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Original ledger classification: Reduction [C]/[O] (continuum lemma narrowed + literature-anchored)

Recorded statement: THE ONE OPEN LEMMA OF SEAM.EQUIV.01, stated precisely and mapped to the recent rigorous literature (2026-06-22): the continuum-limit and OS-reconstruction legs are now CITABLE, the genuinely-open residual narrows to 'the raw collar is a…

Statement, assumptions and sources

Paper context: Origin Theory

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SEAM.EQUIV.CONTINUUM.01

Original ledger classification: Reduction [C]/[O] (continuum lemma narrowed + literature-anchored)

Original claim text

THE ONE OPEN LEMMA OF SEAM.EQUIV.01, stated precisely and mapped to the recent rigorous literature (2026-06-22): the continuum-limit and OS-reconstruction legs are now CITABLE, the genuinely-open residual narrows to 'the raw collar is a gapped quasi-free state whose massless scaling limit is the c=8 (E8)_1 net'. [C] CONTINUUM LEG: Osborne-Stottmeister (CFT from Lattice Fermions, doi:10.1007/s00220-022-04521-8, CMP 398 (2023) 219) give, building on the MMST operator-algebraic (wavelet/Wilson-Kadanoff) renormalization framework (Morinelli-Morsella-Stottmeister-Tanimoto, CMP 387 (2021) 299; PRL 127 (2021) 230601), the chiral-CFT scaling limit of free lattice fermions (Koo-Saleur -> Virasoro, correct c); WZW range rank <= c <= D (D Majorana copies). (E8)_1: c=8, rank 8, D=16 Majoranas (SO(16)_1 -> (E8)_1), 8<=8<=16 IN range; the collar is gapped quasi-free (CAR). [C] OS LEG: Adamo-Moriwaki-Tanimoto (arXiv:2407.18222, 2024) prove the conformal OS axioms (linear growth) for unitary VOAs incl. WZW/lattice VOAs; (E8)_1 is the even self-dual rank-8 lattice VOA. [E] TARGET PINNED: c=g_car+N_fam=8, det Cartan E8=1, the D=16 Majorana SO(16)_1 IS the carrier-16, the mu4 extension IS the seam glue. [O] OPEN RESIDUAL: c=8 alone does NOT pin the net (SO(16)_1 also c=8, det Cartan D8=4), so the open lemma = the holomorphy discriminator det K=1 (v277/v281/v335). A literature-mapping/reduction; does NOT close SEAM.EQUIV.01.

Original classification detail

[C] the two legs are citable as general theorems (not yet specialised to the raw collar in print); [E] the arithmetic target pins; [O] the narrowed residual. Does NOT close SEAM.EQUIV.01.

Dependencies as recorded

SEAM.EQUIV.CHAIN.01; SEAM.EQUIV.UNIFY.01; v308; v277; v335

Open the original ledger source, row 438

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Registered verification modules

  • v336_continuum_limit.pyRegistered verification module · core
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    The ONE open lemma of SEAM.EQUIV.01, stated precisely and mapped to the recent rigorous literature: the continuum-limit and OS-reconstruction legs are now CITABLE, the genuinely-open residual narrows. [C] CONTINUUM LEG: Osborne-Stottmeister (CFT from Lattice Fermions, CMP 398 (2023) 219; legacy internal token MMST, on the MMST/Morinelli-Morsella-Stottmeister-Tanimoto OAR framework, arXiv:2010.11121) give the chiral-CFT scaling limit of free lattice fermions (Koo-Saleur -> Virasoro); WZW range rank<=c<=D. (E8)_1: c=8, rank 8, D=16 Majoranas (SO(16)_1 -> (E8)_1), 8<=8<=16 in range; the collar is gapped quasi-free (CAR). [C] OS LEG: Adamo-Moriwaki-Tanimoto (arXiv:2407.18222) prove the conformal OS axioms for unitary VOAs incl. lattice VOAs; (E8)_1 is the even self-dual rank-8 lattice VOA. [C] CONSTRUCTION + COVARIANCE (Adamo-Giorgetti-Tanimoto): arXiv:2506.01008 builds the 2d conformal net of an even lattice and classifies the 2d Heisenberg-net extensions as lattice nets (the lattice IS a conformal net), and arXiv:2508.07109 shows every DHR rep of such a net is automatically positive-energy + diffeomorphism-covariant (covariance is a consequence, not an input; defuses the v215 BW circularity). [E] TARGET PINNED: c=g_car+N_fam=8, det Cartan E8=1, the D=16 Majorana SO(16)_1 IS the carrier-16 and the mu4 extension IS the seam glue. [O] OPEN RESIDUAL: c=8 alone does NOT pin the net (SO(16)_1 also c=8, det 4), so the open lemma = the holomorphy discriminator det K=1. Does NOT close SEAM.EQUIV.01.

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