Acceptance criterion 1: the toy Lagrangian (OP11)

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.

Status: Alive — met “in the referees’ wording” (#11 §14 item 11; “done to the referees’ wording”, root CLAUDE.md), with the remaining parts tagged [O]. Model I’s checks run in verify/toy_coset.py and verify/toy_s4.py (the toy-model scripts); the toy-model note was graded B in review round 4 and revised to version 2.

The demand#

The referees’ wording, as the toy-Lagrangian worklog records it: “an action in which ρ is a genuine Noether symmetry, spontaneously broken, with η the order parameter” (‘Success criteria’). It has four parts:

  • An explicit action. A specified field theory, not an argument about one; “a toy model suffices”, in the wording of #14 §6.
  • A genuine Noether symmetry. The action is invariant under ρ. This is the point the referees pressed: in ordinary quantum mechanics ρ is carried out by the Fourier transform, a unitary change from the position description of a state to its momentum description. Such a change of description is available in any quantum theory, so by itself it does not make ρ a symmetry of the dynamics (unitary equivalence versus Noether symmetry). #14 conceded the point at its outset: “as usually encountered, rho is a unitary equivalence between representations of the canonical commutation relations, not a Noether symmetry of a specified action” (§1).
  • Spontaneously broken. The vacua are not invariant under ρ, which maps one vacuum to another (spontaneous symmetry breaking).
  • η the order parameter. The quantity that distinguishes the vacua is the neutral metric η, whose choice fixes which Lagrangian subspace of phase space counts as position (Lagrangian polarizations).

These are the program’s own claims put in testable form: the postulates say that the laws are ρ-invariant (P3), that η is not rigid but emergent, the order parameter of the breaking (P2′), and that the vacuum selects a polarization (P5). In the toy models η is built from a real structure K, η = ωK; #11’s statement of P2′ writes η = HJ instead, “with the real structure J chosen spontaneously” (§4). No document reconciles the two notations (the order parameter).

Where it came from#

  • Rounds 1 and 2. The handoff records the demand among the valid hits of round 1: “toy-Lagrangian demand adopted as acceptance criterion #1” (HANDOFF.md §6). Its open-problems list: “Toy Lagrangian in which ρ is a genuine Noether symmetry, spontaneously broken, η the order parameter also selecting the electric sublattice (acceptance criterion #1; both reviewers independently named it)” (§2). See review rounds 1 and 2 and the handoff.
  • #14’s statement. The soliton manuscript, now withdrawn, listed it as the first of three acceptance criteria: “(1) exhibition of an action - a toy model suffices - in which rho is a genuine Noether symmetry, spontaneously broken, with eta the order parameter selecting both the position polarization and the hidden sector’s electric description; deriving hypothesis H from that action would make the coupling a prediction” (§6).
  • The queue. The handoff ranked it fourth: “Toy Lagrangian (acceptance criterion #1) — the item both independent reviewers named as the gate to verdict A. Hardest; highest value. Start in 0+1 or 1+1 dimensions on the Siegel disk.” (§8 item 4).
  • Round 3. The third referee added the deeper objection that the framework does no work. The adjudication’s Decision 4 made the toy Lagrangian the next task: “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.” (round-3 decisions).

From the demand to tests#

Before building anything, the worklog fixed six pass-or-fail criteria in two tiers: “The first tier is the referees’ letter; the second is what would answer round 3.” (‘Success criteria’). Tier 1 is S0, prior work cited; S1, a symmetric action; S2, vacua that ρ permutes; and S3, an order parameter built from η. Tier 2 is S4, survival under quantum fluctuations, and S5, a consequence for matter’s locality. The criteria and the forecasts made with them are on success criteria S0–S5.

The verdict#

#11 §14 item 11, current since version 2.2 (2026-10-20):

Toy Lagrangian (acceptance criterion 1 of every program proposal). Met in the referees’ wording by the coset model of (H, K) with a bounded ρ-preserving potential, in papers/reciprocity-toy-models.md:

  • ρ is a symmetry of an explicit action, and η = ωK is the order parameter;
  • the vacuum is stable with H dynamical;
  • there is an ordered ℤ₂ phase in 1+1 in the frozen-H reduction (Monte Carlo).

Each line rests on a part of model I:

  • The action and the order parameter. Model I’s fields are a phase-space metric H and a real structure K on a 1+1-dimensional base. Its action is invariant under the whole symplectic group Sp(2, ℝ), “checked for generic symplectic S, so for ρ”, and under a separate flip K → −K (toy-model note §2; action and symmetries). At H = 1 the vacua form a circle K(φ), ρ maps K(φ) to −K(φ), and “O = K₁₁ flips sign under ρ” (§2; the vacua).
  • Stability with H dynamical. A definite polarization needs a potential that keeps ρ and reduces the continuous circle to two vacua, ±K_ref. The first such potential was unbounded below once H moves, and died in round 4 (obituary: the naive ρ-preserving potential). Its replacement V_b is bounded, and “With H dynamical, (1, K_ref) is a stable minimum: Hessian eigenvalues 0, 0.764, 2, 5.236” (§4; the bounded potential V_b). The zero eigenvalue belongs to a flat line that, in version 2’s words, “is not symmetry-protected”; #11’s line does not mention it.
  • The ordered phase. In 1+1 dimensions the continuous angle cannot order (Mermin–Wagner–Coleman), but the two vacua ±K_ref can. A lattice Monte Carlo of the polarization angle at H = 1 finds ⟨|m|⟩ = 0.954 ± 0.0001 “at every size” (L = 16, 32 and 64; see flag 13 in review/wiki-build-flags.md) with the potential, and no true order without it (§4; the frozen-H reduction; the Monte Carlo).

Version 2.2 of #11 adds two qualifications in its refinement of P2′. “A definite polarization needs a ρ-preserving potential on that angle — a rigid, unordered pair of axes, so more structure than ω alone, though less than a rigid η.” And “the breaking of ρ is physical only once matter couples to the sign of K (P4): without such a coupling, a twisted element ρ∘(K → −K) leaves the vacuum invariant” (#11 §4; the twisted element; P4).

Model II, the self-dual Grosse–Wulkenhaar model, was tried first and does not meet the criterion: its complex version breaks the position–momentum duality only classically (quantum restoration), and its real version has no duality symmetry to break (Model II).

The review behind it#

The note’s first version was graded B in round 4: “Referee: B (flawed but salvageable), confidence ~0.75. I agree.” The reason: “The only claim of actual quantum symmetry breaking (§4, model I in 1+1) rests on a potential that is unbounded below once H is dynamical”; and “It is repairable.” (review/round4-adjudication.md, ‘Verdict’). The pre-registration had placed the weakest link there, on the rigid background the potential introduces, and the adjudication scored it “Partial: right location, missed severity” (pre-registration). Version 2 applies the accepted revisions, except the comparison with Langmann, Szabo and Zarembo’s exact solution of a related model that revision 4 asked for, which it says it has not carried out (§4; quantum restoration). #11’s item ends “Round-4 blind review: verdict B; v2 applies the accepted revisions”, and the adjudication and the worklog record every accepted revision as applied (flag 42 in review/wiki-build-flags.md).

The referee’s assessment of how spontaneous the 1+1 breaking is: with H frozen, an Ising-type breaking of the remaining ℤ₂, “But the potential fixes the unordered axis pair by hand, so the vacuum chooses only which of two pre-set axes is ‘position’.” With H dynamical, as the model was first written, the breaking was “not established” (audit notes, Step 3). The bounded potential answers the second point for the vacuum’s stability; a simulation with H dynamical has not been run.

How far it answers round 3#

The worklog’s assessment, written before round 4: “The round-3 critique (’the framework does no work’) is only partly answered. The toy makes the program’s structures cohere (η as order parameter; τ as the polarization angle), but matter’s x-locality still enters through the P4-type coupling.” It assigns the idea to prior work: “Prior art: FLM have the idea. The explicit realization appears new per S0.” (‘Assessment against the predictions’; FLM is Freidel–Leigh–Minic, broken Born duality). Here τ is the program’s dark-energy field, which it places on the polarization circle (τ quintessence). The round-4 referee’s searches found no earlier explicit model “with eta=omega K as a dynamical order parameter”, and judged that “Model I appears new as a field theory, but it is a simple coset” (audit notes, Step 3).

The forecast written before construction: if S1–S4 passed and S5 failed, “acceptance criterion 1 as the referees worded it is too weak to answer ’the framework does no work’”. It was scored “Hit”: “Criterion 1 as worded is now met (coset + τ potential). The round-3 critique is only partly answered” (‘Assessment against the predictions’). The τ potential of that scoring is the naive ρ-preserving potential, now dead, and the verdict built on it is superseded (below). The matter-locality side is on locality from an (H, K)-built coupling and Open: forced locality and the P4 lock.

Open: the parts not yet met#

#11 §14 item 11: “Still open [O]: a Monte Carlo with H dynamical; a derivation of the potential; locality of matter beyond the weak (Yukawa-like) form, which in the toy reduces to P4.” The root CLAUDE.md lists the open ends of the toy Lagrangian as “a Monte Carlo with H dynamical; a derivation of the ρ-preserving potential; the strong form of matter locality; the real Grosse–Wulkenhaar model’s one-loop preference for the position-sharp vacuum, a lead.”

All four are registered at open problems. The queue in the root CLAUDE.md adds one optional item for this work: “a round-5 review of the toy note, but only before any external use.”

The electric-sublattice clause#

Two older wordings of the criterion ask more of η. The handoff: “η the order parameter also selecting the electric sublattice” (HANDOFF.md §2). #14: η “selecting both the position polarization and the hidden sector’s electric description” (§6). The clause ties the order parameter to the program’s identification of electric–magnetic duality frames with phase-space polarizations (the symplectic charge lattice). The worklog’s statement of the referees’ wording, its criteria S0–S5 and #11 §14 item 11 omit the clause, and #11 declares the criterion met. The root CLAUDE.md wording, “done to the referees’ wording”, governs (source conflicts). Nothing in the record addresses the clause, and no current document lists it as open.

Superseded: the worklog’s first verdict#

The first statement. Before round 4 the worklog declared: “Verdict on acceptance criterion 1. Met, in the referees’ wording, by the coset toy plus a τ-type potential”, with “It is spontaneously broken (Ising class in 1+1, Monte Carlo verified).” (‘Assessment against the predictions’).

What replaced it. #11 §14 item 11 in version 2.2: met “by the coset model of (H, K) with a bounded ρ-preserving potential”, with “an ordered ℤ₂ phase in 1+1 in the frozen-H reduction”.

Why. The “τ-type potential”, −μ[Tr(K K_ref)]², is unbounded below once H is dynamical, so its intended vacuum is a saddle and the Monte Carlo had simulated a reduction with H frozen, not model I (round-4 findings F1 and F2, both upheld; obituary: the naive ρ-preserving potential). Data at one temperature deep in the ordered phase establish order, not a universality class, so revision 1 replaced “Ising class” with “ordered ℤ₂ phase” (the frozen-H reduction). The adjudicator had also checked the potential only at frozen H (‘Rulings’ F2). Round-4 lesson 1 became a rule of the program’s method: when numerics run on a reduced model, check the unreduced model’s stability (reduced-model stability). The register of superseded statements lists the “Ising class” wording.

#14’s other criteria#

#14 §6 listed two more criteria: “(2) either a gauged hidden charge conjugation, or a demonstration that the millicharged psi window survives existing constraints” and “(3) the strong-coupling relic freeze-out calculation of Sec. 3, in the hidden plasma”. Both served the superheavy soliton candidate, which round 3 retired and which stays dead (obituary: the soliton candidate). Criterion 1 outlived the manuscript that stated it: round 3 made it “acceptance criterion 1 for anything the program proposes” (Decision 4), since it tests the program’s hard core rather than any one candidate.

Sources#

  • papers/broken-born-checkpoint-v2.md §14 item 11 — the current verdict and its open parts [O]
  • papers/broken-born-checkpoint-v2.md §4 — the P2′ refinement: the potential’s rigid pair of axes, the twisted element
  • CLAUDE.md ‘Since the handoff’ — “done to the referees’ wording”; the open ends, including the real-model lead; the optional round 5
  • papers/reciprocity-toy-models.md §2, §4, §5, §6 — model I’s symmetry, vacua and order parameter; the bounded potential and the Monte Carlo; the Yukawa analogy; “Not shown”
  • review/round3-adjudication.md ‘Decisions’ 4 — why the toy Lagrangian came next
  • review/round4-adjudication.md ‘Verdict’, ‘Rulings’ — the B grade; findings F1 and F2 and the adjudicator’s account
  • review/round4-preregistration.md ‘Predictions’ — the predicted weakest link
  • review/round4/referee/audit-notes.md Step 3 — how spontaneous the 1+1 breaking is; the novelty of model I
  • notes/toy-lagrangian.md ‘Success criteria’, ‘Assessment against the predictions’ — the referees’ wording and the two tiers; the first verdict, its prior-art statement and the scored forecast
  • HANDOFF.md §2, §6, §8 item 4 — the electric-sublattice wording; the criterion’s adoption in round 1; the original queue item
  • notes/knot-dark-matter-standalone.md §1, §6 — #14’s concession and its acceptance criteria (1)–(3), in a withdrawn document
  • review/round4-adjudication.md ‘Revisions for v2 of the note’; notes/toy-lagrangian.md ‘Write-up and review (2026-10-19/20)’; review/wiki-build-flags.md item 42 — every accepted revision recorded as applied, against the comparison v2 did not make; awaiting Paul’s ruling