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Original ledger classification: torus GSD=1 / modular uniqueness -- the genus-1 holomorphy discriminator closes the v344 hard part [E]

Recorded statement: S3 closure stack (2026-06-23): the GENUS-1 discriminator -- the torus ground-state degeneracy of the seam net is 1 (holomorphic = (E8)_1), via the KLM index condensation and the modular invariance of the single (E8)_1 character. This is…

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SEAM.S3.MODULAR.01

Original ledger classification: torus GSD=1 / modular uniqueness -- the genus-1 holomorphy discriminator closes the v344 hard part [E]

Original claim text

S3 closure stack (2026-06-23): the GENUS-1 discriminator -- the torus ground-state degeneracy of the seam net is 1 (holomorphic = (E8)_1), via the KLM index condensation and the modular invariance of the single (E8)_1 character. This is exactly the 'hard part' v344 flagged: the genus-0 arguments (a unique plane vacuum) are necessary but NOT sufficient (a plane vacuum is always 1, v87); det K=1 is the torus (genus-1) statement. [E] KLM mu-additivity: the carrier A=(D5)_1 x (A3)_1 has mu(A)=det Cartan(D5)*det Cartan(A3)=4*4=16; the order-4 mu4 glue (|L|=|mu4|=4) gives mu(E8)=16/4^2=1 (torus GSD=1); the index-2 glue gives the rival mu(SO16)=16/2^2=4. [E] |det Cartan(E8)|=1 (one primary), |det Cartan(D8)|=4 (four primaries). [E] the single (E8)_1 character has modular T-eigenvalue e^{2 pi i(h-c/24)}=e^{-2 pi i/3} (h=0,c=8), consistent with v377's q^{-1/3}; a holomorphic theory has a 1-dim modular S,T rep => exactly one character => torus GSD=1. [O] the abstract continuum EXISTENCE (v336).

Original classification detail

[E] the KLM condensation 16->1, the det-Cartan #primaries (1 vs 4), the modular T-eigenvalue / 1-dim rep, and the genus-0-vs-genus-1 distinction; [O] the continuum existence (v336). Pins the net at genus-1; does NOT close SEAM.EQUIV.01.

Dependencies as recorded

SEAM.EQUIV.01; v344; v154; v369; v83; v87

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Registered verification modules

  • v378_seam_s3_modular.pyRegistered verification module · core
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    S3 closure stack: the GENUS-1 discriminator -- torus ground-state degeneracy = 1 (holomorphic = (E8)_1), the 'hard part' v344 flagged (genus-0 unique vacuum is necessary not sufficient). [E] KLM condensation: mu(carrier)=det Cartan(D5)*det Cartan(A3)=4*4=16; order-4 mu4 glue (|L|=4) => mu(E8)=16/16=1 (torus GSD=1); index-2 => mu(SO16)=16/4=4. [E] |det Cartan(E8)|=1 vs |det Cartan(D8)|=4. [E] the single (E8)_1 character has modular T-eigenvalue e^{-2 pi i/3} (h=0,c=8), 1-dim rep => one character => torus GSD=1. [O] continuum existence (v336). Pins the net at genus-1; does NOT close SEAM.EQUIV.01.

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