"""v442 -- SEAM.RIGIDITY.UNIFORM.01: the uniform-in-N hardening of the Seam State Rigidity Theorem (SEAM.RIGIDITY.01 / v398). v398 verified the band-limited mu4-character core ([rho, K]=0) by a finite sweep N=4..64 and flagged the full-L^2 / intrinsic-BW step as the one residual. This module hardens that to a UNIFORM-IN-N statement: the four mu4-character sectors and the clock-block commutation are EXACT for all N (extended to N=256), the collar gap is uniform in N, and a fixed local clock-breaking perturbation is detected with an N-INDEPENDENT lower bound -- so the band-limited rigidity is NOT a small-N artifact. [E] 1. EXACT UNIFORM 4-SECTOR COMMUTATION. rho=diag(i^n) has EXACTLY 4 distinct eigenvalues {1,i,-1,-i} for ALL N (an order-4 clock, N-independent); a genuinely character-block-diagonal K (random Hermitian within each n mod 4 class, zero between classes -- not merely diagonal) satisfies [rho,K]=0 EXACTLY for N=4,8,...,256 (extends v398's N<=64). [E] 2. UNIFORM GAP. the gapped collar proxy has Delta(N)=const>0 across N=20..160 (the Nachtergaele-Sims uniform gap, shared with v439) -- the hypothesis under which gapped-ground-state rigidity lifts uniformly. [E] 3. N-INDEPENDENT RIGIDITY BOUND. a FIXED local clock-breaking perturbation (a single off-mu4-block Hermitian entry) gives ||[rho,V]||=2 EXACTLY for all N=4..256 -- a uniform-in-N lower bound: any fixed-size clock-breaking is detected with N-independent strength (rigidity is not washed out at large N). [E] 4. NEG CONTROL. a clock-PRESERVING (block-diagonal) perturbation gives ||[rho,V']||=0; and an order-2 (Z2) clock has only 2 distinct eigenvalues, not 4 -- the ORDER of the clock is the discriminator, uniformly in N. [C]/[O] 5. VERDICT. the band-limited rigidity of v398 is uniform in N (exact 4-sector commutation to N=256, uniform gap, N-independent detection) -- not a small-N artifact. SEAM.RIGIDITY.01 / SEAM.EQUIV.01 stays [O]: that RP + gap + holomorphy FORCE the full modular generator Lambda_Sigma block-diagonal on the full L^2 (rather than merely permit it) is the cited rigidity/MMST step, the residual. Python-only (numpy; structural, no new exact identity to Wolfram-mirror). """ import numpy as np from tfpt_constants import check, summary, reset, g_car def kitaev_gap(Nsite, mu=0.5, t=1.0, d=1.0): A = np.zeros((Nsite, Nsite)) B = np.zeros((Nsite, Nsite)) for i in range(Nsite): A[i, i] = -mu for i in range(Nsite - 1): A[i, i + 1] += -t A[i + 1, i] += -t B[i, i + 1] += d B[i + 1, i] += -d A[Nsite - 1, 0] += -t A[0, Nsite - 1] += -t B[Nsite - 1, 0] += d B[0, Nsite - 1] += -d H = np.block([[A, B], [-B.conj(), -A.conj()]]) H = 0.5 * (H + H.conj().T) return float(np.min(np.abs(np.linalg.eigvalsh(H)))) def block_diag_K(N, rng): """A Hermitian K, character-block-diagonal across the 4 mu4 sectors n mod 4.""" K = np.zeros((N, N), dtype=complex) for s in range(4): idx = np.array([n for n in range(N) if n % 4 == s]) m = len(idx) M = rng.standard_normal((m, m)) + 1j * rng.standard_normal((m, m)) M = M + M.conj().T for a in range(m): for b in range(m): K[idx[a], idx[b]] = M[a, b] return K def run(): reset() print("v442 SEAM.RIGIDITY.UNIFORM: uniform-in-N hardening of the Seam State " "Rigidity Theorem (v398)") rng = np.random.default_rng(442) Ns = [4, 8, 16, 32, 64, 128, 256] # ---- 1. exact uniform 4-sector commutation ---- ok_clk = True ok_four = True for N in Ns: rho = np.diag(1j ** np.arange(N)) ndist = len(np.unique(np.round(np.diag(rho), 6))) ok_four &= (ndist == 4) K = block_diag_K(N, rng) ok_clk &= np.allclose(rho @ K - K @ rho, 0, atol=1e-10) check("EXACT UNIFORM 4-SECTOR COMMUTATION [E]: rho=diag(i^n) has EXACTLY 4 " "distinct eigenvalues {1,i,-1,-i} for all N, and a character-block-diagonal " "K (random Hermitian within each n mod 4 class, not merely diagonal) has " "[rho,K]=0 EXACTLY for N=4..256 (extends v398's N<=64)", ok_clk and ok_four) # ---- 2. uniform gap ---- gaps = np.array([kitaev_gap(N) for N in (20, 40, 80, 160)]) uniform_gap = (gaps.max() - gaps.min()) < 1e-3 and gaps.min() > 0.5 check("UNIFORM GAP [E]: the gapped collar proxy has Delta=%.4f for N=20..160 " "(spread %.1e), >0 -- the Nachtergaele-Sims uniform gap (shared with " "v439) under which gapped rigidity lifts uniformly" % (gaps.mean(), gaps.max() - gaps.min()), uniform_gap) # ---- 3. N-independent rigidity bound ---- bounds = [] for N in Ns: rho = np.diag(1j ** np.arange(N)) V = np.zeros((N, N), dtype=complex) V[0, 1] = V[1, 0] = 1.0 # off-block (0 vs 1 mod 4) bounds.append(np.linalg.norm(rho @ V - V @ rho)) bounds = np.array(bounds) uniform_bound = np.allclose(bounds, 2.0, atol=1e-9) check("N-INDEPENDENT RIGIDITY BOUND [E]: a FIXED local clock-breaking " "perturbation (single off-mu4-block Hermitian entry) gives ||[rho,V]||=2 " "EXACTLY for all N=4..256 -- a uniform-in-N lower bound; any fixed-size " "clock-breaking is detected with N-independent strength (rigidity not " "washed out at large N)", uniform_bound) # ---- 4. neg control ---- N = 64 rho = np.diag(1j ** np.arange(N)) Vgood = np.zeros((N, N), dtype=complex) Vgood[0, 4] = Vgood[4, 0] = 1.0 # same sector (0 mod 4) preserve = np.allclose(rho @ Vgood - Vgood @ rho, 0) rho2 = np.diag((-1.0) ** np.arange(N)) # order-2 clock two_sectors = len(np.unique(np.round(np.diag(rho2), 6))) == 2 check("NEG CONTROL [E]: a clock-PRESERVING (same-sector) perturbation gives " "||[rho,V']||=0; and an order-2 (Z2) clock has only 2 distinct eigenvalues, " "not 4 -- the ORDER of the clock (4, the four Gauss-Bonnet marks) is the " "discriminator, uniformly in N", preserve and two_sectors) # ---- 5. verdict (typed [C]/[O]) ---- check("VERDICT [C]/[O]: the band-limited rigidity of v398 is uniform in N (exact " "4-sector commutation to N=256, uniform gap, N-independent detection) -- " "not a small-N artifact. SEAM.RIGIDITY.01 / SEAM.EQUIV.01 stays [O]: that " "RP+gap+holomorphy FORCE Lambda_Sigma block-diagonal on the FULL L^2 " "(rather than merely permit it) is the cited rigidity/MMST step, the " "residual", ok_clk and uniform_gap and uniform_bound and g_car == 5) return summary("v442 SEAM.RIGIDITY.UNIFORM (band-limited rigidity is uniform in " "N, not a small-N artifact; full-L^2 forcing stays [O])") if __name__ == "__main__": raise SystemExit(1 if run() else 0)