#!/usr/bin/env python3 """v615 -- WOIT.GAMMA.TOY.01: the interacting gamma slice at toy level -- on the RP-surviving interaction (the alignment bit, v534), the mirror stays excluded sector-exactly, the generation stays multiplicity-free, and the mark transport stays PD over the whole coupling grid. v608 executed the FREE gamma slice (kill tests 3/7 cannot fire at the free level). The gamma milestone proper lives on the interacting algebra A_hol; its only concrete interacting model in the corpus is the 16-Majorana Fidkowski-Kitaev seam toy (v529), where reflection positivity survives for EXACTLY ONE member x sign of the equivariant mark family: delta = pi/2 with positive coupling -- the alignment bit (v534). This probe runs the gamma battery ON that surviving interaction: (I1) TWO PARENTS [float, 1e-9]: the chiral parent (vacuum 2-point I + iC, the exact v519 chiral NS kernel) and the MIRROR parent (2-point I - iC, the conjugated kernel; the quartet interaction is conjugation-invariant, so the mirror system is -H_F + g H_int), both with gap 1. (I2) SURVIVOR ANCHOR [float, v534 regression]: the chiral interacting OS Grams at the forced twist eta = +i are PD (37,0,0) on all four admissible cuts {3,7,11,15} for g in {1/32, 1/4, 1, 8}. (I3) THE INTERACTING MIRROR FLIP [float, the central datum]: at the SAME eta = +i the mirror state's Grams split sector-exactly on every cut and every g: even (29,0,0) PD, odd (0,8,0) STRICTLY NEGATIVE DEFINITE with max odd eigenvalue <= -1e-3 (bounded away from zero over the whole grid) -- every mirror fermion vector has strictly negative OS norm AT THE INTERACTING LEVEL: in the doubled (vector-like) system any OS-positive subspace meets the mirror odd sector in {0} -- kill test (3) cannot fire on the surviving interaction class (upgrade of v608 G3 from g = 0 to the full coupling grid). (I4) CONVENTION CONTROL [float]: eta = -i swaps the roles exactly (chiral flips, mirror PD) -- the datum is the RELATIVE chirality; the doubled-system argument is convention-free. (I5) MARK TRANSPORT INTERACTING [exact + float]: the quarter turn alpha_4 maps the mark-A quartet algebra (16 monomials on sites {2,3,4,5} of the positive half of cut 3) exactly into the mark-B sites {6,7,8,9} (16/16, exact combinatorics), and both the mark-algebra OS Gram G_A and the transport form T_4[a,b] = are Hermitian PD (16,0,0) at every g -- the incidence-compatible mark transport EXISTS on the interacting survivor: kill test (7) does not fire at the interacting toy level. (I6) MULTIPLICITY-FREE [float]: the generalized odd-block transfer spectrum (T_4 vs G_A on the 8 fermionic mark monomials) consists of 8 SIMPLE positive eigenvalues at every g in {0, 1/32, 1/4, 1, 8} (min pairwise gap > 1e-5); honest finding: at strong coupling g = 8 the eigenvalues pair up to gaps ~2e-5 WITHOUT closing -- a strong-coupling near-doubling tendency, strictly simple on the declared grid. The g = 0 baseline is locked (top eigenvalue 1.641438, bottom 0.018673); observation (no check): the top value coincides numerically with the v524 compressed-clock top (1.6414) -- the quarter-turn transfer datum recurs across N. (I7) DEAD-MEMBER CONTROL [float]: on the RP-dead member (m = 2, s = +, g = 1/2) the straddled cut 1 is indefinite (21,16,0) while the avoiding cut 9 stays PD -- off the survivor the gamma battery has no RP home: the dynamical selection of the bit and the chirality data CO-LOCATE on the same member. (I8) FIREWALL [C]: one toy, one interaction class, deg <= 2 cut bases + the full 16-dim mark algebra, finite coupling grid -- NO continuum claim, NO statement about A_hol beyond this model class; kill tests (3)/(6)/(7) stay formally live on A_hol; kill test (6) (internal SU(2)) is UNTOUCHED by this slice; WOIT.OS.TWISTOR.01 does not move. Verdict enums (frozen): GAMMA-TOY-LANDED (all pass), GAMMA-TOY-FAILS, MIXED. FIREWALL: WOIT.OS.TWISTOR.01 does not move; no marker changes. PROVENANCE: discovery probe gamma_toy_interacting_probe.py (2026-08-01, 11/11, verdict GAMMA-TOY-LANDED). Machinery: v529/v534 verbatim (Cl(16) monomial calculus, JW Fock track, flat-band parent, quartets, cut Grams). Python-only, counted per GATE.WOLFRAM.02. """ from fractions import Fraction as Fr from itertools import combinations import numpy as np import sympy as sp N, DIM = 16, 256 TOL_HERM = 1e-8 TOL_ZERO = 1e-9 TOL_ND = 1e-3 # mirror odd sector: bounded away from zero TOL_GAP = 1e-5 # multiplicity-free: min pairwise eigenvalue gap GRID = (1.0 / 32, 0.25, 1.0, 8.0) CHECKS = [] def check(name, ok, detail=""): CHECKS.append((name, bool(ok))) print("[%s] %s%s" % ("PASS" if ok else "FAIL", name, (" -- " + detail) if detail else "")) # ---------------------------------------------------------------- Cl(16) 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 cmul(x, y): out = {} for m1, c1 in x.items(): for m2, c2 in y.items(): s, m = mono_mul(m1, m2) c = out.get(m, Fr(0)) + s * c1 * c2 if c: out[m] = c elif m in out: del out[m] return out def gam(*idx): return {tuple(idx): Fr(1)} ONE = {(): Fr(1)} def refl_map(k, n=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=N): if k % 2 == 0: f1 = (k // 2) % n return [(f1 + j) % n for j in range(1, n // 2)] b = (k + 1) // 2 return [(b + j) % n for j in range(n // 2)] def theta_mono_num(mono, r, s, eta): imgs = [r(a) for a in reversed(mono)] coeff = complex(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 return coeff * sign, tuple(lst) 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 return c * sgn, tuple(lst) TW = tower_maps(N, 1, N) def c_of(d, n=N): if d % 2 == 0: return sp.Integer(0) return sp.Rational(2, n) / sp.sin(sp.pi * sp.Rational(d, n)) CNUM = np.zeros((N, N)) for _a in range(N): for _b in range(N): if _a != _b: CNUM[_a, _b] = float(sp.N(c_of(_a - _b), 20)) def build_gammas(): X = np.array([[0, 1], [1, 0]], dtype=complex) Y = np.array([[0, -1j], [1j, 0]], dtype=complex) Z = np.array([[1, 0], [0, -1]], dtype=complex) E = np.eye(2, dtype=complex) gams = [] for l in range(8): for P in (X, Y): ops = [Z] * l + [P] + [E] * (7 - l) M = ops[0] for o in ops[1:]: M = np.kron(M, o) gams.append(M) return gams GAM = build_gammas() _MM = {(): np.eye(DIM, dtype=complex)} def mono_mat(m): if m not in _MM: M = GAM[m[0]] for a in m[1:]: M = M @ GAM[a] _MM[m] = M return _MM[m] def dict_to_mat(H): M = np.zeros((DIM, DIM), dtype=complex) for m, c in H.items(): M += float(c) * mono_mat(m) return M def build_hfree(): M = np.zeros((DIM, DIM), dtype=complex) for a in range(N): for b in range(a + 1, N): if CNUM[a, b]: M += 0.5j * CNUM[a, b] * (GAM[a] @ GAM[b]) return M def quartet(b): q = ONE for j in (b - 2, b - 1, b, b + 1): q = cmul(q, gam(j % N)) return q def ground(Hm): w, Q = np.linalg.eigh(Hm) return Q[:, 0], float(w[1] - w[0]) def basis_of(k): P = half_of(k) return [()] + [(a,) for a in P] + list(combinations(P, 2)) def gram_forms(x, k, eta, basis, alpha_pm=None): """OS Gram (and optionally the alpha-transport form) on basis.""" r, s = refl_map(k) th = [theta_mono_num(m, r, s, eta) for m in basis] nb = len(basis) G = np.zeros((nb, nb), dtype=complex) T = np.zeros((nb, nb), dtype=complex) if alpha_pm is not None else None for a, (ca, ta) in enumerate(th): wa = mono_mat(ta).conj().T @ x for b in range(nb): G[a, b] = ca * np.vdot(wa, mono_mat(basis[b]) @ x) if alpha_pm is not None: c2, m2 = alpha_mono(basis[b], alpha_pm) T[a, b] = ca * c2 * np.vdot(wa, mono_mat(m2) @ x) return G, T def herm_dev(M): return float(np.max(np.abs(M - M.conj().T))) def inertia(Gh, tol=TOL_ZERO): evs = np.linalg.eigvalsh(Gh) return (int(np.sum(evs > tol)), int(np.sum(evs < -tol)), int(np.sum(np.abs(evs) <= tol))) CUTS = (3, 7, 11, 15) MARKA_SITES = (2, 3, 4, 5) MARKB_SITES = (6, 7, 8, 9) MARKA = [()] for _r in range(1, 5): MARKA += list(combinations(MARKA_SITES, _r)) # ================================================================== I1 print("=" * 72) print("I1: the two parents (chiral and mirror kernels)") print("=" * 72) HF0 = build_hfree() parents = {} for sgn in (1.0, -1.0): x, gap = ground(sgn * HF0) M2 = np.zeros((N, N), dtype=complex) for a in range(N): for b in range(N): M2[a, b] = np.vdot(x, GAM[a] @ GAM[b] @ x) if np.max(np.abs(M2 - (np.eye(N) + 1j * CNUM))) < 1e-9: parents['chiral'] = (sgn, gap) if np.max(np.abs(M2 - (np.eye(N) - 1j * CNUM))) < 1e-9: parents['mirror'] = (sgn, gap) check("I1.1 chiral parent (2-point I + iC, the v519 chiral NS kernel) and " "MIRROR parent (2-point I - iC) identified on opposite sign branches, " "both with gap 1", set(parents) == {'chiral', 'mirror'} and parents['chiral'][0] == -parents['mirror'][0] and abs(parents['chiral'][1] - 1) < 1e-9 and abs(parents['mirror'][1] - 1) < 1e-9) HF = parents['chiral'][0] * HF0 HINT = sum(dict_to_mat(quartet(b)) for b in (0, 4, 8, 12)) # m = 4 survivor HINT2 = sum(dict_to_mat(quartet(b)) for b in (0, 2, 8, 10)) # m = 2 control # ================================================================== I2 + I3 print("=" * 72) print("I2/I3: survivor anchor + the interacting mirror flip") print("=" * 72) anchor_ok = True flip_ok = True nd_worst = -np.inf # largest (closest-to-zero) mirror odd eigenvalue seen dev_worst = 0.0 for g in GRID: xc, _ = ground(HF + g * HINT) xm, _ = ground(-HF + g * HINT) for k in CUTS: basis = basis_of(k) oidx = [i for i, m in enumerate(basis) if len(m) % 2 == 1] eidx = [i for i, m in enumerate(basis) if len(m) % 2 == 0] Gc, _ = gram_forms(xc, k, 1j, basis) Gm, _ = gram_forms(xm, k, 1j, basis) dev_worst = max(dev_worst, herm_dev(Gc), herm_dev(Gm)) Gch = (Gc + Gc.conj().T) / 2 Gmh = (Gm + Gm.conj().T) / 2 anchor_ok &= inertia(Gch) == (37, 0, 0) ie = inertia(Gmh[np.ix_(eidx, eidx)]) io = inertia(Gmh[np.ix_(oidx, oidx)]) flip_ok &= (ie == (29, 0, 0) and io == (0, 8, 0)) nd_worst = max(nd_worst, float(np.linalg.eigvalsh( Gmh[np.ix_(oidx, oidx)]).max())) print(" g = %-6s: chiral PD %s | mirror flip %s (worst odd max %.2e)" % (g, anchor_ok, flip_ok, nd_worst)) check("I2.1 survivor anchor (v534 regression): chiral Grams PD (37,0,0) at " "eta = +i on all four cuts for g in {1/32, 1/4, 1, 8} (Hermiticity " "dev < 1e-8)", anchor_ok and dev_worst < TOL_HERM, "max herm dev = %.1e" % dev_worst) check("I3.1 THE INTERACTING MIRROR FLIP: at the SAME eta the mirror state " "splits sector-exactly on EVERY cut and EVERY g -- even (29,0,0) PD, " "odd (0,8,0) STRICTLY ND", flip_ok) check("I3.2 the mirror odd sector is BOUNDED AWAY from zero over the whole " "grid (max odd eigenvalue <= -1e-3): every mirror fermion vector has " "strictly negative OS norm at the interacting level -- in the doubled " "vector-like system any OS-positive subspace meets the mirror odd " "sector in {0}: kill test (3) cannot fire on the surviving " "interaction class", nd_worst <= -TOL_ND, "worst mirror odd max eigenvalue = %.2e" % nd_worst) # ================================================================== I4 print("=" * 72) print("I4: convention control (eta = -i swaps the roles)") print("=" * 72) g = 0.25 xc, _ = ground(HF + g * HINT) xm, _ = ground(-HF + g * HINT) basis = basis_of(3) oidx = [i for i, m in enumerate(basis) if len(m) % 2 == 1] Gc, _ = gram_forms(xc, 3, -1j, basis) Gm, _ = gram_forms(xm, 3, -1j, basis) Gch = (Gc + Gc.conj().T) / 2 Gmh = (Gm + Gm.conj().T) / 2 check("I4.1 at eta = -i the roles swap exactly: chiral odd flips to (0,8,0) " "and the mirror Gram is PD (37,0,0) -- the datum is the RELATIVE " "chirality; the doubled-system exclusion is convention-free", inertia(Gch[np.ix_(oidx, oidx)]) == (0, 8, 0) and inertia(Gmh) == (37, 0, 0)) # ================================================================== I5 + I6 print("=" * 72) print("I5/I6: interacting mark transport + multiplicity-free spectrum") print("=" * 72) shift_ok = all(set(alpha_mono(m, TW[4])[1]) <= set(MARKB_SITES) for m in MARKA) check("I5.1 the quarter turn maps the mark-A quartet algebra exactly into " "the mark-B sites (alpha_4: {2,3,4,5} -> {6,7,8,9}, 16/16 monomials, " "exact combinatorics)", shift_ok) oidxA = [i for i, m in enumerate(MARKA) if len(m) % 2 == 1] transport_ok = True simple_ok = True min_gap_seen = np.inf spec0 = None for g in (0.0,) + GRID: x, _ = ground(HF + g * HINT) GA, T4 = gram_forms(x, 3, 1j, MARKA, alpha_pm=TW[4]) dev = max(herm_dev(GA), herm_dev(T4)) GAh = (GA + GA.conj().T) / 2 T4h = (T4 + T4.conj().T) / 2 transport_ok &= (dev < TOL_HERM and inertia(GAh) == (16, 0, 0) and inertia(T4h) == (16, 0, 0)) Go = GAh[np.ix_(oidxA, oidxA)] To = T4h[np.ix_(oidxA, oidxA)] L = np.linalg.cholesky(Go) Li = np.linalg.inv(L) lam = np.sort(np.linalg.eigvalsh(Li @ To @ Li.conj().T)) gapmin = float(np.diff(lam).min()) min_gap_seen = min(min_gap_seen, gapmin) simple_ok &= (lam.min() > 0 and gapmin > TOL_GAP) if g == 0.0: spec0 = lam print(" g = %-6s: transport PD %s | odd spectrum min %.4f max %.4f " "min gap %.2e" % (g, transport_ok, lam.min(), lam.max(), gapmin)) check("I5.2 the mark-algebra Gram G_A AND the transport form T_4 are " "Hermitian PD (16,0,0) at EVERY g in {0, 1/32, 1/4, 1, 8}: the " "incidence-compatible mark transport EXISTS on the interacting " "survivor -- kill test (7) does not fire at the interacting toy " "level", transport_ok) check("I6.1 the generalized odd-block transfer spectrum has 8 SIMPLE " "positive eigenvalues at every g (min pairwise gap %.1e > 1e-5); " "honest finding: at g = 8 the eigenvalues pair up to ~2e-5 without " "closing -- a strong-coupling near-doubling tendency, strictly " "simple on the declared grid" % min_gap_seen, simple_ok, "min gap over grid = %.2e" % min_gap_seen) check("I6.2 the g = 0 baseline is locked: top eigenvalue 1.641438, bottom " "0.018673 (observation, no check: the top coincides numerically with " "the v524 compressed-clock top 1.6414 -- the quarter-turn transfer " "datum recurs across N)", abs(spec0[-1] - 1.641438) < 1e-5 and abs(spec0[0] - 0.018673) < 1e-5, "spec0 = [%s]" % ", ".join("%.6f" % v for v in spec0)) # ================================================================== I7 print("=" * 72) print("I7: dead-member control (the gamma battery has no RP home off the " "survivor)") print("=" * 72) x2, _ = ground(HF + 0.5 * HINT2) res = {} for k in (1, 9): basis = basis_of(k) best = None for eta in (1j, -1j): G, _ = gram_forms(x2, k, eta, basis) if herm_dev(G) < TOL_HERM: io = inertia((G + G.conj().T) / 2) if best is None or io[1] < best[1]: best = io res[k] = best check("I7.1 dead-member control (m = 2, s = +, g = 1/2): the straddled cut " "1 is INDEFINITE (21,16,0) while the avoiding cut 9 stays PD " "(37,0,0) -- off the survivor the gamma battery has no RP home: the " "dynamical selection of the bit (v534) and the chirality data " "CO-LOCATE on the same member", res[1] == (21, 16, 0) and res[9] == (37, 0, 0), "cut 1: %s, cut 9: %s" % (res[1], res[9])) # ================================================================== I8 print("=" * 72) print("I8: firewall") print("=" * 72) check("I8.1 [C] FIREWALL: one toy, one interaction class (FK quartets on " "the flat-band parent), deg <= 2 cut bases + the full 16-dim mark " "algebra, finite coupling grid -- NO continuum claim, NO statement " "about A_hol beyond this model class; kill tests (3)/(6)/(7) stay " "formally live on A_hol; kill test (6) (internal SU(2)) is UNTOUCHED " "by this slice; WOIT.OS.TWISTOR.01 does not move", True) # ================================================================== summary print("=" * 72) n_pass = sum(1 for _, ok in CHECKS if ok) print("%d/%d checks passed" % (n_pass, len(CHECKS))) if n_pass == len(CHECKS): print("ALL CHECKS PASSED") print("VERDICT: GAMMA-TOY-LANDED -- on the RP-surviving interaction (the") print("alignment bit) the mirror stays excluded sector-exactly, the") print("generation stays multiplicity-free, and the mark transport stays PD") print("over the whole coupling grid.") else: print("SOME CHECKS FAILED") def run(): """run_all.py entry point; the checks execute at import time above.""" return len([1 for _, ok in CHECKS if not ok]) if __name__ == "__main__": raise SystemExit(1 if run() else 0)