A registered result in its source context

SEAM.DETK.02

Original ledger classification: Execution of R3 [E]/[O] (combinatorial core complete: homology-sphere link via SNF + 2I perfect = Poincare sphere; geometric bridge stays SEAM.EQUIV.01)

Original claim text

EXECUTING THE COMBINATORIAL CORE OF ROUTE R3 (the hypergraph-homotopy route to det K=1, v344) -- the most promising fresh angle, now carried out as far as it computably goes (2026-06-22). Computes explicitly the topology of the (2,3,5)-rewrite attractor graph's plumbing link: it is the POINCARE homology sphere (H_1=0, the det K=1 face), only E8 among ADE. R3's combinatorial half is thereby [E]-complete; the one remaining step is the geometric realisation bridge (= SEAM.EQUIV.01), NOT closed. Standard plumbing topology (Brieskorn/Hirzebruch): a tree plumbing with the ADE Cartan as intersection form has link H_1 = coker(Cartan) (sign-independent: coker(M)=coker(-M)). [E] SMITH NORMAL FORM => coker = H_1(link): the SNF of the E8 Cartan matrix is the identity (all elementary divisors 1), so coker(E8)=0 -- the E8-graph plumbing boundary is an INTEGRAL HOMOLOGY SPHERE, the genus-1/topological meaning of det E8=1 (the discrete graph fixes the link's H_1). [E] ONLY E8 AMONG ADE: coker(A_n)=Z/(n+1), coker(D_n)=Z/4, coker(E6)=Z/3, coker(E7)=Z/2 -- all nontrivial; ONLY E8 has coker=0, so only the E8 rewrite skeleton has a homology-sphere link (= only E8 gives det K=1). [E] pi_1=2I PERFECT => THE POINCARE SPHERE: the link's fundamental group is the binary icosahedral group 2I=SL(2,5) (order 120); the commutator subgroup [2I,2I]=2I (|[G,G]|=120=|G|, computed directly), so 2I is perfect, its abelianisation (=H_1) is trivial, and the link is THE Poincare homology sphere (the unique homology 3-sphere with finite pi_1). [E] (2,3,5) TIE-IN: 2I is the binary (2,3,5) icosahedral group, |2I|=120=|mu4|*h(E8)=4*30, tying the link to the rewrite (2,3,5) data and the order-30 Coxeter cycle (v55/v219/v312). [O] HONEST RESIDUAL: does NOT close det K=1 -- the remaining step is the geometric realisation bridge ('the raw (2,3,5)-rewrite seam attractor IS this E8 plumbing / du Val singularity'), exactly the SEAM.EQUIV.01 content. R3's combinatorial half is now [E]-complete; the geometric half stays [O].

Original classification detail

[E] the SNF/cokernel computation (= H_1 of the plumbing link), the ADE contrast, and the explicit perfectness of 2I=SL(2,5) (commutator subgroup = whole group); [O] the geometric realisation bridge (= SEAM.EQUIV.01). Executes R3's combinatorial core; closes nothing.

Dependencies as recorded

SEAM.DETK.01; SEAM.EQUIV.UNIFY.01; v232; v312; v344; v55; v219