The nontriviality theorem

The nontriviality theorem states that the Broken-Born program predicts anything beyond general relativity plus one model of self-interacting dark matter only if the neutral metric η, the structure that marks which directions count as position, is a property of the vacuum rather than a fixed part of the background. It is the argument that forced the program to change its second axiom. With η fixed, the position–momentum splittings compatible with both fixed structures include no Maslov circle, the loop of splittings on which the program’s dark-energy field lives, and the geometric action depends only on the ordinary spacetime metric, which leaves no second, momentum-space graviton. With η a property of the vacuum, all of these return. The result is a structural argument of the program, tagged derived [D]; the compatibility theorem it starts from is a known result of Born geometry and is machine-checked.

Status: Alive — [D] in #11 §4; the note that derived it tags its section “[D — structural]” — not machine-checked as a whole; its input, the compatibility theorem, is [T][V] (battery test T1, the verification battery).

Statement#

The checkpoint states the result twice. In the postulate list, under P2′: “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]” (#11 §4). In the kinematic core: “Under strict P2 the compatible polarizations are discrete, so no Grassmannian, no Maslov circle, no dark-energy field, and S_H is a functional of g alone: nontriviality theorem — the program has content iff η is emergent. Hence P2 → P2′.” (#11 §5; on “discrete”, see flag 1 in review/wiki-build-flags.md). The Debt-1 note, where it was derived, puts it as:

Program-nontriviality theorem: BBT has content beyond GR + SIDM iff η is emergent. P2 → P2′ is not optional. (#5 §B)

GR is general relativity, SIDM self-interacting dark matter, and BBT the Broken-Born program.

Two terms need unpacking.

  • “Content beyond GR + SIDM.” Under strict P2, #5 §B finds no dark-energy field τ, no momentum graviton X, and no geometric origin for the velocity gate of conjugate matter, and without them “the program has no content of its own”. What remains is general relativity, dark energy as a constant with w ≡ −1, and one self-interacting dark-matter model whose gate is “unexplained” (Branch R, #5 §B).
  • “η is emergent.” The neutral metric η is not a background structure but a property of the vacuum state: the order parameter η = HJ of P2′, with the real structure J “chosen spontaneously” (#11 §4). The alternative, η fixed in advance alongside ω, is the original P2, called “strict P2” in the record.

The ingredients#

Three structures and their algebra [T]#

Born geometry puts three structures on phase space (the structures ω, η, H): the symplectic form ω, the neutral metric η, and the positive generalized metric H. “With ω, η standard, I = ω⁻¹H, J = η⁻¹H, K = η⁻¹ω satisfy I² = −1, J² = K² = +1, IJ = K — a split-quaternion algebra (Born geometry [3†]; a unique compatible connection exists [7†])” (#11 §5). I is a complex structure, a quarter-turn of phase space; J and K are real structures, maps that square to the identity.

The compatibility theorem [T][V]#

Holding both ω and η in standard form leaves almost no freedom in H:

Positive H satisfying both (η⁻¹H)² = 1 and (ω⁻¹H)² = −1 are exactly
H = diag(g, g⁻¹),  g ∈ Sym⁺(d).                                     (#11 §5)

Proof (#11 §5; #5 §A), writing H = [[A, C], [Cᵀ, B]]:
  (2,1) blocks of the two conditions:  ACᵀ + CA = 0  and  CA − ACᵀ = 0   ⇒  ACᵀ = 0
  positivity                                                           ⇒  C = 0
  diagonal blocks                                                      ⇒  B = A⁻¹

The battery’s test T1 checks the theorem by machine: a matrix diag(g, g⁻¹) satisfies both conditions, a randomly drawn H violates at least one, and the two (2,1) block identities used in the proof hold (verify/battery.py; the verification battery). The round-3 audit re-derived the theorem independently, with sympy and a Gröbner basis at d = 2, and “reports that the positivity step is not needed for C = 0” (#11 §5). Since v2.1 the statement is credited to Freidel, Leigh and Minic [26S, §2.1]; the block proof and the machine check are the program’s (#11 §5; the compatibility theorem).

Its corollaries (#5 §A; the corollaries) are det H = 1 identically; no Kalb–Ramond analog, since “The off-diagonal block — DFT’s B-field — is frozen out by the extra symplectic compatibility” (DFT: double field theory); and “The momentum metric is slaved: g̃ = g⁻¹ pointwise. No independent momentum-space geometry survives.” The nontriviality argument starts from the third.

The argument#

With η fixed: what strict P2 removes [D — structural]#

“Corollary 3 collides with the theory as documented” (#5 §B). The note draws two consequences.

  1. No Maslov circle. A polarization is a choice of Lagrangian subspace, the directions that count as position (Lagrangian polarizations). Under strict P2 it must respect both fixed structures: “The space of polarizations compatible with both structures (ω-Lagrangian and η-null) is discrete — the 2^d coordinate-type splittings, GL(d)-rigid. No Lagrangian Grassmannian, no Maslov circle, no τ field.” The program’s dark-energy field τ is the angle around the Maslov circle, the non-contractible loop of the Lagrangian Grassmannian Λ(d), whose fundamental group is π₁ = ℤ (#1 §2 [T]; τ as the Maslov phase). In #5’s reading the compatible polarizations form a discrete set, which has no such loop, so τ cannot be defined: “The dark-energy sector as constructed contradicts P2.” Flag 1 in review/wiki-build-flags.md queries the word “discrete”. For d ≥ 2, any subspace V of the position directions, taken together with its orthogonal complement V^⊥ placed in the momentum directions, gives a polarization V ⊕ V^⊥ that is Lagrangian for ω and null for η, so the compatible polarizations form continuous Grassmannian families Gr(k, d). The Maslov phase det² is constant (±1) on each family, so no family winds around the Maslov circle, the x-polarization (V the whole of position space) is isolated, and the conclusion stands. The flag’s executed check, at d = 2 and 3, is review/wiki-build-checks/verifier_null_lagrangians.py. The flag suggests the wording “no continuous family through the x-polarization; det² is constant, so no Maslov circle”, and it awaits Paul’s ruling.
  2. A pure metric theory. “With H = diag(g, g⁻¹), S_H is a functional of g alone: a pure metric theory. No momentum graviton, hence no geometric origin for the χ gate either.” S_H is the geometric action of H. If the spacetime metric g is the only free field in H, there is no second tensor to serve as the momentum graviton X. The mechanism that the same note finds for the velocity gate of conjugate matter χ is exchange of X (#5 §E; the gate mechanism), so it has nothing to exchange.

The note calls what is left Branch R: “BBT reduces to GR + one bespoke SIDM model, with w ≡ −1 and the gate unexplained. On Branch R the program has no content of its own” (#5 §B). The v0.3 addendum of the lab notebook condenses it: “A theory with strict P2 is GR plus one bespoke SIDM model.”

With η emergent: what the repair restores#

The repair holds ω alone fixed: “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). The possible choices form the full Grassmannian:

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

“the Lagrangian Grassmannian — π₁ = ℤ, Maslov circle intact, τ restored”. The metric opens up in the same step. The positive H compatible with ω alone form the Siegel domain Sp(2d, ℝ)/U(d), coordinatized by Z = X + iY with Y > 0 (#5 §C). Im Z = Y = g is the spacetime metric, and Re Z = X is “a new symmetric tensor: the momentum graviton. It is exactly the fluctuation that breaks η-compatibility” (#5 §C; the modular field Z = X + iY). The rigid case survives inside the emergent one as a slice: “At X = 0, H = diag(g, g⁻¹): the η-compatible locus of Theorem A is precisely the vacuum slice” (#5 §C).

So each item that strict P2 removes is present under P2′. The polarization ranges over Λ(d), τ lives on its Maslov circle, and X exists as Re Z. With X present, the gate has a mediator again (#5 §E).

The two halves of “iff”#

The “only if” half is the first subsection: with η fixed, the program reduces to Branch R. The “if” half rests on what the repair restores: with η emergent, the structures that carry the program’s own content exist. The Debt-1 note tags both §B and the modular-field section §C as structural ("[D — structural]"; §C adds “math facts standard”). Neither half is a line-by-line derivation, and neither is machine-checked as a whole. The machine check covers the compatibility theorem underneath.

Scope and caveats#

  • A theorem in a structural section. The note calls the result a theorem inside a section tagged “[D — structural]”, and #11 carries it as [D]. It is an argument from the structure of the constructions, not a formal proof.
  • What “content” covers. The result concerns the program’s own sectors: what it lists as lost under strict P2 are τ, X and the gate’s geometric origin (#5 §B).
  • Circular pointers in v1. The program paper v1 (#9) states the result in two places that point at each other: its §2 says the correction is “forced by the theorems of §3 and §5”, and its §3 ends “Hence the nontriviality theorem of §2”. #11 places the theorem in §5. #9 is superseded by #11.
  • The v2.2 refinement. The toy models later added structure between the two cases the theorem compares. With only ω rigid the polarization is a Goldstone angle, and “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” (#11 §4). #11 v2.2 states the refinement beside the theorem; no source revisits the theorem in its light. See P2′ and the polarization as a Goldstone angle.

Consequences for the program#

  • P2 → P2′. The theorem is why the program’s second axiom changed. #5 §B: “P2 → P2′ is not optional.” The replaced axiom is kept as the original P2.
  • The modular field. The Siegel-domain geometry of #11 §6, with its momentum graviton and the dark-energy field on the domain’s boundary, exists only under P2′: “With only ω rigid, H ranges over Sp(2d,ℝ)/U(d)” (#11 §6).
  • The next computation. The v0.3 addendum records that, with X restored, “Open problem 1 sharpens to a single decisive computation: the quadratic-order classification of X (healthy / auxiliary / ghost)”. That computation found X to be a second graviton that must be massive (why X must be massive).
  • The hard core. #11 §15 lists the program’s hard core as “spontaneously broken reciprocity, ω-only rigidity” and calls it “intact, twice sharpened by forced corrections (P2′; the w-dictionary)”. P2′, which the theorem forced, is the first of those two sharpenings (Lakatos’s methodology).

History#

  • Derived in the overnight Debt-1 note, Chunk A (#5), whose header reads “Status: two rigorous results, one axiom correction forced, one mechanism identified, one decisive computation defined but not completed.” Its closing tally, headed “Paid”, includes “discovery and repair of an axiom-level inconsistency (P2 → P2′, with a nontriviality theorem making the repair mandatory)” (#5 §F).
  • Recorded in the lab notebook’s v0.3 addendum, which found “with it an axiom-level inconsistency: strict P2 makes the polarization choice discrete, eliminating the Lagrangian Grassmannian, the Maslov circle, and hence the entire τ/dark-energy sector, while also slaving g̃ = g⁻¹ and removing any geometric origin for the χ gate” (on “discrete”, see flag 1 in review/wiki-build-flags.md).
  • Carried forward in the program paper v1 (#9 §2–§3, tagged [D], with the circular pointers above) and in the handoff, which lists among the alive results “P2′ correction: strict rigidity trivializes the theory (nontriviality theorem); η must be emergent” (HANDOFF.md §2).
  • Current form in #11 §4–§5 (v2.0 to v2.2), tagged [D]. In v2.1 the compatibility theorem underneath was re-tagged from derived to [T][V] and credited to Freidel, Leigh and Minic after the round-3 review flagged it (review/round3-adjudication.md); the nontriviality theorem kept its [D].

Sources#

  • papers/broken-born-checkpoint-v2.md §4–§5 — the current statement, tagged [D], and the compatibility theorem with its attribution; §6 — the modular field that exists only under P2′; §15 — the hard core
  • notes/debt1a-geometric-core.md §A — the compatibility theorem and its corollaries; §B — the derivation ("[D — structural]"), Branch R and the repair; §C — the Siegel domain and X; §E — the gate as X exchange; §F — the closing tally
  • notes/broken-born-theory.md §2 — the Grassmannian and its Maslov circle; ‘Addendum (v0.3)’ — the axiom correction as first recorded
  • notes/broken-born-program-v1.md §2–§3 — the v1 statement and its circular pointers (superseded by #11)
  • HANDOFF.md §2 — the result as listed at the handoff
  • review/round3-adjudication.md ‘Rulings’ — the referee’s identification of the compatibility theorem with Freidel–Leigh–Minic’s result
  • verify/battery.py T1 — the compatibility-theorem checks
  • review/wiki-build-flags.md item 1 and review/wiki-build-checks/verifier_null_lagrangians.py — the query on “discrete”: continuous families V ⊕ V^⊥ for d ≥ 2, with det² constant on each, so the conclusion stands; awaiting Paul’s ruling