P2′: only the symplectic form is rigid

P2′ is the program’s corrected second axiom: of the structures on phase space, only the symplectic form ω is fixed in advance, and holding it fixed is the program’s way of saying that Planck’s constant does not change. The neutral metric η, which marks which directions count as position, is not part of the background. It is a property of the vacuum: the order parameter of the broken symmetry, the quantity whose value tells a vacuum from its mirror image, as the direction of magnetization tells a magnet pointing one way from one pointing the other. The original P2 had fixed η as well, and the nontriviality theorem showed that this left the program with no content of its own. Checkpoint v2.2 added a refinement from the toy models: with only ω fixed, the polarization is a continuous angle. Two consequences follow. Pinning the angle to a definite value needs an additional potential, which brings back a little fixed structure, and the choice has physical consequences only once matter couples to it.

Status: Alive — [P]; the correction from the original P2 is forced [D] by the nontriviality theorem; the v2.2 Goldstone-angle sentence is [D][V], checked by the toy-model scripts. The toy models show that the structure is consistent; they do not test the postulate.

Statement#

P2′  Only ω is rigid (symplectic rigidity ≡ constancy of ℏ). The neutral
     metric η is emergent — the order parameter, η = HJ, with the real
     structure J chosen spontaneously.                                 [P]

This is #11 §4’s text. Its bracket adds two things: “the correction from the original P2 is forced: a nontriviality theorem (§5) shows the theory has content beyond GR + SIDM iff η is emergent. [D]” (GR: general relativity; SIDM: self-interacting dark matter), and a note on credit, below. Since v2.2 a refinement follows the statement, also below. The program paper v1 states the same axiom as “P2′ (Rigidity — corrected). Only ω is non-dynamical: symplectic rigidity is the constancy of ℏ. The neutral metric η is not fundamental; it is the order parameter, η = HJ, with the real structure J chosen spontaneously.” (#9 §2).

The axiom has two halves: what is fixed (ω) and what is left to the vacuum (η). Both refer to the structures of P1.

What is rigid: ω and the constancy of ℏ#

A rigid, or non-dynamical, structure is fixed background: it has no equation of motion and takes the same value in every state of the theory [T]. P2′ as stated allows exactly one such structure, the symplectic form ω, the antisymmetric pairing of position with momentum behind the commutator [x, p] = iℏ (the canonical commutator).

The physical reading is the one the original P2 already gave ω: “the symplectic structure is the constancy of ℏ — phase-space cells do not stretch” (#1 §1). Holding ω fixed holds the size of a phase-space cell fixed, which is the statement that ℏ does not change (#1 §6). The repair keeps exactly this part of the old axiom: “Freeze ω only — that is genuinely ℏ” (#5 §B). Two consequences in the record follow from ω alone:

  • Unimodularity. The compatibility theorem’s corollary det H = 1 is, in the Debt-1 note’s words, “the geometric face of ℏ-rigidity” (#5 §F). #11 §5 adds that det H = 1 “persists across the whole modular family of §6” and marks this as machine-verified; the battery’s test T2 checks det H_Z = 1 for one randomly sampled member of the family (verify/battery.py; the verification battery), so the statement for the whole family is [D], not [V].
  • The cosmological constraint. “ω-rigidity fixes the comoving phase-space cell: a·ã = 1 (ℏ does not run)” [D] (#11 §10), where a and ã are the scale factors of position space and momentum space (the cosmological constraint a·ã = 1).

What is emergent: η as the order parameter#

The construction#

The Debt-1 note gives the repair in one sentence: “the fundamental pair is (ω, H); the derived complex structure is I = ω⁻¹H; the vacuum spontaneously selects a real structure J (an involution anticommuting with I), and η ≡ HJ” (#5 §B). In order:

  1. The fundamental structures are ω, which is rigid, and the generalized metric H, which is dynamical.
  2. From them comes the complex structure I = ω⁻¹H, with I² = −1, a quarter-turn of phase space.
  3. The vacuum selects a real structure J: a linear map with J² = 1 (an involution [T]) that anticommutes with I.
  4. The neutral metric is then defined from the vacuum’s choice, η ≡ HJ.

So η is no longer an input. It is built from the dynamical H and the vacuum’s J, and it differs from one vacuum to another.

Why η is an order parameter#

In spontaneous symmetry breaking the laws are symmetric but the ground state is not, and an order parameter is a quantity that the symmetry would change and that takes a definite value in the ground state, as magnetization does in a magnet [T]. At the standard forms of the structures, the Born rotation ρ: (x, p) → (p, −x) leaves ω and H alone but not η. The lab notebook found that it “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). A vacuum with a definite η is therefore carried by ρ to a different vacuum, with −η. Under P2′ the Debt-1 note reads the sign flip this way: “the earlier observation that reciprocity flips η now reads correctly as the broken symmetry moving the order parameter” (#5 §B). The note calls the assignment “philosophically right” as well, since η “is the structure that says which directions are position”, and on this reading the vacuum state, not the background, fixes those directions (#5 §B). See the order parameter.

The space of vacua#

The choices of J form a familiar manifold:

{J} ≅ U(d)/O(d) = Λ(d)                                              (#5 §B)

“the Lagrangian Grassmannian — π₁ = ℤ, Maslov circle intact, τ restored” (#5 §B). The Lagrangian Grassmannian Λ(d) is the space of all Lagrangian subspaces, all possible choices of “position” directions. It is compact, of dimension d(d+1)/2, which is 10 for d = 4 (#1 §2 [T]). Its non-contractible loop, the Maslov circle, carries the program’s dark-energy field τ (τ as the Maslov phase). The metric opens up at the same time: with only ω rigid, “H ranges over Sp(2d,ℝ)/U(d): Z = X + iY” (#11 §6), the Siegel domain, whose real part X is the momentum graviton (the modular field Z = X + iY).

Why the correction was forced#

The original P2 held η fixed as well. The compatibility theorem [T][V] then allows only H = diag(g, g⁻¹), and the theory reduces to “GR + one bespoke SIDM model, with w ≡ −1 and the gate unexplained” (#5 §B, “Branch R”). The nontriviality theorem states the general point: the program has content beyond that “iff η is emergent”. The lab notebook’s v0.3 addendum records the change: “P2 is therefore replaced by P2′”.

Precedent and credit#

#11 §4: “That η should be dynamical rather than fixed was argued independently in metastring theory [26S]; the reading of η as the order parameter of broken reciprocity is this program’s.” The toy-model note’s survey of prior work says the same in more detail: “Freidel, Leigh and Minic make η dynamical and describe spacetime as a choice of Lagrangian [1]”, and they describe a preferred polarization as breaking Born duality, “but give no potential for η and no symmetry-breaking analysis” (toy note §1). See broken Born duality.

The v2.2 refinement#

Checkpoint v2.2 (2026-10-20) appended to P2′ what the toy models had found:

Refinement (v2.2, from the toy models): with only ω rigid, ρ sits inside a continuous U(1), and the polarization is a Goldstone angle rather than a discrete choice [D][V]. Two things follow. 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 toy models work with phase space ℝ² (d = 1) and fields on a 1+1-dimensional base. In the toy-model note the order parameter is written with a real structure K, η = ωK, with K² = 1 and Kᵀ ω K = −ω (toy note §1); the naming question is below.

The polarization is a Goldstone angle [D][V]#

A Goldstone direction is a direction along the vacuum manifold of a broken continuous symmetry: moving along it costs no energy [T] (Goldstone modes and the Higgs mechanism). In the coset toy model (model I) the vacua at H = 1 are

K(φ) = [[cos φ, sin φ], [sin φ, −cos φ]]                            (toy note §2)

a whole circle of them, and “ρ acts as Z → −1/Z and K(φ) → K(φ + π) = −K(φ)” (toy note §2). So ρ moves each vacuum by π along a continuous circle of vacua. The worklog names the continuous group: “the U(1) stabilizer of H that rotates K”, and in the complex Grosse–Wulkenhaar model, the other toy, “the left phase rotations e^{−iαN}, with the duality at α = π” (worklog, ‘Chunk 2 results’). The potential’s Hessian at the vacuum has rank 1, leaving three Goldstone directions: two non-compact directions of the upper half-plane and “the compact angle φ, which parametrizes the Lagrangian Grassmannian Λ(1)” (toy note §2; checked in verify/toy_coset.py). #11 §10 draws the link to dark energy: “In the coset model that angle parametrizes Λ(1): it is τ’s circle”, and adds “None of this tests the cosmology; it only shows the structure is consistent.” See the polarization as a Goldstone angle.

A definite polarization needs a ρ-preserving potential#

A Goldstone angle is not pinned to any value, and in 1+1 dimensions it cannot order at all: “Mermin–Wagner [13, 14] applies to the compact SO(2) containing ρ” (toy note §4; the Mermin–Wagner–Coleman theorem). To obtain a definite polarization the toy adds “a potential that is invariant under ρ and under K → −K, and reduces the U(1) to a discrete group” (toy note §4).

  • The first form tried, −μ[Tr(K K_ref)]², is “unbounded below once H is dynamical”: along a family of boosts it reaches “−8.9 × 10⁶ μ at boost parameter 4”, and the reference vacuum is a saddle (toy note §4; obituary of the naive potential).
  • The normalized form V_b = −μ[Tr(K K_ref)]² / (2 Tr KᵀK) is bounded, “−μ ≤ V_b ≤ 0 by Cauchy–Schwarz”, and with H dynamical the reference vacuum is “a stable minimum: Hessian eigenvalues 0, 0.764, 2, 5.236”. The zero mode is a flat line that “is not symmetry-protected” (toy note §4; the bounded potential V_b).
  • At H = 1, V_b = −(μ/2)(1 + cos 2φ), and with H dynamical the vacuum manifold is “{diag(y, 1/y)} × {±K_ref}” (toy note §4). The two signs ±K_ref are the vacua that ρ exchanges, since ρ sends K to −K at H = 1 (toy note §2).

This is what #11 means by “a rigid, unordered pair of axes”. V_b contains K_ref only inside the square [Tr(K K_ref)]², so K_ref and −K_ref define the same potential. The potential fixes the pair of axes along which K_ref points, but not which of the two is position; the worklog calls the price “a rigid, unordered pair of axes (K_ref up to sign)” (‘Chunk 3 results’). That places the structure a definite polarization needs between two extremes: more than ω alone, which fixes no axes, and less than a rigid η, which would also fix which axis is position. The worklog draws the same conclusion: 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’).

The breaking is physical only through the sign of K#

Model I has a second symmetry besides Sp(2, ℝ): “There is a separate ℤ₂, K → −K” (toy note §2). Combined with ρ it gives the twisted element ρ∘(K → −K), which “maps (1, K₀) to itself. So within the full symmetry group, the vacuum keeps an element that acts on H exactly as ρ does” (toy note §2). Round-4 ruling F3 upheld this and checked it (review/round4-adjudication.md; verify/toy_coset.py, “round-4 F3”). The consequence, in the ruling’s words: “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.” That is why the refinement points to P4. See the twisted element and locality from an (H, K)-built coupling.

Notation: J or K#

The sources do not agree on which real structure to name. #11 §4 names J as “the real structure J chosen spontaneously”, with η = HJ, as in the Debt-1 note and the lab notebook’s v0.3 addendum. #11 §5 defines both, J = η⁻¹H and K = η⁻¹ω. The toy-model note and #11 §14 item 11 call K the real structure, with η = ωK, “K a real structure, K² = 1” (toy note abstract) and Kᵀ ω K = −ω (toy note §1). The lab notebook’s v0.1 called J = η⁻¹H the order parameter (#1 §2). No source reconciles the names; this page quotes each in its own notation. The order-parameter page writes K for the real structure, as the toy-model note does.

Open: the extra structure the refinement needs#

The refinement says what a definite polarization needs; it does not say where that comes from.

  • The potential’s origin. No derivation of the ρ-preserving potential exists. #11 §14 item 11 lists “a derivation of the potential” as still open [O], the toy-model note lists “a derivation of the ρ-preserving potential, which introduces a rigid, unordered pair of axes” under “Not shown” (§6), and CLAUDE.md names it as an open end. Before the toy models, the Debt-1 note had listed “the origin of τ’s potential under P2′ (what explicitly breaks Λ(d) rotations — candidate: the section choice itself, or matter loops)” as open (#5 §F). The lab notebook’s v0.1 had attributed the potential’s existence to “the explicit breaking of Grassmannian rotations by the rigid ω (P2)” (#1 §3); the current position is the refinement’s, that ω alone leaves the angle free and a definite polarization needs “more structure than ω alone” (#11 §4). See Open: the origin of the ρ-preserving potential.
  • The unreduced model. The ordered phase has been simulated only with H frozen; “a Monte Carlo with H dynamical” is open (#11 §14 item 11; Open: a Monte Carlo with H dynamical). So is “whether ρ’s SO(2) breaks in the coupled model without a potential” (toy note §6; Open: does ρ’s SO(2) break in the coupled model?).
  • Beyond the toy. Everything above is for d = 1; “anything beyond d = 1” is listed as not shown (toy note §6).

These items are registered in open problems.

Sources#

  • papers/broken-born-checkpoint-v2.md §4 — the statement, the forced correction, the credit and the v2.2 refinement; §5 — the definitions of J and K and the det H = 1 corollary; §6 — the Siegel domain under P2′; §10 — τ’s circle and “None of this tests the cosmology”; §14 item 11 — what is still open
  • notes/debt1a-geometric-core.md §B — the repair as derived: the pair (ω, H), the real structure J, Λ(d), and the order-parameter reading; §C — the modular field; §F — unimodularity as the face of ℏ-rigidity, and the potential’s origin as open
  • notes/broken-born-theory.md §1 — the original P2’s reading of ω; §2 — how ρ acts on ω, η and H, and the dimension of Λ(d); §3 — the v0.1 attribution of the potential to the rigid ω; ‘Addendum (v0.3)’ — the axiom correction as first recorded
  • notes/broken-born-program-v1.md §2 — the v1 statement
  • papers/reciprocity-toy-models.md Abstract, §1 — K, η = ωK and the prior work; §2 — model I’s vacua, Goldstone directions and twisted element; §4 — Mermin–Wagner, the naive and bounded potentials; §6 — what is not shown
  • notes/toy-lagrangian.md ‘Chunk 2 results’ — the continuous U(1) in both toys; ‘Chunk 3 results’ — the rigid, unordered pair of axes
  • review/round4-adjudication.md ‘Rulings’ F3 — the twisted element and the K-odd coupling
  • verify/toy_coset.py — the Goldstone count, V_b’s stability and the twisted-element check; verify/battery.py T2 — det H_Z = 1 for a sampled Z
  • CLAUDE.md ‘Since the handoff’ — the derivation of the ρ-preserving potential as an open end
Backward The nontriviality theorem P3: a reciprocity-invariant action Forward