Original ledger status
SEAM.EQUIV.GEOM.01
SEAM.EQUIV.GEOM.01
THE GEOMETRIC BRIDGE AS AN EXPLICIT SIX-LINK CHAIN, WITH THE ONE OPEN ARROW PINNED (2026-06-22) -- the honest capstone of the SEAM.EQUIV.01 investigation (v335/v336/v344/v345). The chain raw-seam => det K=1: L1 [E/C] raw seam -> mu4 clock + four marks (Gauss-Bonnet on the flat pillowcase S^2(2,2,2,2), v216); L2 [O] mu4 + the carrier (2,3,5) atoms -> the binary icosahedral 2I action C^2/Gamma -- THE ONE OPEN ARROW, the geometric content of SEAM.EQUIV.01; L3 [E] 2I -> affine E8 McKay graph (computed from the group via Dixon characters, v219); L4 [E] 2I -> the E8 du Val singularity C^2/2I (Brieskorn/Slodowy, v232); L5 [E] C^2/2I -> the Poincare link S^3/2I with H_1=0 (Smith normal form + 2I perfect, v345); L6 [E] H_1=0 -> det K=1 (holomorphy, v345/v235). [E] FIVE OF SIX LINKS ARE THEOREMS: of L1..L6 exactly ONE (L2, the orbifold realisation) is open; L1 is [E/C] and L3..L6 are [E] cited theorems, so the keystone reduces to the single arrow L2. [E] THE GROUP IS FORCED BY THE SEAM'S OWN ATOMS: the exceptional finite SU(2) subgroups have axis-order signatures (2,3,3)->2T->E6, (2,3,4)->2O->E7, (2,3,5)->2I->E8; the signature (2,3,5) is UNIQUE to 2I, and the TFPT atoms (|Z2|,N_fam,g_car)=(2,3,5) ARE these icosahedral axes (v219/v313), so IF the seam realises a finite SU(2) orbifold its group is FORCED to be 2I -- the seam cannot 'choose' a different ADE type. [E] EVERYTHING DOWNSTREAM OF L2 IS THEN FORCED: given 2I, L3..L6 give E8 (h=2*3*5=30, rank=g_car+N_fam=8), the Poincare link and det K=1 by theorems; so det K=1 <=> the seam realises C^2/2I (L2), the group already pinned. [O] THE IRREDUCIBLE OPEN CONTENT: L2 ('the raw quasi-free seam normal bundle IS an ADE orbifold point C^2/Gamma') is the geometric content of SEAM.EQUIV.01; this does NOT prove it but shows the keystone is EXACTLY this one geometric-realisation arrow, with the group forced by (2,3,5) and all downstream consequences proven. The bridge is one arrow, not a programme.
Recorded status: Capstone/roadmap [E]/[O] (5 of 6 links theorems; the group forced by the icosahedral atoms; the keystone reduced to the single orbifold-realisation arrow L2 = SEAM.EQUIV.01)
Dependencies, status detail and source
Canonical status: [E] the chain bookkeeping (5 of 6 links are theorems) + the (2,3,5)->2I axis selection among exceptional SU(2) subgroups; [O] the single orbifold-realisation arrow L2 (= SEAM.EQUIV.01). A capstone/roadmap; closes nothing, fabricates nothing -- it reduces the keystone to one geometric-realisation statement with the group pinned by the seam's own (2,3,5) atoms.
Dependencies: SEAM.EQUIV.UNIFY.01; SEAM.DETK.02; v216; v219; v232; v345; v313
Scripts: v346_seam_geometric_bridge.py
Location: origin_theory
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