Follow the source behind the result.

The registered Python source is shown as text. Displaying it does not execute the code.

"""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)

b4c9cbcc7f0730a82c23b4741bfa980653b9e13d774af6c2ef7c654cf5f887da

The public registry includes 116 document views, 6,693 section records and 985 physical LaTeX sources. Active publication documents, research material and archives retain their recorded roles. This is source navigation, not a new semantic review or proof run.