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Original ledger classification: S3 realised as an explicit gapped chiral-Majorana lattice model (numerical Chern |C|=1, c_-=8, det K 4->1) [E]/[C]/[O]

Recorded statement: Track 1 (2026-06-23): the EXPLICIT lattice realisation of the S3 input -- a concrete gapped chiral-Majorana (p+ip) lattice model whose edge is the 16-Majorana SO(16)_1 carrier, mu4-condensed to (E8)_1. Turns the abstract S3 residual of…

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SEAM.S3.LATTICE.01

Original ledger classification: S3 realised as an explicit gapped chiral-Majorana lattice model (numerical Chern |C|=1, c_-=8, det K 4->1) [E]/[C]/[O]

Original claim text

Track 1 (2026-06-23): the EXPLICIT lattice realisation of the S3 input -- a concrete gapped chiral-Majorana (p+ip) lattice model whose edge is the 16-Majorana SO(16)_1 carrier, mu4-condensed to (E8)_1. Turns the abstract S3 residual of v356/v366 ('the collar realised as a genuine lattice chiral free-fermion invertible phase') into a RUNNABLE model with a numerically computed topological invariant, not a re-assertion of det K / c_-. [E] BULK GAPPED: h(k)=sin kx sx + sin ky sy + (M-cos kx-cos ky) sz at M=1 has band gap 2 min|d(k)|=2>0. [E] NON-TRIVIAL CHERN (numerical Fukui-Hatsugai-Suzuki plaquette): |C|=1 at M=1 (topological) and C=0 at M=3 (trivial neg control). [E] c_-=8: each chiral Majorana edge carries c_-=C/2=1/2; N_Maj=dim S^+=2^(g_car-1)=16 copies => c_-=16/2=8=g_car+N_fam (the edge gravitational anomaly). [E] EDGE K-MATRIX: the 16-Majorana edge is SO(16)_1 (det Cartan D8=4) condensed to (E8)_1 (det Cartan E8=1) by the order-4 mu4 clock (v351/v369). [C] the index-4 mu4 simple-current condensation SO(16)_1 -> (E8)_1 (v154/v351) is the seam glue on top. [O] residual: the continuum scaling-limit existence (the lattice model's massless limit IS the chiral CFT, MMST v336) stays open; S3 is now a concrete lattice model, reduced not closed.

Original classification detail

[E] the gapped Chern model + the c_-=8 counting + the det K discriminators (numerical Chern + lattice arithmetic); [C] the mu4 condensation to (E8)_1; [O] the continuum scaling limit. Reduces S3 to a runnable lattice model; does NOT close SEAM.EQUIV.01.

Dependencies as recorded

SEAM.EQUIV.01; v356; v366; v351; v369; v301; v302; v154

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Registered verification modules

  • v367_seam_s3_lattice.pyRegistered verification module · core
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    Track 1: the EXPLICIT lattice realisation of the S3 input -- a concrete gapped chiral-Majorana (p+ip) lattice model whose edge is the 16-Majorana SO(16)_1 carrier, mu4-condensed to (E8)_1; turns the abstract S3 residual (v356/v366) into a RUNNABLE model with a numerically computed topological invariant. [E] bulk gapped (band gap 2>0 at M=1). [E] non-trivial Chern number (Fukui-Hatsugai-Suzuki: |C|=1 at M=1, C=0 at M=3 trivial control). [E] c_-=8: each chiral Majorana edge carries c_-=C/2=1/2, N_Maj=dim S^+=2^(g_car-1)=16 copies => c_-=16/2=8=g_car+N_fam. [E] edge K-matrix SO(16)_1 (det Cartan D8=4) -> (E8)_1 (det Cartan E8=1) by the order-4 mu4 clock (v351/v369). [C] the mu4 condensation SO(16)_1 -> (E8)_1 (v154/v351). [O] residual: the continuum scaling-limit existence (MMST v336). Reduces S3 to a runnable lattice model; does NOT close SEAM.EQUIV.01.

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