A registered result in its source context

SEAM.DETK.01

Original ledger classification: Synthesis/roadmap [E]/[C]/[O] (one bit/six faces; genus-1 obstruction; three access routes; hypergraph the fresh one; no closure)

Original claim text

THE BIRD'S-EYE SYNTHESIS OF THE ONE OPEN KEYSTONE BIT det K=1 (2026-06-22): a full-spectrum map of SEAM.EQUIV.01's only open residual -- it is ONE statement with six equivalent faces; it is hard for ONE precise reason (the genus-1 obstruction); and there are exactly three genus-1 access routes, of which the hypergraph-homotopy route is the most promising fresh angle. [E] SIX FACES: det K=1 <=> holomorphy (1 primary) <=> homology-sphere link (H_1=0) <=> one 1-dim irrep (Gamma perfect) <=> det Cartan=1 <=> the full mu4 condensation 16->4->1 <=> the tau=i CM rigidity (chi_E8=j^{1/3}); unified by |det Cartan| = |H_1(link)| = |Gamma^ab| = #(1-dim irreps) across ADE (A_n->n+1, D_n->4, E6->3, E7->2, E8->1), ONLY E8 -> 1 (2I the unique perfect ADE group). [E] THE GENUS-1 OBSTRUCTION: every easy argument (unique plane vacuum, gap, free-fermion CAR invertibility) is GENUS-0 -- necessary but NOT sufficient (a plane vacuum is always 1 for any chiral net, v87); det K is the TORUS ground-state degeneracy = #primaries, invisible to genus-0 data; so any closing route MUST access genus-1/modular/topological info -- the precise reason det K=1 is the residual. [E] THREE GENUS-1 ACCESS ROUTES: R1 continuum-modular (scaling limit + chi_E8=j^{1/3}, v336); R2 anyon-condensation-torus (degeneracy = #anyons = |det K|, tower 16->4->1, v281); R3 hypergraph-homotopy (torus degeneracy = discrete H_1 of the substrate complex; a Poincare-sphere link gives 1). [C] THE HYPERGRAPH ROUTE = MOST PROMISING FRESH ANGLE: the (2,3,5) rewrite ATTRACTOR is the affine-E8 Dynkin graph (Kac marks = Perron vector, golden 5-fold, v312) and the E8 du Val resolution graph's link is the Poincare sphere (H_1=0, v232), so det K=1 becomes 'the attractor's link homotopy type is the Poincare sphere' -- a finite COMBINATORIAL statement (no continuum limit), unlike R1/R2. [O] HONEST RESIDUAL: none of R1/R2/R3 closes det K=1; the hypergraph route's remaining gap is the combinatorial-to-geometric bridge ('the rewrite attractor (a graph) IS the raw seam's geometric realisation (the du Val singularity)') -- which is exactly the SEAM.EQUIV.01 content. The synthesis SHARPENS the target to one combinatorial statement and ranks the routes; it does NOT prove it. UPDATE 2026-07-14 (v479/SEAM.KRONHEIMER.01): R3's combinatorial-to-geometric bridge is DISCHARGED to Kronheimer 1989 (ALE hyper-Kaehler quotient + Torelli; all finite input data compiler outputs) -- the [O] residual transforms to 'the raw seam supplies the ALE/orbifold datum', the same single arrow as SEAM.EQUIV.01; status unchanged.

Original classification detail

[E] the six-faces equivalence (|det Cartan|=|H_1| ADE arithmetic, computed) + the genus-1 obstruction + the three-route map; [C] the hypergraph route's alignment (attractor=E8 graph v312 + E8 link=Poincare sphere v232); [O] the unclosed combinatorial-to-geometric bridge (= SEAM.EQUIV.01). A synthesis/roadmap; closes nothing, fabricates nothing.

Dependencies as recorded

SEAM.EQUIV.UNIFY.01; v232; v235; v219; v281; v282; v312; v336; v87