Conjugate special relativity

Conjugate special relativity is Einstein’s special relativity rewritten for the momentum polarization, the program’s mirror description in which momentum plays the role that position plays for ordinary matter. In quantum mechanics position and momentum are two descriptions of one state, related as a waveform and its spectrum; if the two are equal partners, the speed limit in space has a twin, a limit on how fast momentum can change, and a rate of change of momentum is a force. One working paper, #13, develops the consequences by transcribing special relativity with force in place of velocity: a maximum force of about 10⁴⁴ newtons, force boosts, force dilation, an invariant length, and free fall as the dual state of rest. #13 tags most of this as derived, but the material is unreviewed: no review round and no script has checked it, and the checkpoint #11 does not mention it. Its status is alive and unreviewed, and two of its construction problems are open.

Stands: special relativity and the maximum-force and maximal-acceleration conjectures as literature (the maximum-force conjecture itself contested); two causal cones, the maximum force, force boosts, force dilation, the invariant-length shell, dual inertia and the equivalence principle as dual relativity as the program’s claims, alive and unreviewed (all in claims that stand) · Died: nothing · Open: the two dual cones, the four-dimensional Born map and dual dynamics; folded into the claims they qualify, the factor of 4, merger-band corrections, which clocks respond and the invariant length against the checkpoint’s kinematics (all in open problems) · Conjectures: momentum-space black holes (chat-only), with #12’s mirror object as compact dark matter (popsci-only) folded on the same page; the dual second law, marked “conjecture-tier” by #13, on dual inertia; and #12’s unqualified force dilation of all clocks (popsci-only) (all in conjectures)

Map#

Apart from the chat-only conjecture on the last page, every page in this section rests on one document, #13, papers/conjugate-special-relativity.md, titled “Special Relativity in the Broken-Born Framework”: seven sections, the first six opening with an italic plain-language passage, and four references, all marked † (cited from model memory and not yet checked against the source). HANDOFF.md §3 lists it as “final”. It has not been revised since its import on 2026-10-19, so the checkpoint’s later revisions never reached it: it still tags the compatibility theorem [D], where #11 v2.1 re-tagged it [T][V]. No blind review round has covered it, it carries no attribution line, and its preamble inherits the checkpoint’s Euclidean/Lorentzian caveat (Open: Lorentzian dressing of the Euclidean skeleton). Its method throughout is transcription: each piece of special relativity is copied with position x replaced by momentum p, velocity by force F = dp/dt, and c by the maximum force c̃, and #13 states each result without steps beyond the substitution. The tags are #13’s own ([D], [D — structural], [O] and the untagged items listed on each page). #11 v2.2 never mentions the maximum force or dual special relativity, so the section’s status rests on HANDOFF.md §2, which lists both among the alive results without a machine-check mark, and on CLAUDE.md, which lists #13 as a current reference: alive, unreviewed, nothing machine-verified, and none of #13’s numbers reproduced by a verify/ script (the iron rule). #13’s closing line points readers to the checkpoint for the geometric sector and to the popular article #12 for the framework at large (the popular articles); #12 presents this material as settled and is never a source for status.

Established physics comes first. Special relativity: the template sets out the parts of Einstein’s kinematics that #13 copies (the two postulates, boosts and rapidity, the composition of velocities, the interval and the light cone, simultaneity, time dilation and length contraction, the mass shell and E = mc², the Galilean limit and Rindler frames, the 3+1 split) and ends with a table of #13’s transcriptions and the tag #13 gives each. Maximum force and maximal acceleration in the literature covers the published conjectures #13 ties itself to, Gibbons and Schiller’s maximum force c⁴/4G with its companion luminosity c⁵/4G and Caianiello’s maximal acceleration from a phase-space metric, together with the Planck units #13 writes them in and the reason c⁴/G is the one force that c and G can form.

The program’s claims follow in #13’s order. Two causal cones is the foundation: on the vacuum slice the phase-space metric has the block form H = diag(g, g⁻¹), each block defines a light cone (the theorem behind the block form is proven for positive-definite metrics, which have no cone; flag 30 in review/wiki-build-flags.md), and the momentum cone bounds |dp/dt| by c̃ = ε/(λ/c) = ℏc/λ², which is a bound on force; the cone belongs to the slaved geometry, and the massive field X modifies it, in #13’s words, “only at short range in momentum”, a reading the cited checkpoint section does not state (flag 29 in review/wiki-build-flags.md). The maximum force c̃ = c⁴/G gives that bound a value by setting the Born length λ equal to the Planck length, an untagged step; the result, about 1.2 × 10⁴⁴ N in #13’s figure, is a factor of 4 above the literature’s c⁴/4G, and the factor is open. Force boosts and force composition builds the dual Lorentz group on (t, p): the force boost and its factor γ̃, simultaneity that depends on momentum, forces that compose like relativistic velocities and saturate at c̃, and the Galilean limit that makes force look additive, with the correction of order 10⁻⁶⁵ at neutron-star forces and black-hole mergers as the one arena near the limit. Force dilation is the counterpart of time dilation, restricted by #13 to conjugate-polarized systems, together with its mirror, a contraction of momentum spread. The invariant-length shell and Planck closure is the counterpart of the mass shell: conjugate quanta carry an invariant length λ_χ, a length times the maximum force at the Planck length is the Planck energy, and the gate scale of the dark-matter sector is read as a dual-relativistic invariant. #13 §4 writes that scale as “v* = √2 μ/m_χ”, the lab notebook’s definition, which #11 v2.2 does not restate; review/wiki-build-flags.md item 17 records a suspected factor of √2 in μ between the two gate conventions #13 combines there; the flag’s audit found the factor present only if calc note #6’s gate is read at the level of the amplitude, and #13 consistent under the convention the other documents use. #13’s description of conjugate quanta by a length is not reconciled with #11 §8, which treats the χ candidate in ordinary kinematics and governs by the wiki’s order of precedence. Dual inertia: free fall as rest identifies the dual-inertial frames, those of constant force, with free fall and with uniformly accelerated (Rindler) frames, and records the dual first law, untagged, and the dual second law at conjecture tier. The equivalence principle as dual relativity reads the equivalence principle as the dual principle of relativity, and states what the two polarizations share (time and the massless graviton) and what the breaking makes different (a long-range cone and a short-range one).

Two open problems follow the claims they qualify. Open: the two dual cones records that #13 uses one momentum-space cone in §1, in (t, p) with slope c̃, and a different one in §6, E = c|p| in (E, p) with slope c, and that §6’s argument from photon dispersion depends on which governs; Paul raised it on 2026-10-19, and it is scheduled nowhere. Open: the four-dimensional Born map and dual dynamics is the construction that would also exchange time and energy, which #13 calls the natural home of its kinematics and which must exist before the dual field equations can be written. Four smaller open items sit on the pages of the claims they qualify: the factor of 4 on the maximum force, the merger-band corrections on force boosts, which clocks respond on force dilation, and the unreconciled invariant length on the invariant-length shell (review/v3-worklog.md, item O3: “The papers were never reconciled.”). The last page, Momentum-space black holes (chat-only), records a conjecture from the original chat, partly repeated in #12: horizons read as maximum-force surfaces. No document derives it, and its status is conjecture. #12’s further claim, that a mirror object trapped in momentum would look like compact dark matter, appears in no other document and is kept on the same page as a popsci-only section.

#13 §7, “Summary of tethers”, is the paper’s own checklist. Fixed [D]: “dual interval, force boosts, γ̃, force-composition law, force dilation, invariant-length shell, dual-inertial = free-fall identification.” Numerical anchors: “c̃ = c⁴/G; E(l_P) = E_Planck; γ̃ − 1 ~ 10⁻⁶⁵ at neutron-star forces”, none of them reproduced by a verify/ script. Literature tethers: “maximum-tension conjecture [1†], maximal acceleration [2†], Born’s self-dual program [3†, 4†].” Open [O]: “the factor 4; the 4D covariant construction; dual dynamics; whether merger-band gravitational-wave data can bound dual-kinematic corrections.” The fixed list leaves out several claims made in the body (simultaneity across momentum, the contraction of momentum spread, the gate-scale reading, the equivalence-principle claim, the refinement by X and the photon-dispersion claim), and the open list leaves out §3’s question of which clocks respond.

Boundaries#