The frozen-H reduction
The frozen-H reduction is model I with its metric field held fixed at H = 1, so that the only remaining field is the polarization angle φ of the real structure K(φ). In that form the model is a two-dimensional spin model of one angle, with a stiffness term and a ρ-preserving potential, and it can be simulated on a lattice. The ordered ℤ₂ phase the simulation finds is a result about this reduction; with H dynamical, ordering is not established. Round 4 required the text to say so, after showing that the first potential tried was stable only in this reduction. It also replaced the claim of “Ising-class” breaking with “an ordered ℤ₂ phase”, because data at one temperature establish order but not a universality class.
Status: Alive, as the stated scope of the lattice result — untagged — required by round-4 revision 1 (“Present the Monte Carlo as the frozen-H reduction, explicitly”), stated in version 2, §4 and §6, and in #11 §14 item 11; simulated in verify/toy_s4.py (the toy-model scripts).
What the reduction is#
H = 1 is the point Z = i of the upper half-plane, the one metric that Born reciprocity ρ leaves in place (model I vacua). Holding H there removes the metric sector, and restricting K to its circle of compatible values K(φ) = [[cos φ, sin φ], [sin φ, −cos φ]] removes the compatibility term. What is left is one angle per point of the base. The pieces, as the sources give them:
K kinetic term at H = 1: (f_K²/4) Tr(∂Kᵀ ∂K) = (f_K²/2)(∂φ)² an XY stiffness
V_b at H = 1: −(μ/2)(1 + cos 2φ)
ρ at H = 1: φ → φ + π
lattice energy: E = −J Σ⟨ij⟩ cos(φ_i − φ_j) − h Σ_i cos 2φ_iThe stiffness is the referee’s check, “(f_K^2/4)Tr(dK^T dK) = (1/2) f_K^2 (dphi)^2 -> matches (f_K^2/2)(dphi)^2” (review/round4/referee/audit-notes.md, Step 2(a)), and verify/toy_s4.py calls it “an XY stiffness”. The potential is the bounded potential V_b on the circle (toy-model note §4). The action of ρ is the worklog’s: “at H = 1 the polarization field is the angle φ of K(φ), and ρ acts as φ → φ + π” (‘Chunk 3 results’).
Version 2 states the simulated model and its scope in two sentences: “We simulate the lattice model E = −J Σ cos(φ_i − φ_j) − h Σ cos 2φ_i: model I’s polarization angle at H = 1, with V_b up to a constant and normalization. H is not simulated.” (§4). On the lattice the first sum runs over neighbouring sites. For small differences, 1 − cos(φ_i − φ_j) is about half the squared difference, so the first sum is the lattice form of the stiffness term, with J setting the stiffness [T]. The second sum is the potential, with h setting its strength: −h cos 2φ “equals the bounded V_b at H = 1 up to a constant and a factor (both depend on phi only through cos 2 phi)” (verify/toy_s4.py). The lattice coupling J has nothing to do with the J that #11’s postulate P2′ uses for the real structure.
In statistical mechanics this is the XY model, a lattice of planar spins, with a two-fold symmetry-breaking field [T]: the referee’s “textbook XY + two-fold-field model” (Step 3, point C3), the first version’s “two-fold anisotropic planar model”. The cos 2φ term has two minima per turn of the angle, at φ = 0 and φ = π.
Symmetries of the reduced model#
The shift φ → φ + π, which is ρ, leaves both sums unchanged and reverses the sign of cos φ = K₁₁, the order parameter. With h = 0 every shift φ → φ + α is a symmetry, the continuous U(1) of the XY model; with h ≠ 0 only the shift by π survives among the shifts [T]. So the potential reduces the continuous symmetry of the circle to a discrete ℤ₂. A discrete symmetry lies outside the Mermin–Wagner–Coleman theorem and can break in two dimensions, as the two-dimensional Ising model shows [T].
Why a two-dimensional lattice tests 1+1 dimensions#
Model I’s Lagrangian is Euclidean. The Euclidean action of a field theory in 1+1 dimensions has the form of the energy of a classical system in two dimensions, so sampling the lattice model at a temperature T samples the quantum fluctuations of the 1+1-dimensional theory, with T in the role of the coupling [T]. verify/toy_s4.py describes its lattice as “a square lattice (Euclidean 2D)”. That is why a classical Monte Carlo answers criterion S4, “The breaking survives quantum fluctuations in the stated dimension” (worklog, ‘Success criteria’).
Why the scope had to be stated#
The first version of the note made the reduction without naming it: “At H = 1 the polarization field is φ, with XY stiffness (f_K²/2)(∂φ)².” (v1 §4, commit d9a06ac). With the potential it used then, H frozen was the only setting in which the intended vacua exist. Round-4 finding F2, upheld: “So (1, K_ref) is a saddle, and the Monte Carlo simulates a frozen-H reduction, not model I” (review/round4-adjudication.md). The referee: “The ’two vacua K=+-K_ref’ exist only if H is frozen at 1, which the manuscript asserts (‘At H = 1 the polarization field is phi’) but does not justify; H is a dynamical hyperbolic sigma field.” (Step 2(a)). The failure is recorded at Obituary: the naive ρ-preserving potential.
Revision 1 required: “Present the Monte Carlo as the frozen-H reduction, explicitly.” Version 2 says “H is not simulated” (§4), and the script’s docstring now reads: “This is the frozen-H reduction of the coset toy (H = 1 held fixed); the unreduced model’s stability with the bounded potential V_b is checked in toy_coset.py.” (verify/toy_s4.py; capitals lowered here). The second half of that sentence is the check that the reduced-model stability rule now demands: with V_b the unreduced vacuum is a stable minimum with H dynamical (verify/toy_coset.py). The simulation itself still covers only the angle.
What it establishes#
Version 2, §6, under “Shown”: “Model I: an ordered ℤ₂ phase in the frozen-H reduction in 1+1.” Checkpoint #11 §14 item 11 lists the same result as one of three parts of acceptance criterion 1: “there is an ordered ℤ₂ phase in 1+1 in the frozen-H reduction (Monte Carlo)”, next to “the vacuum is stable with H dynamical”. The numbers, error bars and controls are at Monte Carlo: an ordered ℤ₂ phase.
How spontaneous the breaking is#
The referee asked whether “spontaneous breaking” is a fair description of model I in 1+1, and answered (Step 3; its notes abbreviate spontaneous symmetry breaking as SSB):
With H frozen: yes, standard Ising-type SSB of the residual Z2 =
/<rho^2> (the potential’s SO(2) residue is {+-1, +-rho}, sympy). But the potential fixes the unordered axis pair by hand, so the vacuum chooses only which of two pre-set axes is “position”. The manuscript says this in Sec. 6 (“rigid unordered pair of axes”); the abstract’s “rho is spontaneously broken” should carry the qualifier and the frozen-H caveat. With H dynamical as written: not established (C3).
So in the reduction one thing is put in and one is chosen. The potential fixes the unordered pair {x-axis, p-axis}, which is K_ref up to sign; the vacuum chooses which member of the pair counts as position. The worklog had named the same cost: “The price is a rigid, unordered pair of axes (K_ref up to sign). That is more rigid structure than P2′’s ‘only ω is rigid’, though less than a rigid η.” (‘Chunk 3 results’). #11 carries it into the 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.” (§4).
The pre-registration had predicted this as round 4’s weakest link: “in 1+1, the ‘spontaneous’ breaking of ρ rests on an explicit, rigid background. The ρ-preserving potential needs a fixed unordered pair of axes K_ref. A referee can reasonably call model I’s 1+1 result an ordinary Z₂ breaking of an explicitly chosen axis pair. Model I without it is a coset construction by definition.” (review/round4-preregistration.md). The adjudication scored it “Partial: right location, missed severity”: the referee found a worse defect at the same spot, the unbounded potential.
The worklog’s revised conclusion describes what the toy supports: “‘broken reciprocity’ in the form ‘a pseudo-Goldstone polarization angle with a ρ-symmetric potential’, not ‘only ω rigid, nothing else’.” (‘Chunk 3 results’). A pseudo-Goldstone mode is a flat direction given a small energy cost by a small explicit breaking of the symmetry [T] (the polarization as a Goldstone angle).
Superseded: Ising-class breaking in model I#
The first version of the note listed, among the results “Established here”, “Ising-class breaking of ρ in model I in 1+1, once a ρ-preserving potential is added” (v1 §6). Its §4 called the lattice model “the two-fold anisotropic planar model, whose ordered phase is Ising-like [12]”. The worklog had written “Then ρ breaks as a Z₂ (Ising class), with vacua ±K_ref: ‘x is position’ and ‘p is position’” (‘Chunk 3 results’), and its verdict on acceptance criterion 1 says “It is spontaneously broken (Ising class in 1+1, Monte Carlo verified)” (‘Assessment against the predictions’).
Round 4 narrowed the claim in two ways.
- From model I to the reduction. “Sec. 6’s ‘Ising-class breaking of rho in model I in 1+1’ is therefore not established.” (referee, Step 3, point C3). The simulation never let H move.
- From a universality class to order. “‘Ising-class’: data at a single T deep in the ordered phase establish order only, not universality class. Minor wording.” (referee, Step 2(c)). Revision 1: “Say ‘ordered Z₂ phase’, not ‘Ising class’.” Version 2: “We call the first row an ordered ℤ₂ phase. We did not measure critical exponents.” (§4).
The distinction [T]: a universality class is fixed by the critical exponents of a phase transition, which depend on the dimension and the symmetry rather than on details; the two-dimensional Ising model is exactly solved (Onsager 1944†), and its exponents are known exactly. A simulation at one temperature, here T = 0.6, deep in the ordered phase shows that the ℤ₂ is broken there, but says nothing about the exponents at the transition. The current statement is “an ordered ℤ₂ phase in the frozen-H reduction in 1+1” (version 2, §6; #11 §14 item 11). The change is listed among the superseded statements.
Open: a Monte Carlo with H dynamical#
#11 §14 item 11 keeps “a Monte Carlo with H dynamical” open [O]; version 2 lists “a Monte Carlo of model I with H dynamical” under “Not shown” (§6), and the root CLAUDE.md names it first among the toy work’s open ends (‘Since the handoff’). The ordered ℤ₂ phase on this page has been simulated only in the reduction; for the full model there is the classical check that V_b’s vacuum is a minimum with all four field directions free (the bounded potential V_b). The item, what the sources record about it and what would close it are at Open: a Monte Carlo with H dynamical. The separate question whether ρ’s SO(2) breaks in the coupled model without a potential is Open: does ρ’s SO(2) break in the coupled model?. Both are registered at open problems.
The rule it produced#
Round-4 lesson 1: “When numerics run on a reduced model (some fields frozen), check the stability of the unreduced model — boundedness, and the Hessian with all fields dynamical — and state the reduction in the text.” (review/round4-adjudication.md, ‘Lessons’). The root CLAUDE.md carries it in its Method section, adding “Round 4 found a potential that was bounded only at frozen H.” That sentence holds for the reduction as this page defines it, with K kept on its circle K(φ). With H held at 1 and K free on its hyperboloid, the naive potential is unbounded below at λ = μ = 1, the couplings of verify/toy_coset.py, and bounded only for λ ≥ 4μ (flag 40 in review/wiki-build-flags.md, with the check review/wiki-build-checks/frozen_h_naive.py); the flag’s audit rated it as depending on how “frozen H” is read. The adjudicator, too, had checked the potential only at frozen H (‘Rulings’ F2). See the reduced-model stability rule.
Sources#
papers/reciprocity-toy-models.md§4, §6 — the reduction as simulated; “Shown” and “Not shown”; “We did not measure critical exponents”papers/reciprocity-toy-models.mdv1 (commit d9a06ac) §4, §6 — the unstated reduction and the superseded “Ising-class” claimpapers/broken-born-checkpoint-v2.md§4, §14 item 11 — P2′’s price; the result as the checkpoint records it, and the open simulation [O]review/round4-adjudication.md‘Rulings’ F2, ‘Revisions’ 1, ‘Lessons’, ‘Pre-registration scored’ — the frozen-H finding, the required reframing, the rule and the scored weakest linkreview/round4-preregistration.md‘Predictions’ — the predicted weakest linkreview/round4/referee/audit-notes.mdStep 2(a), Step 2(c), Step 3 — the stiffness check, the universality-class point, point C3 and the reading of how spontaneous the breaking isverify/toy_s4.py— the reduction as coded, and its docstringverify/toy_coset.py— the unreduced model’s stability with V_bnotes/toy-lagrangian.md‘Success criteria’, ‘Chunk 3 results’, ‘Assessment against the predictions’ — criterion S4; the reduction as first made; the Ising-class wording and the price of the potentialCLAUDE.md‘Method’, ‘Since the handoff’ — the reduced-model rule; the open simulationreview/wiki-build-flags.mditem 40 andreview/wiki-build-checks/frozen_h_naive.py— “bounded only at frozen H” with only H held fixed, and the audit’s verdict; awaiting Paul’s ruling