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Original ledger classification: Lattice [L] (index arithmetic + reframing) + [P]/[A] (seam-side identification)

Recorded statement: CARRIER INDEX LEMMA [L] - the Gate-A theorem gets an equivalent index-theoretic form (sharper as a target): by Kawahigashi-Longo-Mueger (mu_A=[B:A]^2 mu_B; lattice nets mu=|discriminant|=det Cartan) the carrier inclusion has Jones index…

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Paper context: TFPT — Introduction and reading guide

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GATE.METRIC.06

Original ledger classification: Lattice [L] (index arithmetic + reframing) + [P]/[A] (seam-side identification)

Original claim text

CARRIER INDEX LEMMA [L] - the Gate-A theorem gets an equivalent index-theoretic form (sharper as a target): by Kawahigashi-Longo-Mueger (mu_A=[B:A]^2 mu_B; lattice nets mu=|discriminant|=det Cartan) the carrier inclusion has Jones index [(E8)_1:(D5)_1x(A3)_1]=sqrt(16/1)=4=|mu_4|=|Z2|^2 - the glue-group order IS the inclusion index. The extension is the mu_4 simple-current extension at level 1: adjoint branching 248=(45,1)+(1,15)+(16,4b)+(16b,4)+(10,6)=45+15+64+64+60; the three non-vacuum glue sectors are CURRENTS (h=1 exactly: 1/2+1/2, 5/8+3/8, 5/8+3/8); 1+3=4 sectors = the same Z4 glue as the lattice construction (v1). mu-additivity: an index-4 simple-current extension of the carrier has mu=16/16=1, so HOLOMORPHY FOLLOWS from the index statement - it need not be assumed. CONSEQUENCE: (H) 'seam-Calderon net holomorphic with c=8' <=> (H') 'seam net contains the carrier net with Jones index 4 as its mu_4 simple-current extension'. Why sharper: both ends of (H') are constructed objects (v77), finite index is controllable by Longo index theory, and the gap Delta_eff>0 (v76) supports finiteness. Negative control: mu(SO(16)_1)=4!=1 (v83/v87). Honest scope: the seam-side index computation is NOT done here; the boundary-net identification stays [P]/[A]

Original classification detail

Lattice [L]: Gate A <=> index-4 mu_4 simple-current extension of the carrier net

Dependencies as recorded

GATE.METRIC.05; GATE.METRIC.04; E8.GLU.01; v77

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  • v89_carrier_index_lemma.pyRegistered verification module · registry
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    Carrier Index Lemma: KLM μ_A=[B:A]²μ_B ⇒ Jones index [(E₈)₁:(D₅)₁×(A₃)₁] = 4 = |μ₄| — the glue-group order IS the inclusion index; all three glue sectors are h=1 currents (248=45+15+64+64+60); holomorphy FOLLOWS from μ-additivity 16/4²=1 ⇒ Gate A ⇔ an index computation

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