"""v426 -- SEAM.EQUIV.LTORP.KMS.01: the INVERTIBLE beta=1-KMS variant of the (LTO-RP) axiom -- the one corner Naaijkens-Penneys-Wallick [NPW26, arXiv:2605.10693] leave open. NPW26 prove (LTO-RP) only for TOPOLOGICALLY ORDERED commuting-projector models (Toric Code via a TRACE state; Levin-Wen anyonic). The TFPT seam is the opposite corner: invertible (holomorphic (E8)_1, no anyons) AND a beta=1 KMS state (nontrivial modular flow, NOT a trace). This module exhibits CONSTRUCTIVELY that on this corner the (LTO-RP)/Bisognano-Wichmann condition u_Theta=J is satisfiable and NATURAL, by direct Tomita-Takesaki -- advancing sub-step (ii) of v424. It does NOT close SEAM.EQUIV.01 (the continuum/abstract realisation, sub-step (i), stays open). One-particle / quasi-free dictionary (the seam collar): modular operator Delta = e^{-K}, modular conjugation J, reflection Theta; (LTO-RP)/BW <=> u_Theta = J <=> Theta K Theta^{-1} = -K (J Delta J = Delta^{-1}); the mu4 clock rho is a modular SYMMETRY <=> [rho, K] = 0 (NOT sigma^psi=rho). [E] 1. THE INVERTIBLE-KMS COLLAR. Build K reflection-odd, mu4-even, gapped (gap 2k=6 ln(3/2)); C=1/(1+e^K) is the beta=1 KMS covariance and satisfies the beta=1 KMS relation C/(1-C)=e^{-K} -- a genuine KMS (not trace) state. [E] 2. (LTO-RP)/BW HOLDS. Theta K Theta^{-1} = -K (reflection = modular conjugation, u_Theta=J) AND [rho,K]=0 (mu4 a modular symmetry) -- both hold for the constructed collar, so (LTO-RP) is satisfied on the invertible KMS corner by direct Tomita-Takesaki, no skein/UFC machinery. [E] 3. INVERTIBLE, OUTSIDE NPW26. |det Cartan(E8)|=1 (one primary, invertible, no anyons) vs the anyonic |det|=4 of NPW26's worked models; and the state is NOT a trace (K!=0 => Delta=e^{-K}!=I, nontrivial modular flow) -- so it is in NEITHER proved NPW26 bucket. [E] 4. NEG CONTROLS (teeth). A side-EVEN reflection gives Theta K Theta=+K!=-K (BW fails); an off-mu4-block K mixing two marks gives [rho,K]!=0 (the mu4 symmetry fails). Not every reflection / not every K works. [C] 5. VERDICT. On the gapped invertible beta=1-KMS reflection-symmetric collar, (LTO-RP) reduces to (Theta K Theta=-K) and ([rho,K]=0), both structural -- the invertible-KMS variant NPW26 leave open is constructively satisfiable. Advances v424 sub-step (ii); SEAM.EQUIV.01 stays [O] (continuum sub-step (i) remains). A finite-dim Tomita-Takesaki witness, NOT the continuum theorem. Python-only (numpy linear algebra; det discriminator mirrored via v89/v281/v422/v424). """ import numpy as np from tfpt_constants import check, summary, reset, g_car I2 = np.eye(2) SX = np.array([[0.0, 1.0], [1.0, 0.0]]) # side-flip (reflection) SZ = np.array([[1.0, 0.0], [0.0, -1.0]]) # boost / modular direction def cartan_E8(): return np.array([ [2, 0, -1, 0, 0, 0, 0, 0], [0, 2, 0, -1, 0, 0, 0, 0], [-1, 0, 2, -1, 0, 0, 0, 0], [0, -1, -1, 2, -1, 0, 0, 0], [0, 0, 0, -1, 2, -1, 0, 0], [0, 0, 0, 0, -1, 2, -1, 0], [0, 0, 0, 0, 0, -1, 2, -1], [0, 0, 0, 0, 0, 0, -1, 2]], dtype=float) def run(): reset() print("v426 SEAM.EQUIV.LTORP.KMS: the invertible beta=1-KMS (LTO-RP) variant " "NPW26 leaves open") k = 3.0 * np.log(1.5) # half-gap; gap 2k = 6 ln(3/2) marks = np.diag([1j**m for m in range(4)]) # mu4 clock on the 4 marks rho = np.kron(marks, I2) # rho on mark sector (order 4) K = np.kron(np.eye(4), k * SZ) # reflection-odd, mu4-even, gapped Theta = np.kron(np.eye(4), SX) # the seam reflection (side-flip) # ---- 1. invertible-KMS collar: beta=1 KMS relation + gap ---- eK = np.diag(np.exp(np.diag(K).real)).astype(complex) C = np.linalg.inv(np.eye(8) + eK) # Fermi-Dirac covariance, beta=1 kms = np.allclose(C @ np.linalg.inv(np.eye(8) - C), np.linalg.inv(eK)) # C/(1-C)=e^{-K} gap = 2 * k rho4 = np.allclose(np.linalg.matrix_power(rho, 4), np.eye(8)) rho2 = not np.allclose(np.linalg.matrix_power(rho, 2), np.eye(8)) check("INVERTIBLE-KMS COLLAR [E]: K reflection-odd/mu4-even/gapped (gap " "2k=6 ln(3/2)=%.4f>0); C=1/(1+e^K) is the beta=1 KMS covariance and " "satisfies C/(1-C)=e^{-K} (a genuine KMS, not trace, state); rho order 4" % gap, kms and gap > 1.6 and rho4 and rho2) # ---- 2. (LTO-RP)/BW holds: Theta K Theta = -K and [rho,K]=0 ---- bw = np.allclose(Theta @ K @ Theta.conj().T, -K) # u_Theta = J sym = np.allclose(rho @ K - K @ rho, 0) # mu4 modular symmetry check("(LTO-RP)/BW HOLDS [E]: Theta K Theta^{-1} = -K (reflection = modular " "conjugation, u_Theta=J) AND [rho,K]=0 (mu4 a modular symmetry) -- both " "hold, so (LTO-RP) is satisfied on the invertible KMS corner by direct " "Tomita-Takesaki", bw and sym) # ---- 3. invertible, not a trace, outside NPW26 ---- det_E8 = int(round(np.linalg.det(cartan_E8()))) # 1 (invertible) det_toric = abs(int(round(np.linalg.det(np.array([[0, 2], [2, 0]]))))) # 4 (anyonic) not_trace = not np.allclose(eK, np.eye(8)) # Delta=e^{-K} != I check("INVERTIBLE, OUTSIDE NPW26 [E]: |det Cartan(E8)|=%d (one primary, " "invertible, no anyons) vs anyonic |det|=%d of NPW26's worked models; and " "the state is NOT a trace (Delta=e^{-K}!=I, nontrivial modular flow) -- " "neither proved NPW26 bucket" % (det_E8, det_toric), det_E8 == 1 and det_toric == 4 and not_trace) # ---- 4. neg controls (teeth) ---- Theta_even = np.kron(np.eye(4), I2) # side-EVEN reflection bw_fails = not np.allclose(Theta_even @ K @ Theta_even.T, -K) Koff = K.copy() Koff[0, 2] = Koff[2, 0] = 0.5 # mix mark 0 and mark 1 sym_fails = not np.allclose(rho @ Koff - Koff @ rho, 0) check("NEG CONTROLS [E]: a side-EVEN reflection gives Theta K Theta=+K!=-K (BW " "fails); an off-mu4-block K mixing two marks gives [rho,K]!=0 (mu4 " "symmetry fails) -- the conditions have teeth", bw_fails and sym_fails) # ---- 5. verdict (typed [C]) ---- check("VERDICT [C]: on the gapped invertible beta=1-KMS reflection-symmetric " "collar, (LTO-RP) reduces to (Theta K Theta=-K) and ([rho,K]=0), both " "structural -- the invertible-KMS variant NPW26 leave open is " "constructively satisfiable (advances v424 sub-step (ii)). SEAM.EQUIV.01 " "stays [O]: the continuum/abstract realisation (sub-step (i)) remains; " "this is a finite-dim Tomita-Takesaki witness, not the continuum theorem", bw and sym and kms and det_E8 == 1 and g_car == 5) return summary("v426 SEAM.EQUIV.LTORP.KMS (invertible beta=1-KMS (LTO-RP) " "constructively satisfiable; advances v424 (ii); SEAM.EQUIV.01 " "stays [O])") if __name__ == "__main__": raise SystemExit(1 if run() else 0)