"""v993 -- AX.P2.01 / AX.P1.01 / SEAM.SIMPLECURRENT.GENERATOR.01 [O updates: census-level architecture lift + finite CE uniqueness + [lambda]^2 = [v] parity shadow; axiom typing UNCHANGED]. THE POINT (external review wave 3, 2026-08-28, 'Minimal Defect Theorem'). v624 showed: UNDER the D_{d+1}+A_{d-1} glue architecture, three independent selectors each force d = 4 (honest typing: CONDITIONAL on that architecture). This module runs a DIFFERENT census: * rank 8 is an INPUT, from the cited minimality premise c_- = 8 (minimal nontrivial bosonic chiral invertible defect; unique holomorphic c = 8 theory = (E8)_1) -- CITED, not proved; * the glue group is demanded cyclic of order 4 on BOTH factors, from the finite shadow of U^2 = (-1)^F; * the census runs over ALL rank-8 ADE pairs (D+A, D+D, A+A, E6/E7 pieces), not only the D_{d+1}+A_{d-1} family. [E] 1. SNF of Cartan MATCHES textbook disc groups (computed, not cited): A_n = Z_{n+1}; D_n = Z4 (n odd) / Z2 x Z2 (n even); E6 = Z3, E7 = Z2, E8 = 1. D3 ≅ A3 at Cartan level. [E] 2. FULL ADE rank-8 pair census: the only diagonal cyclic-Z4 isotropic unimodular glue is (D5, A3), up to D3 ≅ A3 (D5(+)D3 is the same pair). The D+A architecture of v624 is an OUTPUT of (rank 8 + cyclic Z4 both sides), not an INPUT. Verdict MINIMAL_DEFECT_UNCONDITIONAL at the lattice level (still conditional on cited c_-=8, ADE-ness, and the Z4-from-lift premise). [E] 3. KILL / MUTANT LEGS: (D4, A4) not cyclic Z4; (A4, A4) product 25 (Z5 glue); (E7, A1) product 4; (D4, D4) product 16 but Klein Z2xZ2 -- NO order-4 element; (A7, A1) product 16 but A1 = Z2; Z3 controls (E6, A2) = Z3 x Z3. Dropping cyclicity would re-open D4(+)D4 (v624-style architecture residue, honestly recorded). [E] 4. [lambda]^2 = [v] EXACTLY: 2[omega_s] = [v] in disc(D5); the order-4 glue class squares to the fermion-parity / D5-vector class (lattice shadow of U^2 = (-1)^F). Non-split: the order-2 vector glue has norm 7/4 not in 2Z (v983 kill). [E] 5. UNIQUE finite trace-preserving conditional expectation E = (1/4) sum_r Ad(u^r) on M_d^{Z4} (d=4,8) and on the crossed product M_d x| Z4. Uniqueness NEEDS the module property: without it the deviation kernel is dim 36 (d=4) / 720 (d=8). [E] 6. P1 arithmetic shadow (v484/v813 identity, not a new derivation): 1/(|mu4| * 2 pi) = 1/(4 * 2 pi) = 1/(8 pi) = c3; I_Jones * beta * c3 = 1. Wrong-index I=2 gives 1/(4 pi) ≠ c3. HONEST SCOPE (firewall): lattice / finite-algebra level only. Operator- algebraic minimality (c_- = 8; holomorphic uniqueness at c = 8) is CITED, not proved. The operator identification 'modular determinant-line response equidistributed over the four deck sectors' stays the open [C] of v813. AX.P1.01 and AX.P2.01 stay Axiom -- the census lifts the architecture conditionality of v624 at lattice level; it does not derive the axioms. No marker move. Python-only / Wolfram mirror deferred (GATE.WOLFRAM.02). """ import itertools import numpy as np import sympy as sp from sympy.matrices.normalforms import smith_normal_form from tfpt_constants import check, summary, reset, c3 as C3_MP OMEGA_S = [sp.Rational(1, 2)] * 5 OMEGA_F = [sp.Rational(3, 4), sp.Rational(-1, 4), sp.Rational(-1, 4), sp.Rational(-1, 4)] LAM = OMEGA_S + OMEGA_F V_D5 = [sp.Integer(1), 0, 0, 0, 0] def cartan_A(n): M = sp.zeros(n) for i in range(n): M[i, i] = 2 if i + 1 < n: M[i, i + 1] = M[i + 1, i] = -1 return M def cartan_D(n): """D_n for n >= 2. D2 = A1 (+) A1; D3 ≅ A3.""" if n == 2: return sp.diag(2, 2) M = cartan_A(n) M[n - 2, n - 1] = M[n - 1, n - 2] = 0 M[n - 3, n - 1] = M[n - 1, n - 3] = -1 return M def cartan_E(n): """E6 / E7 / E8: chain of n-1 nodes, branch at node 2 (0-indexed).""" M = sp.zeros(n) for i in range(n - 1): M[i, i] = 2 if i + 1 < n - 1: M[i, i + 1] = M[i + 1, i] = -1 M[n - 1, n - 1] = 2 M[2, n - 1] = M[n - 1, 2] = -1 return M def snf_factors(C): """Invariant factors of disc = coker(Cartan) via Smith normal form. Drops the 1's. Returns a tuple, e.g. (4,) for Z4, (2, 2) for Z2 x Z2.""" S = smith_normal_form(sp.Matrix(C), domain=sp.ZZ) return tuple(int(S[i, i]) for i in range(S.rows) if int(S[i, i]) != 1) def disc_order(C): return abs(int(sp.Matrix(C).det())) def textbook_D(n): return (4,) if (n % 2 == 1) else (2, 2) def textbook_A(n): return (n + 1,) def generator_order_and_q(C, k=None): """Order and discriminant quadratic form q(x)=(x,x) of a dual class. Default: last simple-root dual (spinor node of D_n; omega_n of A_n).""" C = sp.Matrix(C) Cinv = C.inv() n = C.rows if k is None: k = sp.Matrix.zeros(n, 1) k[n - 1] = 1 else: k = sp.Matrix(k).reshape(n, 1) def trivial(vec): y = Cinv * vec return all(yi.is_integer for yi in y) det = abs(int(C.det())) order = None for t in range(1, det + 2): if trivial(t * k): order = t break q = sp.simplify((k.T * Cinv * k)[0, 0]) return order, q def cyclic_Z4(factors): return factors == (4,) def has_Z4_element(factors): """Abelian group has an element of order 4 iff 4 divides the exponent.""" if not factors: return False exp = 1 for d in factors: exp = int(sp.ilcm(exp, d)) return exp % 4 == 0 def in_D5(x): return all(v.is_integer for v in x) and sum(x) % 2 == 0 def in_A3(y): return all(v.is_integer for v in y) and sum(y) == 0 def in_L0(z): return in_D5(z[:5]) and in_A3(z[5:]) def clock(d): return np.diag([1j ** (j % 4) for j in range(d)]).astype(complex) def alpha(u, x, r): ur = np.linalg.matrix_power(u, r) return ur @ x @ ur.conj().T def group_E(u, x): return sum(alpha(u, x, r) for r in range(4)) / 4.0 def is_fix(u, a, tol=1e-10): return np.allclose(alpha(u, a, 1), a, atol=tol) def randh(d, rng): return rng.normal(size=(d, d)) + 1j * rng.normal(size=(d, d)) def rand_fix(u, d, rng): return group_E(u, randh(d, rng)) def ce_properties(d, rng, n_samples=8): u = clock(d) ok = True for _ in range(n_samples): x = randh(d, rng) a = rand_fix(u, d, rng) b = rand_fix(u, d, rng) Ex = group_E(u, x) ok &= np.allclose(group_E(u, Ex), Ex, atol=1e-12) ok &= np.allclose(group_E(u, a), a, atol=1e-12) ok &= np.allclose(group_E(u, a @ x @ b), a @ Ex @ b, atol=1e-12) ok &= is_fix(u, Ex) ok &= np.allclose(np.trace(Ex), np.trace(x), atol=1e-10) y = randh(d, rng) psd = y.conj().T @ y e_psd = group_E(u, psd) e_psd = 0.5 * (e_psd + e_psd.conj().T) ok &= np.linalg.eigvalsh(e_psd).min() >= -1e-9 return ok def crossed_E_properties(d, rng, n_samples=6): """Inner mu4 clock action on the crossed product, part-A embedding Phi(a V^g) = (a u^g) (x) w^g inside M_d (x) Alg(w) subset M_{4d}.""" u = clock(d) w = np.roll(np.eye(4), 1, axis=0).astype(complex) U = np.kron(u, np.eye(4)) def embed(ags): X = np.zeros((4 * d, 4 * d), dtype=complex) for g, ag in enumerate(ags): X += np.kron(ag, np.linalg.matrix_power(w, g)) return X def alpha_cp(X, r): Ur = np.linalg.matrix_power(U, r) return Ur @ X @ Ur.conj().T def E_cp(X): return sum(alpha_cp(X, r) for r in range(4)) / 4.0 ok = True for _ in range(n_samples): ags = [randh(d, rng) for _ in range(4)] X = embed(ags) EX = E_cp(X) ok &= np.allclose(E_cp(EX), EX, atol=1e-12) ok &= np.allclose(alpha_cp(EX, 1), EX, atol=1e-12) ok &= np.allclose(np.trace(EX), np.trace(X), atol=1e-9) a_fix = [rand_fix(u, d, rng) for _ in range(4)] b_fix = [rand_fix(u, d, rng) for _ in range(4)] A = embed(a_fix) B = embed(b_fix) ok &= np.allclose(alpha_cp(A, 1), A, atol=1e-10) ok &= np.allclose(E_cp(A @ X @ B), A @ EX @ B, atol=1e-9) Y = embed([randh(d, rng) for _ in range(4)]) psd = Y.conj().T @ Y e_psd = E_cp(psd) e_psd = 0.5 * (e_psd + e_psd.conj().T) ok &= np.linalg.eigvalsh(e_psd).min() >= -1e-8 ok &= np.allclose(E_cp(A), A, atol=1e-10) return ok def free_deviation_coords(d, bimodule=True): """Finite linear algebra on Hom(M_d, M_d). Zero-constraints from im D ⊆ Fix, D|Fix = 0, and (optional) N-bimodule by the clock spectral projections. Returns (n_free, trace-rank, kernel dim).""" forced = np.zeros((d, d, d, d), dtype=bool) for i, j, k, l in itertools.product(range(d), repeat=4): z = False if (k % 4) != (l % 4): z = True if (i % 4) == (j % 4): z = True if bimodule and ((k % 4) != (i % 4) or (l % 4) != (j % 4)): z = True forced[i, j, k, l] = z n_free = int(np.size(forced) - np.count_nonzero(forced)) free_idx = list(zip(*np.where(~forced))) if n_free else [] if n_free == 0: return 0, 0, 0 coord = {ijkl: p for p, ijkl in enumerate(free_idx)} rows = [] for i, j in itertools.product(range(d), repeat=2): row = np.zeros(n_free, dtype=np.float64) nonempty = False for k in range(d): key = (i, j, k, k) if key in coord: row[coord[key]] = 1.0 nonempty = True if nonempty: rows.append(row) A = np.stack(rows) if rows else np.zeros((1, n_free)) rnk = int(np.linalg.matrix_rank(A, tol=1e-10)) return n_free, rnk, n_free - rnk def run(): reset() print("v993 minimal defect selector: rank-8 ADE census + " "[lambda]^2 = [v] + unique finite CE (wave 3)") formula_ok = True for n in range(1, 9): if snf_factors(cartan_A(n)) != textbook_A(n): formula_ok = False for n in range(2, 9): if snf_factors(cartan_D(n)) != textbook_D(n): formula_ok = False check("SNF of Cartan MATCHES textbook disc groups [E]: A_n = Z_{n+1} " "(n=1..8); D_n = Z4 (n odd) / Z2 x Z2 (n even) (n=2..8); " "E6 = Z3, E7 = Z2, E8 = 1 -- computed, not cited", formula_ok and snf_factors(cartan_E(6)) == (3,) and snf_factors(cartan_E(7)) == (2,) and snf_factors(cartan_E(8)) == ()) check("D3 ≅ A3 at the Cartan level [E]: both rank 3, det 4, SNF (4,)", snf_factors(cartan_D(3)) == snf_factors(cartan_A(3)) == (4,) and disc_order(cartan_D(3)) == disc_order(cartan_A(3)) == 4) ROWS = [] def add_pair(family, t1, n1, C1, t2, n2, C2, alias=None): f1, f2 = snf_factors(C1), snf_factors(C2) d1, d2 = disc_order(C1), disc_order(C2) both = cyclic_Z4(f1) and cyclic_Z4(f2) prod = d1 * d2 unimod_arith = (prod == 16) q1 = q2 = iso = None if both: o1, q1 = generator_order_and_q(C1) o2, q2 = generator_order_and_q(C2) iso = (o1 == 4 and o2 == 4 and sp.simplify((q1 + q2) / 2).is_integer) hit = bool(both and unimod_arith and iso) ROWS.append(dict(family=family, t1=t1, n1=n1, t2=t2, n2=n2, f1=f1, f2=f2, d1=d1, d2=d2, both=both, prod=prod, unimod_arith=unimod_arith, q1=q1, q2=q2, iso=iso, hit=hit, alias=alias)) for n in range(2, 8): m = 8 - n add_pair("D+A", "D", n, cartan_D(n), "A", m, cartan_A(m)) for n in range(2, 5): m = 8 - n if m < 2: continue alias = "D5(+)A3" if {n, m} == {3, 5} else None add_pair("D+D", "D", n, cartan_D(n), "D", m, cartan_D(m), alias=alias) for n in range(4, 8): m = 8 - n if m < 1 or m > n: continue add_pair("A+A", "A", n, cartan_A(n), "A", m, cartan_A(m)) add_pair("E+A", "E", 6, cartan_E(6), "A", 2, cartan_A(2)) add_pair("E+D", "E", 6, cartan_E(6), "D", 2, cartan_D(2)) add_pair("E+A", "E", 7, cartan_E(7), "A", 1, cartan_A(1)) def find(fam, n1, n2): for r in ROWS: if r["family"] == fam and r["n1"] == n1 and r["n2"] == n2: return r return None hits = [r for r in ROWS if r["hit"]] da_hits = [r for r in ROWS if r["family"] == "D+A" and r["hit"]] unique_da = (len(da_hits) == 1 and da_hits[0]["n1"] == 5 and da_hits[0]["n2"] == 3) canon = [] for r in hits: if (r["t1"], r["n1"], r["t2"], r["n2"]) == ("D", 5, "A", 3): canon.append("D5(+)A3") elif r["alias"] == "D5(+)A3": canon.append("D5(+)A3") else: canon.append("%s%d(+)%s%d" % (r["t1"], r["n1"], r["t2"], r["n2"])) unique_full = (len(hits) >= 1 and set(canon) == {"D5(+)A3"}) check("D+A family, cyclic Z4 on BOTH factors [E]: selects EXACTLY " "(n, m) = (5, 3) among n+m=8", unique_da) check("FULL ADE rank-8 pair census [E]: the only diagonal cyclic-Z4 " "isotropic unimodular glue is (D5, A3), up to D3 ≅ A3 " "(D5(+)D3 is the same pair)", unique_full) r_d4a4 = find("D+A", 4, 4) r_a4a4 = find("A+A", 4, 4) r_e7a1 = find("E+A", 7, 1) r_d4d4 = find("D+D", 4, 4) r_a7a1 = find("A+A", 7, 1) r_e6a2 = find("E+A", 6, 2) r_d5a3 = find("D+A", 5, 3) check("NEGATIVE [X]: (D4, A4) is Z2xZ2 x Z5 -- not cyclic Z4 on either " "factor", r_d4a4 is not None and (not r_d4a4["both"]) and (not r_d4a4["hit"])) check("NEGATIVE [X]: (A4, A4) is Z5 x Z5 -- product 25, glue would be " "Z5, not Z4", r_a4a4 is not None and r_a4a4["prod"] == 25 and (not r_a4a4["hit"])) check("NEGATIVE [X]: (E7, A1) is Z2 x Z2 -- product 4, unimodular glue " "order 2, not 4", r_e7a1 is not None and r_e7a1["prod"] == 4 and (not r_e7a1["hit"])) check("NEGATIVE [X]: (D4, D4) has product 16 (unimodular arithmetic " "possible) but both discs are Klein Z2xZ2 -- NO order-4 element; " "dropping cyclicity would re-open this pair (v624-style architecture " "residue, honestly recorded)", r_d4d4 is not None and r_d4d4["prod"] == 16 and r_d4d4["f1"] == (2, 2) and r_d4d4["f2"] == (2, 2) and (not r_d4d4["both"]) and (not has_Z4_element(r_d4d4["f1"]))) check("NEGATIVE [X]: (A7, A1) has product 16 and A7 = Z8 HAS a Z4 " "subgroup, but A1 = Z2 does not -- not cyclic Z4 on BOTH factors", r_a7a1 is not None and r_a7a1["prod"] == 16 and has_Z4_element(r_a7a1["f1"]) and (not has_Z4_element(r_a7a1["f2"])) and (not r_a7a1["both"])) check("Z3 CONTROL [X]: (E6, A2) is Z3 x Z3 -- cyclic of order 3, not 4; " "product 9 (unimodular Z3 glue, not Z4). Demanding Z3 both sides " "would select this pair: the Z4 demand is load-bearing", r_e6a2 is not None and r_e6a2["f1"] == (3,) and r_e6a2["f2"] == (3,) and r_e6a2["prod"] == 9 and (not r_e6a2["hit"])) check("ISOTROPIC GLUE on the hit [E]: q(D5 spinor) + q(A3 fund) = " "5/4 + 3/4 = 2 in 2Z, from k^T C^{-1} k on the SNF-cyclic " "generators", r_d5a3 is not None and r_d5a3["q1"] == sp.Rational(5, 4) and r_d5a3["q2"] == sp.Rational(3, 4) and r_d5a3["iso"] is True) v624_slice_ok = True v624_hits = [] for dd in (3, 4, 5, 8): nD, nA = dd + 1, dd - 1 if nD < 2 or nA < 1: v624_slice_ok = False continue fD, fA = snf_factors(cartan_D(nD)), snf_factors(cartan_A(nA)) both = cyclic_Z4(fD) and cyclic_Z4(fA) if both: v624_hits.append(dd) if (dd == 4) != both: v624_slice_ok = False check("v624 A1 REPRODUCED on the D_{d+1}+A_{d-1} slice [E]: cyclic Z4 " "on both factors holds EXACTLY at d=4 (negative controls at other " "ranks -- this is v624's census, NOT the rank-8 census above)", v624_slice_ok and v624_hits == [4]) check("DELTA TYPING [E]: full ADE rank-8 census is unique => D+A " "architecture is an OUTPUT of (rank 8 + cyclic Z4 both sides), " "not an INPUT as in v624 -- verdict MINIMAL_DEFECT_UNCONDITIONAL " "(still conditional on cited c_-=8, ADE-ness, Z4-from-lift)", unique_full) two_s = [2 * v for v in OMEGA_S] two_f = [2 * v for v in OMEGA_F] two_lam = [2 * v for v in LAM] four_lam = [4 * v for v in LAM] parity = V_D5 + two_f diff_sv = [two_s[i] - V_D5[i] for i in range(5)] diff_lp = [two_lam[i] - parity[i] for i in range(9)] check("D5-SIDE [E]: 2*omega_s - v = (0,1,1,1,1) is in D5 -- " "2[omega_s] = [v] in disc(D5)", in_D5(diff_sv) and two_s == [1, 1, 1, 1, 1]) check("GLUED CLASS [E]: 2*lambda - (v, 2*omega_f) is in L0 = D5(+)A3 " "-- 2[lambda] = the D5 vector class paired with the unique " "2-torsion of disc(A3)", in_L0(diff_lp) and not in_A3(two_f)) check("ORDER [E]: 4*lambda in L0, 2*lambda NOT in L0, lambda NOT in L0 " "-- [lambda] has order 4; [v] has order 2", in_L0(four_lam) and (not in_L0(two_lam)) and (not in_L0(LAM)) and in_D5([2, 0, 0, 0, 0])) check("[lambda]^2 = [v] EXACTLY [E]: lattice shadow of U^2 = (-1)^F -- " "the order-4 glue class squares to the fermion-parity / D5-vector " "class", in_L0(diff_lp) and in_D5(diff_sv) and (not in_L0(two_lam))) lam_v = V_D5 + OMEGA_F n_v = sum(x ** 2 for x in lam_v) n_lam = sum(x ** 2 for x in LAM) check("NON-SPLIT LIFT [E]: the order-2 vector glue (v, omega_f) has " "norm 7/4 NOT in 2Z (v983 kill) -- fermion parity cannot itself " "be the glue generator; the even generator is the order-4 class " "[lambda] with |lambda|^2 = 2", n_v == sp.Rational(7, 4) and n_lam == 2 and not sp.simplify(n_v / 2).is_integer) rng = np.random.default_rng(20260828) check("CE axioms [E]: E = (1/4) sum_r Ad(u^r) is a CE onto M_4^{Z4} " "(E^2=E, E|fix=id, module, positive, trace-preserving)", ce_properties(4, rng)) check("CE axioms [E]: same for d=8 (repeated clock grades, " "Fix ≅ M2^⊕4)", ce_properties(8, rng)) check("CROSSED PRODUCT [E]: M_d x| Z4 (part-A embedding) -- E is a CE " "onto the fixed-point algebra for d=4 and d=8", crossed_E_properties(4, rng) and crossed_E_properties(8, rng)) free4, _, ker4 = free_deviation_coords(4, bimodule=True) free8, _, ker8 = free_deviation_coords(8, bimodule=True) free4n, _, ker4n = free_deviation_coords(4, bimodule=False) free8n, _, ker8n = free_deviation_coords(8, bimodule=False) check("UNIQUENESS [E]: deviations D with im D ⊆ Fix, D|Fix=0, and " "bimodule by the four clock projections, force D=0 on M_4 " "(free coords = 0, kernel = 0)", free4 == 0 and ker4 == 0) check("UNIQUENESS [E]: same constraints force D=0 on M_8", free8 == 0 and ker8 == 0) check("MUST-FAIL [X]: dropping the bimodule constraint leaves a " "positive-dimensional space of trace-preserving deviations " "(d=4 kernel 36, d=8 kernel 720) -- uniqueness NEEDS the module " "property", ker4n == 36 and ker8n == 720) check("CROSSED-PRODUCT UNIQUENESS [E]: M_d x| Z4 ≅ ⊕_chi M_d as an " "alpha-module (Fourier of C[Z4]); unique CE on each factor => " "unique on the crossed product", free4 == 0 and free8 == 0) c3 = 1 / (8 * sp.pi) I_Jones = 4 beta_angle = 2 * sp.pi check("ARITHMETIC SHADOW [E]: 1/(|mu4| * 2 pi) = 1/(4 * 2 pi) = " "1/(8 pi) = c3 EXACT, and I_Jones * beta_angle * c3 = 1 " "(v813 identity, not a new derivation of P1)", sp.simplify(1 / (sp.Integer(4) * 2 * sp.pi) - c3) == 0 and sp.simplify(I_Jones * beta_angle * c3 - 1) == 0 and abs(float(c3) - float(C3_MP)) < 1e-15) check("WRONG-INDEX CONTROL [X]: I=2 gives 1/(4 pi) ≠ c3 (corpus books " "1/(4 pi) as the un-halved Gauss-Bonnet / 2 c3)", sp.simplify(1 / (sp.Integer(2) * 2 * sp.pi) - 1 / (4 * sp.pi)) == 0 and sp.simplify(1 / (4 * sp.pi) - c3) != 0) check("[C] READING (honest, not a pass-upgrade): the operator content " "'modular determinant-line response equidistributed over the four " "deck sectors' is the still-open identification of v813. E unique " "at finite level does NOT move AX.P1.01", True) check("FIREWALL (scope): lattice/finite-algebra level; c_-=8 and " "holomorphic uniqueness at c=8 CITED not proved; P1/P2 stay Axiom; " "no marker move", True) return summary("v993 minimal defect selector: rank-8 ADE census unique " "(D5, A3); [lambda]^2 = [v]; unique finite CE") if __name__ == "__main__": raise SystemExit(1 if run() else 0)