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Original ledger classification: Contract [O] (lemma + verdict) + [E] (spectral/heat uniqueness) + [C] (Troyanov route)

Recorded statement: THE FLAT-AWAY LEMMA AS A NAMED MINI-THEOREM (2026-06-18). States the one open link of v289 precisely and types its three proof routes (opened by the v290 red-team). LEMMA: given RawRPSeam with the mass gap, chirality and the four mu4…

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Paper context: tfpt research contracts

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FLATAWAY.RP.01

Original ledger classification: Contract [O] (lemma + verdict) + [E] (spectral/heat uniqueness) + [C] (Troyanov route)

Original claim text

THE FLAT-AWAY LEMMA AS A NAMED MINI-THEOREM (2026-06-18). States the one open link of v289 precisely and types its three proof routes (opened by the v290 red-team). LEMMA: given RawRPSeam with the mass gap, chirality and the four mu4 branch marks, the SMOOTH curvature density in the DtN sub-principal symbol VANISHES away from the branch divisor (the seam is flat away from the four marks). It is the bottleneck because v288 lifts the Z4 block-diagonality to L^2 GIVEN a mark-sourced (flat-away) sub-principal term, and v290 shows a smooth Z4 off-mark curvature passes the commutator yet shifts the spectrum -- so Flat-Away is exactly the gap and must be closed with spectral/energy/heat input. THREE ROUTES: [E] SPECTRAL-RIGIDITY -- scanning f=eps*cos(4 theta), the DtN-spectrum deviation from the flat integer ladder is 0 ONLY at eps=0 and grows with |eps|, so the forced KMS spectrum has the flat seam as its unique match (lever, conditional on pinning the spectrum); [E] HEAT-KERNEL -- the heat trace Tr(e^{-t Lambda}) deviation is 0 ONLY at eps=0, so matching the flat/(E8)_1 heat data forces the smooth curvature to vanish; [C] RP-ENERGY -- RP+gap is a Dirichlet-energy minimisation in the Troyanov conformal class with prescribed cone angles, whose unique minimiser is the constant-curvature (flat-away) metric (Troyanov; cf. v284). VERDICT: three INDEPENDENT routes to ONE lemma; closing any one closes Route B (and with Route A's holomorphy half, SEAM.EQUIV.01). All three remain open handles, not proofs.

Original classification detail

[O] the lemma + the verdict (three open handles); [E] the spectral-rigidity + heat-kernel uniqueness demonstrations (flat is the unique match in the smooth Z4 family); [C] the RP-energy/Troyanov route. A contract turning Flat-Away into a precise, attackable mini-theorem; does NOT prove it.

Dependencies as recorded

SEAM.ADVERSARY.01; MARKLOCAL.RAW.01; SEAM.EQUIV.B01; SEAM.EQUIV.01; QGEO.ROUTEI.01

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  • v291_flataway_contract.pyRegistered verification module · frontier
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    FLATAWAY.RP.01 — the Flat-Away lemma as its own named mini-theorem with three proof routes opened by v290. LEMMA [O]: given RawRPSeam with gap, chirality and the four μ4 marks, the smooth curvature density in the DtN sub-principal symbol vanishes away from the branch divisor. [E] SPECTRAL-RIGIDITY route: scanning f=ε·cos(4θ), the spectrum deviation from the flat ladder is 0 only at ε=0 and grows with |ε| — flat is the unique spectral match. [E] HEAT-KERNEL route: the heat-trace deviation is 0 only at ε=0 — smooth off-mark curvature adds heat coefficients. [C] RP-ENERGY route: RP+gap = Dirichlet-energy minimisation in the Troyanov conformal class ⇒ constant-curvature minimiser (Troyanov; cf. v284). [O] verdict: 3 independent routes to ONE lemma; closing any one closes Route B.

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