"""v160 -- Seam Gaussianity from the admissible fixed point: the CONDITIONAL theorem, and the one premise it really rests on. This module formalises the proposed closure of premise (A) ("the seam bulk is a reflection-positive free/Gaussian field") as a fixed-point statement: RP seam kernel + gapped PF transport + QBL (one kernel = the net) => quasi-free seam (all higher cumulants vanish). The structure of the argument is exact and machine-checkable in the finite model; what it does NOT do, on the current evidence, is close (A). It RELOCATES (A) to a single, sharper, internal statement -- the honest content of this module: [P] THE SCOPE PREMISE. The gapped Perron-Frobenius uniqueness proven in v56 holds on the THREE-dimensional scalar transfer spectrum {1,(2/3)^6,(1/3)^6} (the flavor gap / horizon Page-recovery rates), NOT on the cone of OS-positive Schwinger functionals (the states whose higher connected cumulants kappa_{2n} this theorem must contract). Closing (A) requires extending v56's primitivity + spectral gap from that 3-dim readout space to the full correlation cone. That extension is NOT established here; it is the entire remaining load. So v160 is a [P]/[C] reframing module in the spirit of v157/v158: the parts that PASS are the exact algebra (quasi-free => kappa_{2n}=0; Wick/Pfaffian functoriality => the quasi-free state is a transport fixed point; the PF dichotomy on a toy cone; the dimension gap between v56's space and the cone); the part that is recorded-not-proven is the scope premise. Python-only (numpy/sympy + stdlib); no exact-identity content for the Wolfram mirror. [I] 1. QUASI-FREE => ALL HIGHER CONNECTED CUMULANTS VANISH. For a fermionic Gaussian (quasi-free) state with covariance Gamma the 2n-point moment is the Pfaffian of the 2n x 2n submatrix; the moment<->cumulant (set-partition Moebius) inversion then gives kappa(S) = 0 for every |S| >= 3 and kappa(S) = Gamma_ij for |S| = 2 -- verified exactly on a concrete pure Majorana covariance for ALL 4- and 6-point subsets. "The Wick state is Gaussian" is therefore not an assumption but an identity of the one 2-point kernel (the v113 'one kernel = net' property, restated at the level of cumulants). [I] 2. THE QUASI-FREE STATE IS A TRANSPORT FIXED POINT (functoriality). A kernel-preserving transport O.Gamma.O^T = Gamma fixes every correlator, because each correlator is a Pfaffian of (a submatrix of) Gamma -- so if the degree-2 datum K_Sigma is fixed, the whole quasi-free state is fixed. Checked: an O commuting with Gamma leaves all 4- and 6-point functions invariant. [I] 3. THE PERRON-FROBENIUS DICHOTOMY (toy cone). A primitive positive map with a spectral gap has a UNIQUE fixed ray; any state component off the fixed ray decays geometrically at the sub-leading rate. Hence a non-Gaussian piece kappa_{2n} != 0 is impossible: if it were fixed it would be a SECOND fixed ray (kills uniqueness); if not fixed it lies in the gapped complement and decays to 0. Demonstrated on a 5-dim toy transport: gapped => 1-dim fixed space (unique); a degenerate variant with a second unit eigenvalue => 2-dim fixed space (a non-Gaussian fixed datum <=> loss of uniqueness). [I] 4. THE QUALITATIVE CLAIM NEEDS ONLY gap > 0. kappa_{2n} -> 0 follows from ANY sub-leading rate < 1; the specific value Delta = 6 log(3/2) (i.e. lambda2 = (2/3)^6) only sets the DECAY RATE. Carrying that scalar value into the higher-cumulant sectors is an over-extension unless those sectors are shown to sit in the gap -- see point 5. [P] 5. THE SCOPE PREMISE (recorded, NOT proven -- the whole remaining load). v56 establishes primitivity + gap on a 3-dim space (len(spec) = 3). The quasi-free seam state already has C(16,4) = 1820 degree-4 and C(16,6) = 8008 degree-6 connected-cumulant directions: the cone the transport must act on (primitively, with a gap) to force kappa_{2n}=0 has dimension >> 3. PF uniqueness on the 3-dim readout spectrum does NOT, as a matter of dimension, imply PF uniqueness on the correlation cone. NON-CIRCULARITY: assumption "unique fixed ray" must hold on the FULL cone (incl. non-Gaussian functionals); if it held only on the quasi-free sub-cone the argument would assume what it proves. THIS is the equivalent open statement (A) reduces to: "the admissible TFPT seam transport is a primitive, spectrally gapped, RP-cone-preserving map on the full Schwinger cone -- not only on the scalar readout spectrum of v56." Smaller and sharper than 'prove Gaussianity', but unproven. """ import math from itertools import combinations import numpy as np from tfpt_constants import check, summary, reset, g_car, N_fam def _pure_cov(n, seed): """A pure fermionic Gaussian covariance Gamma = Q J Q^T (real antisymmetric, i*Gamma a contraction with Gamma^2 = -I).""" rng = np.random.default_rng(seed) Q, _ = np.linalg.qr(rng.standard_normal((n, n))) J = np.kron(np.eye(n // 2), np.array([[0.0, 1.0], [-1.0, 0.0]])) G = Q @ J @ Q.T return (G - G.T) / 2 def _pfaffian(G, idx): """Pfaffian of the antisymmetric submatrix on the ordered index tuple idx.""" if len(idx) == 0: return 1.0 if len(idx) % 2 == 1: return 0.0 head, rest = idx[0], idx[1:] total = 0.0 for t, j in enumerate(rest): sub = tuple(x for x in rest if x != j) total += (-1) ** t * G[head, j] * _pfaffian(G, sub) return total def _set_partitions(items): """Yield every set partition of the tuple `items`; each block is an ascending tuple (items must be ascending).""" items = list(items) if not items: yield [] return first, rest = items[0], items[1:] for part in _set_partitions(rest): yield [(first,)] + part # first as its own block for i in range(len(part)): # first prepended to a block yield part[:i] + [(first,) + part[i]] + part[i + 1:] def _concat_sign(S, concat): """Parity (+/-1) of the permutation taking the ascending tuple S to `concat` (a reordering of S's elements) -- the fermionic Pfaffian/Grassmann sign.""" pos = {x: i for i, x in enumerate(S)} seq = [pos[x] for x in concat] inv = sum(1 for i in range(len(seq)) for j in range(i + 1, len(seq)) if seq[i] > seq[j]) return -1.0 if inv % 2 else 1.0 def _cumulants(G, max_size): """Connected (fermionic) cumulants kappa(S) from the moments m(S) = Pf(G[S]) by the SIGNED set-partition moment<->cumulant inversion m(S) = sum_pi sgn(pi) prod_B kappa(B). Odd blocks give kappa = 0 (no odd correlators); even blocks commute so the block order is immaterial. Returns {S (ascending tuple): value}.""" kap = {} def kappa(S): if S in kap: return kap[S] val = _pfaffian(G, S) for part in _set_partitions(S): if len(part) == 1: # the trivial {S} block = kappa(S) continue blocks = sorted(part, key=lambda b: b[0]) concat = tuple(x for b in blocks for x in b) prod = 1.0 for b in blocks: prod *= kappa(b) val -= _concat_sign(S, concat) * prod kap[S] = val return val n = G.shape[0] for size in range(1, max_size + 1): for S in combinations(range(n), size): kappa(S) return kap def run(): reset() print("v160 seam gaussianity from the admissible fixed point " "(conditional theorem; the load is the scope premise)") # ---- 1. quasi-free => higher connected cumulants vanish (exact) ---- n = 8 G = _pure_cov(n, 7) pure = np.allclose((1j * G) @ (1j * G), np.eye(n)) kap = _cumulants(G, 6) k2_ok = all(abs(kap[p] - G[p[0], p[1]]) < 1e-9 for p in combinations(range(n), 2)) k4_max = max(abs(kap[s]) for s in combinations(range(n), 4)) k6_max = max(abs(kap[s]) for s in combinations(range(n), 6)) check("QUASI-FREE => kappa_{2n}=0 for n>=2: with the 2n-point moment = " "Pf(Gamma submatrix), the set-partition moment<->cumulant inversion " "gives kappa(|S|=2) = Gamma_ij and kappa(|S|=4) = kappa(|S|=6) = 0 " "EXACTLY (concrete pure Majorana covariance, all subsets) -- 'the " "Wick state is Gaussian' is an identity of the one 2-point kernel", pure and k2_ok and k4_max < 1e-9 and k6_max < 1e-9) # ---- 2. kernel-preserving transport => the quasi-free state is fixed ---- # an orthogonal O commuting with Gamma fixes the kernel, hence every Pfaffian w, V = np.linalg.eig(1j * G) # i*Gamma is Hermitian, real spectrum +-1 theta = 0.37 # rotation inside each conjugate eigichpair keeps i*Gamma (hence Gamma) fixed O = np.real(V @ np.diag(np.exp(1j * theta * np.sign(w.real))) @ np.linalg.inv(V)) Gt = O @ G @ O.T kernel_fixed = np.allclose(Gt, G, atol=1e-9) inv4 = max(abs(_pfaffian(Gt, s) - _pfaffian(G, s)) for s in combinations(range(n), 4)) inv6 = max(abs(_pfaffian(Gt, s) - _pfaffian(G, s)) for s in combinations(range(n), 6)) check("FIXED POINT BY FUNCTORIALITY: a kernel-preserving transport " "O.Gamma.O^T = Gamma fixes EVERY correlator (each is a Pfaffian of a " "Gamma submatrix) -- so fixing the degree-2 datum K_Sigma fixes the " "whole quasi-free state; all 4- and 6-point functions invariant", kernel_fixed and inv4 < 1e-9 and inv6 < 1e-9) # ---- 3. the Perron-Frobenius dichotomy on a toy cone ---- lam2 = (2.0 / 3.0) ** 6 spec = np.array([1.0, lam2, lam2 ** 2, lam2 ** 3, lam2 ** 4]) # gap: 1 then <1 M = np.array([[1, 1, 1, 1, 1], [0, 1, 2, 3, 4], [0, 0, 1, 3, 6], [0, 0, 0, 1, 4], [0, 0, 0, 0, 1]], float) # invertible T = M @ np.diag(spec) @ np.linalg.inv(M) fixed_dim = 5 - np.linalg.matrix_rank(T - np.eye(5), tol=1e-9) # degenerate variant: a SECOND unit eigenvalue (a 'fixed kappa_{2n}') Tdeg = M @ np.diag([1.0, 1.0, lam2 ** 2, lam2 ** 3, lam2 ** 4]) @ np.linalg.inv(M) fixed_dim_deg = 5 - np.linalg.matrix_rank(Tdeg - np.eye(5), tol=1e-9) check("PF DICHOTOMY: a gapped primitive transport has a UNIQUE fixed ray " "(fixed-space dim 1); a non-Gaussian fixed datum would be a SECOND " "fixed ray (degenerate variant => fixed-space dim 2) and break " "uniqueness -- 'kappa_{2n} fixed => contradiction'", fixed_dim == 1 and fixed_dim_deg == 2) # off-fixed-ray components (the non-fixed cumulant modes) live in the gapped # complement; in the eigenbasis they are scaled by spec[1:] (all < 1) and # decay geometrically at the sub-leading rate lambda2 = (2/3)^6 nonfixed = spec[1:] # [lam2, lam2^2, lam2^3, lam2^4] x = np.ones(4) after = (nonfixed ** 12) * x decayed = np.linalg.norm(after) <= (lam2 ** 12) * np.linalg.norm(x) * (1 + 1e-6) check("PF DICHOTOMY: an off-fixed-ray component (a non-fixed kappa_{2n}) " "lies in the gapped complement (eigenvalues spec[1:] all < 1) and " "decays geometrically at the sub-leading rate lambda2 = (2/3)^6 -> 0 " "in the IR -- 'kappa_{2n} not fixed => contracted away by the gap'", decayed and float(nonfixed.max()) < 1.0) # ---- 4. the qualitative claim needs only gap > 0 (not the value) ---- decays_for_any_gap = True for lam in (0.9, 0.5, 0.1): nf = np.array([lam, lam ** 2, lam ** 3, lam ** 4]) decays_for_any_gap &= (np.linalg.norm((nf ** 200) * x) < 1e-6) check("QUALITATIVE GAUSSIANITY NEEDS ONLY gap>0: kappa_{2n}->0 for ANY " "sub-leading rate lambda2 in (0,1); the specific Delta=6 log(3/2) " "((2/3)^6) only sets the DECAY RATE -- so carrying that scalar value " "into the higher-cumulant sectors is an over-extension unless they " "are shown to sit in the gap", decays_for_any_gap) # ---- 5. THE SCOPE PREMISE: v56 acts on 3 scalars, the cone is huge ---- v56_dim = 3 # len({1,(2/3)^6,(1/3)^6}) seam_majoranas = 2 * (g_car + N_fam) # 16 = 2 * rank E8 deg4 = math.comb(seam_majoranas, 4) # 1820 degree-4 cumulant directions deg6 = math.comb(seam_majoranas, 6) # 8008 degree-6 check("THE SCOPE GAP IS REAL: v56's gapped PF uniqueness lives on a " "3-dim scalar transfer spectrum {1,(2/3)^6,(1/3)^6}; the quasi-free " "16-Majorana seam state already has C(16,4)=1820 degree-4 and " "C(16,6)=8008 degree-6 connected-cumulant directions -- the " "correlation cone the transport must contract has dimension >> 3, so " "PF uniqueness on the readout space does NOT imply it on the cone", v56_dim == 3 and seam_majoranas == 16 and deg4 == 1820 and deg6 == 8008 and deg4 > v56_dim) check("THE ONE REMAINING PREMISE [P] (recorded, NOT proven): (A) is " "equivalent to 'the admissible TFPT seam transport is a primitive, " "spectrally gapped, RP-cone-preserving map on the FULL Schwinger " "cone, not only on the scalar readout spectrum of v56'. Given it, " "steps 1-3 close (A) (quasi-free fixed point unique => kappa_{2n}=0 " "=> carrier net A => (E8)1, v113/v154/v156); non-circular only if " "uniqueness holds on the full cone (incl. non-Gaussian functionals). " "This is a strictly smaller, internal, exact-testable statement than " "'prove Gaussianity' -- but it is the entire remaining load, and v56 " "as it stands does not supply it", True) return summary("v160 seam gaussianity from the admissible fixed point") if __name__ == "__main__": raise SystemExit(1 if run() else 0)