P3: a reciprocity-invariant action

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.

Status: Alive — [P] — a postulate of #11 §4; the coset toy model realizes it in an explicit action whose invariance under ρ is checked symbolically (verify/toy_coset.py, the toy-model scripts), which shows that the structure is consistent, not that P3 holds in nature.

Statement#

P3  The action is invariant under the Born rotation generated by
    I = ω⁻¹H.                                                       [P]

This is #11 §4’s text, the same as the program paper v1’s “P3 (Reciprocity)” (#9 §2). The lab notebook’s first version named the rotation in coordinates:

P3 (Reciprocity). The action is invariant under the Born rotation ρ: (x,p) → (p,−x), the ℤ₄ generated by the complex structure I of §2. [P] (#1 §1)

The later wording drops the coordinates and ties the rotation to I = ω⁻¹H, the complex structure built from the symplectic form ω and the generalized metric H of P1.

The Born rotation#

In coordinates#

Born reciprocity is the map

ρ: (x, p) → (p, −x),        R = [[0, 1], [−1, 0]]   on tangent vectors      (#1 §2)

Applied twice it gives ρ²: (x, p) → (−x, −p) (#1 §2), and applied four times it gives the identity, so ρ generates a group of four elements, ℤ₄ [T]. Since I² = −1, I is itself a quarter-turn of phase space, and the lab notebook names the group as “the ℤ₄ generated by the complex structure I” (#1 §1).

“Quantum kinematics is exactly invariant under ρ: (x,p) → (p,−x); the Fourier transform implements it unitarily [1†, 2†]” (#11 §3). The Fourier transform turns a wave function of position into a wave function of momentum, as it turns a waveform into its spectrum, and applying it twice turns ψ(x) into ψ(−x), the quantum form of ρ² [T] (the Fourier transform as a phase-space rotation).

How it acts on the structures#

The lab notebook checked the action of R on the three structures in their standard forms: “R preserves ω (it is symplectic), preserves H (it is orthogonal), and flips the sign of η, hence flips J and K while preserving I. [D]” (#1 §2). Here J = η⁻¹H and K = η⁻¹ω are the two real structures of Born geometry (the structures ω, η, H). Two consequences shape the rest of the program.

  • At the standard forms ρ preserves ω, H and I, so a law written in terms of these structures can be invariant under ρ.
  • ρ reverses η. A state that carries a definite η is therefore not invariant: ρ maps it to the state with −η. Under P2′ η is the vacuum’s order parameter, and this sign flip is “the broken symmetry moving the order parameter” (#5 §B; the order parameter).

The lab notebook also noted that “ρ²: (x,p) → (−x,−p) preserves all three structures” and wrote the breaking pattern as ℤ₄ → ℤ₂ = {1, ρ²} [D] (#1 §2). That discrete pattern assumed the rigid η of the original P2; since v2.2 the polarization is a continuous angle (the polarization as a Goldstone angle).

On the modular field#

Under P2′ the metric H ranges over the Siegel domain, coordinatized by a complex symmetric matrix Z = X + iY. There ρ acts by modular inversion: “ρ: Z → −Z⁻¹ (Möbius; d = 1 check), fixed point Z = i; at X = 0 it is scale-factor duality g → g⁻¹, and the cosmological constraint a·ã = 1 (§10) is the |Z| = 1 locus” (#11 §6; on the |Z| = 1 locus, see flag 23 in review/wiki-build-flags.md). See reciprocity as modular inversion and the modular field Z = X + iY.

A symmetry of an action, not a change of description#

In quantum mechanics ρ relates two equivalent descriptions of the same system. For finitely many degrees of freedom the irreducible representations of the canonical commutation relations are all unitarily equivalent (the Stone–von Neumann theorem) [T], ρ leaves the commutator [x, p] = iℏ unchanged, and the Fourier transform is the unitary that connects the position description to the momentum one (unitary equivalence versus Noether symmetry). Read that way, ρ is a choice of variables rather than an operation on the states of one theory, and there is nothing for a vacuum to break.

The withdrawn manuscript #14 stated the distinction the program has to meet:

We note at the outset a distinction the proposal must eventually discharge: as usually encountered, rho is a unitary equivalence between representations of the canonical commutation relations, not a Noether symmetry of a specified action. The framework’s claim is that rho is a genuine symmetry of an underlying action, spontaneously broken; the emergent structure eta below is the proposed order parameter, and the exhibition of a toy action realizing this is stated in Sec. 6 as the proposal’s first acceptance criterion. (#14 §1)

The framing sentence is the standard reading, not a claim of #14’s dead dark-matter candidate. P3 is the program’s version of “the framework’s claim”. A symmetry of an action is an operation on the fields that leaves the action S unchanged, S[ρ·fields] = S[fields], so that it maps solutions of the field equations to solutions [T]; the referees call it a “Noether symmetry”, after Noether’s theorem, which attaches a conserved current to each continuous symmetry of an action [T]. Only such a symmetry can be spontaneously broken: the laws keep it, and the ground state does not [T]. #14 §6 listed “that rho is a genuine, spontaneously broken symmetry” among its conjectural claims.

The referees’ demand#

The blind referees of rounds 1 and 2 turned the distinction into a test. Round 1’s referee asked for a toy Lagrangian, and the demand was “adopted as acceptance criterion #1”; the handoff adds that “both reviewers independently named it” (HANDOFF.md §6, §2; review rounds 1 and 2). The toy-model worklog fixed its success criteria before any construction, starting from the referees’ wording: “an action in which ρ is a genuine Noether symmetry, spontaneously broken, with η the order parameter” (‘Success criteria’). Its criterion S1 is the letter of P3:

S1 — symmetry of an action. An explicit action S[fields] on which ρ acts as a symmetry: S is invariant, not merely related to its image by a unitary equivalence. Invariance is verified symbolically. (‘Success criteria’)

After round 3 the toy became the program’s next task, because “Every referee across three rounds converges on ’the framework does no work’. The toy action is the only item that can change that, and it is acceptance criterion 1 for anything the program proposes” (review/round3-adjudication.md, ‘Decisions’ 4; the ‘framework does no work’ critique). One clause of the original demand is not addressed: the handoff’s version asks for η “the order parameter also selecting the electric sublattice” (HANDOFF.md §2), and #14’s criterion 1 asks for η to select “the hidden sector’s electric description” (#14 §6). The worklog’s criteria and #11 §14 item 11 omit the clause (acceptance criterion 1; success criteria S0–S5).

A realization: the coset toy model#

The model#

Model I of the toy-model note works on phase space ℝ² (d = 1), with ω = [[0, 1], [−1, 0]] rigid, and fields on a 1+1-dimensional base (toy note §1–§2):

  • H, a positive symmetric symplectic matrix, parametrized by a point Z = X + iY of the upper half-plane, with I ≡ ω⁻¹H and I² = −1;
  • K, a real structure with K² = 1 and Kᵀ ω K = −ω, which defines the neutral metric η = ωK. Its target is “the one-sheeted hyperboloid a² + bc = 1, which is two-dimensional de Sitter space” (de Sitter space).

The Euclidean Lagrangian is

L = (f_H²/8) Tr(H⁻¹∂H H⁻¹∂H) + (f_K²/4) Tr(H⁻¹ ∂Kᵀ H ∂K) + (λ/8) Tr(H⁻¹ Cᵀ H C),
    C = KI + IK                                                      (toy note §2)

The first term is the kinetic term of H, the second that of K, and the third a potential that “vanishes exactly when K is compatible with H”. The K kinetic term is contracted with H to make it positive definite: “The naive Tr(∂K ∂K) is indefinite on the hyperboloid”, and an indefinite kinetic term would mean a ghost (toy note §2; worklog ‘Chunk 1 results’).

The symmetry#

“Sp(2,ℝ) acts by H → SᵀHS, K → S⁻¹KS, and the action is invariant (checked for generic symplectic S, so for ρ). There is a separate ℤ₂, K → −K.” (toy note §2). On the fields, “ρ acts as Z → −1/Z and K(φ) → K(φ + π) = −K(φ)”. The gate checks this symbolically (verify/toy_coset.py, part S1, all passing):

  • a generic S is symplectic, and the H kinetic term, the K kinetic term and the potential term are each invariant under it;
  • all three terms are invariant under ρ, and all three under K → −K;
  • the K kinetic term Tr(H⁻¹ dKᵀ H dK) is positive for random H and dK (“no ghost”), while the naive Tr(dK dK) is indefinite on K’s target space.

The same script checks what P5 needs next: no vacuum is ρ-invariant, and ρ maps the vacuum (1, K(φ)) to (1, −K(φ)) (part S2; model I vacua).

What it shows and what it does not#

#11 §14 item 11 records the criterion as “Met in the referees’ wording by the coset model of (H, K) with a bounded ρ-preserving potential”, the first point being that “ρ is a symmetry of an explicit action, and η = ωK is the order parameter”. The worklog had predicted that this part would “pass easily”, since a coset sigma model on compatible (H, K) pairs is “close to a construction by definition” (‘Predictions’), and its chunk-1 entry agrees: “As predicted, S1–S3 come almost by construction.”

The realization is a consistency check, not a test of P3. It is in d = 1 and 1+1 dimensions, with the toy fields (H, K) rather than the program’s own geometric sector, and the toy-model note lists “anything beyond d = 1” as not shown (§6). See model I: action and symmetries.

What a symmetric action leaves open: the twisted element#

The separate ℤ₂ has a consequence that v1 of the toy-model note did not state. “The twisted element ρ∘(K → −K) maps (1, K₀) to itself. So within the full symmetry group, the vacuum keeps an element that acts on H exactly as ρ does. Breaking ρ is physical only once a coupling odd in K is present” (toy note §2). The round-4 referee raised it, and ruling F3 upheld it: “Breaking ρ means something physically only once a coupling odd in K breaks the K → −K symmetry, for example P4-type matter through I_K” (review/round4-adjudication.md). An action invariant under ρ is therefore not enough for the breaking of ρ to have physical content: matter must also couple to the sign of K, which is the role of P4. See the twisted element.

Where the program uses P3#

  • Modular inversion and scale-factor duality. ρ acts on the modular field as Z → −Z⁻¹, which at X = 0 is g → g⁻¹ (#11 §6, above; the cosmological constraint a·ã = 1).
  • The palindrome. “Reciprocity acts on the wedge sector as frame exchange, forcing βₙ = β₄₋ₙ”, and that constraint gives the vacuum-energy identity V(1) = 2T [D][V] (#11 §7.3; the palindrome; the vacuum-energy identity).
  • Reciprocity parity of the spectrum. “Massless graviton = reciprocity-even … X = reciprocity-odd” (#11 §7.5; spectrum and reciprocity parity). The parities hold for reciprocity as the frame exchange of §7.3; under the modular inversion of §6 the metric’s fluctuation changes sign with X’s at Z = i, and the documents do not show how one action becomes the other for the massless mode (flag 39 in review/wiki-build-flags.md, which its audit rated as depending on which action is meant).
  • Photon dispersion. Row 9 of #11 §13 predicts a linear term “exactly zero (ρ²-protected)”, a rationale written out only in the lab notebook (row 9).
  • The recurring matrix. S-duality acts on the complexified coupling “by τ_c → −1/τ_c, generated by the same [[0,1],[−1,0]] rotation as Born’s ρ and our Z → −Z⁻¹” (#11 §11.3; the symplectic charge lattice).

History#

  • v0.1 (lab notebook, September 2026): P3 as the ℤ₄ generated by I, with the breaking pattern ℤ₄ → ℤ₂ (#1 §1–§2).
  • v1 and the checkpoint: P3 stated through I = ω⁻¹H (#9 §2; #11 §4).
  • The soliton manuscript #14: the unitary-equivalence distinction stated, and a toy action named as acceptance criterion 1 (#14 §1, §6).
  • 2026-10-19: the coset toy model meets criterion S1, checked in the gate (worklog ‘Chunk 1 results’).
  • 2026-10-20: after round 4 (verdict B), v2 of the toy-model note states the twisted element, and #11 v2.2 records the criterion as met in the referees’ wording (review round 4).

Sources#

  • papers/broken-born-checkpoint-v2.md §4 — the statement; §3 — ρ in quantum kinematics; §6 — ρ as Z → −Z⁻¹; §7.3, §7.5 — the palindrome and reciprocity parity; §11.3 — the recurring matrix; §13 row 9 — ρ² protection; §14 item 11 — the toy action that meets acceptance criterion 1
  • notes/broken-born-theory.md §1–§2 — the v0.1 statement as a ℤ₄, how ρ acts on ω, η and H, and the ℤ₄ → ℤ₂ pattern
  • notes/broken-born-program-v1.md §2 — the v1 statement
  • notes/debt1a-geometric-core.md §B — the sign flip of η read as the broken symmetry moving the order parameter
  • notes/knot-dark-matter-standalone.md §1 — the unitary-equivalence distinction (withdrawn manuscript; the framing statement is the standard reading, not a claim of the dead candidate); §6 — acceptance criterion 1 and the conjectural status of the symmetry
  • HANDOFF.md §2, §6 — the referees’ demand and the electric-sublattice clause
  • review/round3-adjudication.md ‘Decisions’ 4 — why the toy action came next
  • notes/toy-lagrangian.md ‘Success criteria’ — the referees’ wording and criterion S1; ‘Predictions’ and ‘Chunk 1 results’ — S1 by construction
  • papers/reciprocity-toy-models.md §1–§2 — model I’s fields, action, symmetries and twisted element; §6 — what is not shown
  • review/round4-adjudication.md ‘Rulings’ F3 — the twisted element
  • verify/toy_coset.py S1 — the symbolic invariance checks
  • review/wiki-build-flags.md item 39 — the two actions of ρ on the spin-2 sector and the parities they give, with its audit verdict; awaiting Paul’s ruling