Model I: action and symmetries

Model I’s action is the integral, over a flat two-dimensional base, of a Lagrangian with three terms: two kinetic terms, which cost action when the fields vary from point to point, and a potential that vanishes exactly when the two fields are compatible. The fields are geometric structures on the phase plane of one particle moving in one dimension: a metric H, parametrized by a point of the upper half-plane, and a real structure K, whose two eigenlines are the candidate position and momentum axes. The action is unchanged by every symplectic (area-preserving) linear map of the phase plane, which includes Born reciprocity ρ, and by the separate flip K → −K. That invariance is criterion S1, checked symbolically, and it supplies the “genuine Noether symmetry” that acceptance criterion 1 asks for.

Status: Alive — [V] (the worklog records “S1 pass [V]”; criterion S1 in verify/toy_coset.py; the toy-model scripts) — confirmed independently with sympy by the round-4 referee (“all three terms Sp(2,R)-invariant”).

The setting: one position and one momentum#

Model I lives on the phase plane of a single degree of freedom, d = 1. Version 2 of the toy-model note sets it up in five sentences (§1):

Phase space ℝ² with symplectic form ω = [[0, 1], [−1, 0]]. ρ = [[0, 1], [−1, 0]] is symplectic and squares to −1. A real structure K with K² = 1 and Kᵀ ω K = −ω defines a neutral metric η = ωK. Its null lines are the two eigenlines L± of K, which are Lagrangian. Choosing K is choosing which line is “position”.

Each object has a plain meaning [T]:

  • ω is the symplectic form: it measures signed area in the (x, p) plane. The linear maps that preserve it form Sp(2, ℝ).
  • ρ is the quarter-turn (x, p) → (p, −x). As a matrix it equals ω, and ρ² = −1 is the half-turn (x, p) → (−x, −p).
  • K squares to 1 and, for d = 1, has determinant −1, so its eigenvalues are +1 and −1 and it has two eigenlines, L+ and L−. The condition Kᵀ ω K = −ω says that K reverses the sign of areas. In d = 1 every line through the origin is Lagrangian, a line on which ω vanishes (Lagrangian subspaces and polarizations).
  • η = ωK is a symmetric metric with one positive and one negative direction, a neutral metric, whose null lines are L+ and L− (the structures ω, η, H). The program reads η, equivalently K, as the order parameter of broken reciprocity.

For the reference structure K₀ = diag(1, −1), L+ is the x-axis and L− the p-axis. Its ρ-image −K₀ swaps them, because “ρ exchanges L+ and L−” (note §5). Choosing K therefore chooses which line plays the role of position, the choice the program calls a polarization.

The fields#

On a Euclidean 1+1-dimensional base the model has two fields (note §2):

H(Z) = [[Y + X²/Y, X/Y], [X/Y, 1/Y]],   Z = X + iY,  Y > 0        I ≡ ω⁻¹H,  I² = −1
K    = [[a, b], [c, −a]],                a² + bc = 1               (traceless, det K = −1)
  • H, “a positive symmetric symplectic matrix”, is a metric on the phase plane: a way of measuring lengths that can mix x and p. Each such H corresponds to one point Z of the upper half-plane, the d = 1 case of the Siegel domain. I = ω⁻¹H squares to −1, so it is a quarter-turn adapted to H, a complex structure [T]. At Z = i the metric is H = 1.
  • K is the real structure. For a traceless 2 × 2 matrix, K² = (a² + bc)·1 and Kᵀ ω K = (det K) ω, so “traceless with det K = −1” is the d = 1 form of K² = 1 and Kᵀ ω K = −ω (both identities are printed in the referee’s check, review/round4/referee/out_model1.txt). “Its target is the one-sheeted hyperboloid a² + bc = 1, which is two-dimensional de Sitter space” (note §2; coset and hyperbolic sigma models; de Sitter space).

Compatibility. The two fields fit together when K anticommutes with I. The worklog writes the potential as the squared size of that anticommutator, “(λ/8)|{K, ω⁻¹H}|²_H” (‘Chunk 1 results’), and version 2 states that the potential “vanishes exactly when K is compatible with H” (§2). At H = 1 the condition is simple: “|C|^2 = 2 (b - c)^2: V = 0 exactly when K is symmetric” (verify/toy_coset.py; the referee: “C at H=1: [b - c, 0, 0, b - c] -> zero iff b=c”).

The Euclidean Lagrangian#

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,   I = ω⁻¹H

The couplings f_H and f_K set the stiffness of the two fields, and λ the strength of the compatibility potential [T]. ∂ stands for a derivative along the base, summed over its two directions. The Lagrangian is Euclidean, so a constant field has only its potential energy.

The H kinetic term#

Tr(H⁻¹∂H H⁻¹∂H) is the standard invariant metric on positive matrices [T]. Under H → SᵀHS with a constant S, the product H⁻¹∂H becomes S⁻¹(H⁻¹∂H)S, and the trace of its square does not change. On the upper half-plane this metric is, up to a constant factor, the hyperbolic metric [T]. The metric sector on its own is therefore a sigma model on the hyperbolic plane, which the round-4 referee identified as “the SO(1,2)/SO(2) hyperbolic sigma model” (review/round4/referee/audit-notes.md, Step 3). That identification matters for whether the model can order in two dimensions (Open: does ρ’s SO(2) break in the coupled model?).

The K kinetic term and the ghost#

The obvious kinetic term for K, Tr(∂K ∂K), would carry a ghost. Version 2: “The K kinetic term is contracted with H and is positive definite. The naive Tr(∂K ∂K) is indefinite on the hyperboloid (2 db dc at K₀ = diag(1, −1)).” (§2). The worklog gives the reason: “K’s own target space (de Sitter dS₂) has an indefinite metric, which would give a ghost” (‘Chunk 1 results’). A kinetic term of indefinite sign gives one direction of fluctuation negative energy (coset and hyperbolic sigma models).

The contracted term Tr(H⁻¹ ∂Kᵀ H ∂K) is positive for every nonzero ∂K [T]: it equals the sum of the squares of the entries of H^{1/2} ∂K H^{−1/2}, since H is symmetric and positive. The sources state the positivity and check it numerically:

  • the referee, with sympy: “naive Tr(dK dK) at K0 = ‘2dbdc’ (indefinite); H-contracted at H=1 = ‘db2 + dc2’”; and over random samples of H and ∂K, the smallest ratio of the term to the sum of the squared entries of ∂K was 0.0598 (out_model1.txt; audit notes Step 2(a));
  • the gate: “the K kinetic term Tr(H^-1 dK^T H dK) is positive for random H and dK (no ghost)” and “the naive Tr(dK dK) is indefinite on K’s target space (de Sitter): 2 db dc” (verify/toy_coset.py).

The contraction keeps the symmetry. Under S, ∂K → S⁻¹∂K S and H → SᵀHS, so H⁻¹ ∂Kᵀ H ∂K becomes S⁻¹(H⁻¹ ∂Kᵀ H ∂K)S and its trace is unchanged [T]. It also couples the two sectors: K’s kinetic energy now depends on H.

On the circle of vacua at H = 1, K = K(φ), this term reduces to the stiffness of one angle: “(f_K^2/4)Tr(dK^T dK) = (1/2) f_K^2 (dphi)^2” (referee, Step 2(a)), “an XY stiffness” (verify/toy_s4.py). That is the starting point of the frozen-H reduction.

The compatibility potential#

The third term is the squared size of C = KI + IK, measured with H, so it is never negative [T]; the gate checks that “V = (lam/8)|C|^2_H is a sum of squares at H = 1” (verify/toy_coset.py). It vanishes exactly on the compatible pairs, which are therefore the vacua (Model I vacua). It shares the symmetry [T]: a symplectic S preserves ω, so I = ω⁻¹H becomes S⁻¹IS, C transforms like K, and the same algebra as for the K kinetic term applies.

The symmetries (criterion S1)#

“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.” (note §2). The same S acts at every point of the base, so this is a global symmetry [T]. The referee wrote the full group as “Sp(2,R) x Z2(K->-K)” (Step 2(a), finding F3).

The flip K → −K is a symmetry because the two terms that involve K are quadratic, one in ∂K and the other in C, and C changes sign with K. ρ belongs to Sp(2, ℝ), so invariance under a generic symplectic S includes invariance under ρ; the gate also checks ρ by itself.

The checks [V]. verify/toy_coset.py takes a symbolic S = [[s₁, s₂], [s₃, (1 + s₂s₃)/s₁]], whose determinant is 1 by construction, and symbolic H, ∂H, K, ∂K. Its S1 block passes these checks:

  • “generic S is symplectic”;
  • “H kinetic term invariant under generic symplectic S”, and the same for the K kinetic term and the potential term;
  • “all three terms invariant under rho”;
  • “all three terms invariant under K -> -K”.

The worklog records the result as “S1 pass [V]: all three terms are invariant under a generic symplectic S (H → SᵀHS, K → S⁻¹KS), under ρ, and under K → −K.” (‘Chunk 1 results’). The round-4 referee repeated the computation in its own conventions and found “transformed - original = 0” for each term, and invariance under K → −K (out_model1.txt; audit notes Step 2(a)).

What S1 asks for. Criterion S1 demands “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.” (worklog, ‘Success criteria’; success criteria S0–S5). In quantum mechanics ρ is carried out by the Fourier transform, a unitary change of description that any theory admits; that does not make ρ a symmetry of the dynamics (unitary equivalence versus Noether symmetry). In model I, ρ acts on the fields H and K, and the action is unchanged. Checkpoint #11 §14 item 11 counts this as the first part of acceptance criterion 1: “ρ is a symmetry of an explicit action, and η = ωK is the order parameter”.

What the symmetries imply#

History#

The worklog fixed the success criteria on 2026-10-19 before building anything, and predicted this part would be easy: “S1–S3 will pass easily. A coset sigma model on the space of compatible (H, K) pairs does it classically, which is close to a construction by definition.” (‘Predictions’). Model I was built first as “A baseline … It is cheap, and it fixes what the minimal construction gives” (‘Plan’), and passed: “As predicted, S1–S3 come almost by construction. This is a coset model of Sp(2,R) → Z₂.” (‘Chunk 1 results’). The assessment scored the prediction “Hit”.

The prior-art survey found that “no existing action has Born reciprocity exactly symmetric and spontaneously broken with η or K as order parameter”, while crediting the idea to Freidel–Leigh–Minic (‘Chunk 1 results’; broken Born duality). The round-4 referee agreed with qualification: “Model I appears new as a field theory, but it is a simple coset” (Step 3). The adjudication’s verdict: “The algebra and numerics of §§2, 3 and 5 reproduced under the referee’s own conventions and code.” (review/round4-adjudication.md, ‘Verdict’; review round 4).

Notation#

These pages follow the toy-model note: K is the real structure and η = ωK, as in #11 §14 item 11. Checkpoint #11’s postulate P2′ writes the real structure as J and the order parameter as η = HJ (§4). These are not two names for one operator: #11 §5 defines both, J = η⁻¹H and K = η⁻¹ω, and the documents differ in which one they call the structure chosen spontaneously. The sources do not reconcile the two (the order parameter; notation; P2′; conflicts between sources).

Sources#

  • papers/reciprocity-toy-models.md §1, §2 — the setting, the fields, the Lagrangian and the symmetries, quoted above; §5 for ρ exchanging L+ and L−
  • verify/toy_coset.py S1, S2 — the gated invariance, positivity and compatibility checks
  • verify/toy_s4.py — the docstring’s XY stiffness at H = 1
  • review/round4/referee/audit-notes.md Step 2(a), Step 3 — the referee’s independent symbolic confirmation; the full symmetry group; the hyperbolic H-sector; the novelty remark
  • review/round4/referee/out_model1.txt — the referee’s printed identities and kinetic-term checks
  • review/round4-adjudication.md ‘Verdict’ — the algebra reproduced
  • notes/toy-lagrangian.md ‘Success criteria’, ‘Predictions’, ‘Chunk 1 results’, ‘Assessment against the predictions’ — criterion S1, the prediction, the S1 pass [V], the reason for the H-contracted kinetic term, the prior-art survey
  • papers/broken-born-checkpoint-v2.md §4, §5, §14 item 11 — P2′’s J notation; the definitions J = η⁻¹H and K = η⁻¹ω; the criterion as met