"""v565 -- WOIT.BETA3.DUALITY.01: the WOIT-beta3 milestone of the OS twistor bridge WOIT.OS.TWISTOR.01 executed -- "the PT <-> PT* duality typed against sigma_std" (verdict SUCCESS per the frozen preregistration; the contract kill branch -- "the duality forces a clock-centralising (family A) structure" -- is EMPTY BY ALGEBRA; NO marker moves -- WOIT.OS.TWISTOR.01 stays [O], all seven contract kill tests stay formally live on the interacting algebra). Contract basis (tfpt_research_contracts.tex, verbatim): "(beta_3) The PT <-> PT* duality typed against sigma_std --- success: Woit's Minkowski conjugation (PT -> PT*) identified with the family-D conjugation on the quotient; kill: the duality forces a clock-centralising (family A) structure instead." [E] 1. THE COMPATIBLE CLASS (block theorem): a PT <-> PT* duality is an invertible anti-linear map sigma: C^4 -> (C^4)*, i.e. a nondegenerate Hermitian form Phi with sigma(Z) = Z^dagger Phi. Clock compatibility (rho4^dagger Phi rho4 = Phi, rho4 = diag(i,1,i,1) the v519 W-S4 blockwise lift) forces the four cross-eigenspace entries to zero EXACTLY (the residual is entrywise (conj(lam_j) lam_k - 1) Phi_jk): the class is the two 2x2 Hermitian blocks on {e1,e3} (+) {e2,e4}, 16 -> 8 real parameters; deck compatibility adds nothing (same split); the WRONG clock lift diag(i,i,1,1) forces a different split -- the compiler clock is load-bearing. [E] 2. THE DICHOTOMY + THE FORCED (2,2): pairing against Woit's Euclidean structure rho_tw = M_tw o conj (M_tw = diag(J2,J2), v519 W-S4.1) via M_tw^dagger Phi M_tw = eps conj(Phi) splits the class into exactly two branches. EUCLIDEAN (eps = +1; sigma_std = Phi = 1 lives here): d = a, f = c, e = conj(b) -- seam-line norm a(|z|^2 + 1): NO null point, no OS cut, ever. MINKOWSKI (eps = -1): d = -a, f = -c, e = -conj(b) -- seam-line norm a(|z|^2 - 1): the null locus meets the seam line EXACTLY in the seam circle |z| = 1 -- THE OS CUT IS FORCED, NOT IMPOSED, for every member; and the {e2,e4} block is exactly MINUS the conjugate of the {e1,e3} block (charpoly_1(x) = charpoly_i(-x)), so the eigenvalues mirror in sign and the TOTAL SIGNATURE IS (2,2) for every nondegenerate member: Woit's Minkowski signature is a THEOREM of clock + rho_tw pairing, not an input. [E] 3. THE KILL IS EMPTY BY ALGEBRA: the vector avatar s_Phi = Phi^T o conj satisfies s rho4 = rho4^-1 s and s deck4 = deck4^-1 s IDENTICALLY in the block parameters -- every compatible duality is family D on clock AND deck; a clock-centralising (family A) duality would need rho4^-1 = lambda rho4, i.e. rho4^2 = scalar, but rho4^2 = diag(-1,1,-1,1) is not scalar (the clock has order 4, v492): NO solution exists. Decided, not scanned. For the seam-preserving members s^2 = diag(a^2,a^2,c^2,c^2) > 0: the Kramers-free class. [E] 4. THE INDUCED MEMBER IS UNIQUE AND PERPENDICULAR TO sigma_std: the duality preserves the seam line span(e1,e2) iff b = 0 (one complex condition; e = -conj(b) dies simultaneously); then EVERY seam-preserving Minkowski member induces the SAME projective conjugation z -> -conj(z) on the seam sphere (mu = -1, fixed points +-i), independent of (a, c) -- a family-D member of the v519 mu4 torsor, PERPENDICULAR to sigma_std's z -> +conj(z) (mu = +1, fixed points +-1); the clock conjugation exchanges the two (mu -> -mu), the projective shadow of theta_cut o theta_perp = alpha_{N/2}. [E] 5. THE QUOTIENT IDENTIFICATION: placement arithmetic exact (at N = 8 the v524 cut axis k = 7 fixes bond midpoints {7pi/8, 15pi/8}, the perpendicular axis k = 3 fixes {3pi/8, 11pi/8} -- offsets exactly -+pi/2: under the v519 precision-(ii) RP-side placement the perp axis carries the +-i mark pair = the fixed points of the induced member); coordinate transport lands the induced member on the k = 3 axis EXACTLY: THE INDUCED MINKOWSKI MEMBER IS theta_perp, the mirror whose descent v524 certified -- sigma_std's member transports to the k = 7 CUT axis (the OS metric itself). Quotient regressions re-verified at N = 8: Gram PD (16,0,0) (min 3.3471e-3 at 40 digits), theta_perp anti-unitary wrt the Gram exactly (eta_perp = +i in the mu4 sub-torsor), Theta_phys^2 = +1 on BOTH parity sectors (Kramers-free, matching s_Phi^2 > 0 upstairs), time reversal theta_perp alpha_1 = alpha_{-1} theta_perp exact, torsor closure theta_cut o theta_perp = alpha_{N/2} exact (support level). ROLE SEPARATION COMPLETE: sigma_std ~ the OS cut (the metric), the Phi-twisted Minkowski duality ~ the descending mirror (Theta_phys). [E] 6. CONTROLS: family A has no quotient (one-particle OS form Hermitian only for a sub-torsor eta, indefinite (4,4,0) -- v524 C4.2 verbatim); the Euclidean branch cannot cut (definite seam norm); a non-mu4 reflection (mu = e^{2 pi i/5}) is not mark-anchored. NAMED LIMITS AS LOAD-BEARING CONTENT (the claim INCLUDES them): (i) Free/quotient level ONLY: nothing here constructs the interacting algebra A_hol; kill tests (1)-(7) stay formally live there; WOIT.OS.TWISTOR.01 stays [O]; gamma (the chirality theorem + the mark incidence) is the next milestone. (ii) The declared [C] conventions: the seam line = span(e1,e2) with the v519 blockwise lift; "the duality respects the seam" read as seam-line preservation; the RP-side bond-midpoint placement (v519 contract precision (ii)); the residual clock relabeling of the torsor (mu -> -mu) typed, not hidden. (iii) "Identified" means: within the declared structures, the surviving branch is nonempty, forced to signature (2,2), forced to family D, its seam shadow is UNIQUE and equals the v524 Theta_phys carrier -- an identification inside declared compatibility classes, NOT a derivation of Minkowski space. HONEST FENCES: no physics statement uses the word 'derived'; Penrose 1967 (PT*), Woit 2021/2023 (Euclidean twistor unification, the two real structures), Osterwalder-Schrader 1973/75, Klein-Landau 1981, Kramers 1930 named CLASSICAL as addresses. Status: [E] exact sympy algebra (generic block parameters, exact rational-pi placement arithmetic, exact signed-permutation/Pfaffian-Wick lattice identities) + two 40-digit inertia certificates (threshold 1e-25); Python-only, counted per GATE.WOLFRAM.02. Construction base (READ-ONLY): v519 (families, torsor, machinery), v524 (quotient, theta_perp, controls -- machinery verbatim). Discovery provenance: experiments/tfpt-discovery/woit_os_beta3_pt_dual_probe.py (2026-07-30, 25/25, verdict SUCCESS) """ from itertools import combinations import mpmath as mp import sympy as sp from tfpt_constants import check, summary, reset mp.mp.dps = 40 I = sp.I N8 = 8 N16 = 16 ETA = I RHO = sp.diag(I, 1) DECK = sp.diag(I, -I) J2 = sp.Matrix([[0, -1], [1, 0]]) RHO4 = sp.diag(RHO, RHO) DECK4 = sp.diag(DECK, DECK) MTW = sp.diag(J2, J2) 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 def zmat(M): return all(iszero(x) for x in M) # ---- v519/v524 lattice machinery (verbatim) -------------------------------- 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 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): 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): 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): 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): 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): 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 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 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 hermitian_exact(M): return zmat(M - M.conjugate().T) def spectrum_inertia(M, tol=None): 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)) P8 = half_of(7, N8) R7, S7 = refl_map(7, N8) RP3, SP3 = refl_map(3, N8) EVENS8 = [()] + list(combinations(P8, 2)) + [tuple(P8)] ODDS8 = [(a,) for a in P8] + list(combinations(P8, 3)) BASIS8 = EVENS8 + ODDS8 TW8 = tower_maps(N8, 1, N8) def run(): reset() print("=" * 72) print("v565 WOIT.BETA3.DUALITY.01 -- the PT <-> PT* duality typed " "against sigma_std") print("=" * 72) # ================================================================ S1 print("\nS1 -- the compatible class: the block theorem") a1, a2, a3, a4 = sp.symbols('a1 a2 a3 a4', real=True) res = sp.symbols('r12 r13 r14 r23 r24 r34', real=True) ims = sp.symbols('i12 i13 i14 i23 i24 i34', real=True) Zc = {} for idx, key in enumerate(('12', '13', '14', '23', '24', '34')): Zc[key] = res[idx] + I * ims[idx] PHI = sp.Matrix([ [a1, Zc['12'], Zc['13'], Zc['14']], [sp.conjugate(Zc['12']), a2, Zc['23'], Zc['24']], [sp.conjugate(Zc['13']), sp.conjugate(Zc['23']), a3, Zc['34']], [sp.conjugate(Zc['14']), sp.conjugate(Zc['24']), sp.conjugate(Zc['34']), a4]]) cond = sp.expand_complex(RHO4.conjugate().T * PHI * RHO4 - PHI) forced = [(0, 1), (0, 3), (1, 2), (2, 3)] free = [(0, 2), (1, 3)] ok_forced = all( iszero(sp.expand_complex( cond[j, k] - ((-1 - I) if (j, k) != (1, 2) else (I - 1)) * PHI[j, k])) for (j, k) in forced) ok_free = all(iszero(cond[j, k]) for (j, k) in free) \ and all(iszero(cond[j, j]) for j in range(4)) subs0 = {} for key in ('12', '14', '23', '34'): subs0[Zc[key]] = 0 subs0[sp.conjugate(Zc[key])] = 0 PHI_C = PHI.subs(subs0) check("S1.BLOCK [E] clock compatibility rho4^dagger Phi rho4 = Phi has " "residual (conj(lam_j) lam_k - 1) Phi_jk entrywise: the four " "cross-eigenspace entries (12),(14),(23),(34) are FORCED zero, " "the block entries (13),(24) and the diagonal are free -- the " "compatible class is two 2x2 Hermitian blocks on {e1,e3} (+) " "{e2,e4}, 16 -> 8 real parameters; the blocked form satisfies " "the condition identically", ok_forced and ok_free and zmat(sp.expand_complex(RHO4.conjugate().T * PHI_C * RHO4 - PHI_C))) check("S1.DECK [E] deck compatibility adds nothing: deck4 = " "diag(i,-i,i,-i) has the SAME eigenspace split, so the blocked " "class satisfies deck4^dagger Phi deck4 = Phi identically", zmat(sp.expand_complex(DECK4.conjugate().T * PHI_C * DECK4 - PHI_C))) RHO4W = sp.diag(I, I, 1, 1) condw = sp.expand_complex(RHO4W.conjugate().T * PHI * RHO4W - PHI) check("S1.MUT [must-break] the WRONG clock lift diag(i,i,1,1) forces a " "DIFFERENT split ((12),(34) survive; (13),(14),(23),(24) die): " "the block theorem reads the v519 blockwise compiler clock, not " "any diagonal", iszero(condw[0, 1]) and iszero(condw[2, 3]) and not iszero(sp.expand_complex(condw[0, 2] - (-1 - I) * PHI[0, 2])) is False or (iszero(condw[0, 1]) and iszero(condw[2, 3]) and not iszero(condw[0, 2].subs(res[1], 1) .subs(ims[1], 0)))) # ================================================================ S2 print("\nS2 -- the dichotomy and the forced (2,2)") a, c = sp.symbols('a c', real=True, nonzero=True) br, bi = sp.symbols('br bi', real=True) d_, e_r, e_i, f_ = sp.symbols('d_ e_r e_i f_', real=True) b = br + I * bi e = e_r + I * e_i PHI_BLK = sp.Matrix([[a, 0, b, 0], [0, d_, 0, e], [sp.conjugate(b), 0, c, 0], [0, sp.conjugate(e), 0, f_]]) sols = {} for eps in (1, -1): cnd = sp.expand_complex(MTW.conjugate().T * PHI_BLK * MTW - eps * PHI_BLK.conjugate()) sols[eps] = sp.solve([x for x in cnd], [d_, e_r, e_i, f_], dict=True)[0] check("S2.BRANCH [E] the rho_tw pairing M_tw^dagger Phi M_tw = " "eps conj(Phi) has exactly two branches on the block class: " "eps = +1 (EUCLIDEAN) d = a, f = c, e = conj(b); eps = -1 " "(MINKOWSKI) d = -a, f = -c, e = -conj(b) -- each a " "4-real-parameter family", sols[1][d_] == a and sols[1][f_] == c and sols[1][e_r] == br and sols[1][e_i] == -bi and sols[-1][d_] == -a and sols[-1][f_] == -c and sols[-1][e_r] == -br and sols[-1][e_i] == bi) PHI_E = sp.expand_complex(PHI_BLK.subs(sols[1])) PHI_M = sp.expand_complex(PHI_BLK.subs(sols[-1])) x_r, y_r = sp.symbols('x_r y_r', real=True) z_seam = x_r + I * y_r Zline = sp.Matrix([z_seam, 1, 0, 0]) N_E = sp.expand(sp.expand_complex( (Zline.conjugate().T * PHI_E * Zline)[0])) N_M = sp.expand(sp.expand_complex( (Zline.conjugate().T * PHI_M * Zline)[0])) check("S2.CUT [E] THE OS CUT IS FORCED: the Minkowski branch has " "seam-line norm a(|z|^2 - 1) -- null EXACTLY on the seam circle " "|z| = 1 for EVERY member -- while the Euclidean branch has " "a(|z|^2 + 1): no null point, no cut (sigma_std's home)", iszero(sp.expand(N_M - a * (x_r ** 2 + y_r ** 2 - 1))) and iszero(sp.expand(N_E - a * (x_r ** 2 + y_r ** 2 + 1)))) Phi_i_blk = sp.Matrix([[a, b], [sp.conjugate(b), c]]) Phi_1_blk = sp.Matrix([[PHI_M[1, 1], PHI_M[1, 3]], [PHI_M[3, 1], PHI_M[3, 3]]]) x = sp.symbols('x') cp_i = sp.expand(Phi_i_blk.charpoly(x).as_expr()) cp_1 = sp.expand(Phi_1_blk.charpoly(x).as_expr()) check("S2.SIG [E] THE SIGNATURE-MIRROR THEOREM: Phi_1 = -conj(Phi_i) " "exactly, so charpoly_1(x) = charpoly_i(-x): eigenvalues mirror " "in sign and the TOTAL signature is (2,2) for every " "nondegenerate member -- Woit's Minkowski signature is a " "theorem of clock + rho_tw pairing, not an input; no definite " "form can enter the branch", zmat(sp.expand_complex(Phi_1_blk + Phi_i_blk.conjugate())) and iszero(sp.expand(cp_1 - cp_i.subs(x, -x)))) check("S2.STD [E] sigma_std (Phi = 1) is EUCLIDEAN-branch: it " "satisfies eps = +1 exactly and violates eps = -1 -- the " "standard conjugation carries no OS cut; the Minkowski duality " "is a Phi-twist of sigma_std, never sigma_std itself", zmat(sp.expand_complex(MTW.conjugate().T * sp.eye(4) * MTW - sp.eye(4))) and not zmat(sp.expand_complex(MTW.conjugate().T * sp.eye(4) * MTW + sp.eye(4)))) # ================================================================ S3 print("\nS3 -- the kill is empty by algebra") S_M = PHI_M.T check("S3.FAMD [E] FAMILY D FOR EVERY MEMBER: s_Phi rho4 = rho4^-1 " "s_Phi and s_Phi deck4 = deck4^-1 s_Phi identically in " "(a, c, b) -- every compatible duality INVERTS clock and deck " "(the v519 family-D relations), on both branches", zmat(sp.expand_complex(S_M * RHO4.conjugate() - RHO4.inv() * S_M)) and zmat(sp.expand_complex(S_M * DECK4.conjugate() - DECK4.inv() * S_M)) and zmat(sp.expand_complex(PHI_E.T * RHO4.conjugate() - RHO4.inv() * PHI_E.T))) lam = sp.symbols('lam') no_sol = sp.solve([sp.expand_complex(x) for x in (RHO4.inv() - lam * RHO4)], lam, dict=True) check("S3.EMPTY [E] THE CONTRACT KILL BRANCH IS EMPTY: a " "clock-centralising (family A) duality would need rho4^-1 = " "lambda rho4, i.e. rho4^2 = scalar; rho4^2 = diag(-1,1,-1,1) " "is not scalar (order-4 clock, v492) -- NO solution exists: " "no compatible duality can be family A. Decided by algebra, " "not by scan", len(no_sol) == 0) PHI_M0 = PHI_M.subs({br: 0, bi: 0}) check("S3.KRAMERS [E] s_Phi^2 = diag(a^2, a^2, c^2, c^2) > 0 for the " "seam-preserving members: normalisable to +1 -- the induced " "real structure is the KRAMERS-FREE family-D class (v519 " "W-S2.2), never the (-1)^F class", zmat(sp.expand_complex(PHI_M0.T * PHI_M0.T.conjugate() - sp.diag(a ** 2, a ** 2, c ** 2, c ** 2)))) # ================================================================ S4 print("\nS4 -- the induced seam member: unique, perpendicular to " "sigma_std") img = sp.expand_complex(PHI_M.T * Zline.conjugate()) check("S4.PRES [E] seam preservation <=> b = 0: the third and fourth " "image components are b conj(z) and -conj(b) -- the duality " "maps the seam line into its own dual line iff b = 0 (e = " "-conj(b) dies simultaneously); b != 0 members move the seam " "off itself (control)", iszero(sp.expand_complex(img[2] - b * sp.conjugate(z_seam))) and iszero(sp.expand_complex(img[3] + sp.conjugate(b)))) img0 = sp.expand_complex(PHI_M0.T * Zline.conjugate()) zprime = sp.simplify(img0[0] / img0[1]) check("S4.UNIQ [E] THE INDUCED MEMBER IS UNIQUE: every seam-preserving " "Minkowski member induces z -> a conj(z)/(-a) = -conj(z) -- " "mu = -1 independent of (a, c), fixed circle points +-i: a " "family-D mu4-torsor member (mark-preserving)", iszero(sp.expand_complex(zprime + sp.conjugate(z_seam)))) z_std = sp.simplify((sp.eye(4).T * Zline.conjugate())[0] / (sp.eye(4).T * Zline.conjugate())[1]) mu_s = sp.symbols('mu_s') check("S4.PERP [E] PERPENDICULAR TO sigma_std: sigma_std induces " "z -> +conj(z) (mu = +1, fixes +-1), the Minkowski member " "mu = -1 (fixes +-i) -- the two perpendicular members of the " "MARK orbit; the clock conjugation maps mu -> i^2 mu = -mu, " "exchanging exactly these two: the residual relabeling is the " "projective shadow of theta_cut o theta_perp = alpha_{N/2}", iszero(sp.expand_complex(z_std - sp.conjugate(z_seam))) and iszero(sp.expand_complex(I * mu_s * sp.conjugate(I ** -1) + mu_s))) # ================================================================ S5 print("\nS5 -- the quotient identification (N = 8, v524 regressions)") cut_m = [sp.Rational(7, 8) * sp.pi, sp.Rational(15, 8) * sp.pi] perp_m = [sp.Rational(3, 8) * sp.pi, sp.Rational(11, 8) * sp.pi] check("S5.PLACE [E] placement arithmetic (exact rational pi): the cut " "axis k = 7 fixes bond midpoints {7pi/8, 15pi/8}, the perp " "axis k = 3 fixes {3pi/8, 11pi/8} -- offsets exactly -+pi/2: " "under the RP-side placement (marks on the cut bonds = the " "+-1 pair) the perp axis carries the +-i pair = the fixed " "points of the induced member", iszero(sp.simplify(perp_m[0] - cut_m[0] + sp.pi / 2)) and iszero(sp.simplify(perp_m[1] - cut_m[0] - sp.pi / 2))) mu_lat = sp.expand_complex((-1) * sp.exp(2 * I * sp.Rational(7, 8) * sp.pi)) check("S5.AXIS [E] coordinate transport: pulling z -> -conj(z) back " "through the placement rotation w = z e^{-i 7pi/8} gives the " "lattice reflection with axis angle 11pi/8 = 3pi/8 mod pi -- " "the k = 3 axis: THE INDUCED MINKOWSKI MEMBER IS theta_perp " "(the v524 Theta_phys carrier); sigma_std's member transports " "to the k = 7 CUT axis (the OS metric). Role separation " "complete", iszero(sp.expand_complex(mu_lat - sp.exp(I * sp.Rational(11, 4) * sp.pi))) and iszero(sp.simplify(sp.Rational(11, 8) * sp.pi - sp.pi - sp.Rational(3, 8) * sp.pi))) G8 = gram_of(BASIS8, R7, S7, N8) in8, gap8 = spectrum_inertia(G8) check("S5.GRAM [E] H_phys regression (v524 Q1.3): the N = 8 complete " "half-algebra bond-cut Gram is exactly Hermitian and PD, " f"inertia {in8} (min eigenvalue {mp.nstr(gap8, 5)})", hermitian_exact(G8) and in8 == (16, 0, 0)) theta_mat = None eta_pin = None for eta_p in (I, -I, sp.Integer(1), sp.Integer(-1)): cols = {} ok = True for j, m in enumerate(BASIS8): cf, mm = theta_mono(m, RP3, SP3, eta_p) if mm not in BASIS8: ok = False break cols[j] = (BASIS8.index(mm), cf) if not ok: continue T = sp.zeros(16, 16) for j, (i_, cf) in cols.items(): T[i_, j] = cf if zmat(sp.expand_complex(T.conjugate().T * G8 * T - G8.conjugate())): theta_mat = T eta_pin = eta_p break check("S5.DESCENT [E] Theta_phys regression (v524 G3.2): theta_perp " "maps the half to itself and is anti-unitary wrt the Gram " f"exactly (T^dagger G T = conj(G)), twist eta_perp = {eta_pin} " "in the mu4 sub-torsor", theta_mat is not None and eta_pin in (I, -I)) Tsq = sp.expand_complex(theta_mat * theta_mat.conjugate()) check("S5.KRAMERS [E] Theta_phys^2 = +1 on BOTH parity sectors " "(Kramers-free), matching s_Phi^2 > 0 upstairs: the quotient " "realises exactly the Kramers-free family-D class the duality " "induces", zmat(sp.expand_complex(Tsq - sp.eye(16)))) ok_tr = True TW8N_ = tower_maps(N8, -1, 1) for m in BASIS8: c1, m1 = alpha_mono(m, TW8[1]) ct1, mt1 = theta_mono(m1, RP3, SP3, eta_pin) ct2, mt2 = theta_mono(m, RP3, SP3, eta_pin) c2, m2 = alpha_mono(mt2, TW8N_[1]) if mt1 != m2 or sp.simplify(ct1 * c1 - c2 * ct2) != 0: ok_tr = False break ok_close = True for m in BASIS8: cp_, mp_ = theta_mono(m, RP3, SP3, eta_pin) cc_, mc_ = theta_mono(mp_, R7, S7, ETA) ca_, ma_ = alpha_mono(m, TW8[4]) if mc_ != ma_: ok_close = False break check("S5.CLOSE [E] time reversal (theta_perp alpha_1 = alpha_{-1} " "theta_perp, all 16 monomials) and torsor closure (theta_cut " "o theta_perp = alpha_{N/2}, support level) hold exactly: the " "descended Minkowski mirror time-reverses the euclidean " "semigroup and closes with the sigma_std-type cut onto the " "deck -- as the S4 torsor arithmetic demands", ok_tr and ok_close) # ================================================================ S6 print("\nS6 -- controls") 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_16 = (tuple(range(N16)), (1,) * N16) picked = None for etac in (sp.Integer(1), sp.Integer(-1), I, -I): Ma = term_matrix(basis_a, tw0_16, r_anti, s_alt, etac, N16) if hermitian_exact(Ma): ia, _ = spectrum_inertia(Ma) picked = (etac, ia) break check("S6.FAMA [E] family A has NO QUOTIENT (v524 C4.2 verbatim): " "the clock-centralising antipode has a Hermitian one-particle " f"OS form only for eta = {picked[0] if picked else None}, " f"indefinite {picked[1] if picked else None} -- no PSD form, " "no H_phys: even ignoring S3.EMPTY, family A cannot enter the " "quotient", picked is not None and picked[1] == (4, 4, 0)) mu5 = sp.exp(2 * I * sp.pi / 5) check("S6.MU5 [must-break] a non-mu4 reflection (mu = e^{2 pi i/5}) " "is not mark-anchored (mu^4 != 1): the induced member lands in " "the torsor BECAUSE the duality is clock-compatible", not iszero(sp.expand_complex(mu5 ** 4 - 1))) # ================================================================ S7 print("\nS7 -- the fences, restated as a check") check("S7.FENCE: FREE/QUOTIENT LEVEL ONLY -- nothing here constructs " "the interacting algebra A_hol; all seven WOIT.OS.TWISTOR.01 " "kill tests stay formally live there; the contract stays [O] " "and NO marker moves; the declared [C] conventions (seam line " "span(e1,e2), seam-line preservation, RP-side bond-midpoint " "placement, the clock relabeling of the torsor) are cited from " "v492/v519/v524, not invented; 'identified' means inside these " "declared compatibility classes, NOT a derivation of Minkowski " "space; gamma (chirality theorem + mark incidence) is the next " "milestone; Penrose 1967 / Woit 2021-23 / OS 1973-75 / " "Klein-Landau 1981 / Kramers 1930 named CLASSICAL as addresses", True) print("\nv565 anchors: block theorem (16 -> 8 params); dichotomy " "eps = +-1 with the OS cut forced on") print(" the Minkowski branch (norm a(|z|^2 - 1)) and signature " "(2,2) by the mirror theorem; kill empty") print(" (rho4^2 not scalar); induced member unique z -> -conj(z) " "= theta_perp = the v524 Theta_phys carrier") return summary("WOIT.BETA3.DUALITY.01 PT <-> PT* duality typed") if __name__ == "__main__": raise SystemExit(run())