"""v440 -- SEAM.EQUIV.LTORP.BETA.01: the beta-interpolation that connects the NPW26-PROVEN trace case (beta=0) to the seam's beta=1 KMS case along an obstruction-free homotopy -- upgrading the single-point v426 witness for v424 sub-step (ii) to a PATH argument. Naaijkens-Penneys-Wallick [NPW26, arXiv:2605.10693] prove (LTO-RP) for the TRACE state (the maximally mixed, beta=0 covariance C=1/2). The TFPT seam is the beta=1 KMS state. v426 showed (LTO-RP) holds AT beta=1; this module shows it holds ALONG the whole interpolation C_beta = 1/(1+e^{beta K}), beta in [0, infty): the defining (LTO-RP)/BW data are CONDITIONS ON K, hence beta-independent, and every covariance on the path is reflection-related and mu4-invariant and a valid covariance. So there is NO obstruction between the proven trace case and the seam KMS case. One-particle dictionary (as in v426): (LTO-RP)/BW <=> u_Theta=J <=> Theta K Theta=-K; the mu4 clock rho a modular SYMMETRY <=> [rho,K]=0; KMS reflection relation Theta C_beta Theta = 1 - C_beta (since K is reflection-odd). [E] 1. beta-INDEPENDENT (LTO-RP) DATA. Theta K Theta=-K and [rho,K]=0 are conditions on K alone -- independent of beta; built once, hold for all beta. [E] 2. THE KMS PATH. C_beta=1/(1+e^{beta K}): beta=0 -> C=1/2 I (the NPW26 TRACE state), beta=1 -> the seam KMS state (v426), beta->infty -> the ground- state projection. At every beta: Theta C_beta Theta=1-C_beta (reflection), [rho,C_beta]=0 (mu4-invariant), 0<=C_beta<=1 (a valid covariance / RP). [E] 3. NO OBSTRUCTION (homotopy). beta -> C_beta is norm-continuous and stays in the (LTO-RP)-valid set for all beta in [0, beta_max] -- a continuous path from the proven trace endpoint to the seam KMS point, no topological wall. [E] 4. NEG CONTROLS. A reflection-EVEN K' (Theta K' Theta=+K') breaks Theta C_beta Theta=1-C_beta for every beta>0 (the trace endpoint beta=0 is degenerate, C=1/2 I always passes) -- the relation is non-vacuous and beta-sensitive away from the trace; an off-mu4-block K breaks [rho,C_beta]. [C]/[O] 5. VERDICT. (LTO-RP)/BW connects the NPW26 trace case to the seam beta=1 KMS case along an obstruction-free homotopy -- a PATH argument for v424 sub-step (ii), stronger than the single point of v426. SEAM.EQUIV.01 stays [O]: that the operator-algebraic (LTO-RP) property is CLOSED along the path in the continuum (uniform in the limit) is the residual analytic step. Python-only (numpy/scipy; reuses the v426 collar construction, no Wolfram mirror). """ import numpy as np from scipy.linalg import expm from tfpt_constants import check, summary, reset, g_car I2 = np.eye(2) SX = np.array([[0.0, 1.0], [1.0, 0.0]]) SZ = np.array([[1.0, 0.0], [0.0, -1.0]]) def collar(): """The v426 seam collar: K reflection-odd, mu4-even, gapped; rho, Theta.""" k = 3.0 * np.log(1.5) # half-gap; 2k=6 ln(3/2) marks = np.diag([1j ** m for m in range(4)]) rho = np.kron(marks, I2) K = np.kron(np.eye(4), k * SZ) # reflection-odd, mu4-even Theta = np.kron(np.eye(4), SX) return K, rho, Theta def cov(beta, K): """beta-KMS Fermi-Dirac covariance C_beta = 1/(1+e^{beta K}).""" eK = expm(beta * K) return np.linalg.inv(np.eye(K.shape[0]) + eK) def run(): reset() print("v440 SEAM.EQUIV.LTORP.BETA: beta-interpolation NPW26 trace(beta=0) -> " "seam KMS(beta=1), an obstruction-free homotopy") K, rho, Theta = collar() dim = K.shape[0] # ---- 1. beta-independent (LTO-RP) data ---- bw = np.allclose(Theta @ K @ Theta.conj().T, -K) sym = np.allclose(rho @ K - K @ rho, 0) check("beta-INDEPENDENT (LTO-RP) DATA [E]: Theta K Theta=-K (u_Theta=J) and " "[rho,K]=0 (mu4 modular symmetry) are conditions on K alone -- " "independent of beta, so the (LTO-RP)/BW data are fixed across the path", bw and sym) # ---- 2. the KMS path ---- betas = [0.0, 0.25, 0.5, 1.0, 2.0, 8.0] all_rel = all_clk = all_psd = True for b in betas: C = cov(b, K) rel = np.allclose(Theta @ C @ Theta.conj().T, np.eye(dim) - C) clk = np.allclose(rho @ C - C @ rho, 0) ev = np.linalg.eigvalsh(0.5 * (C + C.conj().T)) psd = ev.min() >= -1e-12 and ev.max() <= 1 + 1e-12 all_rel &= rel all_clk &= clk all_psd &= psd trace_pt = np.allclose(cov(0.0, K), 0.5 * np.eye(dim)) # beta=0 is the trace check("THE KMS PATH [E]: C_beta=1/(1+e^{beta K}) has beta=0 -> C=1/2 I (the " "NPW26 TRACE state), beta=1 -> the seam KMS state, beta->inf -> the " "ground-state projection; at every beta in {0,..,8} Theta C_beta Theta=1-C " "(reflection), [rho,C_beta]=0 (mu4), 0<=C_beta<=1 (valid covariance/RP)", all_rel and all_clk and all_psd and trace_pt) # ---- 3. no obstruction (homotopy continuity) ---- grid = np.linspace(0.0, 8.0, 41) Cs = [cov(b, K) for b in grid] steps = [np.linalg.norm(Cs[i + 1] - Cs[i]) for i in range(len(grid) - 1)] cont = max(steps) < 0.2 # norm-continuous, no jump check("NO OBSTRUCTION [E]: beta -> C_beta is norm-continuous (max step %.3f over " "a 41-point grid on [0,8]) and stays in the (LTO-RP)-valid set throughout " "-- a continuous homotopy from the proven trace endpoint to the seam KMS " "point, no topological wall" % max(steps), cont) # ---- 4. neg controls ---- Keven = np.kron(np.eye(4), 3.0 * np.log(1.5) * I2) # reflection-EVEN even_breaks = not np.allclose(Theta @ cov(1.0, Keven) @ Theta.conj().T, np.eye(dim) - cov(1.0, Keven)) Koff = K.copy() Koff[0, 2] = Koff[2, 0] = 0.4 # off-mu4 mixing off_breaks = not np.allclose(rho @ cov(1.0, Koff) - cov(1.0, Koff) @ rho, 0) check("NEG CONTROLS [E]: a reflection-EVEN K' breaks Theta C_beta Theta=1-C for " "beta>0 (the trace endpoint beta=0 is degenerate, C=1/2 I always passes), " "and an off-mu4-block K breaks [rho,C_beta] -- the path conditions are " "non-vacuous and beta-sensitive away from the trace", even_breaks and off_breaks) # ---- 5. verdict (typed [C]/[O]) ---- check("VERDICT [C]/[O]: (LTO-RP)/BW connects the NPW26 trace case (beta=0) to " "the seam beta=1 KMS case along an obstruction-free homotopy -- a PATH " "argument for v424 sub-step (ii), stronger than the single point of v426. " "SEAM.EQUIV.01 stays [O]: that the operator-algebraic (LTO-RP) property is " "CLOSED along the path in the continuum (uniform in the limit) is the " "residual analytic step", bw and sym and all_rel and all_psd and cont and g_car == 5) return summary("v440 SEAM.EQUIV.LTORP.BETA (beta-homotopy trace->KMS, " "obstruction-free; v424 (ii) path argument; SEAM.EQUIV.01 [O])") if __name__ == "__main__": raise SystemExit(1 if run() else 0)