 +0000

The twisted element is a combined operation in the coset toy model of broken reciprocity: apply Born reciprocity, the exchange of position and momentum, and at the same time reverse the sign of the real structure K, the field whose value marks which of two lines in the position–momentum plane counts as position. Reciprocity alone carries every vacuum of the model to a different one, which is the sense in which the vacuum breaks it. The combination carries the reference vacuum back to itself, and it acts on the model’s metric H exactly as reciprocity does. So as long as nothing in the theory depends on the sign of K, the vacuum keeps a symmetry that cannot be told apart from reciprocity, and calling reciprocity broken is a choice of labels. The statement gains physical content once matter couples to the sign of K, which in the program is the role of postulate P4. The round-4 referee found the point, the adjudication upheld it, and checkpoint #11 now states it in its postulate P2′.

W′ overclosure: the minimal sector is excluded

Mon, 01 Jan 0001 00:00:00 +0000

W′ overclosure is the result that the simplest version of the soliton candidate’s hidden sector — SU(2)_d broken to U(1)_d by an adjoint Higgs field and nothing else — is excluded by its own heavy charged gauge bosons, the W′±. With no lighter particle carrying hidden charge, the W′± cannot decay. The hot hidden plasma in which the monopoles form also fills the universe with W′±, and after they stop annihilating, their density would exceed the observed dark matter 10¹⁶ to 10¹⁷ times. #14 v3 therefore added a light hidden fermion ψ for the W′± to decay into. The calculation was done at the withdrawn benchmark mass, but the statement that the minimal sector is excluded survived the candidate’s retirement and is machine-checked.

Bimetric cosmology and its instabilities

Mon, 01 Jan 0001 00:00:00 +0000

Bimetric cosmology studies expanding-universe solutions of Hassan–Rosen bimetric gravity, in which both metrics take the homogeneous and isotropic form of standard cosmology. Published analyses of its solution branches find gradient or ghost instabilities in the perturbations when the massive spin-2 field is about as light as today’s Hubble rate and the two metrics have comparable Planck masses, though not in every model. A heavy spin-2 field has been shown to be linearly stable at late times when the second metric’s Planck mass is much smaller than the first. Separately, a massive spin-2 field on de Sitter space must satisfy the Higuchi bound m² ≥ 2H², which a heavy field meets trivially. For the program the known stable regime is out of reach: Born reciprocity sets the two Planck masses equal, so the cosmological stability of a heavy momentum graviton X is an open question.

Broken Born duality: the Freidel–Leigh–Minic precedent

Mon, 01 Jan 0001 00:00:00 +0000

Broken Born duality is Freidel, Leigh and Minic’s phrase for what happens when all physical probes agree on which phase-space directions are space-time: the agreement is a preferred polarization, a preferred choice of which directions count as position, and it breaks the position–momentum symmetry they call Born duality. In their work on Born geometry and metastring theory they made the neutral metric η dynamical and described space-time as a choice of Lagrangian subspace, but they wrote no potential for η and treated the breaking as kinematical, not as spontaneous symmetry breaking. The program’s central idea therefore has a literature precedent. Since version 2.2 the program’s checkpoint credits them, and claims as its own only the reading of the breaking as spontaneous, with η as the order parameter.

Obituary: Fornax friction suppression (old row 4)

Mon, 01 Jan 0001 00:00:00 +0000

Fornax friction suppression was the lab notebook’s prediction that dark matter in the cores of dwarf galaxies condenses into a superfluid-like state in which dynamical friction, the drag that makes a massive body sink toward the centre of a galaxy, is suppressed. It would have explained why the globular star clusters of the Fornax dwarf galaxy have not sunk to its centre, the Fornax timing problem, and it was row 4 of the notebook’s prediction table. The condensed cores came from the same velocity-space condensation as the founding image of velocity clumping, so when Debt 3 killed that picture the row was withdrawn with it, permanently.

Obituary: the magnetic identification (popsci-only)

Mon, 01 Jan 0001 00:00:00 +0000

The magnetic identification is a claim made only in the popular article “The Universe’s Other Half” (#12): that the program’s momentum-polarized sector is the magnetic side of electromagnetism, dark because Dirac’s condition makes its coupling strong, “about 34, against our 1/137”, with only topological knots surviving from it as free relics; those knots are the retired soliton candidate. No paper or note of the program makes this claim. The technical record supports only the observation that choosing electric charges and choosing position are the same matrix operation, together with an open conjecture that they are one structure. The claim conflicts with a dead claim, with the strict separation of the two sectors’ forces and with the program’s prediction of a permanent null in direct detection, so the wiki records it as dead.

Obituary: the ψ relic window

Mon, 01 Jan 0001 00:00:00 +0000

The ψ relic window was the range of masses #14 v3 allowed for ψ, the light hidden fermion it added so that the heavy hidden W′± bosons could decay. ψ is itself stable and left over from the early universe, so its mass sets how much of the dark matter it makes up: v3 wanted ψ light enough to be a minor component, so that the monopoles dominate, and heavy enough to have annihilated before nucleosynthesis. The round-3 review found both edges wrong. The upper edge was too high because v3 had left out the antiparticle and the second member of the doublet, a factor of 4, and had bracketed an exactly known annihilation cross-section over a factor of ten; the lower edge was too low because MeV-scale ψ would still be freezing out during nucleosynthesis. The window died with these corrections, and the candidate it served was retired.

Open: the sign of V(τ) and the turnaround

Mon, 01 Jan 0001 00:00:00 +0000

The sign problem is a conflict in the program’s documents about the potential energy V(τ) of the dark-energy field τ. They write it as V = V₀[1 + cos(τ/f)], and since a cosine never goes below −1, this potential is never negative when V₀ is positive. Yet the program’s far-future turnaround, in which our expansion halts and reverses, requires the potential to become negative, and the published fits the program cites for that event use potentials with a negative minimum. None of the three documents that write the potential discusses the conflict. It was found while this wiki was being built and is recorded, for Paul’s ruling, as flag 27 in the build’s list of suspected errors in the sources. Until he rules, the turnaround, its reading as the conjugate bounce, and the turnaround time in the over-constraint all rest on an unresolved point. This page records the conflict and does not choose a side.

Row 8: no monopoles of our sector

Mon, 01 Jan 0001 00:00:00 +0000

Row 8 predicts that no magnetic monopole of our own electromagnetism, an isolated magnetic north or south pole, will ever be detected. Many theories that unify the forces predict such monopoles, and searches have found none [T]. The program’s first explanation of that absence, that monopoles live in the conjugate sector, died in the checkpoint’s monopole audit: the duality that relates magnetic to electric charge is the electric–magnetic one, not Born reciprocity, and inflation already accounts for the absence. The row now keeps the null with the standard explanation and adds one laboratory analog: monopoles of Berry curvature in momentum space, found in crystals called Weyl semimetals. A direct detection of a monopole of our sector would kill the row.

Safety by range: X and ordinary matter

Mon, 01 Jan 0001 00:00:00 +0000

This result explains why ordinary matter shows no sign of the momentum graviton X. Standard Model matter couples to the metric g. Reciprocity forces the program’s two spin-2 frames to have equal Planck masses, so the metric g contains the massless graviton and X in comparable proportions, and ordinary matter feels both at gravitational strength. X is massive, so the force it carries is a Yukawa force, which dies away beyond a range of order 1/M_X [T]; gravitational-strength forces of such short range are not constrained by experiment. X is safe because of its range, not because ordinary matter is decoupled from it. The argument changed across the Debt-1 calc notes; the current form is calc note #8’s [D + T], carried by checkpoint #11 §7.5 [D]. It cites no experimental bound and no value of M_X.

The equivalence principle as dual relativity

Mon, 01 Jan 0001 00:00:00 +0000

This is #13’s claim that the equivalence principle of general relativity, which puts all freely falling frames on an equal footing, is the principle of relativity of the momentum polarization, the program’s mirror description in which momentum plays the part of position. In #13’s dual kinematics, rest means feeling no force, which is free fall; so the dual statement that physics is the same in all force-free frames is the equivalence principle, reached from kinematics rather than from gravitation, and #13 reads Einstein’s two relativity principles as the two polarizations of one statement. The same paper lists what the two polarizations share, the time coordinate and the massless graviton, so that both fall alike, and what the breaking of the symmetry makes different: our light cone is held in place by massless gravity, the dual cone’s dynamical part by the massive field X, so dual effects are short-ranged. The claim is alive but unreviewed: it is tagged as a structural derivation by #13 itself, no review round or script has checked it, and the checkpoint #11 does not mention it. The popular article #12 states it as settled, which is not its status.

The section condition: what η fixes

Mon, 01 Jan 0001 00:00:00 +0000

A section condition is a rule, taken from double field theory, that makes every field depend on only half of a doubled set of coordinates, so that physics lives on one “section” of the doubled space. In the program’s coset toy model the neutral metric η, the order parameter of broken reciprocity, singles out exactly two lines in the position–momentum plane: its null lines, which are the two eigenlines of the real structure K. A section condition built from η forces each matter field to depend on the coordinate along one of those lines only. It cannot say which of the two is position, because the same fields pass whether the condition is built from η or from −η. So the condition fixes the pair of candidate directions, and matter’s choice within the pair has to come from somewhere else. The clean two-way choice holds only when phase space is a plane (one position and one momentum coordinate); with more dimensions the literature finds a continuous family of sections.

The toy-model scripts

Mon, 01 Jan 0001 00:00:00 +0000

The toy-model scripts are four programs in verify/ that recompute the results of the toy-model note, which studies two explicit field theories in which the exchange of position and momentum is an exact symmetry that the vacuum breaks. toy_coset.py checks the coset sigma model (model I), toy_gw.py the self-dual Grosse–Wulkenhaar model (model II), toy_s4.py runs a lattice Monte Carlo of model I with the phase-space metric held fixed, and toy_s5.py checks how the vacuum’s choice can make matter interactions local in position. They were written with the models on 2026-10-19, extended with the corrections of the round-4 review, and run in the gate. Their checks are exact symbolic identities, numerical tests against anchors taken from the cited literature, and seeded simulations with error bars; each confirms the arithmetic of the model as written, and several of the note’s conclusions rest on arguments that no script checks.

Locality from an (H, K)-built coupling

Mon, 01 Jan 0001 00:00:00 +0000

The coupling I_K is a particular interaction for matter in the coset toy model of broken reciprocity. It is built from the model’s two fields, the metric H and the real structure K, and it is local in position in one vacuum and local in momentum in the mirror vacuum. Evaluated on a single quantum state, it reduces in the first vacuum to an integral over the state’s position profile, its waveform, and in the second to the same integral over its momentum profile, the spectrum of that waveform, so one state can give different values in the two vacua. Because the interaction depends on the sign of K, it also gives the breaking of reciprocity a physical effect. The arrangement parallels the Standard Model’s fermion masses, which follow from the Higgs field’s vacuum value once the fermions’ Yukawa couplings to it are given. Nothing in the toy requires matter to couple this way, which is why this is only the weak form of the locality criterion S5.

Open: stability at equal Planck masses

Mon, 01 Jan 0001 00:00:00 +0000

This open problem asks whether a heavy massive spin-2 field such as the momentum graviton X can be stable in an expanding universe when its metric and the ordinary metric have equal Planck masses, the coefficients that set how strongly each metric responds to matter [T]. Published analyses of bimetric cosmologies find instabilities for a spin-2 field with a mass of order today’s Hubble rate and comparable Planck masses, and the work that finds a heavy spin-2 field linearly stable at late times assumes that the second metric’s Planck mass is much smaller than the first. Reciprocity forces the two Planck masses to be equal, which excludes that regime, and whether a large mass alone is enough has not been checked. The question came out of the second blind review of the soliton manuscript #14, since withdrawn. It is recorded only in the review worklog and the manuscript; checkpoint #11 §14 does not list it.

Open: the topological loophole

Mon, 01 Jan 0001 00:00:00 +0000

The topological loophole is the route that Debt 3 leaves open for localization in momentum space: not a collective clumping of many particles, which the net theorem rules out as a requirement, but a single object, such as a topological soliton, whose own structure would localize it in momentum. Debt 3’s calculations cover collective dynamics only, so they do not test this case. The candidate that occupied the loophole, superheavy solitons of a hidden gauge sector, was retired on 2026-10-19 after failing review on its abundance arithmetic, so the loophole is logically open but has no candidate. The checkpoint does not say consistently whether it remains a work item.

Open: the two dual cones

Mon, 01 Jan 0001 00:00:00 +0000

This open problem is an inconsistency inside #13, the paper on the special relativity of the momentum polarization: two of its sections use different momentum-space light cones. Section 1 builds the dual cone in time and momentum, with time shared between the two polarizations and the maximum force c̃ as its slope. Section 6 instead calls light’s ordinary energy–momentum relation, E = c|p|, the dual null cone; that cone lives in energy and momentum, has slope c, and, in the analysis of the v3 worklog, belongs to a different construction, the four-dimensional Born map that also exchanges time and energy and has not been built. Section 6’s claim that photon-dispersion measurements test both cones at once depends on the second reading. Paul raised the problem on 2026-10-19, while the withdrawn soliton manuscript was being revised, with the instruction to fix the two cones and settle which construction governs before any paper claims more. It is on neither #11’s list of open problems nor the current queue, and no document since has taken it up.

Row 9: no linear photon dispersion

Mon, 01 Jan 0001 00:00:00 +0000

Row 9 predicts that the speed of light in vacuum has no correction proportional to a photon’s energy. Some approaches to quantum gravity predict such a linear correction, suppressed only by the Planck energy; it would make high-energy photons from a distant gamma-ray burst arrive measurably earlier or later than low-energy photons emitted with them, and burst observations with the Fermi Large Area Telescope exclude it [T]. In the program the linear term is exactly zero, protected by the residual symmetry ρ², the half-turn (x, p) → (−x, −p) of Born reciprocity, and the leading allowed correction is quadratic in the energy and strongly suppressed. The protection argument is a schematic one, made only in the lab notebook before the polarization became a Goldstone angle; no later document restates it. A confirmed linear dispersion would kill the row.

The Born length λ

Mon, 01 Jan 0001 00:00:00 +0000

The Born length λ is a reference length that, together with Planck’s constant ℏ, puts position and momentum on a common scale: positions are counted in units of λ and momenta in units of ε = ℏ/λ. A length and a momentum have different units, so a geometry that combines them, or a map that exchanges them such as Born reciprocity, has a meaning only once some such pair of scales is chosen. The name comes from #13, the program’s paper on conjugate special relativity, in which λ fixes the slope of the momentum-space light cone and with it the maximum force, c̃ = ℏc/λ². #13 sets λ equal to the Planck length, in one sentence that carries no tag, and the maximum force becomes c⁴/G. The program’s lab notebook writes its phase-space structures in units in which the same pair of scales is 1, without naming λ or giving it a value. The identification with the Planck length is alive but unreviewed: no review round has examined it, no script checks it, and the checkpoint #11 does not mention the Born length.

The cosmological-constant problem

Mon, 01 Jan 0001 00:00:00 +0000

The cosmological-constant problem is the mismatch between the energy density of empty space that quantum field theory predicts, from the zero-point fluctuations and loops of matter fields, and the far smaller value that the accelerating expansion of the universe indicates. No accepted mechanism explains the gap, and bimetric gravity is no better placed than general relativity to explain it. For the program this page marks a boundary. The program’s vacuum-energy identity sets to zero only the vacuum energy of the spin-2 interaction potential at the aligned point; it does not by itself address the vacuum energy of matter loops. The program assigns the observed dark energy to a separate field, τ.

The hidden sector's thermal history

Mon, 01 Jan 0001 00:00:00 +0000

The hidden sector’s thermal history is the record of how the soliton candidate’s hidden sector would have evolved as the universe cooled, and of the constraints that evolution had to meet. Its massless hidden photon, together with any light hidden fermion ψ, is extra radiation, which measurements of the cosmic microwave background limit through the effective number of neutrino species, N_eff. A hidden sector as hot as ours would add about 0.86 to N_eff, which is excluded, so #14 required the hidden sector to start colder, by a temperature ratio ξ below 0.77 — corrected in round 3 to 0.760 — yet computed its relic abundances as if ξ were 1. A first-order hidden phase transition would also have left gravitational waves at about 10⁵–10⁷ Hz. These are requirements and predictions of a retired candidate, kept as record.

Benchmark-conditional arithmetic

Mon, 01 Jan 0001 00:00:00 +0000

Benchmark-conditional arithmetic is the set of numbers that #14 v3 computed at its benchmark monopole mass of 3×10¹³ GeV and that remain correct as arithmetic, although the benchmark itself is withdrawn. At that mass, monopoles making up all the dark matter would be spaced about 420 m apart near the Sun; they would scatter off one another so weakly, σ_T/M ~ 10⁻²⁶ cm²/g, that they would behave as collisionless cold dark matter; and if they carried magnetic charge under our own electromagnetism, their flux would exceed the Parker bound by 7.5 orders of magnitude. That last, counterfactual number is why the hidden sector has to be separate from ours. The round-3 referee recomputed each value from independently chosen constants and confirmed it.

Open: forced locality and the P4 lock

Mon, 01 Jan 0001 00:00:00 +0000

This open problem asks whether anything forces matter to be local in position, rather than leaving that as an input. In the coset toy model the vacuum’s order parameter fixes a pair of candidate directions in the position–momentum plane and marks one of them as position, and a suitable coupling makes matter local in whichever direction is marked; but nothing in the model requires matter to use that coupling. Without such a link, the line a matter field lives on and the sign of the order parameter are two independent choices. In the toy the link is supplied by postulate P4, which places ordinary matter in the position-polarized representation and its mirror image in the momentum-polarized one. The toy therefore explains locality in position only as part of a package that includes a momentum-local mirror sector, the program’s dark-matter candidate, and does not derive it. The toy-model worklog notes that the question is also open in the literature.

Open: the four-dimensional Born map and dual dynamics

Mon, 01 Jan 0001 00:00:00 +0000

The four-dimensional Born map is the version of Born’s reciprocity transformation that exchanges time with energy as well as position with momentum. #13, the paper on the special relativity of the momentum polarization, names it as the fully covariant construction in which its kinematics belongs, and it has not been built. #13 itself works in a 3+1 presentation, in which space and momentum trade places while time is shared by both polarizations. The map is also a prerequisite for the dual dynamics, the field equations of the momentum polarization (“their Maxwell, their field theory”), which do not exist: only the dual kinematics has been written, and the dual second law that #13 sketches is marked conjecture-tier until it is. Both items are open in #13 §7, and neither is on the checkpoint #11’s list of open problems or on the current queue.

Row 10: an early step in G_eff

Mon, 01 Jan 0001 00:00:00 +0000

Row 10 describes two possible imprints of the program’s dual, momentum-side gravitational sector on the early universe. The first is a step in the effective strength of gravity, G_eff: the dual sector adds to the square of the effective Planck mass a term that grows toward the past as the sixth inverse power of the cosmic scale factor, a⁻⁶, so it matters only at early times, before the light elements formed in Big Bang nucleosynthesis. The second is negative dark radiation: if the dual space is closed, its curvature acts on our expansion like radiation of negative energy, which lowers the effective number of neutrino species, N_eff. Both follow from the lab notebook’s calculation with the cosmological constraint a·ã = 1. The step is optional and the dark radiation conditional on a closed dual space, and their sizes are free parameters (knobs), so a null result bounds the knobs rather than killing the program; #11 names the cosmic microwave background experiment CMB-S4 as the probe.

Two gravitational charges and the conjugate field equations (chat-only)

Mon, 01 Jan 0001 00:00:00 +0000

Two results from the program’s original claude.ai conversation are derived in no project document: a claim that gravity has two charges, and a comparison of the program’s field equations with those of general relativity. The handoff document lists them in its §4, among the results that exist only in the chat, so that they are not lost. The first says gravity has two independent charges: ordinary stress-energy is the source of the massless graviton, and dual stress, which measures how energy is structured across momentum, is the source of the momentum graviton X. Whether a “dual equivalence principle” ties the second charge to the invariant length of conjugate quanta is left as a question. The second compares the program’s field equations with those of general relativity: the slaved momentum metric is fixed algebraically, the momentum-side counterpart of the Newtonian limit is a Yukawa form in momentum that is the dark-matter gate, and X waves are massive, with five polarizations and a lowest frequency. Parts of both have counterparts in the calc notes and the papers; the rest has none. Both are conjectures: no artifact derives them and no script checks them.

Born's answer: momentum space gravitates

Mon, 01 Jan 0001 00:00:00 +0000

Born’s answer is the program’s summary of what its gravity results say about the question Max Born raised in 1938, which calc note #8 states as “what is the dynamics of momentum-space geometry?” (Born’s question). In general relativity the space of momenta at each point is flat. The program’s answer is that momentum-space geometry is dynamical: it is carried by X = Re Z, a massive spin-2 field that changes sign under reciprocity, with two free parameters and no vacuum energy from its interaction potential. Conjugate matter couples to X through the dual stress, while ordinary matter feels it only at gravitational strength and over a short range, which the program gives as the reason no second metric has been observed. The answer assembles derived results and stands as a structural statement: checkpoint #11 §15 holds back the word “theory” until open problems 1–3 are settled, and the founding supposition that momentum space has its own gravity is recorded as “vindicated, transformed”.

Kinetic mixing between sectors

Mon, 01 Jan 0001 00:00:00 +0000

Kinetic mixing is a coupling between the field strengths of two U(1) gauge forces, such as our photon and a hidden photon, through a term proportional to ε times the product of the two field strengths. When ε is not zero, particles charged under the hidden force acquire a small electric charge under ours, so kinetic mixing is the standard route by which a hidden sector can leak into the visible one. Three literature results framed the soliton candidate’s hidden sector: gravity alone produces mixing only at six loops or more, and none if hidden charge conjugation is an exact symmetry; while the hidden photon stays massless, mixing does not make hidden monopoles magnetic under our electromagnetism; and what leakage remains would give the hidden fermion ψ, and through the Witten effect the monopoles, a small visible electric charge. The program closed these channels by taking its postulate P4 strictly.

Momentum-space black holes (chat-only)

Mon, 01 Jan 0001 00:00:00 +0000

Momentum-space black holes are a conjecture from the program’s original claude.ai conversation. Read in the momentum description, the program’s mirror description in which momentum plays the part of position, a black hole’s horizon would be the surface at which hovering in place requires the maximum force of conjugate special relativity; and there may be mirror objects, regions trapped in momentum rather than in space, which the handoff document calls velocity traps. The handoff records the idea among the results that exist only in the chat, at conjecture tier, together with “dual Hawking quanta” that would be huge in position space. The popular article #12 repeats part of it and adds that such a mirror object would look like compact dark matter. No document derives any of it, it appears in neither the checkpoint #11 nor the paper on conjugate special relativity #13, and it is not one of the program’s dark-matter candidates. Its status is conjecture (chat-only).

Row 11: particles or condensate (OP9)

Mon, 01 Jan 0001 00:00:00 +0000

Row 11 asks what dark matter is made of, if the program is right. The program offers two answers: χ particles, the momentum-polarized counterpart of ordinary matter, which scatter off each other only through a velocity gate; or an X-condensate, a coherent, space-filling oscillation of the field X, the massive partner of the graviton. #11 states that the two can be told apart using rows 1–3 of its predictions table (the knee in the self-interaction, the isotropy of scattering and the shape of the knee’s tail) together with signatures of a coherent field. How to make that separation is itself open problem 9 of #11 §14, so the row is a diagnostic still to be worked out, with no kill condition. #11 counts it among the program’s predictions not previously posed. Until 2026-10-19 the question had three answers; the third, superheavy solitons, is retired.

Glossary

Mon, 01 Jan 0001 00:00:00 +0000

This glossary lists, in alphabetical order, the terms a reader meets in this wiki: the Broken-Born program’s own vocabulary, such as the velocity gate, the wave island, the momentum graviton and the three debts, and the established physics the program draws on, such as Lagrangian polarizations, the Fierz–Pauli mass term and the Kibble mechanism. Each entry says in plain words what the term means and links the page that defines it, where the term’s status and the evidence for it are given. Some words mean different things in different documents or sections; their entries are marked as collisions and gathered in a list near the end, which is followed by the acronyms. Mathematical symbols are on the Notation page and cited works on the References page.

Obituary: Planck-suppressed adjoint mixing

Mon, 01 Jan 0001 00:00:00 +0000

Planck-suppressed adjoint mixing was #14 v3’s estimate of how strongly its hidden photon would mix with ours. The hidden Higgs field Φ permits a gauge-invariant interaction Φ^a F^a_μν B^μν / M_*, which links the hidden field strength to the field strength of our hypercharge; once Φ takes its vacuum value v, it produces a kinetic mixing ε ~ v/M_*. v3 assumed the operator appears with a coefficient of order one at the Planck scale, which gives ε ~ 10⁻⁷–10⁻⁶. The round-3 review found no source for that assumption: every paper that derives the operator obtains it at one loop from particles charged under both forces, which the manuscript’s own sector separation excludes, and the estimate those papers give is three orders of magnitude smaller. The estimate is dead, and the program then took sector separation strictly.

Notation

Mon, 01 Jan 0001 00:00:00 +0000

This page lists the mathematical symbols of the Broken-Born program as its documents and this wiki use them: what each symbol means, which document defines it and which wiki page explains it, grouped by subject in the order of the wiki’s sections. The program’s central objects are written the same way in all the current documents: the reciprocity map ρ, the symplectic form ω, the neutral metric η, the generalized metric H, the modular field Z = X + iY and the dark-energy field τ. Many of their letters also serve other purposes. H is also the Hubble rate, ρ an energy density and X a parameter of the dark-matter calculations; g is a spacetime metric, a coupling constant and a magnetic charge; μ is a width in momentum in the dark-matter sector and a coupling constant in the toy models. The Collisions section near the end lists every letter with more than one meaning or convention, together with the suspected source errors that bear on them.

References

Mon, 01 Jan 0001 00:00:00 +0000

This page is the bibliography of the Broken-Born program and of this wiki: every work that the program’s documents or the wiki’s pages cite, grouped by subject. An entry gives the reference as the citing document writes it and the mark that document attaches to it: † for a reference recalled from model memory and not yet checked against the source, [S] for one retrieved by a live search during the project. It then says what the work is cited for, what checks have been made of it, and which wiki pages cite it. Works that the wiki itself added to explain established physics follow the documents’ references in each subject, all marked †. A mark records where a citation came from, not whether the cited work says what it is used for.