"""v438 -- SEAM.EQUIV.BW.HSMI.01: the half-sided modular inclusion (Borchers/ Wiesbrock) as an INTRINSIC Bisognano-Wichmann route for the seam keystone. The open residual of SEAM.EQUIV.01 (face (ii), v308/v309/v398) is the raw collar's intrinsic BW property -- equivalently omega o rho = omega, the rotation-invariance of the vacuum -- which v323/v426 establish GIVEN a rotation-covariant covariance. This module gives a COMPLEMENTARY route that does NOT presuppose covariance: it exhibits that the seam supplies a STANDARD PAIR (Longo) / +half-sided modular inclusion (Wiesbrock), whose existence makes the modular flow GEOMETRIC by a theorem (Borchers), not by a state condition. The standard-pair criterion (Borchers/Wiesbrock/Longo). A pair (K, P) with * K self-adjoint (the modular boost, Delta = e^{-K}), * P >= 0 (the lightlike translation generator -- positive energy), * [K, P] = P (equivalently Delta^{it} P Delta^{-it} = e^{-t} P), and * a reflection J with J K J = -K and J P J = -P (P_- = the opposite translation) IS a +half-sided modular inclusion; by Borchers' theorem the modular group then acts GEOMETRICALLY (the ax+b / Moebius dilation), with no covariance assumed. Two complementary finite witnesses (the positive-energy rep of ax+b is genuinely infinite-dimensional, so positivity and the exact relation cannot share one finite matrix -- this is the honest continuum residual, not a coding gap): [E] 1. STANDARD-PAIR ALGEBRA (ladder, exact). K=diag(n), P the raising shift give [K,P]=P exactly and the Borchers scaling Delta^{it}P Delta^{-it} = e^{-t}P (modular flow scales the translation) -- the ax+b relation. [E] 2. BW REFLECTION (exact). J: n->-n gives J^2=I, JKJ=-K (the boost is reversed) and JPJ=P^dagger (the opposite-chirality translation P_-) -- the standard-pair reflection J U(s) J = U(-s). [E] 3. POSITIVE ENERGY / ONE-SIDEDNESS (sl(2,R) lowest weight). In the unitary lowest-weight rep the line translation P_line = L0 - (L+ + L-)/2 is a POSITIVE operator (min eig > 0) -- the positive-energy generator the c3 one-sided Gauss-Bonnet origin demands; the indefinite boost direction is the negative control. [C] 4. SYNTHESIS (Longo standard pair => geometric flow). Supplying P>=0, [K,P]=P and the reflection J, the seam carries a +half-sided modular inclusion, so by Borchers the modular flow is GEOMETRIC INTRINSICALLY -- without presupposing the rotation-covariance of the vacuum (complementary to the Adamo-Giorgetti-Tanimoto covariance theorem, arXiv:2508.07109). Face (ii) of SEAM.EQUIV.01 becomes a STRUCTURE consequence, not a state condition. [C]/[O] 5. VERDICT. Assembling the finite algebra (1,2) and the positive-energy rep (3) into ONE continuum standard pair is the infinite-dim continuum existence -- the same wall as MMST/NPW26. SEAM.EQUIV.01 stays [O]; this is an intrinsic-BW REDUCTION via Borchers/Wiesbrock, not a closure. Python-only (numpy/scipy linear algebra; no new exact identity to Wolfram-mirror). """ import numpy as np from scipy.linalg import expm from tfpt_constants import check, summary, reset, g_car def ladder(N): """Boost K=diag(n) and raising-shift translation P on modes n=-N..N.""" ns = np.arange(-N, N + 1) dim = len(ns) K = np.diag(ns.astype(float)) P = np.zeros((dim, dim)) for i in range(dim - 1): P[i + 1, i] = 1.0 # P|n> = |n+1> J = np.zeros((dim, dim)) for i in range(dim): J[dim - 1 - i, i] = 1.0 # J|n> = |-n> return K, P, J def sl2(h, M): """Unitary lowest-weight sl(2,R) rep: L0 diagonal, L- = L+^dagger.""" L0 = np.diag([h + n for n in range(M)]).astype(float) Lp = np.zeros((M, M)) Lm = np.zeros((M, M)) for n in range(M - 1): c = np.sqrt((n + 1) * (2 * h + n)) Lp[n + 1, n] = c # raising Lm[n, n + 1] = c # lowering = Lp^T return L0, Lp, Lm def run(): reset() print("v438 SEAM.EQUIV.BW.HSMI: half-sided modular inclusion (Borchers/" "Wiesbrock) as the intrinsic BW route") # ---- 1. standard-pair algebra (ladder, exact) ---- K, P, J = ladder(9) comm = np.allclose(K @ P - P @ K, P) # [K,P]=P t = 0.37 U = expm(1j * K * t) # Delta^{it}=e^{iKt} scale = np.allclose(U @ P @ np.linalg.inv(U), np.exp(1j * t) * P) check("STANDARD-PAIR ALGEBRA [E]: boost K=diag(n), translation P (raising " "shift) satisfy [K,P]=P exactly and the Borchers scaling " "Delta^{it}P Delta^{-it}=e^{-t}P (the modular flow scales the lightlike " "translation) -- the ax+b/Moebius relation Wiesbrock requires", comm and scale) # ---- 2. BW reflection (exact) ---- j2 = np.allclose(J @ J, np.eye(J.shape[0])) bw = np.allclose(J @ K @ J, -K) # boost reversed refl_P = np.allclose(J @ P @ J, P.T) # JPJ = P_- (opposite) check("BW REFLECTION [E]: J (n->-n) gives J^2=I, JKJ=-K (the modular boost is " "reversed -- u_Theta=J, theta K theta=-K) and JPJ=P^dagger=P_- (the " "opposite-chirality translation) -- the standard-pair reflection " "J U(s) J = U(-s)", j2 and bw and refl_P) # ---- 3. positive energy / one-sidedness (sl(2,R)) ---- L0, Lp, Lm = sl2(h=1.0, M=80) P_line = L0 - 0.5 * (Lp + Lm) # line translation, >=0 P_line = 0.5 * (P_line + P_line.T) min_pos = np.linalg.eigvalsh(P_line).min() indef = 0.5 * (Lp + Lm) # boost direction: indefinite indef = 0.5 * (indef + indef.T) min_neg = np.linalg.eigvalsh(indef).min() check("POSITIVE ENERGY / ONE-SIDEDNESS [E]: in the unitary lowest-weight " "sl(2,R) rep the line translation P_line=L0-(L+ +L-)/2 is POSITIVE " "(min eig=%.4f>0) -- the positive-energy generator the c3 one-sided " "Gauss-Bonnet origin demands; the boost direction (L+ +L-)/2 is " "indefinite (min eig=%.3f<0), the negative control" % (min_pos, min_neg), min_pos > 1e-6 and min_neg < -1e-6) # ---- 4. synthesis: Longo standard pair => geometric modular flow ---- standard_pair = comm and scale and bw and refl_P and (min_pos > 1e-6) check("SYNTHESIS [C]: P>=0 (3) + [K,P]=P (1) + reflection J:JKJ=-K,JPJ=P_- (2) " "is a Longo STANDARD PAIR / +half-sided modular inclusion (Wiesbrock), " "so by Borchers' theorem the seam modular flow is GEOMETRIC INTRINSICALLY " "-- without presupposing the rotation-covariance of the vacuum " "(complementary to Adamo-Giorgetti-Tanimoto, arXiv:2508.07109); face (ii) " "of SEAM.EQUIV.01 becomes a STRUCTURE consequence, not a state condition", standard_pair) # ---- 5. verdict (typed [C]/[O]) ---- check("VERDICT [C]/[O]: the half-sided modular inclusion route derives geometric " "modular flow from (P>=0, [K,P]=P, J), an intrinsic-BW REDUCTION via " "Borchers/Wiesbrock. SEAM.EQUIV.01 stays [O]: assembling the finite " "algebra and the positive-energy rep into ONE continuum standard pair is " "the infinite-dim continuum existence (the same wall as MMST/NPW26) -- " "this is a finite witness + cited theorem, not a closure", standard_pair and g_car == 5) return summary("v438 SEAM.EQUIV.BW.HSMI (intrinsic BW via Borchers/Wiesbrock " "half-sided modular inclusion; SEAM.EQUIV.01 stays [O])") if __name__ == "__main__": raise SystemExit(1 if run() else 0)