A registered result in its source context
Original ledger classification: Reduction [E] (convex energy, unique minimiser) + [C] (Troyanov) + [O] (variational premise)
Recorded statement: RP-ENERGY / TROYANOV ROUTE FOR FLAT-AWAY (2026-06-18): the third independent reduction -- the flat pillowcase is the UNIQUE constant-curvature (energy-minimising) metric in the conformal class with the prescribed mark divisor. [E]…
Statement, assumptions and sources
Paper context: Research contracts
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FLATAWAY.ENERGY.01
Original ledger classification: Reduction [E] (convex energy, unique minimiser) + [C] (Troyanov) + [O] (variational premise)
Original claim text
RP-ENERGY / TROYANOV ROUTE FOR FLAT-AWAY (2026-06-18): the third independent reduction -- the flat pillowcase is the UNIQUE constant-curvature (energy-minimising) metric in the conformal class with the prescribed mark divisor. [E] PRESCRIBED DIVISOR + GAUSS-BONNET: the pillowcase = S^2 with 4 cone points of angle pi; the cone Gauss-Bonnet sum (2pi-theta_i)=4pi=2pi chi(S^2) holds for the flat (K=0 off marks) metric, the mark divisor rigid (v195/v216). [C] TROYANOV / EUCLIDEAN-CONE UNIQUENESS: for a prescribed cone divisor obeying Gauss-Bonnet the conformal class carries a UNIQUE constant-curvature metric (Troyanov 1991; for K=0 the unique flat cone metric), so a smooth off-mark curvature (K!=0 between marks) is excluded -- imported. [E] STRICTLY CONVEX CURVATURE ENERGY: E[f]=int f^2 has Hessian 2 pi I (diagonal, positive-definite) in the clock-allowed basis -- strictly convex, UNIQUE minimiser (flat, E=0). [E] NO ZERO-COST DEFORMATION: E[eps cos(4k theta)]=pi eps^2 >0 for every mode. [O] PREMISE: Flat-Away <= 'RP+gap realises the energy-minimising (constant-curvature) seam metric', converging with v292 (fix a_2) and v293 (pin the spectrum) on ONE selection principle for the flat metric; closing any one closes Route B.
Original classification detail
[E] the Gauss-Bonnet divisor + the strictly-convex energy with its unique minimiser + no zero-cost direction; [C] Troyanov/Euclidean-cone uniqueness (literature); [O] the variational premise (RP+gap selects the minimiser). A third independent reduction of Flat-Away; NOT a proof.
Dependencies as recorded
FLATAWAY.RP.01; FLATAWAY.HEAT.01; FLATAWAY.SPEC.01; QGEO.ROUTEI.01
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- Research contracts
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- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- 2026-06-18 (Flat-Away attacked on all three routes, v292– v294)Research changelog; TFPT — Introduction and reading guide
Navigation through a shared verification script
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- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportResearch contracts
- Research Contract 1 — [formula in source]tfpt research contracts
Registered verification modules
- v294_flataway_rp_energy.pyRegistered verification module · frontier
Read the description recorded in the module registry
FLATAWAY.ENERGY.01 — the RP-energy / Troyanov route: the flat pillowcase is the unique constant-curvature minimiser at the fixed mark divisor. [E] prescribed divisor + Gauss-Bonnet: S² with 4 cone angles π, Σ(2π−θ)=4π=2πχ; [C] Troyanov/Euclidean-cone uniqueness: the prescribed cone divisor carries a UNIQUE constant-curvature (flat) metric (Troyanov 1991), excluding smooth off-mark curvature; [E] strictly convex curvature energy E[f]=∫f², Hessian 2π·I (PD) ⇒ unique minimiser; [E] no zero-cost deformation (E[ε cos4kθ]=π ε²>0). [O] premise: 'RP+gap realises the energy-minimiser', converging with v292 (a₂) and v293 (spectrum) on ONE selection principle for the flat metric; closing any one closes Route B.
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