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A registered result in its source context

Original ledger classification: Identity/Lattice [I]/[L] (exhaustive classification) + [P]/[A] (seam side)

Recorded statement: GLUE UNIQUENESS [I]/[L] - the carrier extension tower completely classified (Gate A loses its last structural freedom): extensions of the carrier net/lattice correspond to isotropic subgroups of the discriminant form (Z4xZ4,…

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GATE.METRIC.07

Original ledger classification: Identity/Lattice [I]/[L] (exhaustive classification) + [P]/[A] (seam side)

Original claim text

GLUE UNIQUENESS [I]/[L] - the carrier extension tower completely classified (Gate A loses its last structural freedom): extensions of the carrier net/lattice correspond to isotropic subgroups of the discriminant form (Z4xZ4, q=(5x^2+3y^2)/8; q reproduces the level-1 sector weights of v87/v89). Exhaustive enumeration of all 15 subgroups: isotropic = {1, Z2, Z4, Z4}. (a) Exactly TWO Lagrangian glues <(1,1)> and <(1,3)>, each det=16/16=1 even unimodular rank 8 => E8 (unique, Minkowski-Siegel v83); they are swapped by EITHER single spinor conjugation (D5 or A3) and fixed by the simultaneous one => the mu_4 glue is unique UP TO THE SHEET Z2 - the two chiralities, matching exactly the two E8-extension modular invariants |chi_0+chi_{s,c}|^2 of v87. (b) The UNIQUE halfway extension {(0,0),(2,2)} has H_perp/H=Z2xZ2 with induced q={0,0,0,1/2} = the D8 discriminant form => SO(16)_1 is the ONLY intermediate of the tower (and is excluded by holomorphy, mu=4!=1). TOWER: carrier (mu=16) -> SO(16)_1 (mu=4) -> (E8)_1 (mu=1, sheet pair) - nothing else exists. CONSEQUENCE: with v89 (index 4 <=> holomorphy) and this row, Gate A is the BARE index statement 'the seam-Calderon inclusion has Jones index 4'; which extension it then is carries NO freedom. Seam-side identification stays [P]/[A]

Original classification detail

Identity/Lattice [I]/[L]: extension tower = carrier -> SO(16)_1 -> E8 (sheet pair); no freedom left in WHICH extension

Dependencies as recorded

GATE.METRIC.06; GATE.METRIC.05; E8.GLU.01; v83; v87

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Registered verification modules

  • v92_glue_uniqueness.pyRegistered verification module · registry
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    Glue uniqueness: exhaustive classification of the carrier discriminant form (Z₄×Z₄, q=(5x²+3y²)/8) — exactly two Lagrangian glues (the two chiralities, swapped by spinor conjugation = sheet Z₂) and exactly one halfway extension whose induced form IS the D₈ form: tower carrier(μ=16) → SO(16)₁(μ=4) → E₈₁(μ=1), nothing else; Gate A = the bare index statement. Now also Lean-formalised (GlueUniqueness.lean, AUDIT: PASS)

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