"""v276 -- QGEO.SYM.03: the flat-pillowcase premise CLOSES the modular-commutator chain to all orders, so the bedrock QGEO.SYM.01 collapses to ONE geometric postulate. This does NOT prove QGEO.SYM.01 (that the raw seam IS the flat tau=i pillowcase) -- it proves that GRANTING the flat geometry discharges everything downstream exactly, upgrading the leading-order results v198 (principal symbol) + v201 (subprincipal) to the FULL operator. The reduction chain (already established): QGEO.SYM.01 == omega_Sigma o rho = omega_Sigma (mu4-invariance of the seam state) quasi-free ==> omega o rho = omega <=> [rho, C] = 0 <=> [rho, H] = 0 (C = (1+e^H)^{-1}, v258; H the DtN-induced modular Hamiltonian) [rho, H] = 0 ==> mark-locality + state invariance (v199/v267) -- all downstream [E]. So the entire bedrock sits on the single operator equation [rho, H] = 0. [E] 1. ISOMETRY => COMMUTATOR (the key step). If the seam carries the flat tau=i pillowcase and H = f(Delta_flat) is a function of its intrinsic (flat) Laplacian -- the DtN of a flat metric, with NO curvature corrections -- then the order-4 deck rho (z -> iz) is an ISOMETRY, so [rho, Delta] = 0 and hence [rho, H] = [rho, f(Delta)] = 0 EXACTLY, to all orders. Verified on the discretised square torus: rho^4 = I, [rho, Delta] = 0, [rho, sqrt(-Delta)] = 0; a generic non-isometry permutation FAILS (negative control). [E] 2. ALL-ORDERS UPGRADE. this upgrades v198 (principal symbol |k| commutes) and v201 (subprincipal block-diagonal) from leading orders to the FULL H: flatness removes the lower-order curvature terms, so there is no residual obstruction -- the commutator is exact, not asymptotic. [C] 3. tau=i IS FORCED (not chosen). the order-4 requirement forces the cross-ratio 2 <=> j = 1728 <=> tau = i (the unique modulus with order-4 CM by Z[i]; v214/v267); the hexagonal tau = omega (j = 0) is order-6, excluded. So IF the seam has the order-4 deck at all, the flat tau=i pillowcase is forced. [O] 4. THE IRREDUCIBLE POSTULATE. QGEO.SYM.01 therefore collapses to the SINGLE geometric statement: 'the raw RP seam state is the flat tau=i pillowcase quasi-free state (H = sqrt(-Delta_flat))'. Everything downstream is [E]; this one statement is the bedrock (the role 'c = const' plays in relativity) -- it needs a human constructive-QFT proof that the raw seam DtN state IS the flat tau=i geometry, or it stays an honest axiom. Sharpest non-circular form. Status: [E] the isometry=>commutator closure (numerically verified, all orders) + the all-orders upgrade of v198/v201; [C] tau=i forced by order-4; [O] the one geometric postulate stays open. Narrows QGEO.SYM.01 to a single flat-geometry statement; does NOT close it. Python (numpy + scipy + sympy). """ import numpy as np import sympy as sp from scipy.linalg import sqrtm from tfpt_constants import check, summary, reset N = 8 # square-torus grid size def _idx(x, y): return (x % N) * N + (y % N) def flat_laplacian(): """5-point periodic discrete Laplacian on the square (tau=i) torus.""" L = np.zeros((N * N, N * N)) for x in range(N): for y in range(N): i = _idx(x, y) L[i, i] = -4 for dx, dy in ((1, 0), (-1, 0), (0, 1), (0, -1)): L[i, _idx(x + dx, y + dy)] += 1 return L def order4_deck(): """the order-4 isometry rho: z -> i z, i.e. the 90-deg rotation (x,y) -> (-y,x).""" R = np.zeros((N * N, N * N)) for x in range(N): for y in range(N): R[_idx(-y, x), _idx(x, y)] = 1 return R def run(): reset() print("v276 QGEO.SYM.03: the flat tau=i pillowcase closes [rho,H]=0 to all orders") L = flat_laplacian() R = order4_deck() # 1. isometry => commutator (the key step) is_order4 = np.allclose(np.linalg.matrix_power(R, 4), np.eye(N * N)) comm_lap = np.allclose(R @ L - L @ R, 0) H = sqrtm(-L + 1e-6 * np.eye(N * N)).real # H = sqrt(-Delta) ~ flat DtN comm_H = np.allclose(R @ H - H @ R, 0, atol=1e-8) check("ISOMETRY => COMMUTATOR [E]: the order-4 deck rho (z->iz) on the flat tau=i " "torus has rho^4=I (%s), [rho,Delta]=0 (%s, isometry commutes with the " "Laplacian), and [rho,H]=0 for H=sqrt(-Delta) (%s) -- so granting the flat " "geometry gives [rho,H]=0 EXACTLY, to all orders" % (is_order4, comm_lap, comm_H), is_order4 and comm_lap and comm_H) # negative control: a generic non-isometry permutation does NOT commute rng = np.random.default_rng(276) P = np.eye(N * N)[rng.permutation(N * N)] ctrl = not np.allclose(P @ L - L @ P, 0) check("NEGATIVE CONTROL [E]: a generic (non-isometry) permutation P does NOT " "commute with Delta ([P,Delta] != 0: %s) -- the commutation is special to " "the deck isometry, not generic" % ctrl, ctrl) # 2. all-orders upgrade: commute with every spectral function of Delta comm_L2 = np.allclose(R @ (L @ L) - (L @ L) @ R, 0) comm_exp = np.allclose(R @ sqrtm(sqrtm(-L + 1e-6 * np.eye(N * N))).real - sqrtm(sqrtm(-L + 1e-6 * np.eye(N * N))).real @ R, 0, atol=1e-7) check("ALL-ORDERS UPGRADE [E]: rho commutes with Delta^2 (%s) and with " "(-Delta)^{1/4} (%s) -- with EVERY spectral function of Delta, so the v198 " "(principal symbol) + v201 (subprincipal) leading-order results upgrade to " "the FULL H; flatness removes the curvature terms, no residual obstruction" % (comm_L2, comm_exp), comm_L2 and comm_exp) # 3. tau=i is forced by order-4 (j=1728), hexagonal tau=omega (j=0) is order-6 lam = sp.symbols("lam") jf = 256 * (lam ** 2 - lam + 1) ** 3 / (lam ** 2 * (lam - 1) ** 2) j_square = sp.simplify(jf.subs(lam, 2)) # cross-ratio-2 config j_hex = sp.simplify(jf.subs(lam, sp.exp(sp.I * sp.pi / 3))) check("tau=i FORCED [C]: order-4 => cross-ratio 2 => j=1728 (%s), the unique " "modulus with order-4 CM by Z[i]; the hexagonal tau=omega has j=0 (order-6, " "excluded) -- so IF the seam has the order-4 deck, the flat tau=i pillowcase " "is forced (v214/v267)" % (j_square == 1728), j_square == 1728 and sp.simplify(j_hex) == 0) # 4. the irreducible postulate check("IRREDUCIBLE POSTULATE [O]: QGEO.SYM.01 collapses to ONE geometric " "statement -- 'the raw RP seam state is the flat tau=i pillowcase quasi-free " "state, H=sqrt(-Delta_flat)'. Everything downstream is [E]; this one " "statement is the bedrock (like c=const) -- needs a human constructive-QFT " "proof that the raw seam DtN state IS the flat tau=i geometry, or stays an " "honest axiom. Sharpest non-circular form", True) return summary("v276 flat tau=i pillowcase => [rho,H]=0 to all orders; QGEO.SYM.01 = one geometric postulate (QGEO.SYM.03)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)