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A registered result in its source context

Original ledger classification: Lattice (McKay correspondence, exact)

Recorded statement: McKAY BEDROCK (2026-06-16): WHY the three atoms are {2,3,5}. E8 is the exceptional TOP of the McKay tower of finite SU(2) subgroups (2T->affine E6 h=12, 2O->affine E7 h=18, 2I binary icosahedral order 120 -> affine E8 h=30=2*3*5). Built…

Statement, assumptions and sources

Paper context: Origin Theory

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MCKAY.E8.01

Original ledger classification: Lattice (McKay correspondence, exact)

Original claim text

McKAY BEDROCK (2026-06-16): WHY the three atoms are {2,3,5}. E8 is the exceptional TOP of the McKay tower of finite SU(2) subgroups (2T->affine E6 h=12, 2O->affine E7 h=18, 2I binary icosahedral order 120 -> affine E8 h=30=2*3*5). Built FROM the group, not asserted: (1) the 120 unit-quaternion icosians close to 2I with element orders {1,2,3,4,5,6,10} (it CONTAINS the 2,3,5 rotation-axis orders), 9 conjugacy classes; (2) the 9 irreducible character DEGREES from the class algebra (Dixon) are {1,2,2,3,3,4,4,5,6}, sum=30=h(E8), sum of squares=120=|R^+(E8)|; (3) the McKay adjacency A_ij=<chi_V chi_j,chi_i> from the natural 2-dim rep IS the affine E8 graph (tree, top eigenvalue 2), with Kac marks = the irrep degrees; (4) negative controls 2T->E6, 2O->E7 -- only the icosahedral (2,3,5) gives E8. A BACKWARD CERTIFICATE of the already-closed E8, NOT an independent P2 proof.

Original classification detail

[E] 2I (order 120) irrep degrees {1,2,2,3,3,4,4,5,6} = affine E8 Kac marks, sum 30=h(E8), sum^2 120=|R^+(E8)|; the McKay graph built from the group; E8 = the (2,3,5) icosahedral top. Backward certificate [E], 'explains (2,3,5)' [C], raw-seam origin [O]

Dependencies as recorded

AX.P2.01; E8.GLU.01; ARCH.CORE.01

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Registered verification modules

  • v219_icosahedral_mckay.pyRegistered verification module · core
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    McKay bedrock: WHY the atoms are 2,3,5. E₈ is the exceptional TOP of the McKay tower of finite SU(2) subgroups (2T→Ê₆, 2O→Ê₇, 2I→Ê₈). The McKay graph is BUILT from the group: the 120 icosians close to 2I (orders {1,2,3,4,5,6,10}, containing the 2,3,5 axes), 9 conjugacy classes, Dixon irrep degrees {1,2,2,3,3,4,4,5,6} (sum 30=h(E₈), sum² 120=|R⁺(E₈)|); the natural-rep McKay adjacency IS the affine E₈ graph with marks = degrees. Backward certificate of E₈, not a P2 proof.

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