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Original ledger classification: Lemma decomposition [E] (3/4 discharged + route coincidence) + [O] (unified lemma)

Recorded statement: ROUTE (ii) (SEAM-NET CONDENSATION) LEMMA DECOMPOSITION + ROUTE-COINCIDENCE (2026-06-18): a 4-lemma chain with 3 discharged and the open lemma shown to COINCIDE with Route (i)'s (the v282 unification at the lemma level; does NOT prove the…

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QGAMB.ROUTEII.01

Original ledger classification: Lemma decomposition [E] (3/4 discharged + route coincidence) + [O] (unified lemma)

Original claim text

ROUTE (ii) (SEAM-NET CONDENSATION) LEMMA DECOMPOSITION + ROUTE-COINCIDENCE (2026-06-18): a 4-lemma chain with 3 discharged and the open lemma shown to COINCIDE with Route (i)'s (the v282 unification at the lemma level; does NOT prove the open premise). M1 [E] no nontrivial abelian sector <=> |det K|=1 <=> holomorphic (mu-index 1) (v234 GATE.HOLO.01: #(1-dim irreps)=|Gamma^ab|=|H1|=1 <=> E8; v235 GATE.HOLO.02 Chern-Simons det K=1). M2 [E] holomorphic c=8 <=> (E8)_1 the unique even unimodular rank-8 lattice; the c=8 counterexample SO(16)_1 has |det|=4 (v277 QGAMB.TIERB.01 + v281 QGAMB.MODULAR.01 #anyons=|det Gram|). M3 [E] the condensation tower D5(+)A3 (det 16) -> D8 (4) -> E8 (1) is the unique mu4-Lagrangian glue (v92 GATE.METRIC.07 + v281 + v235). THE DISCRIMINATOR [E]: det 1 (E8, holomorphic) vs det 4 (SO(16)) at the same c=8. ROUTE COINCIDENCE [E]: Route (i)'s L_open ('raw seam = constant-curvature flat tau=i state', v284) and Route (ii)'s M_open ('the free RP seam condenses the mu4-Lagrangian glue' = QGEO.SYM.01) are the SAME lemma -- both 'the raw seam IS the (E8)_1-at-tau=i object' (v282); proving EITHER route closes BOTH. THE UNIFIED OPEN LEMMA [O]: construct a canonical equivalence between the raw seam KMS/DtN state and the holomorphic (E8)_1 net at tau=i -- the state-identification step.

Original classification detail

[E] 3/4 Route-(ii) lemmas discharged + the discriminator + the route-(i)/(ii) coincidence; [O] the unified open lemma (canonical seam<->(E8)_1 equivalence). Decomposes Route (ii), isolates its single open lemma, and shows it coincides with Route (i)'s -- confirming v282 at the lemma level. Does NOT close it.

Dependencies as recorded

QGEO.SYM.01; GATE.HOLO.01; GATE.HOLO.02; QGAMB.TIERB.01; QGAMB.MODULAR.01; QGAMB.UNIFY.01; QGEO.ROUTEI.01

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