"""v524 -- WOIT.BETA2.OS.01: the WOIT-beta2 milestone of the OS twistor bridge WOIT.OS.TWISTOR.01 executed -- "the OS quotient of the free system made explicit" (verdict SUCCESS per the frozen preregistration kept verbatim below, [C]-typed per contract precision (iii): "the clock acting unitarily" is operationalised as the reconstructed transfer/rotation structure on H_phys; NO marker moves -- WOIT.OS.TWISTOR.01 stays [O], all seven contract kill tests stay formally live on the interacting algebra A_hol). [E] 1. H_phys EXPLICIT: N = 16 (deg <= 2): the 37x37 bond-cut OS Gram is exactly Hermitian, parity-block-diagonal, inertia (37, 0, 0) PD (min eigenvalue 1.7801e-6 at 40 digits), null space {0}, dim 37 = 29 (+) 8; N = 8 (complete half algebra): (16, 0, 0) PD (min 3.3471e-3), dim 16 = 8 (+) 8 = 4^2 -- compact euclidean time reconstructs a THERMAL (KMS) representation; exact certificate sin^2(3pi/8) - sin(pi/8) sin(5pi/8) = 1/2. [E] 2. TRANSFER SEMIGROUP (Klein-Landau): tau_k exactly Hermitian on all domains (k = 1..7, dims 29, 22, 16, 11, 7, 4, 2), chain identities exact, vacuum fixed; the positivity pattern IS the site/bond dichotomy: even k PSD via the exact square identity T(2j) = A_j* A_j ((22,0,0)/(11,0,0)/(4,0,0)), odd k indefinite ((10,12,7)/(5,6,5)/(2,2,3)) -- the one-step transfer is NOT positive (the free chirality datum, kill-test-3 shadow sharpened), its even powers ARE. [E] 3. THE CLOCK = T^{N/4}: N = 16: tau_4 PD (11, 0, 0) (min 6.6821e-5), compressed spectrum [0.0194517, 1.64144], vacuum eigenvalue exactly 1; N = 8: spectrum EXACTLY {1, sqrt(2) - 1} = {1, 1/delta_S} in both parity sectors -- the silver axes of the v519 mu4 torsor return as the clock-transfer eigenvalue; spectral projections ~1e-40; the reconstructed rotation group U(s) = exp(isH) is unitary with the group law. [E] 4. v522 RESOLUTION: the census (745, 96, 0) reproduced; EVERY anti-matching entry is a wrap overlap, every overlap-free entry matches -- the pre-quotient non-Hermiticity is EXACTLY the domain/wrap artifact; on the quotient the clock has spectral calculus. Pre-registered KMS deviation: NO contraction on the compact circle (edge modes C(1)/C(3) = 1 + sqrt(2) = delta_S exactly, det(G - tau_4) < 0). [E] 5. GSO/THETA: the grading survives; the perpendicular torsor mirror descends anti-unitarily (eta_perp in {+-i}), Theta_phys^2 = +1 on every sector (Kramers-free), theta_cut o theta_perp = alpha_{N/2} exactly. [E] 6. CONTROLS: site cut indefinite ((3,3,1)/(2,2,4)) -- the contract kill branch fires there; family A (4,4,0); anti-chiral (8,8,0) no quotient; wrong direction non-Hermitian (mismatches all wrap-overlap). KILL-TEST BOOKKEEPING: (1), (2) strengthened, (3) shadow sharpened, none fires; (4)-(7) untouched; all seven stay live on A_hol. [C] FENCE: "the clock acting unitarily" is operationalised per contract precision (iii) as the reconstructed transfer/rotation structure (compressed clock PD + U(s) = exp(isH) unitary with the group law) -- a [C]-refinement, prose only. [O] beta3 is the next milestone; WOIT.OS.TWISTOR.01 stays [O]; the interacting algebra A_hol untouched. Status: [E] exact sympy trig-algebraic entries, Pfaffian-Wick, exact signed-permutation towers; the definiteness certificates are 40-digit spectra of exact Hermitian matrices (threshold 1e-25). Python; Wolfram-mirrored (the alpha_1^N = (-1)^F wrap arithmetic + the certificate identity sin^2(3pi/8) - sin(pi/8) sin(5pi/8) = 1/2; the one-particle Klein-Landau transfer forms: exact Hermiticity, odd-k zero diagonal, even-k square identity; the exact N = 8 clock spectrum {1, sqrt(2) - 1} + the transparency pi/16 product formula; the KMS silver witnesses C(1)/C(3) = 1 + sqrt(2) and det(G - tau_4) < 0 -- the 37x37/16x16 Grams, 40-digit inertia spectra, spectral projections and the reconstructed rotation group are Python-only), counted per GATE.WOLFRAM.02. Discovery provenance: experiments/tfpt-discovery/woit_os_beta2_os_quotient_probe.py (20/20, 2026-07-23; run 1 scored 18/20 -- a sympy simplify failure on a TRUE pi/16 product formula, fixed by a Laurent-polynomial certificate, no criterion changed -- see TRANSPARENCY below). === PREREGISTRATION (frozen BEFORE the run; criteria not adjusted after) === Contract basis (tfpt_research_contracts.tex, WOIT.OS.TWISTOR.01): beta_2 milestone, verbatim: "The OS quotient of the *free* system made explicit --- success: (H, Omega) constructed from the v519 bond-cut form with the clock acting unitarily; kill: the quotient degenerates (zero or indefinite completion)." Precision (iii) (from v522, verbatim): "the clock is the euclidean rotation itself (timelike: all 16 seam mirror axes invert it, none commutes), and its equivariant statement moves to beta_2 --- OS quotient first, then the clock as reconstructed transfer operator." [C] operationalisations (declared; contract wording prevails where it exists, the attack-shape criteria (a)-(d) are [C]-refinements): [C-1] free system = the v519-pinned 16-Majorana chiral NS seam circle; both v519 levels: N = 16 at deg <= 2 (37 = 1 + 8 + 28 monomials on the half) AND the complete 2^4 = 16-monomial half algebra at N = 8. OS reflection = the v519 bond cut (axis k = 15 resp. k = 7, NS spin signs, twist eta = +i as forced by v519 R3/R5.1). [C-2] H_phys := A_+ / N with N = {a : = 0} = the kernel of the PSD OS Gram; "explicit" = basis + Gram + dimension. Nondegeneracy certificates: exact Hermiticity + 40-digit inertia (mp.mp.dps = 40, zero threshold 1e-25) with the minimum eigenvalue printed; exact 2x2 sub-determinants where cheap. KILL branch (contract verbatim): a negative direction (indefinite completion) or rank <= 1 (zero completion beyond the vacuum). [C-3] transfer = the Klein-Landau LOCAL symmetric semigroup of the euclidean one-step rotation alpha_1 (2 pi / N): T(k)[b] := [alpha_k b] on the domain D_k := monomials supported in {cut+1, ..., far-cut - k} (the shifted support must stay in the half). Success-checks: (a) tau_k(a,b) = omega(theta(a) alpha_k(b)) is EXACTLY Hermitian on D_k x D_k for all 1 <= k <= N/2 - 1 (self-adjointness via Theta on the good domain); (b) semigroup consistency <[alpha_j a],[alpha_k b]> = tau_{j+k}(a,b) exactly, and the vacuum is fixed exactly (tau_k(1,.) = Gram(1,.)); (c) positivity PATTERN (pre-derived, to be machine-confirmed): for EVEN k the exact A*A identity tau_k(a,b) = Gram(alpha_{k/2} a, alpha_{k/2} b) forces tau_k >= 0; for ODD k the effective reflection axis 15 - k is a SITE axis and the chiral checkerboard C(even) = 0 forces an exactly ZERO one-particle diagonal -> indefinite (or degenerate on too-small domains): the one-step transfer is NOT positive -- this is the v519 site/bond dichotomy resurfacing INSIDE the semigroup and is typed as the free chirality datum (kill-test-3 shadow), NOT as a beta_2 failure; (d) CONTRACTION (0 < T <= 1): PRE-DECLARED EXPECTED DEVIATION -- on the compact euclidean circle the reconstruction is THERMAL (KMS; H_phys at N = 8 has dim 16 = 4^2, a doubled/two-sided space) and the local transfer need NOT be a contraction: exact witnesses derived pre-run: the norm drift ||[g_7]||^2 / ||[g_5]||^2 = C(1)/C(3) = 1 + sqrt(2) = the SILVER RATIO delta_S (exact) at N = 8, and at N = 16 the odd one-particle clock block has det(G - tau_4) < 0 exactly, so the compressed clock has an eigenvalue > 1 (predicted ~1.64). Contraction failure with EXACTLY these witnesses is a typed KMS finding and does NOT block success (the contract demands no contraction); any OTHER failure mode -> UNDECIDED. [C-4] the clock: quarter turn = alpha_{N/4} = T(N/4), an EVEN step -> positive by (c). Success: tau_{N/4} on D_{N/4} exactly Hermitian AND positive definite (40-digit inertia; at the one-particle level EXACT PD certificates via 2x2 trace/det with the exact identities C(9) = C(7), C(11) = C(5) at N = 16); the exact form identity tau_{N/4}(a,b) = Gram(alpha_{N/8} a, alpha_{N/8} b) ("quarter turn = T^{N/4}, with square root A_{N/8}"); spectral projections of the compressed clock well-defined (eigendecomposition residuals < 1e-30 at 40 digits) -- EXACTLY what failed pre-quotient in v522 (the non-Hermitian clock average has no spectral calculus); the reconstructed rotation one-parameter group U(s) := exp(i s H_c), H_c := -log(compressed clock) (PD required), unitary with the semigroup law (residuals < 1e-30) -- per precision (iii) this IS the [C]-operationalisation of the contract phrase "the clock acting unitarily". PRE-DERIVED EXACT ANCHOR (to be machine-confirmed): at N = 8 the compressed clock spectrum is EXACTLY {1, sqrt(2) - 1} = {1, 1/delta_S} (each twice: even + odd sector) -- the silver ratio of the v519 silver-midpoint axes; at N = 16 the known free trig kernel gives the spectrum (40-digit certificates, vacuum eigenvalue exactly 1). [C-5] GSO / Theta on H_phys: (i) fermion parity descends to a G-unitary P with P^2 = 1 commuting with all transfer data (grading survives: H_phys = H_even (+) H_odd exactly orthogonal); (ii) the OS Theta itself maps A_+ -> A_- (support census) -- it does NOT act on H_phys (it IS the metric); the anti-unitary that DOES descend is the PERPENDICULAR torsor mirror theta_perp (axis k_cut - N/2, the other clock orbit of the v519 mu4 torsor W-S2.5): Theta_phys[a] := [theta_perp(a)], anti-unitarity = checked exactly entrywise, eta_perp pinned inside mu4 by the census; Theta_phys^2 measured exactly per parity sector (Kramers question IN the quotient); theta_perp alpha_k = alpha_{-k} theta_perp exactly (Theta_phys time-reverses the semigroup); theta_cut o theta_perp = the deck/antipode lift exactly (the torsor closes onto the deck). Success (d): anti-unitary descent exists, GSO grading survives, Theta_phys^2 = +1 at least on H_even. [C-6] negative controls (all mandatory): (i) SITE cut -> the OS form is indefinite (contract kill branch fires for the site placement; negative directions counted at N = 16 one-particle (3,3,1) regression and N = 8 full site-half algebra); (ii) FAMILY A (clock-centralising antipode, alternating dressing) -> Hermitian only for a sub-torsor eta and INDEFINITE ((4,4,0) one-particle regression): no PSD form, no quotient; (iii) DECK candidate Theta_t = Utilde o K (Kramers class (-1)^F): support census shows it maps A_+ to A_- -> it does not act on H_phys at all; the descended real structure is the torsor partner with Theta_phys^2 measured (the Kramers dichotomy resolved in the quotient); (iv) WRONG semigroup direction (alpha_{-1} on the UNSHRUNK domain): Hermiticity must FAIL with wrap-overlap witnesses (the v522 mechanism localised: no-overlap entries must all match, anti-matching entries must all have overlap), while the mirrored domain is Hermitian with the mirror inertia; (v) ANTI-CHIRAL state (chi = -1): the odd block flips to negative definite -> the FULL half-algebra Gram is indefinite: the OS quotient EXISTS ONLY for the chirality matching the twist (the kill-test-3 shadow at quotient level, quantified). [C-7] verdict logic (frozen): KILL iff (K1) a bond-cut Gram at either level is indefinite or rank <= 1 [contract kill], or (K2) the clock step on its domain is non-Hermitian or has a negative direction [clock not implementable on H_phys]. SUCCESS iff no kill AND all of: S1 = H_phys explicit + nondegenerate (both levels); S2 = local transfer exactly Hermitian on all domains + semigroup identities + vacuum fixed; S3 = clock PD + spectral projections + T^{N/4} square identity + reconstructed unitary rotation group; S4 = GSO grading + anti-unitary Theta descent; S5 = all five controls behave as typed; the [C-3](d) contraction deviation counts as SUCCESS-compatible ONLY with exactly the pre-declared witnesses. Otherwise UNDECIDED with the gap named. Runtime target < 20 min. === FINDINGS (from the frozen-prereg run, 2026-07-23; 20/20) === VERDICT: SUCCESS (beta2, [C]-typed per contract precision (iii)); the contract kill branch does NOT fire. Headline numbers: * H_phys explicit and NONDEGENERATE: null space N = {0} at both levels -- N = 16 deg <= 2: 37 = 29 (+) 8, Gram PD (min eigenvalue 1.7801e-6 at 40 digits); N = 8 complete half algebra: 16 = 8 (+) 8 = 4^2, PD (min 3.3471e-3), with exact 2x2 certificates via sin^2(3pi/8) - sin(pi/8) sin(5pi/8) = 1/2. * Local transfer (Klein-Landau): tau_k exactly Hermitian on every domain D_k (k = 1..7 at N = 16, 1..3 at N = 8); semigroup chain identities exact; vacuum fixed exactly. Positivity pattern exactly the site/bond dichotomy: even k PSD via the square identity T(2j) = A_j* A_j ((22,0,0)/(11,0,0)/(4,0,0)); odd k indefinite/ degenerate with exactly zero one-particle diagonal ((10,12,7)/ (5,6,5)/(2,2,3)/(1,0,1) at N = 16) -- the one-step transfer is NOT positive (staggering/chirality datum), its even powers are. * THE CLOCK: quarter turn = T^{N/4} positive self-adjoint on its domain -- N = 16: (11,0,0), min 6.6821e-5, one-particle blocks PD EXACTLY; N = 8: (4,0,0) and the compressed clock spectrum EXACTLY {1, sqrt(2)-1} = {1, 1/delta_S} in both parity sectors (SILVER ratio); compressed spectrum at N = 16: min 0.0194517, max 1.64144 (vacuum eigenvalue exactly 1); spectral projections certified (residuals ~1e-40); reconstructed rotation group U(s) = exp(isH) unitary with group law (~1e-40). * v522 RESOLUTION: the clock-insertion census (745 match, 96 anti, 0 violations) reproduced; EVERY anti-matching entry is a wrap-overlap entry and every overlap-free entry matches; the same insertion restricted to the quotient domain IS tau_4 entrywise. What dissolves in beta2: the ill-posedness of clock positivity (the clock now has a spectral calculus). What remains time-like: the clock is a transfer step, not a symmetry of H_phys; its unitary avatar is the reconstructed rotation group. * KMS DEVIATION (pre-declared, confirmed with exactly the declared witnesses): NO contraction on the compact circle -- the local clock step expands the edge mode by exactly C(1)/C(3) = 1 + sqrt(2) = delta_S (N = 8), det(G - tau_4) < 0 on the odd one-particle block (N = 16, compressed lambda_max = 1.6414 > 1). * GSO/THETA: parity descends G-unitarily, H_even (+) H_odd exact; theta_cut and the deck candidate do NOT act on H_phys (support censuses); the PERPENDICULAR torsor mirror descends anti-unitarily (eta_perp in {+i, -i}), Theta_phys^2 = +1 on EVERY sector (the quotient is KRAMERS-FREE -- the (-1)^F class does not reach H_phys), time-reverses the semigroup, and theta_cut o theta_perp = alpha_{N/2} exactly (the torsor closes onto the deck). * CONTROLS: site cut indefinite ((3,3,1) / (2,2,4) with 2-3 negative directions -- kill branch fires for the site placement); family A indefinite (4,4,0); anti-chiral state (8,8,0) -- no quotient for the wrong chirality; wrong direction non-Hermitian off-domain (census (58,6,0), all mismatches wrap-overlap), mirrored domain Hermitian with mirror inertia. * KILL TESTS: (1), (2) strengthened, (3) shadow sharpened; none fires; (4)-(7) untouched; all seven stay live on A_hol. TRANSPARENCY (tooling only; preregistration text unchanged): run 1 scored 18/20 -- the single substantive miss was a sympy simplify failure on the exact product formula sin(7pi/16) - sin(5pi/16) = 2 cos(3pi/8) sin(pi/16) (true identity; simplify() does not close pi/16 arguments), which cascaded into the verdict check. Fixed by certifying it as a Laurent-polynomial identity in z = e^{i pi/16} (exact term-by-term cancellation, no trig simplification) plus a 40-digit numeric zero; an intermediate exp-rewrite attempt failed because sympy auto-evaluates cos(3pi/8) to radicals. No criterion, tolerance or expectation was changed. Construction base (READ-ONLY): verification/v519_woit_theta_rp_free.py (kernel C(d), Pfaffian-Wick, dihedral machinery, theta_mono, Gram, inertia certificates -- copied verbatim where possible), verification/v522_woit_beta1_gso_gauge.py (tower maps, os_term, Hermiticity census), experiments/tfpt-discovery/ woit_os_beta1_gauge_invariant_theta_probe.py. Exact throughout (sympy trig-algebraic entries, Pfaffian-Wick, exact signed-permutation towers); definiteness certificates are 40-digit spectra of exact Hermitian matrices (mp.mp.dps = 40, threshold 1e-25). """ import time from itertools import combinations import mpmath as mp import sympy as sp from tfpt_constants import check as _check, summary, reset mp.mp.dps = 40 I = sp.I N16 = 16 N8 = 8 ETA = I RESULTS = [] FLAGS = {} T0_WALL = time.perf_counter() def check(name, ok): RESULTS.append(bool(ok)) _check("[%2d] %s" % (len(RESULTS), name), bool(ok)) def iszero(e): e2 = sp.expand(e) if e2 == 0: return True e3 = sp.expand(sp.expand_complex(e2)) if e3 == 0: return True return sp.simplify(e3) == 0 # --------------------------------------------------------------------------- # chiral NS vacuum kernel + Pfaffian-Wick (v519/v522 verbatim, memoised) # --------------------------------------------------------------------------- def c_of(d, n): if d % 2 == 0: return sp.Integer(0) return sp.Rational(2, n) / sp.sin(sp.pi * sp.Rational(d, n)) def g2f(a, b, n, chi): if a == b: return sp.Integer(1) return I * chi * c_of(a - b, n) _WICK = {} def wick(idx, n, chi=1): idx = tuple(idx) if len(idx) == 0: return sp.Integer(1) if len(idx) % 2 == 1: return sp.Integer(0) key = (idx, n, chi) if key in _WICK: return _WICK[key] head, rest = idx[0], idx[1:] tot = sp.Integer(0) for j, b in enumerate(rest): sub = rest[:j] + rest[j + 1:] tot += (-1) ** j * g2f(head, b, n, chi) * wick(sub, n, chi) tot = sp.expand_complex(tot) _WICK[key] = tot return tot # --------------------------------------------------------------------------- # dihedral / tower machinery (v510/v519/v522 verbatim, tower generalised) # --------------------------------------------------------------------------- def shift_matrix(n, k, wrap_sign): M = [[0] * n for _ in range(n)] for a in range(n): b = (a + k) % n M[b][a] = wrap_sign if a + k >= n else 1 return M def refl_matrix(n, k, wrap_sign): M = [[0] * n for _ in range(n)] for a in range(n): idx = (k - a) % (2 * n) M[idx % n][a] = wrap_sign if idx >= n else 1 return M def refl_map(k, n): def r(a): return (k - a) % n def s(a): return -1 if (k - a) % (2 * n) >= n else 1 return r, s def half_of(k, n): if k % 2 == 0: f1 = (k // 2) % n P = [(f1 + j) % n for j in range(1, n // 2)] else: b = (k + 1) // 2 P = [(b + j) % n for j in range(n // 2)] rP = {(k - a) % n for a in P} assert not (rP & set(P)) return P PLUS = lambda a: 1 def tower_maps(n, shift, kmax): """(perm, sign) for alpha^k, k = 0..kmax; NS wrap sign -1 per crossing (v522 verbatim, generalised to arbitrary step count and shift sign).""" maps = [(tuple(range(n)), (1,) * n)] for _ in range(kmax): perm, sign = maps[-1] np_, ns_ = [], [] for a in range(n): p, s0 = perm[a], sign[a] q = p + shift np_.append(q % n) ns_.append(s0 * (-1 if (q >= n or q < 0) else 1)) maps.append((tuple(np_), tuple(ns_))) return maps def alpha_mono(m, pm): """apply the Bogoliubov automorphism (perm, sign) to a sorted monomial; returns (int coeff, sorted tuple). (v522 verbatim)""" perm, sign = pm c = 1 imgs = [] for a in m: c *= sign[a] imgs.append(perm[a]) lst = list(imgs) sgn = 1 for i in range(len(lst)): for j in range(len(lst) - 1 - i): if lst[j] > lst[j + 1]: lst[j], lst[j + 1] = lst[j + 1], lst[j] sgn = -sgn assert len(set(lst)) == len(lst) return c * sgn, tuple(lst) def theta_mono(mono, r, s, eta): """theta(g_{i1}...g_{ik}) = eta^k s_{ik}..s_{i1} g_{r(ik)}..g_{r(i1)}, sorted back; returns (sympy coeff, tuple). (v519 verbatim)""" imgs = [r(a) for a in reversed(mono)] coeff = eta ** len(mono) for a in mono: coeff *= s(a) lst = list(imgs) sign = 1 for i in range(len(lst)): for j in range(len(lst) - 1 - i): if lst[j] > lst[j + 1]: lst[j], lst[j + 1] = lst[j + 1], lst[j] sign = -sign assert len(set(lst)) == len(lst) return coeff * sign, tuple(lst) def mono_mul(m1, m2): """exact Clifford product of sorted monomials (g_a^2 = 1); handles overlapping supports. (v506/v519 verbatim)""" out = list(m1) sign = 1 for g in m2: out.append(g) i = len(out) - 1 while i > 0 and out[i - 1] > out[i]: out[i - 1], out[i] = out[i], out[i - 1] sign = -sign i -= 1 if i > 0 and out[i - 1] == out[i]: del out[i - 1:i + 1] return sign, tuple(out) def os_term(ea, eb, pm, r, s, eta, n, chi=1): """omega(theta(e_a) alpha^k(e_b)), exact. (v522 verbatim)""" ca, ma = theta_mono(ea, r, s, eta) cb, mb = alpha_mono(eb, pm) cs, mm = mono_mul(ma, mb) return sp.expand_complex(ca * cb * cs * wick(mm, n, chi)) def os_lr(ea, eb, pmj, pmk, r, s, eta, n, chi=1): """omega(theta(alpha^j(e_a)) alpha^k(e_b)), exact (theta anti-linear; the alpha coefficients are +-1, so no conjugation subtlety).""" cj, mj = alpha_mono(ea, pmj) ct, mt = theta_mono(mj, r, s, eta) ck, mk = alpha_mono(eb, pmk) cs, mm = mono_mul(mt, mk) return sp.expand_complex(cj * ct * ck * cs * wick(mm, n, chi)) def term_matrix(basis, pm, r, s, eta, n, chi=1): d = len(basis) return sp.Matrix(d, d, lambda i, j: os_term(basis[i], basis[j], pm, r, s, eta, n, chi)) def hermitian_exact(M): d = M - M.conjugate().T return all(iszero(x) for x in d) def spectrum_inertia(M, tol=None): """inertia (n+, n-, n0) of the exact Hermitian matrix via 40-digit spectrum; returns (inertia, min|nonzero|). (v519 verbatim)""" if tol is None: tol = mp.mpf(10) ** (-25) n = M.shape[0] A = mp.matrix(n, n) for i in range(n): for j in range(n): re_, im_ = M[i, j].evalf(45).as_real_imag() A[i, j] = mp.mpc(mp.mpf(str(re_)), mp.mpf(str(im_))) E, _ = mp.eighe(A) evs = [E[i].real for i in range(n)] npos = sum(1 for e in evs if e > tol) nneg = sum(1 for e in evs if e < -tol) nzero = n - npos - nneg nz = [abs(e) for e in evs if abs(e) > tol] return (npos, nneg, nzero), (min(nz) if nz else mp.mpf(0)) def to_mp(M): n, m = M.shape A = mp.matrix(n, m) for i in range(n): for j in range(m): re_, im_ = M[i, j].evalf(45).as_real_imag() A[i, j] = mp.mpc(mp.mpf(str(re_)), mp.mpf(str(im_))) return A def ctr(A): """conjugate transpose of an mp.matrix.""" B = mp.matrix(A.cols, A.rows) for i in range(A.rows): for j in range(A.cols): B[j, i] = mp.conj(A[i, j]) return B def eig_frame(Gm, Tm): """spectrum of the G-compressed operator G^{-1/2} T G^{-1/2} (exact inputs, 40-digit output); returns (sorted eigenvalues, B, QB).""" n = Gm.shape[0] G = to_mp(Gm) T = to_mp(Tm) EG, QG = mp.eighe(G) Gis = QG * mp.diag([1 / mp.sqrt(EG[i]) for i in range(n)]) * ctr(QG) B = Gis * T * Gis B = (B + ctr(B)) * mp.mpf('0.5') EB, QB = mp.eighe(B) return sorted([EB[i].real for i in range(n)]), B, QB def dom_monos(basis, lo, hi): return [m for m in basis if all(lo <= a <= hi for a in m)] # --------------------------------------------------------------------------- # bases and pinned data # --------------------------------------------------------------------------- P16 = half_of(15, N16) # {8..15} P8 = half_of(7, N8) # {4,5,6,7} R15, S15 = refl_map(15, N16) R7, S7 = refl_map(7, N8) BASIS16 = [()] + [(a,) for a in P16] + list(combinations(P16, 2)) # 37 EVENS8 = [()] + list(combinations(P8, 2)) + [tuple(P8)] # 8 ODDS8 = [(a,) for a in P8] + list(combinations(P8, 3)) # 8 BASIS8 = EVENS8 + ODDS8 # 16 TW16 = tower_maps(N16, 1, N16) TW8 = tower_maps(N8, 1, N8) TW16N = tower_maps(N16, -1, N16) TW8N = tower_maps(N8, -1, N8) C1_8, C3_8 = c_of(1, 8), c_of(3, 8) C_16 = {d: c_of(d, 16) for d in range(1, 16, 2)} def gram_of(basis, r, s, n, chi=1): tw0 = (tuple(range(n)), (1,) * n) return term_matrix(basis, tw0, r, s, ETA, n, chi) def block_inertia(M, idx): return spectrum_inertia(M.extract(idx, idx)) # --------------------------------------------------------------------------- # Part Q -- the OS quotient (H_phys, Omega) made explicit # --------------------------------------------------------------------------- def q1_preconditions(): print(" -- Q1.1: one-step euclidean rotation -- exact preconditions") ok_full16 = (TW16[16][0] == tuple(range(N16)) and set(TW16[16][1]) == {-1}) ok_full8 = (TW8[8][0] == tuple(range(N8)) and set(TW8[8][1]) == {-1}) S1_16 = sp.Matrix(shift_matrix(N16, 1, -1)) Cm16 = sp.Matrix(N16, N16, lambda a, b: c_of(a - b, N16)) inv16 = sp.simplify(S1_16 * Cm16 * S1_16.T - Cm16) == sp.zeros(N16, N16) S1_8 = sp.Matrix(shift_matrix(N8, 1, -1)) Cm8 = sp.Matrix(N8, N8, lambda a, b: c_of(a - b, N8)) inv8 = sp.simplify(S1_8 * Cm8 * S1_8.T - Cm8) == sp.zeros(N8, N8) pure8 = sp.simplify(Cm8 * Cm8 + sp.eye(N8)) == sp.zeros(N8, N8) Rm = sp.Matrix(refl_matrix(N16, 15, -1)) dihed = sp.simplify(Rm * S1_16 * Rm.inv() - S1_16.inv()) == sp.zeros(N16, N16) # full state anti-invariance census at N = 8: omega(theta(m)) = # conj(omega(m)) for ALL 256 Cl(8) monomials (v519 R1.3 was 2-pt + # one 4-pt witness; here the complete census) anti_all = True for d in range(N8 + 1): for m in combinations(range(N8), d): ct_, mt_ = theta_mono(m, R7, S7, ETA) lhs = sp.expand_complex(ct_ * wick(mt_, N8)) rhs = sp.expand_complex(sp.conjugate(wick(m, N8))) if not iszero(lhs - rhs): anti_all = False break if not anti_all: break FLAGS['q11'] = (ok_full16 and ok_full8 and inv16 and inv8 and pure8 and dihed and anti_all) check("Q1.1 PRECONDITIONS EXACT: the one-step NS rotation alpha_1 " "(2 pi/N) preserves the chiral vacuum (S1 C S1^T = C at N = 16: " "%s, N = 8: %s; C^2 = -1: %s), theta inverts it (R S1 R^-1 = " "S1^-1: %s), the FULL turn is fermion parity (alpha_1^N = " "(-1)^F as signed permutation, N = 16: %s, N = 8: %s -- lattice " "spin-statistics: the 2 pi seam rotation is the GSO element), " "and omega o theta = conj o omega on ALL 256 Cl(8) monomials " "(%s; full census, beyond the v519 witness)" % (inv16, inv8, pure8, dihed, ok_full16, ok_full8, anti_all), FLAGS['q11']) def q2_gram16(): print(" -- Q1.2: N = 16 deg <= 2 -- Gram, null space, H_phys") G16 = gram_of(BASIS16, R15, S15, N16) FLAGS['G16'] = G16 ev_idx = [i for i, m in enumerate(BASIS16) if len(m) % 2 == 0] od_idx = [i for i, m in enumerate(BASIS16) if len(m) % 2 == 1] blocks0 = all(iszero(G16[i, j]) for i in ev_idx for j in od_idx) herm = hermitian_exact(G16) in_all, gap_all = spectrum_inertia(G16) in_odd, gap_odd = block_inertia(G16, od_idx) in_ev, gap_ev = block_inertia(G16, ev_idx) reg_odd = in_odd == (8, 0, 0) and mp.mpf('1.7e-3') < gap_odd \ < mp.mpf('2.0e-3') reg_ev = in_ev == (29, 0, 0) and mp.mpf('1.5e-6') < gap_ev \ < mp.mpf('2.1e-6') FLAGS['q12'] = (blocks0 and herm and in_all == (37, 0, 0) and reg_odd and reg_ev) FLAGS['q_pd16'] = in_all == (37, 0, 0) check("Q1.2 H_phys EXPLICIT AT N = 16 (deg <= 2) [exact entries, " "40-digit inertia]: the 37x37 bond-cut OS Gram (1 + 8 + 28 " "monomials on the half {8..15}) is Hermitian exactly (%s), " "parity-block-diagonal exactly (%s), inertia %s = POSITIVE " "DEFINITE (min eigenvalue %s) -- NULL SPACE N = {0}: " "H_phys = A_+/N has dimension 37 = 29 (+) 8 at this level, " "with the v519/v522 sub-block regressions (one-particle %s " "min %s ~ 1.888e-3, even 29-basis %s min %s ~ 1.78e-6)" % (herm, blocks0, in_all, mp.nstr(gap_all, 5), in_odd, mp.nstr(gap_odd, 5), in_ev, mp.nstr(gap_ev, 5)), FLAGS['q12']) def q3_gram8(): print(" -- Q1.3: N = 8 complete half algebra -- Gram, null space") G8 = gram_of(BASIS8, R7, S7, N8) FLAGS['G8'] = G8 blocks0 = all(iszero(G8[i, j]) for i in range(8) for j in range(8, 16)) herm = hermitian_exact(G8) in_all, gap_all = spectrum_inertia(G8) # exact garnish: the one-particle block is checkerboard {4,6}(+){5,7}; # 2x2 determinants exactly = (1/32)/(s1 s5 s3^2)-type via # sin^2(3pi/8) - sin(pi/8) sin(5pi/8) = 1/2 EXACTLY s1, s3, s5 = (sp.sin(sp.pi / 8), sp.sin(3 * sp.pi / 8), sp.sin(5 * sp.pi / 8)) half_id = iszero(s3 ** 2 - s1 * s5 - sp.Rational(1, 2)) det_a = sp.simplify(c_of(1, 8) * c_of(5, 8) - c_of(3, 8) ** 2) det_ok = iszero(det_a * 32 * s1 * s5 * s3 ** 2 - 1) FLAGS['q13'] = (blocks0 and herm and in_all == (16, 0, 0) and half_id and det_ok) FLAGS['q_pd8'] = in_all == (16, 0, 0) check("Q1.3 H_phys EXPLICIT AT N = 8 (COMPLETE half algebra) [exact " "entries, 40-digit inertia + exact 2x2 dets]: the 16x16 Gram " "(all 2^4 monomials on {4,5,6,7}) is Hermitian (%s), parity-" "block-diagonal (%s), inertia %s = PD (min eigenvalue %s) -- " "NULL SPACE N = {0} with NO degree truncation: dim H_phys = " "16 = 8 (+) 8 = 4^2 (the DOUBLED/two-sided dimension: compact " "euclidean time reconstructs a THERMAL representation, the " "KMS typing of [C-3](d)); exact certificates: sin^2(3pi/8) - " "sin(pi/8) sin(5pi/8) = 1/2 (%s) makes the one-particle " "checkerboard dets exactly positive (%s)" % (herm, blocks0, in_all, mp.nstr(gap_all, 5), half_id, det_ok), FLAGS['q13']) def q4_kill_branch(): print(" -- Q1.4: contract kill branch") fired = not (FLAGS['q_pd16'] and FLAGS['q_pd8']) FLAGS['q14'] = not fired check("Q1.4 CONTRACT KILL BRANCH ('the quotient degenerates: zero or " "indefinite completion') does NOT fire [assembled]: both " "completions are positive definite with rank 37 resp. 16 >> 1 " "(Q1.2/Q1.3) -- no negative direction, no collapse to the " "vacuum line: the OS quotient of the free system EXISTS and " "is explicit (basis + Gram + dimension)", not fired) # --------------------------------------------------------------------------- # Part T -- the reconstructed transfer operator (Klein-Landau local # symmetric semigroup) and the clock # --------------------------------------------------------------------------- def t1_domain_hermiticity(): print(" -- T2.1: local-domain Hermiticity of the transfer forms") # disjoint-support anti-multiplicativity lemma + overlap failure seed lem_ok = True for dx in range(3): for x in combinations(range(4), dx): # negative half N=8 for dy in range(3): for y in combinations(P8, dy): # positive half cs, mm = mono_mul(x, y) ct1, mt1 = theta_mono(mm, R7, S7, ETA) lhs = (sp.expand_complex(cs * ct1), mt1) cy, my = theta_mono(y, R7, S7, ETA) cx, mx = theta_mono(x, R7, S7, ETA) cr, mr = mono_mul(my, mx) rhs = (sp.expand_complex(cy * cx * cr), mr) if lhs[1] != rhs[1] or not iszero(lhs[0] - rhs[0]): lem_ok = False seed = not iszero(ETA ** 2 - 1) # theta(g a g a)=1 vs theta(ga)^2=-1 taus16, herm16 = {}, {} for k in range(1, 8): D = dom_monos(BASIS16, 8, 15 - k) M = term_matrix(D, TW16[k], R15, S15, ETA, N16) taus16[k] = (D, M) herm16[k] = hermitian_exact(M) taus8, herm8 = {}, {} for k in range(1, 4): D = dom_monos(BASIS8, 4, 7 - k) M = term_matrix(D, TW8[k], R7, S7, ETA, N8) taus8[k] = (D, M) herm8[k] = hermitian_exact(M) FLAGS['taus16'], FLAGS['taus8'] = taus16, taus8 dims16 = {k: len(taus16[k][0]) for k in taus16} FLAGS['t21'] = (lem_ok and seed and all(herm16.values()) and all(herm8.values()) and dims16 == {1: 29, 2: 22, 3: 16, 4: 11, 5: 7, 6: 4, 7: 2}) check("T2.1 SELF-ADJOINTNESS VIA THETA ON THE KLEIN-LANDAU DOMAINS " "[exact]: theta is anti-multiplicative on DISJOINT supports " "(full census across the N = 8 cut, deg <= 2 x deg <= 2: %s) " "but NOT on overlapping ones (seed eta^2 = -1: theta(g_a g_a) " "= 1 vs theta(g_a)theta(g_a) = -1: %s -- the exact v522 " "mechanism); on the shrinking domains D_k (dims %s at N = 16) " "every insertion tau_k = omega(theta(a) alpha_k(b)) is " "Hermitian EXACTLY, k = 1..7 at N = 16 (%s) and k = 1..3 at " "N = 8 (%s): the local transfer T(k)[b] = [alpha_k b] is " "G-symmetric on its domain -- the OS-quotient cure of the " "v522 Hermiticity failure" % (lem_ok, seed, dims16, {k: herm16[k] for k in sorted(herm16)}, {k: herm8[k] for k in sorted(herm8)}), FLAGS['t21']) def t2_semigroup(): print(" -- T2.2: semigroup structure + vacuum fixed") ok_sg8 = True for (j, k) in [(1, 1), (1, 2), (2, 1), (2, 2)]: for ea in BASIS8: for eb in BASIS8: lhs = os_lr(ea, eb, TW8[j], TW8[k], R7, S7, ETA, N8) rhs = os_term(ea, eb, TW8[j + k], R7, S7, ETA, N8) if not iszero(lhs - rhs): ok_sg8 = False break if not ok_sg8: break if not ok_sg8: break sub16 = dom_monos(BASIS16, 8, 11) ok_sg16 = True for (j, k) in [(1, 1), (2, 2), (1, 3)]: for ea in sub16: for eb in sub16: lhs = os_lr(ea, eb, TW16[j], TW16[k], R15, S15, ETA, N16) rhs = os_term(ea, eb, TW16[j + k], R15, S15, ETA, N16) if not iszero(lhs - rhs): ok_sg16 = False break if not ok_sg16: break if not ok_sg16: break vac_ok = True G16, G8 = FLAGS['G16'], FLAGS['G8'] for k in range(1, 8): D, M = FLAGS['taus16'][k] i1 = D.index(()) for j, m in enumerate(D): gi = BASIS16.index(m) if not iszero(M[i1, j] - G16[0, gi]): vac_ok = False for k in range(1, 4): D, M = FLAGS['taus8'][k] i1 = D.index(()) for j, m in enumerate(D): gi = BASIS8.index(m) if not iszero(M[i1, j] - G8[0, gi]): vac_ok = False FLAGS['t22'] = ok_sg8 and ok_sg16 and vac_ok check("T2.2 TRANSLATION SEMIGROUP T^k [exact]: the chain identity " "<[alpha_j a], [alpha_k b]> = tau_{j+k}(a, b) holds EXACTLY -- " "N = 8: all 16x16 monomial pairs for (j,k) = (1,1), (1,2), " "(2,1), (2,2) (%s); N = 16: the 11-element sub-half {8..11} " "for (j,k) = (1,1), (2,2), (1,3) (%s) -- Klein-Landau local " "semigroup structure T(j)T(k) = T(j+k) on nested domains; the " "vacuum is FIXED exactly: tau_k(1, b) = for every " "k and b (%s; omega o alpha = omega), so T(k)Omega = Omega" % (ok_sg8, ok_sg16, vac_ok), FLAGS['t22']) def t3_positivity_pattern(): print(" -- T2.3: positivity pattern of the local semigroup") inert16 = {} for k in range(1, 8): D, M = FLAGS['taus16'][k] inert16[k], _ = spectrum_inertia(M) inert8 = {} for k in range(1, 4): D, M = FLAGS['taus8'][k] inert8[k], _ = spectrum_inertia(M) # exact mechanism: odd k -> one-particle diagonal identically zero diag0 = True for k in (1, 3, 5): D, M = FLAGS['taus16'][k] for i, m in enumerate(D): if len(m) == 1 and not iszero(M[i, i]): diag0 = False D1_8, M1_8 = FLAGS['taus8'][1] for i, m in enumerate(D1_8): if len(m) == 1 and not iszero(M1_8[i, i]): diag0 = False # exact A*A identity for even k: tau_k = Gram(alpha_{k/2}., alpha_{k/2}.) G16 = FLAGS['G16'] bidx16 = {m: i for i, m in enumerate(BASIS16)} aa_ok = True for k in (2, 4, 6): D, M = FLAGS['taus16'][k] for i, ma in enumerate(D): ca, ia = alpha_mono(ma, TW16[k // 2]) for j, mb in enumerate(D): cb, ib = alpha_mono(mb, TW16[k // 2]) if not iszero(M[i, j] - ca * cb * G16[bidx16[ia], bidx16[ib]]): aa_ok = False odd_indef = all(inert16[k][1] > 0 for k in (1, 3, 5)) \ and inert16[7] == (1, 0, 1) and inert8[1][1] > 0 \ and inert8[3] == (1, 0, 1) even_psd = all(inert16[k][1] == 0 for k in (2, 4, 6)) \ and inert8[2][1] == 0 FLAGS['inert16'], FLAGS['inert8'] = inert16, inert8 FLAGS['t23'] = diag0 and aa_ok and odd_indef and even_psd check("T2.3 THE SITE/BOND DICHOTOMY BECOMES THE SEMIGROUP POSITIVITY " "PATTERN [exact mechanism + 40-digit inertia]: EVEN steps obey " "the exact square identity tau_2j(a,b) = <[alpha_j a], " "[alpha_j b]> (T(2j) = A_j* A_j: %s) and are PSD (N = 16 " "inertias k = 2, 4, 6: %s, %s, %s; N = 8 k = 2: %s); ODD steps " "land on SITE axes (axis 15 - k even): the chiral checkerboard " "C(even) = 0 kills the one-particle diagonal EXACTLY (%s) and " "they are indefinite or degenerate (N = 16 k = 1, 3, 5, 7: " "%s, %s, %s, %s; N = 8 k = 1, 3: %s, %s) -- the one-step " "transfer is NOT positive (the free chirality/staggering " "datum, kill-test-3 shadow), its even powers ARE: exactly the " "v519 bond-vs-site RP dichotomy resurfacing INSIDE the " "reconstructed semigroup" % (aa_ok, inert16[2], inert16[4], inert16[6], inert8[2], diag0, inert16[1], inert16[3], inert16[5], inert16[7], inert8[1], inert8[3]), FLAGS['t23']) def t4_clock(): print(" -- T2.4: the mu4 clock as positive transfer step T^{N/4}") D4, M4 = FLAGS['taus16'][4] herm4 = hermitian_exact(M4) in4, gap4 = spectrum_inertia(M4) # exact one-particle PD certificates (2x2 trace/det with C9=C7, C11=C5) c5, c7, c9, c11 = C_16[5], C_16[7], C_16[9], C_16[11] ids = iszero(c9 - c7) and iszero(c11 - c5) det_blk = sp.simplify(c5 * c9 - c7 ** 2) fact_ok = iszero(det_blk - c7 * (c5 - c7)) s5, s7 = sp.sin(5 * sp.pi / 16), sp.sin(7 * sp.pi / 16) prod_expr = s7 - s5 - 2 * sp.cos(3 * sp.pi / 8) * sp.sin(sp.pi / 16) # exact certificate as a Laurent identity in z = e^{i pi/16}: # 2i [sin(7pi/16) - sin(5pi/16) - 2 cos(6pi/16) sin(pi/16)] # = (z^7 - z^-7) - (z^5 - z^-5) - (z^6 + z^-6)(z - z^-1) = 0 z = sp.symbols('z') laurent = sp.expand((z ** 7 - z ** -7) - (z ** 5 - z ** -5) - (z ** 6 + z ** -6) * (z - 1 / z)) == 0 re_p, im_p = prod_expr.evalf(40).as_real_imag() num0 = (abs(mp.mpf(str(re_p))) < mp.mpf(10) ** (-35) and abs(mp.mpf(str(im_p))) < mp.mpf(10) ** (-35)) prodform = laurent and num0 pos_factors = (sp.cos(3 * sp.pi / 8).is_positive and sp.sin(sp.pi / 16).is_positive and sp.sin(5 * sp.pi / 16).is_positive and sp.sin(7 * sp.pi / 16).is_positive) D2_8, M2_8 = FLAGS['taus8'][2] herm2_8 = hermitian_exact(M2_8) in2_8, gap2_8 = spectrum_inertia(M2_8) # exact N = 8 anchor: compressed clock spectrum {1, sqrt(2)-1} exactly idx_o = [i for i, m in enumerate(D2_8) if len(m) == 1] idx_e = [i for i, m in enumerate(D2_8) if len(m) % 2 == 0] G8 = FLAGS['G8'] bidx8 = {m: i for i, m in enumerate(BASIS8)} Go = sp.Matrix(2, 2, lambda i, j: G8[bidx8[D2_8[idx_o[i]]], bidx8[D2_8[idx_o[j]]]]) To = M2_8.extract(idx_o, idx_o) lam = sp.symbols('lam') roots_o = sp.solve(sp.det(To - lam * Go), lam) Ge = sp.Matrix(2, 2, lambda i, j: G8[bidx8[D2_8[idx_e[i]]], bidx8[D2_8[idx_e[j]]]]) Te = M2_8.extract(idx_e, idx_e) roots_e = sp.solve(sp.det(Te - lam * Ge), lam) silver = sp.sqrt(2) - 1 ok_o = (len(roots_o) == 2 and any(iszero(rt - 1) for rt in roots_o) and any(iszero(rt - silver) for rt in roots_o)) ok_e = (len(roots_e) == 2 and any(iszero(rt - 1) for rt in roots_e) and any(iszero(rt - silver) for rt in roots_e)) silver_id = iszero(sp.sin(sp.pi / 8) / sp.sin(3 * sp.pi / 8) - silver) FLAGS['t24'] = (herm4 and in4 == (11, 0, 0) and ids and fact_ok and prodform and bool(pos_factors) and herm2_8 and in2_8 == (4, 0, 0) and ok_o and ok_e and silver_id) FLAGS['clock_herm_pd'] = herm4 and in4[1] == 0 and herm2_8 \ and in2_8[1] == 0 check("T2.4 THE CLOCK IS IMPLEMENTED ON H_phys AS A POSITIVE " "SELF-ADJOINT TRANSFER STEP [exact + 40-digit]: the quarter " "turn alpha_{N/4} is an EVEN step -- at N = 16 tau_4 on the " "11-dim domain {8..11} is Hermitian (%s) and PD, inertia %s " "(min eigenvalue %s), with the one-particle blocks EXACTLY PD " "(C(9) = C(7), C(11) = C(5): %s; det = C7 (C5 - C7): %s; " "sin(7pi/16) - sin(5pi/16) = 2 cos(3pi/8) sin(pi/16) > 0: " "%s/%s); at N = 8 tau_2 on {4,5} is Hermitian (%s), PD %s " "(min %s), and the COMPRESSED clock spectrum is EXACTLY " "{1, sqrt(2) - 1} = {1, 1/delta_silver} in BOTH parity " "sectors (odd: %s, even: %s; sin(pi/8)/sin(3pi/8) = sqrt(2) " "- 1: %s) -- the SILVER-midpoint axes of the v519 mu4 torsor " "return as the exact clock-transfer eigenvalue" % (herm4, in4, mp.nstr(gap4, 5), ids, fact_ok, prodform, bool(pos_factors), herm2_8, in2_8, mp.nstr(gap2_8, 5), ok_o, ok_e, silver_id), FLAGS['t24']) def t5_spectral_rotation(): print(" -- T2.5: spectral projections + reconstructed rotation group") D4, M4 = FLAGS['taus16'][4] bidx16 = {m: i for i, m in enumerate(BASIS16)} idx = [bidx16[m] for m in D4] G4 = FLAGS['G16'].extract(idx, idx) spec, B, QB = eig_frame(G4, M4) n = len(spec) tol = mp.mpf(10) ** (-30) all_pos = all(e > mp.mpf(10) ** (-25) for e in spec) EB, QB2 = mp.eighe(B) res_eig = mp.mnorm(B - QB2 * mp.diag([EB[i] for i in range(n)]) * ctr(QB2), 1) res_orth = mp.mnorm(ctr(QB2) * QB2 - mp.eye(n), 1) lamH = [-mp.log(EB[i].real) for i in range(n)] def U(sv): return QB2 * mp.diag([mp.exp(1j * sv * h) for h in lamH]) \ * ctr(QB2) Ua, Ub, Uc = U(mp.mpf('0.3')), U(mp.mpf('0.4')), U(mp.mpf('0.7')) res_uni = mp.mnorm(ctr(Ua) * Ua - mp.eye(n), 1) res_grp = mp.mnorm(Ua * Ub - Uc, 1) specs = sorted([float(e) for e in spec]) D2_8, M2_8 = FLAGS['taus8'][2] bidx8 = {m: i for i, m in enumerate(BASIS8)} idx8 = [bidx8[m] for m in D2_8] G28 = FLAGS['G8'].extract(idx8, idx8) spec8, _, _ = eig_frame(G28, M2_8) FLAGS['spec_clock16'] = spec FLAGS['spec_clock8'] = spec8 ok = (all_pos and res_eig < tol and res_orth < tol and res_uni < tol and res_grp < tol) FLAGS['t25'] = ok check("T2.5 SPECTRAL PROJECTIONS + RECONSTRUCTED ROTATION GROUP " "[40-digit certificates]: the compressed clock G^{-1/2} tau_4 " "G^{-1/2} at N = 16 (11-dim) is strictly positive (%s; " "spectrum min %.6g, max %.6g, vacuum eigenvalue 1), its " "eigendecomposition is certified (residual %s, orthonormality " "%s < 1e-30) -- SPECTRAL PROJECTIONS ARE WELL-DEFINED on " "H_phys, exactly the calculus that the non-Hermitian " "pre-quotient clock average of v522 could not have; the " "reconstructed rotation one-parameter group U(s) = exp(is H), " "H = -log(compressed clock), is unitary with the group law " "(residuals %s, %s < 1e-30) -- per contract precision (iii) " "this is the [C]-operationalisation of 'the clock acting " "unitarily' (N = 8 compressed clock spectrum: %s)" % (all_pos, specs[0], specs[-1], mp.nstr(res_eig, 3), mp.nstr(res_orth, 3), mp.nstr(res_uni, 3), mp.nstr(res_grp, 3), [mp.nstr(e, 8) for e in FLAGS['spec_clock8']]), FLAGS['t25']) def t6_kms(): print(" -- T2.6: contraction fails with the pre-declared KMS " "witnesses") # N = 8 exact: ||T(2)[g_5]||^2 / ||[g_5]||^2 = C(1)/C(3) = 1 + sqrt(2) G8 = FLAGS['G8'] bidx8 = {m: i for i, m in enumerate(BASIS8)} n57 = G8[bidx8[(7,)], bidx8[(7,)]] / G8[bidx8[(5,)], bidx8[(5,)]] silver_exp = iszero(sp.simplify(n57) - (1 + sp.sqrt(2))) drift = iszero(sp.simplify(C1_8 / C3_8) - (1 + sp.sqrt(2))) # N = 16 exact: det(G - tau_4) < 0 on the odd one-particle clock block c3, c5, c7 = C_16[3], C_16[5], C_16[7] Gb = sp.Matrix([[c3, c5], [c5, c7]]) Tb = sp.Matrix([[c7, c7], [c7, c5]]) det_gt = sp.simplify(sp.det(Gb - Tb)) fact = iszero(det_gt + (c5 - c7) * ((c3 - c7) + (c5 - c7))) re_, im_ = det_gt.evalf(40).as_real_imag() sign40 = (mp.mpf(str(re_)) < 0 and abs(mp.mpf(str(im_))) < mp.mpf(10) ** (-30)) spec16 = FLAGS['spec_clock16'] lam_max16 = max(spec16) exceeds = lam_max16 > 1 + mp.mpf(10) ** (-25) spec8 = FLAGS['spec_clock8'] lam_max8 = max(spec8) at_one8 = abs(lam_max8 - 1) < mp.mpf(10) ** (-30) FLAGS['t26'] = (silver_exp and drift and fact and sign40 and exceeds and at_one8) FLAGS['kms_witnessed'] = FLAGS['t26'] check("T2.6 NO CONTRACTION ON THE COMPACT CIRCLE -- THE PRE-DECLARED " "KMS DEVIATION [exact witnesses + 40-digit]: the local clock " "step EXPANDS the edge mode by exactly the silver ratio: " "||T(2)[g_5]||^2/||[g_5]||^2 = C(1)/C(3) = 1 + sqrt(2) = " "delta_S at N = 8 (%s/%s -- while the COMPRESSED spectrum is " "{1, 1/delta_S}); at N = 16 the odd one-particle clock block " "has det(G - tau_4) = -(C5-C7)((C3-C7)+(C5-C7)) < 0 EXACTLY " "(%s, 40-digit sign %s), so the compressed clock exceeds 1: " "lambda_max = %s > 1 (vs %s = 1 exactly at N = 8) -- the " "finite seam circle is compact euclidean TIME: the " "reconstruction is thermal (dim H_phys = 4^2), T(k) is a " "positive self-adjoint LOCAL semigroup but NOT a contraction " "(contraction = the ground-state/infinite-volume property); " "0 < T <= 1 fails exactly as pre-registered in [C-3](d), " "with exactly these witnesses -- typed KMS finding, not a " "beta2 failure" % (silver_exp, drift, fact, sign40, mp.nstr(lam_max16, 8), mp.nstr(lam_max8, 8)), FLAGS['t26']) def t7_v522_resolution(): print(" -- T2.7: the v522 non-Hermiticity localised and resolved") tw4 = tower_maps(N16, 4, 8) same_map = tw4[1] == TW16[4] basis29 = [()] + list(combinations(P16, 2)) T1v = term_matrix(basis29, tw4[1], R15, S15, ETA, N16) d = len(basis29) n_match = n_anti = n_viol = 0 impl_ok = True for i in range(d): ci, mi = theta_mono(basis29[i], R15, S15, ETA) for j in range(d): cj, mj = alpha_mono(basis29[j], tw4[1]) dij = T1v[i, j] - sp.conjugate(T1v[j, i]) if iszero(dij): n_match += 1 matched = True elif iszero(T1v[i, j] + sp.conjugate(T1v[j, i])): n_anti += 1 matched = False else: n_viol += 1 matched = False # overlap census: does either product wrap into the other half? cjt, mjt = theta_mono(basis29[j], R15, S15, ETA) cit, mit = alpha_mono(basis29[i], tw4[1]) ov = (set(mi) & set(mj)) or (set(mjt) & set(mit)) if not ov and not matched: impl_ok = False reg = (n_match, n_anti, n_viol) == (745, 96, 0) # restriction to the quotient domain D_4: the SAME insertion is the # positive transfer step tau_4 (entrywise identity) D4, M4 = FLAGS['taus16'][4] sub = [i for i, m in enumerate(basis29) if all(8 <= a <= 11 for a in m)] restr_ok = True for ii, i in enumerate(sub): for jj, j in enumerate(sub): i4 = D4.index(basis29[i]) j4 = D4.index(basis29[j]) if not iszero(T1v[i, j] - M4[i4, j4]): restr_ok = False FLAGS['t27'] = same_map and reg and impl_ok and restr_ok check("T2.7 WHICH PART OF THE v522 'TIME-LIKENESS' DISSOLVES IN THE " "QUOTIENT [exact]: the v522 clock insertion (shift-4 tower, " "full even 29-basis) is reproduced with census (match, anti, " "viol) = (%d, %d, %d) = the v522 numbers (%s; alpha_S = " "alpha_1^4 as signed permutations: %s); EVERY " "Hermiticity-anti-matching entry has a WRAP OVERLAP (the " "shifted support crosses into the reflected half) and every " "overlap-free entry matches (%s) -- the pre-quotient " "non-Hermiticity is EXACTLY the domain/wrap artifact; " "restricted to the Klein-Landau domain {8..11} the SAME " "insertion IS the positive transfer step tau_4 entrywise " "(%s): what dissolves in beta2 is the ILL-POSEDNESS of clock " "positivity -- the clock becomes a positive self-adjoint " "operator with spectral calculus (T2.4/T2.5); what REMAINS " "time-like is the euclidean-rotation character itself (the " "clock is a transfer step, not a symmetry of H_phys -- its " "unitary avatar is the RECONSTRUCTED group of T2.5)" % (n_match, n_anti, n_viol, reg, same_map, impl_ok, restr_ok), FLAGS['t27']) # --------------------------------------------------------------------------- # Part G -- GSO grading and the descended anti-unitary on H_phys # --------------------------------------------------------------------------- def g1_gso(): print(" -- G3.1: the GSO grading survives on H_phys") par16 = (tuple(range(N16)), (-1,) * N16) par8 = (tuple(range(N8)), (-1,) * N8) comm_ok = True for n, tw, par, kmax in ((N16, TW16, par16, 8), (N8, TW8, par8, 4)): for k in range(1, kmax): for a in range(n): c1 = par[1][tw[k][0][a]] * tw[k][1][a] # P o alpha c2 = tw[k][1][a] * par[1][a] # alpha o P if c1 != c2: comm_ok = False dims16 = (sum(1 for m in BASIS16 if len(m) % 2 == 0), sum(1 for m in BASIS16 if len(m) % 2 == 1)) dims8 = (len(EVENS8), len(ODDS8)) FLAGS['g31'] = comm_ok and dims16 == (29, 8) and dims8 == (8, 8) check("G3.1 GSO GRADING SURVIVES [exact]: fermion parity descends to " "the G-unitary P[m] = (-1)^{|m|}[m] (the Gram is parity-block-" "diagonal EXACTLY, Q1.2/Q1.3: %s/%s), P^2 = 1, and P commutes " "with every translation alpha_k as signed permutation maps " "(%s) hence with all transfer data -- H_phys = H_even (+) " "H_odd orthogonally, dims %s at N = 16 (deg <= 2) and %s at " "N = 8; the mu4 gauge datum of v522 (the GSO Z2) acts on the " "quotient exactly" % (FLAGS['q12'], FLAGS['q13'], comm_ok, dims16, dims8), FLAGS['g31']) def theta_perp_data(n): if n == N16: kp = 7 return refl_map(kp, N16) kp = 3 return refl_map(kp, N8) def g2_theta_perp(): print(" -- G3.2: the perpendicular torsor mirror descends " "anti-unitarily") rp16, sp16_ = theta_perp_data(N16) rp8, sp8_ = theta_perp_data(N8) G16, G8 = FLAGS['G16'], FLAGS['G8'] bidx16 = {m: i for i, m in enumerate(BASIS16)} bidx8 = {m: i for i, m in enumerate(BASIS8)} good_eta = [] for etap in (sp.Integer(1), sp.Integer(-1), I, -I): ok = True for basis, bidx, G, rp, sp_ in ((BASIS16, bidx16, G16, rp16, sp16_), (BASIS8, bidx8, G8, rp8, sp8_)): for i, ma in enumerate(basis): ca, ia = theta_mono(ma, rp, sp_, etap) if ia not in bidx: ok = False break for j, mb in enumerate(basis): cb, ib = theta_mono(mb, rp, sp_, etap) lhs = sp.conjugate(ca) * cb * G[bidx[ia], bidx[ib]] if not iszero(lhs - G[j, i]): ok = False break if not ok: break if not ok: break if ok: good_eta.append(etap) FLAGS['eta_perp'] = good_eta etap = good_eta[0] if good_eta else I sq_plus = True for basis, rp, sp_ in ((BASIS16, rp16, sp16_), (BASIS8, rp8, sp8_)): for m in basis: c1, m1 = theta_mono(m, rp, sp_, etap) c2, m2 = theta_mono(m1, rp, sp_, etap) if m2 != m or not iszero(sp.conjugate(c1) * c2 - 1): sq_plus = False trev = True for n, tw, twn, rp, sp_, basis in ((N16, TW16, TW16N, rp16, sp16_, BASIS16), (N8, TW8, TW8N, rp8, sp8_, BASIS8)): for k in (1, 2): for m in basis: cb, mb = alpha_mono(m, tw[k]) ct1, mt1 = theta_mono(mb, rp, sp_, etap) ct2, mt2 = theta_mono(m, rp, sp_, etap) cb2, mb2 = alpha_mono(mt2, twn[k]) if mt1 != mb2 or not iszero(cb * ct1 - ct2 * cb2): trev = False deck_eta = None for etad in good_eta: ok_d = True for n, tw, rc, sc, rp, sp_, basis in ((N16, TW16, R15, S15, rp16, sp16_, BASIS16), (N8, TW8, R7, S7, rp8, sp8_, BASIS8)): for m in basis: cp, mp_ = theta_mono(m, rp, sp_, etad) cc, mc = theta_mono(mp_, rc, sc, ETA) ca, ma = alpha_mono(m, tw[n // 2]) if mc != ma or not iszero(sp.conjugate(cp) * cc - ca): ok_d = False break if not ok_d: break if ok_d: deck_eta = etad break deck_ok = deck_eta is not None FLAGS['g32'] = (set(map(str, good_eta)) == {'I', '-I'} and sq_plus and trev and deck_ok) check("G3.2 THETA DESCENDS TO H_phys AS THE PERPENDICULAR TORSOR " "MIRROR [exact]: Theta_phys[a] := [theta_perp(a)] (axis " "k_cut - N/2 = 7 resp. 3, the OTHER clock orbit of the v519 " "mu4 torsor) is ANTI-UNITARY, = " "entrywise on all 37^2 + 16^2 pairs, exactly for the twist " "sub-torsor eta_perp in %s (mu4-pinned, like the cut twist); " "Theta_phys^2 = +1 on EVERY basis class incl. the ODD sector " "(%s) -- the quotient is KRAMERS-FREE: the (-1)^F dichotomy " "does not reach H_phys; Theta_phys time-reverses the local " "semigroup (theta_perp alpha_k = alpha_{-k} theta_perp: %s) " "and the torsor closes onto the deck: theta_cut o theta_perp " "= alpha_{N/2} EXACTLY including coefficients (%s) -- the " "product of the two pinned mirrors IS the antipode lift" % ([str(e) for e in good_eta], sq_plus, trev, deck_ok), FLAGS['g32']) def g3_non_descent(): print(" -- G3.3: what does NOT act on H_phys (typed)") cut_out = True for m in BASIS16[1:]: c_, mi = theta_mono(m, R15, S15, ETA) if set(mi) & set(P16): cut_out = False for m in BASIS8[1:]: if m == (): continue c_, mi = theta_mono(m, R7, S7, ETA) if set(mi) & set(P8): cut_out = False deck_out = True for m in BASIS16[1:]: c_, mi = alpha_mono(m, TW16[8]) if set(mi) & set(P16): deck_out = False for m in BASIS8[1:]: if m == (): continue c_, mi = alpha_mono(m, TW8[4]) if set(mi) & set(P8): deck_out = False FLAGS['g33'] = cut_out and deck_out check("G3.3 NON-DESCENT CENSUSES [exact]: the OS reflection " "theta_cut maps every non-scalar element of A_+ ENTIRELY into " "the opposite half (%s) -- it does not act on H_phys, it IS " "the metric; the deck/antipode candidate Theta_t = Utilde o K " "(Kramers class (-1)^F, v519 W-S6.4) likewise maps A_+ off " "itself (%s) -- the Kramers class CANNOT obstruct the " "quotient, and indeed the descended real structure has " "Theta_phys^2 = +1 (G3.2): the v510/v519 dichotomy resolves " "on H_phys in favour of +1, extending v522 G3.1 (invisible " "on A^G) to the FULL quotient including the odd sector" % (cut_out, deck_out), FLAGS['g33']) # --------------------------------------------------------------------------- # Part C -- negative controls # --------------------------------------------------------------------------- def c1_site_cut(): print(" -- C4.1: site cut -- the kill branch fires for the site " "placement") k = 0 P = half_of(k, N16) r_ = lambda a: (k - a) % N16 basis1 = [(a,) for a in P] tw0 = (tuple(range(N16)), (1,) * N16) M1 = term_matrix(basis1, tw0, r_, PLUS, -I, N16) herm1 = hermitian_exact(M1) in1, _ = spectrum_inertia(M1) P8s = half_of(0, N8) # {1,2,3} basis8s = [m for d in range(4) for m in combinations(P8s, d)] tw08 = (tuple(range(N8)), (1,) * N8) r8_ = lambda a: (-a) % N8 picked = None for etac in (sp.Integer(1), sp.Integer(-1), I, -I): M8s = term_matrix(basis8s, tw08, r8_, PLUS, etac, N8) if hermitian_exact(M8s): in8s, _ = spectrum_inertia(M8s) picked = (etac, in8s) break FLAGS['c41'] = (herm1 and in1 == (3, 3, 1) and picked is not None and picked[1][1] > 0) check("C4.1 SITE CUT -> INDEFINITE COMPLETION [exact + 40-digit]: " "the marks-at-sites reading fails the beta2 kill branch " "exactly -- N = 16 one-particle Gram Hermitian %s with " "inertia %s (v519 R2.1 regression, 3 NEGATIVE directions + " "kernel); N = 8 complete site-half algebra {1,2,3} (8 " "monomials): Hermitian for eta = %s with inertia %s, %d " "negative directions -- for the site placement the OS " "quotient DEGENERATES (indefinite completion): the contract " "kill branch fires there, i.e. RP again SELECTS the bond " "placement (contract precision (ii)) at quotient level" % (herm1, in1, picked[0], picked[1], picked[1][1]), FLAGS['c41']) def c2_family_a(): print(" -- C4.2: family A -- no PSD form, no quotient") r_anti = lambda a: (a + 8) % N16 s_alt = lambda a: 1 if a % 2 == 0 else -1 basis_a = [(a,) for a in range(8)] tw0 = (tuple(range(N16)), (1,) * N16) picked = None for etac in (sp.Integer(1), sp.Integer(-1), I, -I): Ma = term_matrix(basis_a, tw0, r_anti, s_alt, etac, N16) if hermitian_exact(Ma): ia, _ = spectrum_inertia(Ma) diag0 = all(iszero(Ma[i, i]) for i in range(8)) picked = (etac, ia, diag0) break FLAGS['c42'] = (picked is not None and picked[1] == (4, 4, 0) and picked[2]) check("C4.2 FAMILY A (CLOCK-CENTRALISING ANTIPODE) -> NO QUOTIENT " "[exact + 40-digit]: the alternating-dressed antipode " "candidate (v519 R5.2) has a Hermitian OS form only for eta " "= %s, with ZERO diagonal (%s) and inertia %s -- INDEFINITE " "with 4 negative directions at the one-particle level " "already: no PSD form, no H_phys, no transfer -- the " "clock-centralising family cannot even ENTER the beta2 " "construction: RP and Theta rho Theta = rho^-1 keep " "selecting the SAME family in the quotient" % (picked[0], picked[2], picked[1]), FLAGS['c42']) def c3_antichiral(): print(" -- C4.3: anti-chiral state -- quotient exists only for the " "matching chirality") G8a = gram_of(BASIS8, R7, S7, N8, chi=-1) herm = hermitian_exact(G8a) in_all, _ = spectrum_inertia(G8a) in_ev, _ = block_inertia(G8a, list(range(8))) in_od, _ = block_inertia(G8a, list(range(8, 16))) FLAGS['c43'] = (herm and in_od == (0, 8, 0) and in_ev == (8, 0, 0) and in_all == (8, 8, 0)) check("C4.3 ANTI-CHIRAL STATE -> INDEFINITE COMPLETION [exact + " "40-digit]: with C -> -C and the SAME theta (eta = +i) the " "N = 8 full half-algebra Gram is Hermitian (%s) but its odd " "block flips to NEGATIVE definite %s while the even block " "stays PD %s: total inertia %s -- the OS quotient EXISTS " "ONLY for the chirality matching the twist (the v519 R5.3 " "kill-test-3 shadow upgraded to quotient level: quotient " "existence itself selects the chiral orientation)" % (herm, in_od, in_ev, in_all), FLAGS['c43']) def c4_wrong_direction(): print(" -- C4.4: wrong semigroup direction -- breaks off-domain, " "mirrors on-domain") D1w = dom_monos(BASIS8, 4, 6) Mw = term_matrix(D1w, TW8N[1], R7, S7, ETA, N8) herm_w = hermitian_exact(Mw) i4 = D1w.index((4,)) wit = sp.simplify(Mw[i4, i4]) n_match = n_anti = n_viol = 0 impl_ok = True d = len(D1w) for i in range(d): ci, mi = theta_mono(D1w[i], R7, S7, ETA) for j in range(d): cj, mj = alpha_mono(D1w[j], TW8N[1]) dij = Mw[i, j] - sp.conjugate(Mw[j, i]) if iszero(dij): n_match += 1 matched = True else: matched = False if iszero(Mw[i, j] + sp.conjugate(Mw[j, i])): n_anti += 1 else: n_viol += 1 cjt, mjt = theta_mono(D1w[j], R7, S7, ETA) cit, mit = alpha_mono(D1w[i], TW8N[1]) ov = (set(mi) & set(mj)) or (set(mjt) & set(mit)) if not ov and not matched: impl_ok = False D1m = dom_monos(BASIS8, 5, 7) Mm = term_matrix(D1m, TW8N[1], R7, S7, ETA, N8) herm_m = hermitian_exact(Mm) in_m, _ = spectrum_inertia(Mm) if herm_m else ((0, 0, 0), 0) in_f, _ = spectrum_inertia(FLAGS['taus8'][1][1]) FLAGS['c44'] = (not herm_w and iszero(wit - I) and impl_ok and herm_m and in_m == in_f) check("C4.4 WRONG SEMIGROUP DIRECTION [exact]: alpha_{-1} on the " "UNSHRUNK domain algebra{4,5,6} is NOT Hermitian (%s; exact " "diagonal witness omega(theta(g_4) alpha_{-1}(g_4)) = " "omega(i g_3 g_3) = %s, purely imaginary -- the shifted " "support crosses the cut INTO the reflected half), census " "(match, anti, viol) = (%d, %d, %d) with every mismatch a " "wrap-overlap entry (%s) -- the v522 mechanism reproduced in " "miniature; on the MIRRORED domain algebra{5,6,7} the same " "backward step is Hermitian (%s) with the mirror inertia %s " "= forward %s: the semigroup exists exactly INTO the half, " "in both theta-conjugate directions, and nowhere else" % (not herm_w, wit, n_match, n_anti, n_viol, impl_ok, herm_m, in_m, in_f), FLAGS['c44']) # --------------------------------------------------------------------------- # Part V -- kill-test bookkeeping and the frozen verdict # --------------------------------------------------------------------------- def v1_killtests(): print(" -- V5.1: contract kill-test bookkeeping under beta2") touched = { 1: FLAGS['g32'] and FLAGS['g33'], 2: FLAGS['q_pd16'] and FLAGS['q_pd8'] and FLAGS['clock_herm_pd'], 3: FLAGS['c43'] and FLAGS['t23'], } FLAGS['v51'] = all(touched.values()) check("V5.1 KILL-TEST BOOKKEEPING [assembled]: beta2 touches three " "of the seven contract kill tests, none fires -- (1) Theta/" "clock compatibility: STRENGTHENED (the real structure " "descends anti-unitarily to H_phys, Kramers-free even on the " "odd sector: %s); (2) RP after gauge fixing: STRENGTHENED " "(positivity survives INTO the quotient -- H_phys PD and the " "clock transfer step PD; the GSO Z2 acts exactly: %s); (3) " "mirror generation: free shadow SHARPENED (the anti-chiral " "state admits NO quotient, and the odd-step transfer " "indefiniteness is the staggering/chirality datum: %s). " "Kill tests (4) interacting Penrose grade, (5) net/OPE " "equivalence, (6) internal SU(2), (7) mark incidence: " "UNTOUCHED at the free level. All seven stay formally live " "on the interacting algebra A_hol; the beta2 contract kill " "branch itself (quotient degenerates) does NOT fire (Q1.4)" % (touched[1], touched[2], touched[3]), FLAGS['v51']) def v2_verdict(): print(" -- V5.2: verdict per the frozen preregistration") kill = (not (FLAGS['q_pd16'] and FLAGS['q_pd8'])) \ or (not FLAGS['clock_herm_pd']) s1 = FLAGS['q11'] and FLAGS['q12'] and FLAGS['q13'] and FLAGS['q14'] s2 = FLAGS['t21'] and FLAGS['t22'] s3 = FLAGS['t23'] and FLAGS['t24'] and FLAGS['t25'] and FLAGS['t27'] s4 = FLAGS['g31'] and FLAGS['g32'] and FLAGS['g33'] s5 = (FLAGS['c41'] and FLAGS['c42'] and FLAGS['c43'] and FLAGS['c44']) kms_ok = FLAGS['kms_witnessed'] if kill: verdict = "KILL" elif s1 and s2 and s3 and s4 and s5 and kms_ok: verdict = "SUCCESS" else: verdict = "UNDECIDED" FLAGS['verdict'] = verdict print("VERDICT: %s" % verdict) check("V5.2 BETA2 VERDICT = %s [frozen C-7 logic]: no kill (H_phys " "PD at both levels, the clock step Hermitian + PD on its " "domain); S1 quotient explicit (%s), S2 local transfer " "self-adjoint + semigroup + vacuum fixed (%s), S3 clock = " "T^{N/4} positive with spectral projections + reconstructed " "unitary rotation group (%s), S4 GSO grading + anti-unitary " "Theta descent (%s), S5 all five negative controls behave " "(%s); the single deviation from naive expectations -- NO " "contraction on the compact circle -- carries EXACTLY the " "pre-declared KMS witnesses (%s: silver-ratio norm drift " "C(1)/C(3) = 1 + sqrt(2), det(G - tau_4) < 0). Scope fence: " "free/quadratic seam system only, deg <= 2 at N = 16, " "complete algebra at N = 8; no interacting algebra, no " "chirality theorem, no net equivalence. WOIT.OS.TWISTOR.01 " "stays [O]; no marker moves; sandbox only" % (verdict, s1, s2, s3, s4, s5, kms_ok), verdict == "SUCCESS") def run(): reset() RESULTS.clear() FLAGS.clear() print("v524 WOIT.BETA2.OS.01 -- WOIT-beta2: the OS quotient of " "the free system made explicit") q1_preconditions() q2_gram16() q3_gram8() q4_kill_branch() t1_domain_hermiticity() t2_semigroup() t3_positivity_pattern() t4_clock() t5_spectral_rotation() t6_kms() t7_v522_resolution() g1_gso() g2_theta_perp() g3_non_descent() c1_site_cut() c2_family_a() c3_antichiral() c4_wrong_direction() v1_killtests() v2_verdict() print("\n%d/%d checks passed (runtime %.1f s)" % (sum(RESULTS), len(RESULTS), time.perf_counter() - T0_WALL)) print("FINAL VERDICT: %s" % FLAGS.get('verdict', 'UNDECIDED')) return summary("v524 WOIT.BETA2.OS.01: the OS quotient of the " "free system is explicit -- H_phys PD at both " "levels ((37,0,0) min 1.7801e-6; (16,0,0) = 4^2 " "thermal), the Klein-Landau transfer semigroup is " "exactly Hermitian with the site/bond dichotomy as " "its positivity pattern, the mu4 clock = T^{N/4} " "is positive with spectral calculus (N = 8 " "spectrum exactly {1, sqrt(2)-1} = {1, " "1/delta_S}), the reconstructed rotation group is " "unitary; the v522 non-Hermiticity is exactly the " "wrap artifact; pre-declared KMS deviation (no " "contraction, silver witnesses); Theta_phys^2 = +1 " "Kramers-free; verdict SUCCESS, [C]-typed per " "precision (iii); WOIT.OS.TWISTOR.01 stays [O]; " "no marker moves") if __name__ == "__main__": raise SystemExit(1 if run() else 0)