Toy models of broken reciprocity

A toy model is a small, fully specified field theory built to test one idea in isolation. The idea tested here is the program’s hard core: that Born reciprocity, the exchange of position and momentum, is an exact symmetry of the equations that the lowest-energy state does not share. Two models were built and tested on 19–20 October 2026. Model I is a sigma model, a field theory whose fields take values in a curved space; its two fields are a metric on the phase plane, the plane of a particle’s position and momentum, and a real structure, which picks out two lines of that plane as the candidate position and momentum axes. Model II is a field theory on a noncommutative plane, a plane whose two coordinates fail to commute, as position and momentum do in quantum mechanics. Model I, with a bounded potential added (one that cannot fall without limit), answers the referees’ acceptance criterion 1, their demand for an explicit model in which the exchange is a genuine symmetry that the vacuum breaks; checkpoint #11 records it as met in the referees’ wording, with stated open ends. With a complex field, model II’s vacua break the exchange only before quantum effects are included, and quantum fluctuations restore it. With a real field the exchange is not a symmetry of the model, and the first quantum correction (the one-loop term) favours the vacuum that is sharper in position over its mirror image, a preference recorded as a lead.

Stands: acceptance criterion 1, met in the referees’ wording; in model I, the action, the vacua and order parameter, the bounded potential V_b, the frozen-H reduction and its ordered ℤ₂ phase; in model II, the matrix base and the action, the duality lemma, complex versus real fields, the complex-model vacua and quantum restoration (a negative result); the literature pages coset and hyperbolic sigma models and noncommutative field theory · Died: the naive ρ-preserving potential · Open: a Monte Carlo with H dynamical, whether ρ’s SO(2) breaks in the coupled model, the real model’s one-loop preference; folded sections on the parts of criterion 1 not yet met [O] and the comparison with the exact solution, and the one on model I’s index · Superseded: folded sections on the worklog’s first verdict, the reversed sign of the polarization and “Increasing the truncation adds only massive levels”, and the three on model I’s index

Map#

The two documents. The work is recorded in two documents, and this section covers both. papers/reciprocity-toy-models.md, titled “Spontaneous Breaking of Born Reciprocity: Two Toy Models”, version 2 (commit 6606734, 2026-10-20), is the current reference. It is written as an anonymous manuscript for blind review, so it carries no provenance tags, and its † marks are Hermitian adjoints (φ†φ), not citation marks. The version sent to review round 4 (commit d9a06ac, 2026-10-19) was titled “Spontaneous Breaking of Born Reciprocity in Two Toy Models: the Polarization as a Goldstone Angle”; version 2 no longer carries that subtitle. Round 4’s revision 3, among its changes to the note’s section on model II, reads “Restrict ‘Goldstone angle’ to the complex model.” (review/round4-adjudication.md, ‘Revisions’ 3), and version 2’s abstract reads “Where ρ sits inside a continuous U(1) — model I, and model II with complex fields — the polarization is a Goldstone angle.” (the polarization as a Goldstone angle). notes/toy-lagrangian.md is the dated worklog: the criteria and forecasts, four chunks of results, the assessment against the forecasts, and the write-up. Where the two differ, version 2 governs; several of the worklog’s numbers and wordings predate the round-4 review, and the pages here mark each such case. The sequence ran over two days: the criteria were fixed and the four chunks computed on 19 October 2026, the first version of the note was pre-registered and sent to blind review the same evening, and the adjudication and version 2 followed on 20 October (timeline).

The pages run in the order a reader needs them: the demand, the tests, then the two models.

  • Acceptance criterion 1: the toy Lagrangian comes first because it is the demand the models answer: an action in which ρ is a genuine Noether symmetry, spontaneously broken, with the neutral metric η as order parameter. The page traces the demand from the round-1 and round-2 referees through #14 and the handoff to round 3’s decision to build it, gives #11’s verdict and what each part of it rests on, and records what remains open, the worklog’s superseded first verdict, and a clause of an older wording, the electric sublattice, that nothing addresses.
  • Success criteria S0–S5 and the pre-registered predictions are the pass-or-fail tests written down before either model was built, with the forecasts made alongside them. The page explains each criterion, the prior-art survey S0 required, the scorecards, the worklog’s self-scored assessment and the three findings nobody forecast. It is method, not physics, and it defines what the later pages mean by “pass”.
  • Model I: the coset sigma model is a field theory of a phase-space metric H and a real structure K, and it is the model that meets the criterion. Its subsection runs from the sigma-model background through the action, the vacua and the Goldstone spectrum, the failed and the repaired potentials, the frozen-H reduction and its lattice simulation, to the two open questions: a Monte Carlo with H dynamical and ρ’s SO(2) in the coupled model.
  • Model II: the self-dual Grosse–Wulkenhaar model was tested first, as the route in which a field’s own vacuum might select position or momentum with no K put in by hand. Its subsection runs from the noncommutative-field-theory background through the matrix base, the duality lemma and the distinction between complex and real fields, to the complex model’s vacua, their quantum restoration, and the real model’s one-loop lead. The route fails the quantum-stability test, so the coset model’s fallback, a ρ-preserving potential, became the result for that test.

Each model’s subsection opens with one page of established physics, coset and hyperbolic sigma models and noncommutative field theory and the Langmann–Szabo duality, both literature [T]. Every other page in this section is the program’s own work. The numbers are checked by the four gated scripts described at the toy-model scripts, and the note was graded B in review round 4 and revised to version 2. The adjudication and the worklog record every accepted revision as applied, though version 2 says that the comparison with an exactly solved related model, which revision 4 asked for, was not carried out (quantum restoration; flag 42 in review/wiki-build-flags.md). A round-5 review of the note is optional, “but only before any external use” (root CLAUDE.md).

Boundaries#