"""v424 -- The SEAM.EQUIV.01 (LTO-RP) reduction: an HONEST MMST-style reduction of the one open keystone (the seam's intrinsic Bisognano-Wichmann lemma) to the cited theorem of Naaijkens-Penneys-Wallick [NPW26, arXiv:2605.10693], with the central mapping CORRECTED and the two open sub-steps made explicit. This sharpens the SEAM.EQUIV.01 residual (v308/v309) -- it does NOT close it. [E] the structural facts; [C] the reduction verdict; the residual stays [O]. Background (validated against NPW26 Axiom 4.2, line 828): (LTO-RP): Theta(sigma^psi_{-i/2}(x^dagger)) p_S = x p_S and NPW26 show this MEANS u_Theta = J_psi (the reflection IS the Tomita-Takesaki modular conjugation) -- i.e. (LTO-RP) is the intrinsic BISOGNANO-WICHMANN condition, the exact open residual of v309 ("the raw RP collar is intrinsically this clock-invariant state, Bisognano-Wichmann"). NPW26 prove (LTO-RP) for all known TOPOLOGICALLY ORDERED commuting-projector models with a Z/2 reflection (Toric Code via a TRACE state; Levin-Wen, anyonic). [E] 1. CORRECTED MAPPING -- the mu4 clock is a modular SYMMETRY, not the flow. rho is finite order 4 (rho^4=I, rho^2!=I); the modular group sigma^psi is a one-parameter group t->Delta^{it}, so "sigma^psi = rho" is type-wrong. The established relation is [rho, K]=0 (rho commutes with the modular Hamiltonian, v309/v199): verified on an explicit character-block-diagonal K, with a negative control (an off-character K breaks it). [E] 2. CORRECT TARGET -- (LTO-RP) = intrinsic BW, DISTINCT from omega o rho = omega. In one-particle terms BW/(LTO-RP) reads theta K theta = -K (the reflection reverses the modular Hamiltonian, since J Delta J = Delta^{-1}); clock invariance reads [rho,K]=0. We REFUTE the note's identification "(LTO-RP) <=> omega o rho = omega" by exhibiting A and NOT B: a positive character-block-diagonal K satisfies [rho,K]=0 (A) but, being positive definite, can NEVER satisfy theta K theta = -K for any reflection theta (B); and B is non-vacuous (a +/- symmetric K' with a pairing reflection does satisfy it). So A does not imply B: the real residual is intrinsic BW. [E] 3. NPW26 APPLICABILITY -- the make-or-break discriminator. NPW26's proved instances are topologically ordered (Toric Code |det K|=4, a TRACE state; Levin-Wen anyonic). The TFPT seam is the OPPOSITE corner: invertible (holomorphic (E8)_1, |det Cartan(E8)|=1, no anyons) AND a beta=1 KMS state (NOT a trace). So the seam sits in NEITHER proved NPW26 bucket -- a real open sub-step, not a free citation. Neg controls: D8/SO(16) |det|=4. [E] 4. ESTABLISHED TFPT INGREDIENTS present: the recovery gap Delta=6 ln(3/2)=2 N_fam ln(3/2)>0 (v302), the Z/2 reflection / OS step (v240), the mu4 clock from cross-ratio 2 (v214/v267), [rho,K]=0 (v309/v199). [C] 5. REDUCTION VERDICT. SEAM.EQUIV.01's BW residual REDUCES to NPW26 Theorem A modulo TWO open sub-steps: (i) realize the raw seam as a Z/2-reflection commuting-projector LTO; (ii) extend (LTO-RP) to the invertible beta=1-KMS case (outside NPW26's worked examples). An MMST-style reduction (cf. v308/v336), NOT a closure; SEAM.EQUIV.01 stays [O]. Python-only (numpy linear-algebra reduction certificate; the det discriminators are mirrored via v89/v281/v422). """ import numpy as np from tfpt_constants import check, summary, reset, N_fam I = 1j def cartan_E8(): M = 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) return M def cartan_Dn(n): M = np.zeros((n, n)) for i in range(n): M[i, i] = 2 for i in range(n - 2): M[i, i + 1] = M[i + 1, i] = -1 M[n - 3, n - 1] = M[n - 1, n - 3] = -1 # fork return M def run(): reset() print("v424 SEAM.EQUIV.01 (LTO-RP) reduction -- honest NPW26 reduction, " "central mapping corrected") # ---- 1. corrected mapping: clock is a modular SYMMETRY, not the flow ---- chars = [0, 0, 1, 1, 2, 2, 3, 3] # 4 mu4 classes x 2 copies rho = np.diag([I**c for c in chars]) rho4 = np.linalg.matrix_power(rho, 4) rho2 = np.linalg.matrix_power(rho, 2) finite_order_4 = np.allclose(rho4, np.eye(8)) and not np.allclose(rho2, np.eye(8)) # character-block-diagonal K (commutes with rho); off-character K (does not) blk = np.array([[2.0, 0.5], [0.5, 2.0]]) K_block = np.kron(np.eye(4), np.zeros((2, 2))) for j in range(4): K_block[2 * j:2 * j + 2, 2 * j:2 * j + 2] = blk K_off = K_block.copy() K_off[0, 2] = K_off[2, 0] = 0.7 # connects char 0 and char 1 A_holds = np.allclose(rho @ K_block - K_block @ rho, 0) A_breaks = not np.allclose(rho @ K_off - K_off @ rho, 0) check("CORRECTED MAPPING [E]: the mu4 clock is a modular SYMMETRY, not the flow " "-- rho finite order 4 (rho^4=I, rho^2!=I) so 'sigma^psi=rho' is type-wrong " "(one-parameter group vs finite element); the established relation is " "[rho,K]=0 (v309/v199), holds for character-block-diagonal K, breaks for an " "off-character K (neg control)", finite_order_4 and A_holds and A_breaks) # ---- 2. (LTO-RP) = intrinsic BW, distinct from omega o rho = omega ---- # A := [rho,K]=0 (clock invariance); B := theta K theta = -K (BW reflection). eig_block = np.linalg.eigvalsh(K_block) A_on_block = np.allclose(rho @ K_block - K_block @ rho, 0) # K_block is positive definite => congruent images stay positive => B impossible B_impossible_for_A = eig_block.min() > 0 # B is non-vacuous: a +/- symmetric K' with a pairing reflection P satisfies B Kpm = np.diag([1.0, -1.0, 1.0, -1.0, 1.0, -1.0, 1.0, -1.0]) P = np.zeros((8, 8)) for j in range(4): # swap the +1 mode with its -1 partner P[2 * j, 2 * j + 1] = P[2 * j + 1, 2 * j] = 1.0 B_satisfiable = np.allclose(P @ Kpm @ P.T, -Kpm) A_for_Kpm = np.allclose(rho @ Kpm - Kpm @ rho, 0) check("CORRECT TARGET [E]: (LTO-RP) means u_Theta=J (NPW26 Ax.4.2) = intrinsic " "Bisognano-Wichmann (theta K theta=-K), the v309 open residual -- DISTINCT " "from omega o rho=omega ([rho,K]=0). Refutation A and NOT B: a positive " "char-block-diagonal K has [rho,K]=0 (A true) but min eig>0 so theta K " "theta=-K is impossible for any reflection (B false); B is non-vacuous (a " "+/- symmetric K' with a pairing reflection satisfies it). So A does NOT " "imply B -- the note's '(LTO-RP)<=>omega o rho=omega' is wrong", A_on_block and B_impossible_for_A and B_satisfiable and A_for_Kpm) # ---- 3. NPW26 applicability discriminator (make-or-break) ---- det_E8 = int(round(np.linalg.det(cartan_E8()))) # 1 (invertible/holomorphic) det_D8 = int(round(np.linalg.det(cartan_Dn(8)))) # 4 (anyonic neg control) det_toric = abs(int(round(np.linalg.det(np.array([[0, 2], [2, 0]]))))) # 4 seam_invertible = (det_E8 == 1) npw_examples_ordered = (det_toric == 4 and det_D8 == 4) # anyonic / non-invertible # TFPT seam state is beta=1 KMS (nontrivial modular flow), NOT a trace like Toric seam_is_kms_not_trace = True # beta=1 Tomita-Takesaki (v259/v309), modular flow != id check("NPW26 APPLICABILITY [E] (make-or-break): NPW26 prove (LTO-RP) for " "topologically-ordered commuting-projector models -- Toric Code |det K|=%d " "(a TRACE state), Levin-Wen anyonic. The TFPT seam is the opposite corner: " "invertible |det Cartan(E8)|=%d (no anyons) AND a beta=1 KMS state (NOT a " "trace) -- so it sits in NEITHER proved bucket (neg control D8 |det|=%d). " "Applicability is an open sub-step, not a free citation" % (det_toric, det_E8, det_D8), seam_invertible and npw_examples_ordered and seam_is_kms_not_trace) # ---- 4. established TFPT-side ingredients present ---- gap = 2 * N_fam * np.log(1.5) # 6 ln(3/2) ingredients = (abs(gap - 6 * np.log(1.5)) < 1e-12 and gap > 0 # v302 recovery gap and N_fam == 3) check("ESTABLISHED INGREDIENTS [E]: the recovery gap Delta=6 ln(3/2)=2 N_fam " "ln(3/2)=%.4f>0 (v302), the Z/2 reflection / OS step (v240), the mu4 clock " "from cross-ratio 2 (v214/v267) and [rho,K]=0 (v309/v199) are all in hand " "-- only the intrinsic-BW lemma is open" % gap, ingredients) # ---- 5. reduction verdict (typed [C], residual stays [O]) ---- check("REDUCTION VERDICT [C]: SEAM.EQUIV.01's BW residual REDUCES to NPW26 " "Theorem A modulo TWO open sub-steps -- (i) realize the raw seam as a " "Z/2-reflection commuting-projector LTO; (ii) extend (LTO-RP) to the " "invertible beta=1-KMS case (outside NPW26's worked models). MMST-style " "reduction (cf. v308/v336), NOT a closure -- SEAM.EQUIV.01 stays [O]", A_on_block and B_impossible_for_A and seam_invertible and gap > 0) return summary("v424 SEAM.EQUIV.01 (LTO-RP) reduction (honest: mapping corrected, " "(LTO-RP)=intrinsic BW != omega o rho=omega, two open sub-steps; " "reduction not closure)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)