The momentum graviton X on The Broken-Born program

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.

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.

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.

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.

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.

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.

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].

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.

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.

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.

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”.