"""v474 -- GRAV.ENTROPIC.HODGE.01: the operator level of the entropic-action bridge (work packages 1 + 4-algebra of the v473 round). v473 pinned the VECTOR-SPACE identity Omega^{0,1,2}(C^5) ~ S+_{D5} (1+5+10 = 16) and the exact Lambda-channel target Tr Q^2 = 32 c3^4; this module upgrades the Hodge count to the OPERATOR level on the finite carrier Fock space Lambda^* C^5 (dim 2^g_car = 32) and solves/kills the Q-block ansatz by exact enumeration. It closes NO gate; the compression conjecture (AP2) and the R^2 mechanism stay open. [E] 1. CAR => CLIFFORD (the spinor structure EXHIBITED). On the 32-dim Fock space Lambda^* C^5 the ten operators Gamma_{2k} = a_k^dag + a_k, Gamma_{2k+1} = i(a_k^dag - a_k) satisfy {Gamma_a, Gamma_b} = 2 delta_ab exactly -- the so(10) = D5 Clifford algebra acts on the carrier exterior algebra (exact integer matrices). [E] 2. S+ = Lambda^even AS A D5-MODULE (operator level). The 45 bilinears M_ab = [Gamma_a, Gamma_b]/4 all commute with the grading involution gamma = (-1)^N, span a 45-dimensional Lie algebra (= dim so(10)), and preserve the 16-dim even subspace: the carrier half-spinor IS the even exterior algebra with its spinor structure -- the v2/v44/v197 count, now with the algebra action checked. [E] 3. HODGE-DIRAC SYMBOL = CLIFFORD MULTIPLICATION. For symbolic real xi, the Dirac-Kaehler symbol c(xi) = a^dag(xi) - a(xi) (= the principal symbol of Bianconi's D = d + delta) satisfies c(xi)^2 = -|xi|^2 * Id exactly on all 32 dimensions -- her Hodge-Dirac operator is Clifford multiplication at symbol level, the operator content behind check 1 of v473. [E] 4. THE FOLD IS THE 5 -> 5bar CONJUGATION. Under the Cartan of u(5) the traceless weights of Lambda^4 are EXACTLY the negatives of those of Lambda^1, so the Hodge fold *: Lambda^1 ~ Lambda^4 conjugates the vector block: Bianconi's fiber 1 + 5 + 10 becomes the GUT half-spinor 16 = 1 + 5bar + 10 -- the fold supplies precisely the conjugation that the SM carrier decomposition needs (not a mere dimension match). [E] 5. HONEST OBSTRUCTION. Omega^{<=2} is NOT Clifford-invariant (c(xi) maps Lambda^2 into Lambda^3), while Lambda^even is invariant under the even Clifford algebra -- so the bridge identification must go through the Hodge fold, never through naive degree truncation. Recorded as a constraint on AP2, not a failure. [E] 6. Q-TARGET ENUMERATION (AP4-algebra; the v473 conjecture DECIDED). The uniform-block ansatz Q = k c3^2 P_d with integer multiplier k solves Tr Q^2 = d k^2 c3^4 = 32 c3^4 for EXACTLY three supports: (d,k) = (2,4), (8,2), (32,1) -- the sheet pair |Z2|, the rank g_car + N_fam = rank E8, and the FULL Fock space 2^g_car = dim Lambda^* C^5 (the minimal uniform solution: eps_* = e^{-1/alpha} c3^2 PER FOCK MODE, q = c3^2 = 1/(64 pi^2)). The two-form block d = 10 (problem-note conjecture 'the pair sector carries it') and the fiber d = 16 REQUIRE irrational multipliers sqrt(16/5), sqrt(2) -- the naive pair-block reading is KILLED. Bonus identity: Tr Q^2 / delta_top = 32/48 = 2/3 = dim Lambda^* / Omega_adm -- the (2/3) in the v473 target is the mode-count ratio. [C] 7. WHAT STAYS OPEN. Which of the three admissible supports is the physical Q (the uniform-32 reading is minimal, not derived); the AP2 compression P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2}; the R^2 mechanism (v473 kill test). No status marker moves. NET TYPING: [E] the five operator/arithmetic facts; [C] the physical Q assignment; [O] unchanged (AP2 + R^2). Extends GRAV.ENTROPIC.ACTION.01 (v473). Mixed exact: integer Fock matrices (numpy, entries in {0,+-1,+-i}, arithmetic exact in double precision) + symbolic symbol identity and weight / enumeration checks (sympy, Wolfram-mirrored where cheap). """ import itertools import numpy as np import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam pi = sp.pi c3 = sp.Rational(1, 8) / pi N = 5 DIM = 2 ** N # 32 SUBSETS = [frozenset(s) for k in range(N + 1) for s in itertools.combinations(range(N), k)] IDX = {s: i for i, s in enumerate(SUBSETS)} def adag(k): """Creation operator e_k ^ . on the Fock basis (exact integer entries).""" M = np.zeros((DIM, DIM), dtype=complex) for s in SUBSETS: if k not in s: sign = (-1) ** sum(1 for j in s if j < k) M[IDX[frozenset(s | {k})], IDX[s]] = sign return M def run(): reset() print("v474 GRAV.ENTROPIC.HODGE.01: operator level of the entropic-action bridge (AP1 + AP4-algebra)") A_DAG = [adag(k) for k in range(N)] A = [m.conj().T for m in A_DAG] ID = np.eye(DIM) # 1. CAR => Clifford: ten gammas, {Gamma_a, Gamma_b} = 2 delta_ab G = [] for k in range(N): G.append(A_DAG[k] + A[k]) G.append(1j * (A_DAG[k] - A[k])) max_err = 0.0 for i in range(10): for j in range(10): anti = G[i] @ G[j] + G[j] @ G[i] max_err = max(max_err, float(np.abs(anti - 2 * (i == j) * ID).max())) check("CAR => CLIFFORD [E]: ten Gamma's from the CAR pair (a_k^dag, a_k) on " "Lambda^* C^5 (dim 2^g_car = %d) satisfy {Gamma_a, Gamma_b} = 2 delta_ab " "for all 100 pairs (max residual %.1e, exact integer arithmetic) -- the " "so(10) = D5 Clifford algebra acts on the carrier exterior algebra" % (DIM, max_err), max_err == 0.0) # 2. so(10) bilinears: parity-preserving, 45-dim span, Lambda^even invariant parity = np.diag([(-1.0) ** len(s) for s in SUBSETS]) bilinears = [(G[i] @ G[j] - G[j] @ G[i]) / 4 for i in range(10) for j in range(i + 1, 10)] commute_par = max(float(np.abs(M @ parity - parity @ M).max()) for M in bilinears) stack = np.array([np.concatenate([M.real.ravel(), M.imag.ravel()]) for M in bilinears]) span_dim = int(np.linalg.matrix_rank(stack, tol=1e-9)) even_idx = [IDX[s] for s in SUBSETS if len(s) % 2 == 0] odd_idx = [IDX[s] for s in SUBSETS if len(s) % 2 == 1] leak = max(float(np.abs(M[np.ix_(odd_idx, even_idx)]).max()) for M in bilinears) check("S+ = LAMBDA^EVEN AS D5-MODULE [E]: the 45 bilinears [Gamma_a,Gamma_b]/4 " "all commute with gamma = (-1)^N (max residual %.1e), span a %d-dim Lie " "algebra = dim so(10) = 45, and leak NOTHING from the 16-dim even " "subspace (max %.1e) -- the carrier half-spinor IS the even exterior " "algebra with its spinor structure (operator level of v2/v44/v197)" % (commute_par, span_dim, leak), commute_par == 0.0 and span_dim == 45 and leak == 0.0 and len(even_idx) == 16) # 3. Hodge-Dirac symbol = Clifford multiplication (symbolic, exact) xi = sp.symbols('xi0:5', real=True) Adag_s = sp.zeros(DIM, DIM) for k in range(N): Adag_s += xi[k] * sp.Matrix(A_DAG[k].real.astype(int)) C = Adag_s - Adag_s.T # a(xi) = a^dag(xi)^T on the real integer matrices resid = sp.expand(C * C + sum(x ** 2 for x in xi) * sp.eye(DIM)) check("HODGE-DIRAC SYMBOL [E]: c(xi) = a^dag(xi) - a(xi) (the principal symbol " "of Bianconi's D = d + delta) satisfies c(xi)^2 = -|xi|^2 Id exactly on " "all 32 dimensions, for SYMBOLIC real xi -- her Hodge-Dirac operator is " "Clifford multiplication at symbol level", resid == sp.zeros(DIM, DIM)) # 4. the fold is the 5 -> 5bar conjugation (traceless u(5) weights) def traceless_weights(deg): out = [] for s in SUBSETS: if len(s) == deg: w = [sp.Integer(1) if i in s else sp.Integer(0) for i in range(N)] shift = sp.Rational(deg, N) out.append(tuple(wi - shift for wi in w)) return sorted(out) w1, w4 = traceless_weights(1), traceless_weights(4) neg_w1 = sorted(tuple(-x for x in w) for w in w1) dims_fiber = [len(traceless_weights(d)) for d in (0, 1, 2)] dims_even = [len(traceless_weights(d)) for d in (0, 2, 4)] check("THE FOLD IS 5 -> 5bar [E]: traceless u(5) weights of Lambda^4 = " "NEGATIVES of those of Lambda^1 (multisets equal), so the Hodge fold " "*: Lambda^1 ~ Lambda^4 conjugates the vector block -- Bianconi's fiber " "1+5+10 %s becomes the GUT half-spinor 16 = 1 + 5bar + 10 %s (the SM " "carrier decomposition), not a mere dimension match" % (dims_fiber, dims_even), w4 == neg_w1 and dims_fiber == [1, 5, 10] and dims_even == [1, 10, 5]) # 5. honest obstruction: Omega^{<=2} not Clifford-invariant c_e4 = A_DAG[4] - A[4] v_l2 = np.zeros(DIM) v_l2[IDX[frozenset({0, 1})]] = 1.0 # a Lambda^2 basis element img = c_e4 @ v_l2 l3_comp = float(np.abs(img[IDX[frozenset({0, 1, 4})]])) even_prod = c_e4 @ (A_DAG[2] - A[2]) even_leak = float(np.abs(even_prod[np.ix_(odd_idx, even_idx)]).max()) check("HONEST OBSTRUCTION [E]: c(e_5) maps Lambda^2 into Lambda^3 (component " "%.0f != 0), so Omega^{<=2} is NOT Clifford-invariant; the EVEN Clifford " "algebra preserves Lambda^even (leak %.1e) -- the bridge identification " "must go through the Hodge fold, never through naive degree truncation " "(a constraint on AP2, recorded)" % (l3_comp, even_leak), l3_comp == 1.0 and even_leak == 0.0) # 6. Q-target enumeration: d * k^2 = 32, integer k target = 32 * c3 ** 4 sols = [(d, sp.sqrt(sp.Rational(32, d))) for d in range(1, 33) if sp.sqrt(sp.Rational(32, d)).is_rational] int_sols = [(d, int(k)) for d, k in sols if k == int(k)] ten_irr = not sp.sqrt(sp.Rational(32, 10)).is_rational sixteen_irr = not sp.sqrt(sp.Rational(32, 16)).is_rational delta_top = 48 * c3 ** 4 ratio = sp.simplify(target / delta_top) q_min = sp.simplify(c3 ** 2) check("Q-TARGET ENUMERATION [E] (AP4-algebra, the v473 conjecture DECIDED): " "Tr Q^2 = d k^2 c3^4 = 32 c3^4 with INTEGER k has exactly the supports " "%s = {|Z2|, g_car+N_fam = rank E8, 2^g_car = dim Lambda^*} -- minimal " "uniform solution Q = c3^2 Id (q = c3^2 = %s per Fock mode); the " "two-form block d = 10 needs sqrt(16/5) and the fiber d = 16 needs " "sqrt(2) (both irrational) => the naive pair-block reading is KILLED; " "bonus: Tr Q^2 / delta_top = %s = 32/48 = dim Lambda^*/Omega_adm" % (int_sols, q_min, ratio), int_sols == [(2, 4), (8, 2), (32, 1)] and ten_irr and sixteen_irr and ratio == sp.Rational(2, 3) and sp.simplify(target - sp.Rational(1, 128) / pi ** 4) == 0 and 2 ** g_car == 32 and g_car + N_fam == 8) # 7. what stays open check("WHAT STAYS OPEN [C/O]: the physical choice among the three admissible " "Q supports (uniform-32 is minimal, not derived); the AP2 compression " "P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2}; the R^2 mechanism " "(v473 kill test). No status marker moves; v359's equation-of-state " "typing stays [O]", True) return summary("v474 GRAV.ENTROPIC.HODGE.01: operator level of the entropic bridge -- [E] " "CAR=>Clifford (10 gammas exact on 32-dim Fock), so(10) span 45 preserving " "Lambda^even=16 (S+ IS the even exterior algebra, operator level), Hodge-Dirac " "symbol c(xi)^2 = -|xi|^2 symbolically, the fold = 5->5bar conjugation " "(1+5+10 -> 1+5bar+10 GUT 16), Omega^{<=2} NOT Clifford-invariant (honest " "obstruction), Q-target d k^2 = 32 solved EXACTLY by supports {2,8,32} atoms " "with minimal uniform q = c3^2 and the 10/16 blocks KILLED (irrational), " "Tr Q^2/delta_top = 2/3 = 32/48; [C] physical Q assignment; [O] AP2 + R^2 open") if __name__ == "__main__": raise SystemExit(1 if run() else 0)