"""v299 -- the hypergraph / Wolfram-program bridge, hardened. This does NOT claim "P2 is derived" or "E8 emerges from nothing". It proves the precise, honest statement: P2 (the 3-arm (2,3,r) exceptional branch type) is the irreducible seed on which a LOCAL, witness-based spectral growth dynamics REACHES E8 = T(2,3,5) as the LAST FINITE point: it runs E6 -> E7 -> E8 and stops at the affine boundary Ê8 (the critical rho=2 edge). The Wolfram floor and the TFPT floor fall back to the SAME (2,3,5) seed. On the groups (a deliberate precision): the icosahedron GRAPH has automorphism group I_h = A5 x Z2 of order 120 (rotations A5 order 60 + reflections). The McKay group is the binary icosahedral group 2I, the SU(2) double cover of the rotation group A5 -- also order 120 but a DIFFERENT (perfect) group; it is 2I, not the graph automorphism group, whose McKay graph is affine E8 (built from quaternions in v219). Negative controls (the built-in guards against over-reach): a path A_n seed and a D_n seed have the EXACT closed forms rho(A_n)=2cos(pi/(n+1)) and rho(D_n)=2cos(pi/(2n-2)), both < 2 for ALL n -- so rho<=2 selects the ADE world, NOT E8; only the exceptional (1,2,r) arm does. Claims (hard-typed): [E] HYP.ICO.01 icosahedron = 3-uniform hypergraph (12 V, 20 triangular hyperedges, 30 edges = h(E8)); |Aut(graph)| = 120 = |I_h| = |A5 x Z2| (the McKay group is the SU(2) binary lift 2I, a different order-120 group; v219). [E] HYP.REWRITE.01 the 1->4 subdivision rewrite preserves all 120 icosahedral symmetries at every scale (V: 12 -> 42 -> 162). [E] HYP.FUSION.01 the 2I fusion tensor N^k_ij is a nonneg-integer weighted 3-uniform hypergraph on 9 nodes; its 2-dim-rep slice = the affine E8 McKay graph (A.deg = 2 deg, top eigenvalue 2). [E] HYP.SMITH.01 Smith: rho<=2 selects (affine) ADE; rho(A_n,E6,E7,E8) < 2, rho(Ê8) = 2, rho(E10) > 2; E8 = maximal finite exceptional in T(2,3,r). [E] HYP.CW.01 Collatz-Wielandt: rho<=2 <=> a positive witness m with A m <= 2m exists (a LOCAL per-node balance), holds for E8/Ê8, fails for E10. [E*] HYP.AUTO.01 the autonomous rule (self-generated witness via projectively- normalised diffusion + LOCAL gate A m <= 2m, NO global rho) runs E6 -> E7 -> E8 -> Ê8 then stops -- exact UNDER the 3-arm (2,3,*) seed and the grow-longest-leg policy. Critical witness = the Kac marks. [E] HYP.NEG.AN.01 control: rho(A_n)=2cos(pi/(n+1))<2 for ALL n (closed form), so the A-family never reaches E8. [E] HYP.NEG.DN.01 control: rho(D_n)=2cos(pi/(2n-2))<2 for ALL n (closed form), so the D-family never reaches E8; only the (1,2,r) arm does. [O] HYP.SEED.01 the 3-arm (2,3,r) exceptional branch is the SINGLE irreducible input = TFPT's P2 / the icosahedral choice (NOT "P2 derived"). Status: [E] the hypergraph facts (ICO/REWRITE/FUSION), the spectral selection (SMITH/CW), the two negative controls; [E*] the autonomous rule under the stated seed + policy; [O] the seed itself = P2. A second, network-theoretic reading of the SAME (2,3,5)/E8 core, honestly bounded. Builds on v219/v298. Python (numpy; imports v219 for the 2I group + characters). """ import itertools import numpy as np import v219_icosahedral_mckay as mck from tfpt_constants import check, summary, reset, N_fam, g_car PHI = (1 + np.sqrt(5)) / 2 # ---------- icosahedron (3-uniform hypergraph) ---------------------------- def _icosahedron(): V = [] for a, b in itertools.product([1, -1], repeat=2): V += [(0, a, b * PHI), (a, b * PHI, 0), (b * PHI, 0, a)] V = np.array(sorted(set(map(tuple, np.round(V, 9))))) D = np.round(((V[:, None] - V[None]) ** 2).sum(-1), 6) el2 = np.min(D[D > 1e-9]) A = (np.abs(D - el2) < 1e-6).astype(int) faces = [t for t in itertools.combinations(range(len(V)), 3) if A[t[0], t[1]] and A[t[0], t[2]] and A[t[1], t[2]]] return V, A, faces def _graph_autos(A): n = len(A) nbr = [set(np.where(A[i])[0]) for i in range(n)] autos = [] def bt(perm, used): k = len(perm) if k == n: autos.append(tuple(perm)) return for img in range(n): if img in used: continue if all((perm[j] in nbr[img]) == (j in nbr[k]) for j in range(k)): bt(perm + [img], used | {img}) bt([], set()) return autos # ---------- star graphs T_{p,q,r} (Dynkin) ------------------------------- def _star(legs): nodes = 1 adj = [] for L in legs: prev = 0 for _ in range(L): adj.append((prev, nodes)) prev = nodes nodes += 1 A = np.zeros((nodes, nodes)) for a, b in adj: A[a, b] = A[b, a] = 1 return A def _rho(A): return float(max(np.linalg.eigvalsh(A))) def _witness_localgate(A): """Self-generate the Perron witness by projectively-normalised lazy diffusion, then test the LOCAL per-node balance A m <= 2 m (no global rho in the decision).""" d = A.shape[0] M = (A + 2 * np.eye(d)) / 4 rng = np.random.default_rng(0) m = rng.random(d) for _ in range(5000): m = M @ m m = m / m.min() # projective normalisation return m, bool(np.all(A @ m <= 2 * m + 1e-6)) def _grow(seed, policy_longest=True, cap=12): """Grow the longest leg while the LOCAL gate accepts; return the leg-sequence.""" legs = list(seed) seq = [] for _ in range(cap): seq.append(tuple(sorted(legs))) legs2 = legs[:] legs2[legs2.index(max(legs2))] += 1 if _witness_localgate(_star(legs2))[1]: legs = legs2 else: break return seq def run(): reset() print("v299 hypergraph/Wolfram bridge: the (2,3,r) seed reaches E8 as the last finite point of a local ADE rule") # HYP.ICO.01 V, A, faces = _icosahedron() nV, nE, nF = len(V), int(A.sum() // 2), len(faces) naut = len(_graph_autos(A)) check("HYP.ICO.01 [E]: the icosahedron is a 3-uniform hypergraph (V=%d, triangular " "hyperedges=%d, edges=%d=h(E8)); |Aut(graph)|=%d=|I_h|=|A5xZ2| (rotations A5 " "order 60 + reflections). The McKay group is the SU(2) binary lift 2I (order " "120, perfect, != I_h), built from quaternions in v219" % (nV, nF, nE, naut), nV == 12 and nF == 20 and nE == 30 == 2 * N_fam * g_car and naut == 120) # HYP.REWRITE.01: 1->4 subdivision preserves all 120 symmetries autos = _graph_autos(A) Os = [np.linalg.lstsq(V, V[list(s)], rcond=None)[0].T for s in autos] nbr = [set(np.where(A[i])[0]) for i in range(12)] edges = sorted({(min(i, j), max(i, j)) for i in range(12) for j in nbr[i]}) scale = np.linalg.norm(V[0]) mids = [((V[i] + V[j]) / 2) / np.linalg.norm((V[i] + V[j]) / 2) * scale for i, j in edges] Vsub = np.vstack([V, np.array(mids)]) preserved = sum(all(np.min(np.abs(Vsub - w).sum(1)) < 1e-6 for w in (O @ Vsub.T).T) for O in Os) check("HYP.REWRITE.01 [E]: the 1->4 triangle-subdivision rewrite is equivariant -- " "all %d/%d full icosahedral symmetries (I_h, whose rotation subgroup A5 has " "the binary lift 2I) still permute the subdivided hypergraph (V 12->%d->162); " "the icosahedral symmetry is carried to every scale" % (preserved, len(Os), len(Vsub)), preserved == 120 and len(Vsub) == 42) # HYP.FUSION.01: 2I fusion tensor (via v219) is a 3-uniform hypergraph; V-slice = E8 G = mck._icosians() n = len(G) index = {mck._key(q): i for i, q in enumerate(G)} table = np.empty((n, n), int) for i in range(n): for j in range(n): table[i, j] = index[mck._key(mck._quat_mult(G[i], G[j]))] classes = mck._conjugacy_classes(table, n) degs, chars, _ = mck._dixon(table, classes, n) chars = np.array(chars) sizes = np.array([len(c) for c in classes]) nc = len(classes) Nt = np.zeros((nc, nc, nc)) for i in range(nc): for j in range(nc): for k in range(nc): Nt[i, j, k] = np.real(np.sum(sizes * chars[i] * chars[j] * np.conj(chars[k])) / n) Nt = np.round(Nt).astype(int) chiV = np.array([2 * G[classes[c][0]][0] for c in range(nc)]) vi = [i for i in range(nc) if degs[i] == 2 and np.allclose(chars[i], chiV, atol=1e-6)][0] deg_vec = np.array(degs, float) slice_ok = np.allclose(Nt[vi].dot(deg_vec), 2 * deg_vec) check("HYP.FUSION.01 [E]: the 2I fusion tensor N^k_ij is a nonneg-integer weighted " "3-uniform hypergraph on 9 nodes (entries %s); its 2-dim-rep slice = the " "affine E8 McKay graph (A.deg=2deg, top eig 2)" % sorted(set(Nt.flatten().tolist())), (Nt >= 0).all() and slice_ok) # HYP.SMITH.01 rs = {nm: _rho(_star(lg)) for nm, lg in [("A8", [7, 0, 0]), ("E6", [1, 2, 2]), ("E7", [1, 2, 3]), ("E8", [1, 2, 4]), ("Ê8", [1, 2, 5]), ("E10", [1, 2, 6])]} check("HYP.SMITH.01 [E]: Smith -- rho<=2 selects (affine) ADE; rho(A8,E6,E7,E8)<2 " "(%.3f,%.3f,%.3f,%.3f), rho(Ê8)=%.3f, rho(E10)=%.3f>2; E8 = maximal finite " "exceptional in T(2,3,r)" % (rs["A8"], rs["E6"], rs["E7"], rs["E8"], rs["Ê8"], rs["E10"]), rs["E8"] < 2 < rs["E10"] and abs(rs["Ê8"] - 2) < 1e-9) # HYP.CW.01: the LOCAL gate is decided by the DIFFUSION-generated witness (no global # rho); the Perron eigenvector is compared only as an independent audit. cw = {nm: _witness_localgate(_star(lg))[1] for nm, lg in [("E8", [1, 2, 4]), ("Ê8", [1, 2, 5]), ("E10", [1, 2, 6])]} def _perron_gate(lg): Ad = _star(lg).astype(float) m = np.abs(np.linalg.eigh(Ad)[1][:, -1]); m = m / m.min() return bool(np.all(Ad @ m <= 2 * m + 1e-6)) audit_ok = all(_perron_gate(lg) == cw[nm] for nm, lg in [("E8", [1, 2, 4]), ("Ê8", [1, 2, 5]), ("E10", [1, 2, 6])]) check("HYP.CW.01 [E]: Collatz-Wielandt -- the LOCAL gate (diffusion-generated " "witness, A m <= 2m, no global rho in the decision): E8=%s, Ê8=%s, E10=%s; " "the independent Perron-eigenvector audit agrees (%s)" % (cw["E8"], cw["Ê8"], cw["E10"], audit_ok), cw["E8"] and cw["Ê8"] and not cw["E10"] and audit_ok) # HYP.AUTO.01: autonomous rule from the (2,3,*) seed -> E6,E7,E8,Ê8 seq = _grow([1, 2, 2]) expected = [(1, 2, 2), (1, 2, 3), (1, 2, 4), (1, 2, 5)] m_aff, _ = _witness_localgate(_star([1, 2, 5])) marks = sorted(np.round(m_aff).astype(int).tolist()) check("HYP.AUTO.01 [E*]: the autonomous rule (self-generated witness via " "projectively-normalised diffusion + LOCAL gate A m<=2m, no global rho) runs " "E6->E7->E8->Ê8 then stops (seq %s); critical witness = Kac marks %s -- exact " "UNDER the (2,3,*) seed + grow-longest-leg policy" % (seq, marks), seq == expected and marks == [1, 2, 2, 3, 3, 4, 4, 5, 6]) # HYP.NEG.AN.01: path control, EXACT for all n (closed form rho(A_m)=2cos(pi/(m+1))) an_ok = all(abs(_rho(_star([m - 1, 0, 0])) - 2 * np.cos(np.pi / (m + 1))) < 1e-6 for m in range(4, 12)) seq_a = _grow([1, 0, 0], cap=8) check("HYP.NEG.AN.01 [E]: control -- A_n is a path with rho(A_n)=2cos(pi/(n+1))<2 " "for ALL n (closed form verified, n=4..11; finite growth %d steps never " "reaches E8). So rho<=2 selects the ADE world, NOT E8" % len(seq_a), an_ok and (1, 2, 4) not in seq_a) # HYP.NEG.DN.01: D_n control, EXACT for all n (rho(D_m)=2cos(pi/(2m-2))) dn_ok = all(abs(_rho(_star([1, 1, m - 3])) - 2 * np.cos(np.pi / (2 * m - 2))) < 1e-6 for m in range(4, 12)) seq_d = _grow([1, 1, 2], cap=8) check("HYP.NEG.DN.01 [E]: control -- D_n has rho(D_n)=2cos(pi/(2n-2))<2 for ALL n " "(closed form verified, n=4..11; finite growth %d steps never reaches E8). " "Only the exceptional (1,2,r) arm reaches E8" % len(seq_d), dn_ok and (1, 2, 4) not in seq_d) # HYP.SEED.01: the seed is the irreducible input = P2 (NOT derived) check("HYP.SEED.01 [O]: the 3-arm (2,3,r) exceptional branch type is the SINGLE " "irreducible input = TFPT's P2 / the icosahedral choice (the A_n and D_n " "controls confirm rho<=2 alone is not enough) -- this is NOT 'P2 derived', it " "is 'P2 = the minimal seed on which the local rule REACHES E8 as the last " "finite point'", True) return summary("v299 hypergraph bridge: the 3-arm (2,3,r) seed is the irreducible input on which a local witness-based rule REACHES E8 as the last finite point (E6->E7->E8->Ê8; A_n/D_n controls with exact rho=2cos formulas: rho<=2 selects ADE, not E8); a second network reading of the (2,3,5)/E8 core, NOT a derivation of P2") if __name__ == "__main__": raise SystemExit(1 if run() else 0)