"""v289 -- MARKLOCAL.RAW.01: decompose the ONE remaining Route-B residual ('why is the raw seam sub-principal term mark-local?') into an explicit five-lemma chain, discharge the four standard / already-established lemmas, and isolate the SINGLE open analytic lemma (Flat-Away). This is the best closing lever after v288: it does NOT close mark-locality, it locates exactly which analytic statement remains. The chain (raw RP seam => mark-local sub-principal term): L1 [C] CONIC PARAMETRIX. the Steklov/DtN operator of a surface with conic singularities splits as Lambda = |D_theta| (principal) + M_f (sub-principal = multiplication by the boundary curvature f) + delta-sources at the cone points -- the b-calculus / conic-parametrix construction (Melrose); standard microlocal analysis, imported. L2 [O] FLAT-AWAY -- THE ONE OPEN ANALYTIC LEMMA. reflection positivity + the mass gap + the four branch marks force the seam curvature to VANISH away from the four marks (the flat-pillowcase realisation). This is the curvature-locality heart of SEAM.EQUIV.01; not a finite computation, left honestly open. L3 [E] ORBIT-SOURCE. the four marks form a single mu4 orbit {0, pi/2, pi, 3pi/2} (v195 Lefschetz/character marks + the order-4 carrier clock; v216 Gauss-Bonnet gives exactly 4 cone points of deficit pi). L4 [E] FOURIER-SUPPORT. a curvature sourced on the mu4 orbit has discrete Fourier support ONLY on modes m == 0 (mod 4) (v264/v288; verified by FFT here). L5 [E] FULL-OPERATOR. hence M_f is Z4 block-diagonal and [rho, Lambda] = 0 on the full L^2 (v288), so omega o rho = omega (Tomita-Takesaki, v198). [E] 1. CHAIN TYPED. five lemmas; four discharged (L1 conditional-by-literature, L3/L4/L5 exact), exactly ONE open (L2 Flat-Away). [E] 2. L3+L4 ORBIT => SUPPORT. a mu4-orbit curvature has Fourier support only on 4Z (verified); a non-orbit (generic 4-point) source does NOT -- so the orbit structure, not just '4 marks', is what gives mark-locality. [E] 3. L5 FULL-OPERATOR. with that support, Lambda = |D_theta| + M_f commutes with rho = diag(i^n) on the full mode range (||[rho,Lambda]|| ~ 1e-15, v288). [C] 4. L1 CONIC PARAMETRIX. the conic Steklov symbol decomposes as |k| + curvature multiplication + cone sources (structural check: M_f is multiplication by f); the decomposition itself is standard b-calculus, imported as [C]. [O] 5. L2 IS THE ONE OPEN LEMMA. precise statement + negative control: a curvature that does NOT vanish away from the marks breaks Z4-invariance and the whole chain -- so Route B reduces to exactly 'RP + gap + 4 marks => flat away from the marks'. If L2 falls, Route B closes; SEAM.EQUIV.01 then needs only the holomorphy half (Route A). Status: [E] the chain typing + L3/L4/L5 (orbit, support, full operator); [C] L1 (conic parametrix, literature); [O] L2 (Flat-Away), the single open analytic lemma. A finer decomposition of the v288 residual, NOT a closure. Python (numpy FFT). """ import numpy as np from tfpt_constants import check, summary, reset # lemma: (id, statement, status, evidence) CHAIN = [ ("L1", "conic Steklov DtN = |D_theta| + M_f + cone sources", "conditional", "b-calculus / conic parametrix (Melrose) -- imported"), ("L2", "RP + gap + 4 marks => curvature vanishes away from the marks (flat-away)", "open", "THE ONE OPEN ANALYTIC LEMMA (curvature-locality heart of SEAM.EQUIV.01)"), ("L3", "the 4 marks form a single mu4 orbit", "established", "v195/v216 + order-4 clock"), ("L4", "mu4-orbit curvature => Fourier support only on 4Z", "established", "v264/v288 (FFT)"), ("L5", "=> M_f Z4 block-diagonal, [rho,Lambda]=0 on full L^2", "established", "v288/v198"), ] N = 64 def _orbit_curvature(marks): idx = np.arange(N) f = np.zeros(N) for m in marks: d = np.minimum((idx - m) % N, (m - idx) % N) f += np.exp(-(d ** 2) / 2.0) return f def run(): reset() print("v289 MARKLOCAL.RAW.01: decompose raw mark-locality into a 5-lemma chain; isolate Flat-Away") established = [c for c in CHAIN if c[2] == "established"] conditional = [c for c in CHAIN if c[2] == "conditional"] openl = [c for c in CHAIN if c[2] == "open"] # 1. chain typed check("CHAIN TYPED [E]: 5 lemmas -- %d discharged (L1 conditional-by-literature, " "L3/L4/L5 exact), exactly ONE open (L2 Flat-Away)" % (len(established) + len(conditional)), len(established) == 3 and len(conditional) == 1 and len(openl) == 1 and openl[0][0] == "L2") # 2. L3+L4: mu4-orbit => Fourier support on 4Z; generic 4-point does NOT f_orbit = _orbit_curvature([0, N // 4, N // 2, 3 * N // 4]) fhat = np.fft.fft(f_orbit) / N off4 = max(abs(fhat[k]) for k in range(N) if k % 4 != 0) on4 = max(abs(fhat[k]) for k in range(N) if k % 4 == 0) f_generic = _orbit_curvature([0, 7, 19, 33]) # NOT a mu4 orbit ghat = np.fft.fft(f_generic) / N off4_g = max(abs(ghat[k]) for k in range(N) if k % 4 != 0) check("L3+L4 ORBIT => SUPPORT [E]: a mu4-orbit curvature has Fourier support only " "on 4Z (off-4Z %.2e vs on-4Z %.3f); a generic 4-point source does NOT " "(off-4Z %.3f > 0) -- the ORBIT structure, not just '4 marks', gives " "mark-locality" % (off4, on4, off4_g), off4 < 1e-12 and on4 > 0.1 and off4_g > 1e-3) # 3. L5: full-operator commutation ns = np.arange(N) rho = np.diag(1j ** ns) Mf = np.array([[fhat[(a - b) % N] for b in range(N)] for a in range(N)]) k = np.fft.fftfreq(N, d=1.0 / N) Lam = np.diag(np.abs(k)) + Mf comm = np.linalg.norm(rho @ Lam - Lam @ rho) check("L5 FULL-OPERATOR [E]: with 4Z support, Lambda = |D_theta| + M_f commutes " "with rho = diag(i^n) on the full mode range (||[rho,Lambda]|| = %.1e ~ 0, " "v288)" % comm, comm < 1e-12) # 4. L1 conic parametrix: M_f is multiplication by the curvature (the subprincipal # piece); the |k| + M_f + cone-source split is standard b-calculus is_circulant = np.allclose(Mf, np.array([[fhat[(a - b) % N] for b in range(N)] for a in range(N)])) check("L1 CONIC PARAMETRIX [C]: the conic Steklov symbol splits as |k| (principal) " "+ M_f (sub-principal = multiplication by curvature f, circulant in the " "Fourier basis: %s) + cone sources -- standard b-calculus/parametrix " "(Melrose), imported as conditional" % is_circulant, is_circulant) # 5. L2 is the one open lemma + negative control f_offmark = _orbit_curvature([0, N // 4, N // 2, 3 * N // 4]) + 0.3 * _orbit_curvature([11]) hhat = np.fft.fft(f_offmark) / N Mf_bad = np.array([[hhat[(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("L2 IS THE ONE OPEN LEMMA [O]: 'RP + gap + 4 marks => curvature vanishes " "away from the marks'. NEG control: adding curvature OFF the mu4 orbit " "breaks Z4-invariance and the whole chain (||[rho,M_f]|| > 0: %s). Route B " "thus reduces to exactly L2 -- if it falls, Route B closes and SEAM.EQUIV.01 " "needs only the holomorphy half (Route A)" % broken, broken) return summary("v289 MARKLOCAL.RAW.01: raw mark-locality decomposed into 5 lemmas, 4 discharged; the one open analytic lemma is Flat-Away (RP+gap+marks => flat away from the marks)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)