"""v264 -- QGEO.ENERGY.03: the raw-seam => mark-locality REDUCTION (NOT a closure of QGEO.SYM.01). This makes the converse implication explicit on the canonical uniformising metric and isolates the true bedrock: it shows that GIVEN the seam carries the flat pillowcase metric (QGEO.PILLOW.01, v214), the DtN sub-principal symbol is automatically mark-local => Z4-invariant => block-diagonal => [rho, Lambda] = 0 => omega o rho = omega. So the whole bedrock collapses to the single metric-realisation statement "the raw RP seam carries the uniformising flat pillowcase metric" -- which IS QGEO.SYM.01. A sharpening, honestly left [O]; it does NOT prove the seam must carry that metric. Chain (composes v216 + v201 + v198 on the explicit pillowcase): flat pillowcase S^2(2,2,2,2): curvature concentrated at the 4 cone points (marks) => the DtN sub-principal symbol M_f = curvature multiplication is a mu4-orbit sum => f is Z4-invariant (Fourier support 0 mod 4) => M_f block-diagonal in the clock-character basis => [rho, M_f] = 0 (rho = diag(i^n)) => omega o rho = omega (Tomita-Takesaki, v198). [E] 1. GAUSS-BONNET MARKS. the flat pillowcase has exactly 4 cone points, each deficit pi, summing to 4 pi = 2 pi chi(S^2) -- so the curvature lives ONLY at the 4 marks (v216), the geometric source of mark-locality. [N] 2. CURVATURE IS Z4-INVARIANT. a curvature f(theta) sourced at the mu4 orbit {0, pi/2, pi, 3pi/2} has discrete Fourier support ONLY on modes = 0 mod 4 (verified by FFT) -- mark-locality is automatic for the pillowcase curvature. [E] 3. BLOCK-DIAGONAL => CLOCK-INVARIANT. M_f (multiplication by f) then commutes with the carrier clock rho = diag(i^n): [rho, M_f]_{mn} = (i^m - i^n) f_hat(m-n) = 0 since f_hat vanishes off 4Z -- exactly the v201 condition. [E] 4. STATE INVARIANCE. with the principal symbol |k| already commuting (v198), [rho, Lambda] = 0 on the whole DtN, so by Tomita-Takesaki omega o rho = omega -- the operative form of QGEO.SYM.01 follows on this metric. [O] 5. THE RESIDUAL IS THE METRIC (no closure). the chain ASSUMES the raw seam carries the uniformising flat pillowcase metric; proving the raw RP seam MUST carry it (curvature only at the marks) is precisely QGEO.SYM.01 -- not a finite computation, not closed here. NEG control: a generic metric with curvature off the mu4 orbit breaks Z4-invariance and the chain fails. Status: [E] Gauss-Bonnet marks + the block-diagonal/state-invariance chain; [N] the Z4-Fourier support of the pillowcase curvature; [O] the metric-realisation premise itself (the bedrock QGEO.SYM.01) stays open. A genuine reduction/sharpening, NOT a closure. Python-only (numpy FFT + commutator). """ import numpy as np from tfpt_constants import check, summary, reset def run(): reset() print("v264 QGEO.ENERGY.03: raw-seam => mark-locality REDUCTION (not a closure)") # 1. Gauss-Bonnet: 4 marks, deficit pi each, sum = 2 pi chi(S^2) n_marks, deficit = 4, np.pi total_deficit = n_marks * deficit chi_S2 = 2 check("GAUSS-BONNET MARKS [E]: the flat pillowcase has %d cone points, each " "deficit pi, summing to %s = 4 pi = 2 pi chi(S^2) (chi=%d) -- curvature " "lives ONLY at the 4 marks (v216), the geometric source of mark-locality" % (n_marks, "4 pi" if abs(total_deficit - 4 * np.pi) < 1e-9 else "?", chi_S2), n_marks == 4 and abs(total_deficit - 2 * np.pi * chi_S2) < 1e-9) # 2. curvature sourced at the mu4 orbit -> Fourier support on 4Z N = 64 theta = np.arange(N) * 2 * np.pi / N # narrow bumps at the four marks {0, pi/2, pi, 3pi/2} = the mu4 orbit marks = [0, N // 4, N // 2, 3 * N // 4] f = np.zeros(N) for m in marks: f += np.exp(-((np.minimum((np.arange(N) - m) % N, (m - np.arange(N)) % N)) ** 2) / 2.0) fhat = np.fft.fft(f) / N off_4 = max(abs(fhat[k]) for k in range(N) if k % 4 != 0) on_4 = max(abs(fhat[k]) for k in range(N) if k % 4 == 0) check("CURVATURE IS Z4-INVARIANT [N]: a curvature sourced at the mu4 orbit " "{0,pi/2,pi,3pi/2} has discrete Fourier support ONLY on modes = 0 mod 4 " "(max off-4Z amplitude %.2e vs on-4Z %.3f) -- mark-locality is automatic " "for the pillowcase curvature" % (off_4, on_4), off_4 < 1e-12 and on_4 > 0.1) # 3. M_f (multiplication by f) commutes with the clock rho = diag(i^n) n = np.arange(N) rho = np.diag(1j ** n) # carrier clock on Fourier modes # M_f in the Fourier basis: (M_f)_{mn} = fhat[(m-n) mod N] (circulant) Mf = np.array([[fhat[(a - b) % N] for b in range(N)] for a in range(N)]) comm = rho @ Mf - Mf @ rho check("BLOCK-DIAGONAL => CLOCK-INVARIANT [E]: M_f commutes with rho = diag(i^n): " "[rho, M_f]_{mn} = (i^m - i^n) f_hat(m-n) = 0 since f_hat vanishes off 4Z " "(||[rho,M_f]|| = %.2e) -- exactly the v201 condition" % np.linalg.norm(comm), np.linalg.norm(comm) < 1e-12) # 4. state invariance: principal symbol |k| commutes too (v198) => [rho,Lambda]=0 k = np.fft.fftfreq(N, d=1.0 / N) Lam_principal = np.diag(np.abs(k)) # |k| diagonal in Fourier basis comm_principal = rho @ Lam_principal - Lam_principal @ rho Lambda = Lam_principal + Mf # full DtN = |k| + M_f comm_full = rho @ Lambda - Lambda @ rho check("STATE INVARIANCE [E]: the principal symbol |k| commutes with rho " "(||[rho,|k|]||=%.2e, v198), so on the full DtN Lambda = |k| + M_f we get " "[rho, Lambda] = 0 (||.||=%.2e); by Tomita-Takesaki omega o rho = omega -- " "the operative QGEO.SYM.01 holds ON THIS METRIC" % (np.linalg.norm(comm_principal), np.linalg.norm(comm_full)), np.linalg.norm(comm_principal) < 1e-12 and np.linalg.norm(comm_full) < 1e-12) # 5. NEG control + the honest residual f_bad = np.zeros(N); f_bad[3] = 1.0; f_bad[7] = 1.0 # curvature OFF the mu4 orbit fhat_bad = np.fft.fft(f_bad) / N Mf_bad = np.array([[fhat_bad[(a - b) % N] for b in range(N)] for a in range(N)]) broken = np.linalg.norm(rho @ Mf_bad - Mf_bad @ rho) > 1e-6 check("THE RESIDUAL IS THE METRIC [O]: the chain ASSUMES the raw seam carries " "the uniformising flat pillowcase metric (curvature only at the marks). " "A generic metric with curvature OFF the mu4 orbit breaks Z4-invariance " "and the chain fails (||[rho,M_f^bad]|| > 0: %s). Proving the raw RP seam " "MUST carry the pillowcase metric is precisely QGEO.SYM.01 -- a reduction " "to one metric-realisation premise, NOT a closure" % broken, broken) return summary("v264 raw-seam => mark-locality: reduced to the metric premise, QGEO.SYM.01 stays [O] (QGEO.ENERGY.03)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)