"""v298 -- E8 AS A NETWORK ATTRACTOR: a computational-substrate reading of the (2,3,5) hull (the Wolfram-program connection), built on the McKay bedrock v219. It tests HONESTLY which "TFPT is a hypergraph/rewrite system" claims are realised in TFPT's actual structure and which stay open -- it does NOT force TFPT into a hypergraph nor fabricate a rewrite system. Background: v219 already shows E8 is the McKay graph of the binary icosahedral group 2I (the (2,3,5) hull), with the irrep degrees = the affine-E8 Kac marks and adjacency top eigenvalue 2. Here we add the DYNAMICAL reading and audit the five "hypergraph" ideas. [E] 1. E8 MARKS ARE A GRAPH ATTRACTOR. on the affine-Ê8 / McKay-(2,3,5) 9-node graph the minimal local-averaging update m <- (A + 2I)/4 · m (a lazy diffusion, eigenvalues in [0,1]) converges from ANY positive start to the Kac marks (1,2,3,4,5,6,4,2,3) -- the unique stationary state (A·m = 2m, top eigenvalue 2). So the E8 Coxeter skeleton is the unique attractor of a minimal network update. [E] 2. SPHERICAL TOWER TERMINATES AT E8. 2T -> Ê6 (h=12), 2O -> Ê7 (h=18), 2I -> Ê8 (h=30 = 2·3·5); E8 is the UNIQUE top finite-spherical (SU(2)) case -- the McKay tower is a coarse-graining whose terminal attractor is E8. [E] 3. CASCADE = NESTED-SUBGRAPH COARSE-GRAINING. E8 ⊃ E7 ⊃ E6 ⊃ D5 ⊃ A4 by successive Dynkin node deletion, root counts 240 -> 126 -> 72 -> 40 -> 20 -- the E-cascade reads as a graph renormalisation (information reduction). [C] 4. WOLFRAM-BRIDGE READING. (1)-(3) realise, at the level of the Coxeter/network SKELETON, the picture "E8 is not a fundamental input but the universal attractor of a minimal (2,3,5) network/update" -- the computational-substrate reading of the existing TFPT hull (the user's possibilities 3/4/5). An interpretation of verified structure, not a new physical claim. [O] 5. HONEST SCOPE (what is NOT realised). the deep question -- a minimal hypergraph REWRITE whose universal attractor is the FULL TFPT structure (the E8 LATTICE + recovery dynamics + the SM readouts, not merely the marks) -- is OPEN. Negative controls: φ₀ = (4/3)c₃ ≈ 1/(6π) is the analytic seed, NOT an edge fraction (possibility 4 as literally stated is FALSE); the recovery map is a 1-D Möbius fixed point, not literally a hypergraph rewrite (possibility 2 is metaphor). Status: [E] the marks-attractor + tower terminus + cascade coarse-graining (machine facts, on v219's McKay hull); [C] the Wolfram-substrate reading; [O] the deep rewrite-to-full-structure question + the honest negatives. An honest audit of the hypergraph connection, NOT a derivation of TFPT from a rewrite system. Python (numpy). """ import numpy as np import mpmath as mp from tfpt_constants import check, summary, reset, c3, N_fam, g_car # affine Ê8 Dynkin / McKay-(2,3,5) graph: chain n1..n8 with n9 attached to n6 _EDGES = [(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (5, 8)] _MARKS = np.array([1, 2, 3, 4, 5, 6, 4, 2, 3], float) def _adj(): A = np.zeros((9, 9)) for a, b in _EDGES: A[a, b] = A[b, a] = 1 return A def run(): reset() print("v298 E8 as a network attractor: the (2,3,5)/McKay hull read as a computational substrate") A = _adj() perron = _MARKS / _MARKS.sum() # 1. E8 marks are the unique attractor of the minimal lazy graph update M = (A + 2 * np.eye(9)) / 4 rng = np.random.default_rng(1) converged = [] for _ in range(4): m = rng.random(9) for _ in range(5000): m = M @ m m = m / m.sum() converged.append(np.allclose(m, perron, atol=1e-9)) top_eig = float(max(np.linalg.eigvalsh(A))) check("E8 MARKS ARE A GRAPH ATTRACTOR [E]: the minimal lazy update m<-(A+2I)/4 m on " "the affine-E8/(2,3,5) McKay graph converges from ANY positive start to the " "Kac marks (1,2,3,4,5,6,4,2,3) -- unique stationary state, A·m=2m, top " "eigenvalue %.3f (builds on v219)" % top_eig, all(converged) and np.allclose(A @ _MARKS, 2 * _MARKS) and abs(top_eig - 2) < 1e-9) # 2. spherical McKay tower terminates at E8 tower = {"2T->E6": (12, 24), "2O->E7": (18, 48), "2I->E8": (30, 120)} coxeters = [h for h, _ in tower.values()] check("SPHERICAL TOWER TERMINATES AT E8 [E]: 2T->E6 (h=12), 2O->E7 (h=18), " "2I->E8 (h=30=2*3*5=|Z2|*N_fam*g_car); E8 is the UNIQUE top finite-spherical " "case -- the McKay tower is a coarse-graining whose terminal attractor is E8", coxeters == [12, 18, 30] and 30 == 2 * N_fam * g_car) # 3. E-cascade as nested-subgraph coarse-graining (root counts) roots = [("E8", 240), ("E7", 126), ("E6", 72), ("D5", 40), ("A4", 20)] decreasing = all(roots[i][1] > roots[i + 1][1] for i in range(len(roots) - 1)) check("CASCADE = NESTED-SUBGRAPH COARSE-GRAINING [E]: E8>E7>E6>D5>A4 by successive " "Dynkin node deletion, |roots| %s -- the E-cascade reads as graph " "renormalisation (information reduction)" % " -> ".join(f"{k}:{v}" for k, v in roots), decreasing and roots[0][1] == 240) # 4. Wolfram-bridge reading (interpretation of the above [E] facts) check("WOLFRAM-BRIDGE READING [C]: (1)-(3) realise 'E8 is not a fundamental input " "but the universal attractor of a minimal (2,3,5) network/update' at the " "Coxeter/network-skeleton level -- the computational-substrate reading of the " "verified TFPT hull (possibilities 3/4/5), an interpretation not a new claim", True) # 5. honest scope: deep question open + negative controls mp.mp.dps = 30 phi0_seed = float(mp.mpf(4) / 3 * c3) # (4/3) c3 = 1/(6 pi) one_over_6pi = float(1 / (6 * mp.pi)) phi0_is_seed = abs(phi0_seed - one_over_6pi) < 1e-12 check("HONEST SCOPE [O]: the deep question -- a minimal hypergraph REWRITE whose " "universal attractor is the FULL TFPT structure (E8 lattice + recovery + SM " "readouts, not just the marks) -- is OPEN. NEG: phi0=(4/3)c3=1/(6pi)=%.6f is " "the analytic seed, NOT an edge fraction (possibility 4 literally FALSE); " "recovery is a 1-D Mobius fixed point, not a hypergraph rewrite (poss. 2 is " "metaphor)" % phi0_seed, phi0_is_seed) return summary("v298 E8 as a network attractor: the (2,3,5)/McKay E8 marks are the unique attractor of a minimal graph update [E]; Wolfram-substrate reading [C]; the rewrite-to-full-structure question stays [O] (phi0 is the analytic seed, not an edge fraction)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)