The equivalence principle as dual relativity

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.

Status: Alive, unreviewed — [D — structural] (#13 §5); the long-range/short-range asymmetry [D — checkpoint §7] (#13 §6), whose derived form is #11 §7.5’s (flag 29 in review/wiki-build-flags.md). The tags are #13’s own: no review round has covered #13, no verify/ script checks the claim, it is not on #13 §7’s fixed list, and #11 v2.2 does not mention it.

Background: two relativity principles [T]#

The principle of relativity. The laws of physics take the same form in every inertial frame, and no experiment done inside a closed laboratory can detect its uniform motion. Galileo stated it for mechanics; Einstein’s special relativity of 1905 extended it to all of physics, with Lorentz boosts relating the frames (special relativity: the template).
The equivalence principle. In its weak form it is the universality of free fall: a test body’s trajectory depends only on where it starts and how fast it moves, not on what it is made of. In Einstein’s form, within a small enough region a freely falling frame is an inertial frame, so local physics in it is that of special relativity, and a uniformly accelerated frame cannot be told apart from one at rest in a uniform gravitational field. Einstein used it as the step from special to general relativity, in which the freely falling frames are the local inertial frames (general relativity and the graviton).

The two principles have different subjects: the first relates frames in uniform motion, the second relates free fall to inertial motion and acceleration to gravity.

The claim#

#13 §5, after defining the dual-inertial frames:

“The handshake worth recording [D — structural]: the dual principle of relativity — equivalence of all force-free frames — is the equivalence principle, arrived at from kinematics rather than gravitation. In this framework Einstein’s two relativity principles are the two polarizations of one statement.”

The argument has three steps, each taken from #13:

The dual-inertial frames. In the dual kinematics the rate of motion through momentum space, dp/dt, is a force, and the dual state of rest is dp/dt = 0, “force-free motion — our free fall” (§5; dual inertia: free fall as rest). This identification is on #13 §7’s list of fixed results.
The dual principle of relativity. It is the transcription of special relativity’s principle of relativity, which #13 words as the “equivalence of all force-free frames”.
The reading. Since the force-free frames are the freely falling ones, the dual principle states that physics is the same in all freely falling frames, which #13 identifies with the equivalence principle. Read in the position description, the statement that physics is the same in all inertial frames is Einstein’s principle of relativity; read in the momentum description, which Born reciprocity exchanges with it, the same statement is the equivalence principle: hence “the two polarizations of one statement”.

What kind of result this is:

#13 tags it [D — structural] and calls it “the handshake worth recording”; no further derivation is given. The step rests on the identification in step 1, which uses the general-relativistic reading of free fall as force-free (dual inertia: which reading of free fall).
#13 does not say which form of the equivalence principle it means. Its wording names the force-free frames; its dual-inertial class (§5) also contains the uniformly forced frames, “our Rindler frames”, and #13 does not say whether the dual principle extends to them.
The claim is not on §7’s list of “Kinematic structure fixed [D]”, which ends with the “dual-inertial = free-fall identification” it rests on.

#13 §6 opens with what the breaking leaves common to both polarizations:

“Shared: the time coordinate (in the 3+1 presentation used here) and the massless graviton, so both polarizations fall identically.”

The time coordinate. #13 works in a 3+1 presentation, in which space and momentum trade places and time is left alone; the construction that also exchanges time and energy is open (Open: the four-dimensional Born map and dual dynamics). The shared time is the t in the dual-inertial condition dp/dt = 0.
The massless graviton. In the checkpoint the massless graviton is even under reciprocity and the massive field X is odd: “Massless graviton = reciprocity-even”, “X = reciprocity-odd” (#11 §7.5; spectrum, parameters and reciprocity parity; Row 6: the exactly massless graviton; on the parities under #11 §6’s Z → −Z⁻¹, see flag 39 in review/wiki-build-flags.md).
Falling alike. In the checkpoint, that both kinds of matter fall alike is a postulate. P4 [P] says that x-polarized matter (the Standard Model) and p-polarized matter (χ) “both couple universally to H” (#11 §4), and #11 §8 has conjugate matter χ propagate “on g (equivalence principle; Bullet-Cluster-safe)” (conjugate matter and the velocity gate; the Bullet Cluster). The lab notebook’s version of P4 said it outright: “the equivalence principle holds across polarizations” (#1 §1, superseded as a reference). #13 §6’s “so both polarizations fall identically” is untagged and states the same consequence.

So the program contains two statements about the equivalence principle: the postulated universal coupling of P4, which makes both polarizations fall alike in our description, and #13’s structural reading of the equivalence principle as the momentum polarization’s relativity principle.

What the breaking makes different#

The statement#

#13 §6 continues:

“Broken: the long-range/short-range asymmetry — our null structure is policed by massless gravity, the dual structure’s dynamical part by the massive X, so dual-geometric effects are Yukawa-cut in momentum [D — checkpoint §7].”

Its plain-language passage: “our light cone is enforced by a massless field and theirs by a massive one, so ours reaches across the universe and theirs is short-ranged.”

Background: range and mass [T]#

A force carried by a massless field, like gravity or electromagnetism, falls off as an inverse square and has no limit to its range. A force carried by a field of mass M is cut off exponentially: its potential has the Yukawa form e^{−r/R}/r, with range R = ℏ/(Mc). In units with ℏ = c = 1 the range is 1/M.

What the checkpoint derives#

#13 §1 had already described the dual cone as “the causal structure of the slaved geometry g⁻¹”, with the massive field X as “the independently dynamical part of momentum-space geometry”, which “modifies the dual cone only at short range in momentum [D — §7 of the checkpoint]” (two causal cones; the momentum graviton X). Both citations point to #11 §7, the section on X. What #11 §7.5 states there [D] is a range in position for X’s couplings to ordinary matter: “SM couplings gravitational-strength and Yukawa-cut at 1/M_X” (SM: Standard Model; safety by range).

The reading “in momentum” has a counterpart only in the original chat, where the handoff records “dual Newtonian limit = Yukawa in momentum = the gate” (HANDOFF.md §4; the gate mechanism; two gravitational charges and the conjugate field equations (chat-only)). By the wiki’s order of precedence #11’s wording is the derived statement (conflicts between sources). #13’s citation of §7 for a claim §7 does not make is flag 29 in review/wiki-build-flags.md.

A different question: the dual equivalence principle#

The handoff’s list of results that exist only in the chat contains a question with a similar name. Its entry on two gravitational charges ends: “whether invariant length λ and dual charge lock together = ‘dual equivalence principle’ open question” (HANDOFF.md §4). The ordinary equivalence principle locks a body’s gravitational charge to its inertial mass; the chat’s question is whether the charge that sources X is locked in the same way to the invariant length λ_χ of a conjugate quantum (the invariant-length shell). It is chat-only, open, and separate from #13’s claim on this page.

What is open#

The derivation. #13 gives the claim as a structural match, without a derivation from an action; the dual dynamics in which it could be derived are unwritten (Open: the four-dimensional Born map and dual dynamics).
The presentation. The claim uses #13’s 3+1 presentation, with shared time, and #13 uses two different dual cones in different sections (Open: the two dual cones).
The signature. #13’s preamble: “The Euclidean/Lorentzian caveat of the main checkpoint is inherited throughout.” Free fall, light cones and Rindler frames are Lorentzian notions, and the checkpoint’s geometric results are proven for the Euclidean case (Open: Lorentzian dressing of the Euclidean skeleton (OP3)).

The popular version#

The popular article #12 (“A second speed of light”): “The momentum description’s principle of relativity, that physics is identical in all unforced frames, is a statement already accepted under another name: the equivalence principle, the foundation Einstein chose for general relativity. In this framework his two relativity principles are one principle, read in the two descriptions.” #12 presents the framework as if confirmed and is never a source for status (the popular articles).

Basis of the status#

HANDOFF.md §2 lists “dual inertial frames = free fall ⇒ equivalence principle as the dual relativity principle” among the alive results, without the ✓ it uses for machine-verified items. #13 has not been revised since its import on 2026-10-19, no review round has covered it, and #11 v2.2 does not mention it (conjugate special relativity).

Sources#

papers/conjugate-special-relativity.md §5 — the claim and the dual-inertial frames it rests on; §6 for what is shared and what is broken, with the plain-language passage; §1 for the slaved dual cone and X’s short-range modification; §7 for the fixed list, which omits the claim; preamble for the inherited Euclidean/Lorentzian caveat.
papers/broken-born-checkpoint-v2.md §7.5 — the massless graviton as reciprocity-even and “Yukawa-cut at 1/M_X”; §4 for P4; §8 for conjugate matter on g; §13 row 6 for the massless graviton’s test.
HANDOFF.md §2 — the alive listing; §4 for the chat-only “dual equivalence principle” and “Yukawa in momentum”.
notes/broken-born-theory.md §1 — the notebook’s P4, “the equivalence principle holds across polarizations” (superseded as a reference).
review/wiki-build-flags.md item 29 — #13’s “[D — checkpoint §7]” for a reading in momentum that #11 §7 does not state; awaits Paul’s ruling.
popsci/universes-other-half.md ‘A second speed of light’ — the popular statement (not a status source).