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A registered result in its source context

Original ledger classification: Bridge [E] (hypergraph facts, Smith, Collatz-Wielandt, controls) + [E*] (autonomous rule under seed+policy) + [O] (the seed = P2)

Recorded statement: HYPERGRAPH/WOLFRAM BRIDGE, HARDENED (2026-06-19): the 3-arm (2,3,r) 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…

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Paper context: TFPT — Introduction and reading guide

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HYP.0D.01

Original ledger classification: Bridge [E] (hypergraph facts, Smith, Collatz-Wielandt, controls) + [E*] (autonomous rule under seed+policy) + [O] (the seed = P2)

Original claim text

HYPERGRAPH/WOLFRAM BRIDGE, HARDENED (2026-06-19): the 3-arm (2,3,r) 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 E8-hat (Ê8) -- E8 the maximal FINITE point, Ê8 the critical rho=2 edge. EXPLICITLY NOT 'P2 derived': the seed itself is the irreducible input (= P2). A second, network-theoretic reading of the SAME (2,3,5)/E8 core (builds on v219/v298). Sub-claims: HYP.ICO.01 [E] icosahedron = 3-uniform hypergraph (12 vertices, 20 triangular hyperedges, 30 edges = h(E8)=2*3*5), graph automorphism group I_h = A5 x Z2 of order 120 (rotations A5 order 60 + reflections); the McKay group is the SU(2) binary lift 2I, a different order-120 (perfect) group whose McKay graph is affine E8 (v219). HYP.REWRITE.01 [E] the 1->4 triangle-subdivision rewrite is equivariant: all 120 icosahedral isometries permute every subdivision level (V 12->42->162); the full icosahedral symmetry I_h carried to all scales. HYP.FUSION.01 [E] the 2I fusion tensor N^k_ij (n_i (x) n_j = sum N^k_ij n_k) is a nonneg-integer weighted 3-uniform hypergraph on 9 nodes (entries 0..3), commutative-associative; its 2-dim-rep slice = the affine E8 McKay graph (A.deg=2deg, top eigenvalue 2). HYP.SMITH.01 [E] Smith's theorem: a connected graph has rho<=2 iff it is an (affine) ADE diagram; rho(A_n,E6,E7,E8)<2, rho(Ê8)=2, rho(E10)>2; E8 = maximal finite exceptional in the T(2,3,r) arm. HYP.CW.01 [E] Collatz-Wielandt: rho = min_{m>0} max_i (Am)_i/m_i, so rho<=2 <=> a positive witness m with A m <= 2m exists (a LOCAL per-node balance); holds for E8/Ê8, fails for E10. HYP.AUTO.01 [E*] the autonomous rule -- self-generated witness via projectively-normalised lazy diffusion m<-(A+2I)/4 m, growth gated LOCALLY by A m <= 2m (no global rho in the decision) -- runs E6 -> E7 -> E8 -> Ê8 then stops, with critical witness = the Kac marks; exact UNDER the (2,3,*) seed and the grow-longest-leg policy. HYP.NEG.AN.01 [E] negative control: rho(A_n)=2cos(pi/(n+1))<2 for ALL n (closed form verified), so the A-family NEVER reaches E8. HYP.NEG.DN.01 [E] negative control: rho(D_n)=2cos(pi/(2n-2))<2 for ALL n (closed form verified), so the D-family NEVER reaches E8 -- so rho<=2 selects the ADE world, not E8; only the exceptional (1,2,r) arm reaches E8. HYP.SEED.01 [O] the 3-arm (2,3,*) exceptional seed is the single irreducible input = TFPT's P2 / the icosahedral choice (NOT derived). The diffusion is projectively normalised; the acceptance gate is the local witness inequality, with rho<=2 as audit only.

Original classification detail

[E] the hypergraph facts (icosahedron 3-uniform, |Aut|=120; equivariant subdivision; 2I fusion tensor with E8 slice), the spectral selection (Smith rho<=2 -> ADE; Collatz-Wielandt locality), and the two negative controls (A_n, D_n never reach E8); [E*] the autonomous local rule (self-generated witness + local gate) running E6->E7->E8->Ê8, exact under the (2,3,*) seed and growth policy; [O] the seed itself = P2 (NOT 'P2 derived'). A second network-theoretic reading of the (2,3,5)/E8 core; does NOT close P2 and adds no physics.

Dependencies as recorded

MCKAY.E8.01; E8.NETWORK.01; QBL.GATE.01

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Registered verification modules

  • v299_hypergraph_0d_autonomous.pyRegistered verification module · frontier
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    The hypergraph/Wolfram bridge, hardened: **the 3-arm (2,3,·) branch type** is the irreducible seed on which a LOCAL witness-based spectral growth REACHES E8 as the last finite point (E6→E7→E8, then stops at the affine boundary Ê8; E8 = maximal FINITE point) — NOT 'P2 derived'. [E] HYP.ICO.01 icosahedron = 3-uniform hypergraph (12 V, 20 hyperedges, 30 edges=h(E8)), |Aut(graph)|=120=|I_h|=A5×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 is equivariant (120 symmetries at every scale, 12→42→162); [E] HYP.FUSION.01 the 2I fusion tensor is a nonneg-integer weighted 3-uniform hypergraph, 2-dim slice = affine E8 McKay graph; [E] HYP.SMITH.01 ρ≤2 selects (affine) ADE (Smith); [E] HYP.CW.01 Collatz–Wielandt: ρ≤2 ⟺ positive witness m with A m≤2m (local per-node), gate decided by the diffusion witness (Perron only as audit); [E*] HYP.AUTO.01 self-generated witness (projectively-normalised diffusion) + LOCAL gate A m≤2m runs E6→E7→E8→Ê8 (critical witness = Kac marks) under the (2,3,·) seed + grow-longest-leg policy; [E] HYP.NEG.AN.01 ρ(A_n)=2cos(π/(n+1))<2 for all n, never reaches E8; [E] HYP.NEG.DN.01 ρ(D_n)=2cos(π/(2n−2))<2 for all n, never reaches E8; [O] HYP.SEED.01 the (2,3,·) seed is the single irreducible input = P2. A second network reading of the (2,3,5)/E8 core, honestly bounded; builds on v219/v298.

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