"""v294 -- FLATAWAY.ENERGY.01: the RP-Energy / Troyanov route for Flat-Away, the third independent attack. It does NOT prove Flat-Away; it casts the lemma as a variational selection -- the flat pillowcase is the UNIQUE constant-curvature (energy-minimizing) metric in the conformal class with the prescribed mark divisor -- so Flat-Away reduces to the single premise 'RP + gap realises that energy-minimiser'. Geometry: the seam is the sphere with four cone points of angle pi (the pillowcase). A smooth Z4 off-mark curvature is a curvature density f != 0 away from the four marks. [E] 1. PRESCRIBED DIVISOR + GAUSS-BONNET. the pillowcase = S^2 with 4 cone points of angle pi; the cone Gauss-Bonnet sum sum_i (2pi - theta_i) = 4*pi = 2pi*chi(S^2) (chi=2) is satisfied by the FLAT (K=0 away from marks) metric -- the mark divisor is rigid (v195/v216) and forces curvature 4pi total, concentrated at the marks. [C] 2. 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). The flat pillowcase IS that metric; any smooth off-mark curvature (K != 0 between marks) is NOT constant-curvature, hence not the Troyanov metric -- imported as conditional. [E] 3. STRICTLY CONVEX CURVATURE ENERGY. the curvature energy E[f] = int f^2 over the smooth-Z4 deformations has Hessian 2*pi*I (diagonal, positive-definite) in the clock-allowed Fourier basis -- strictly convex, so a UNIQUE minimiser (flat, E=0); the variational form of the uniqueness above. [E] 4. NO ZERO-COST DEFORMATION. E[eps*cos(4k theta)] = pi*eps^2 > 0 for every mode and grows quadratically -- there is no flat (energy-free) smooth Z4 direction; E[f] = 0 <=> f = 0 off the marks. [O] 5. THE PREMISE + CONVERGENCE. Flat-Away <= 'RP + gap realises the energy-minimising (constant-curvature) seam metric'. This is the route's one open input -- a variational characterisation of the RP+gap state -- and it converges with the heat route (v292, fix a_2) and the spectral route (v293, pin the Steklov spectrum): all three reduce Flat-Away to ONE selection principle for the flat metric. Closing any one closes Route B. Status: [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. Python (numpy). """ import numpy as np from tfpt_constants import check, summary, reset NTH = 720 def _curv_energy(coeffs): """E[f] = int_0^2pi f^2 dtheta for f = sum_k coeffs[k] cos(4k theta).""" th = np.linspace(0, 2 * np.pi, NTH, endpoint=False) f = sum(c * np.cos(4 * k * th) for k, c in coeffs.items()) return float(np.sum(f ** 2) * (2 * np.pi / NTH)) def run(): reset() print("v294 FLATAWAY.ENERGY.01: the RP-energy / Troyanov route -- flat is the unique constant-curvature minimiser") # 1. prescribed divisor + Gauss-Bonnet n_marks, angle = 4, np.pi cone_sum = n_marks * (2 * np.pi - angle) # sum (2pi - theta_i) check("PRESCRIBED DIVISOR + GAUSS-BONNET [E]: the pillowcase = S^2 with %d cone " "points of angle pi; the cone Gauss-Bonnet sum (2pi-theta) = %s = 2pi*chi " "(chi=2) holds for the FLAT metric -- the mark divisor is rigid (v195/v216), " "forcing 4pi curvature concentrated at the marks" % (n_marks, "4pi" if abs(cone_sum - 4 * np.pi) < 1e-9 else "?"), abs(cone_sum - 2 * np.pi * 2) < 1e-9) # 2. Troyanov uniqueness (literature) check("TROYANOV / EUCLIDEAN-CONE UNIQUENESS [C]: 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) -- the flat " "pillowcase IS it; a smooth off-mark curvature (K!=0 between marks) is not " "constant-curvature, hence excluded (imported)", True) # 3. strictly convex curvature energy: Hessian 2*pi*I (diagonal PD) eps = 0.01 modes = (1, 2, 3) Hdiag = [(_curv_energy({k: eps}) + _curv_energy({k: -eps}) - 2 * _curv_energy({k: 0.0})) / eps ** 2 for k in modes] # off-diagonals offdiag = [] for i, ki in enumerate(modes): for kj in modes[i + 1:]: mixed = (_curv_energy({ki: eps, kj: eps}) - _curv_energy({ki: eps}) - _curv_energy({kj: eps}) + _curv_energy({})) / eps ** 2 offdiag.append(abs(mixed)) check("STRICTLY CONVEX CURVATURE ENERGY [E]: E[f]=int f^2 has Hessian ~ 2pi*I " "(diagonal entries %s ~ 2pi=%.3f, off-diagonals < %.1e) in the clock-allowed " "basis -- strictly convex => UNIQUE minimiser (flat, E=0)" % ([round(h, 3) for h in Hdiag], 2 * np.pi, max(offdiag) + 1e-12), all(abs(h - 2 * np.pi) < 1e-2 for h in Hdiag) and max(offdiag) < 1e-6) # 4. no zero-cost deformation energies = {k: _curv_energy({k: 0.05}) for k in modes} check("NO ZERO-COST DEFORMATION [E]: E[eps cos(4kθ)] = pi eps^2 > 0 for every mode " "(%s at eps=0.05, vs pi eps^2 = %.4f) and grows quadratically -- no flat " "smooth Z4 direction; E[f]=0 <=> f=0 off the marks" % ({k: round(v, 4) for k, v in energies.items()}, np.pi * 0.05 ** 2), all(v > 0 for v in energies.values())) # 5. the premise + convergence check("THE PREMISE + CONVERGENCE [O]: Flat-Away <= 'RP + gap realises the " "energy-minimising (constant-curvature) seam metric' -- the one open " "variational input, converging with the heat route (v292, fix a_2) and the " "spectral route (v293, pin the Steklov spectrum): all three reduce Flat-Away " "to ONE selection principle for the flat metric; closing any one closes " "Route B", True) return summary("v294 FLATAWAY.ENERGY.01: the flat pillowcase is the unique constant-curvature / energy-minimising metric at the fixed mark divisor (Troyanov + strictly convex int f^2); Flat-Away reduces to 'RP+gap selects the minimiser'") if __name__ == "__main__": raise SystemExit(1 if run() else 0)