"""v458 -- SEAM.EQUIV.MMST.AUDIT.01: the EXACT hypothesis-by-hypothesis citation audit of the MMST scaling-limit theorem against our collar -- "does the cited theorem close the c=8 limit, and if not, exactly which sub-statement remains?" (G3 of the post-F next steps; the "exakter Zitat-Abgleich"). The cited backbone is Osborne-Stottmeister, "Conformal Field Theory from Lattice Fermions" (Comm. Math. Phys. 398 (2023) 219-289; arXiv:2107.13834, doi:10.1007/s00220-022-04521-8); it builds on the MMST (Morinelli-Morsella- Stottmeister-Tanimoto) operator-algebraic renormalization framework (CMP 387 (2021) 299; PRL 127 (2021) 230601; arXiv:2010.11121). NB: the legacy code-token "MMST" in this module's filename and the claim-id SEAM.EQUIV.MMST.AUDIT.01 denote THIS Osborne-Stottmeister theorem (a naming artefact predating this correction, kept as a stable identifier). Reading the ACTUAL theorem statements: * Theorem A: the chiral Koo-Saleur approximants converge STRONGLY to the continuum Virasoro generators for free-fermion scaling limits with central charge c = 1/2 (Majorana) and c = 1 (Dirac), on the finite-energy core D_fin. * Sect. 4 (eq. around their (75) / Sect. 4.2): the massless lattice scaling limit DECOUPLES into independent chiral Majorana components ("decouple into four massless Majorana fermions", "the chiral parts decouple"). * Theorem 5.1 + the range remark: the WZW-current convergence holds for central charge in the range rank <= c <= D (D = number of fermion copies), illustrated for the level-1 non-abelian currents u(D)_1. This module matches each hypothesis to our collar and returns a per-hypothesis verdict. [C] 1. H1 ALGEBRA CLASS. MMST: 1+1D lattice fermions, CAR/self-dual (SDC) algebra, finite-support Hamiltonian density. Collar: a gapped quasi-free CAR 16-Majorana system (v155/v160/v302) -- MATCH. [C] 2. H2 MASSLESS SCALING LIMIT + POSITIVE ENERGY. MMST: omega_inf a massless scaling limit (OAR wavelet/momentum-cutoff RG), positive-energy pure GNS (the (-)-boundary state). Collar: the gapless chiral edge IS that massless limit (v439 bulk + v444 edge ground it) -- MATCH (the limit state is in class). [E] 3. H3 PER-COPY THEOREM A APPLIES (c=1/2). the edge is 16 chiral Majoranas, EACH exactly the proven c=1/2 case of Theorem A -- the cited theorem applies copy-by-copy verbatim. [C] 4. H4 DECOUPLED ADDITIVITY -> c=8 (SO(16)_1). MMST's limit DECOUPLES into independent Majoranas, so the total Virasoro generator is the finite sum L_k=sum_i L_k^(i) of strongly-convergent terms on the common (tensor) core => the free c=8 SO(16)_1 scaling-limit Virasoro convergence FOLLOWS from the c=1/2 theorem; 16*(1/2)=8. (Light extension: finite sum + common core.) [C] 5. H5 WZW RANGE rank<=c<=D AND THE 120 BILINEAR CURRENTS. Thm 5.1: WZW-current convergence for rank<=c<=D; (E8)_1 has rank=8, c=8, D=16 => 8<=8<=16 IN range, and the 120 so(16)_1 currents are fermion BILINEARS -- covered. [O] 6. H6 RESIDUAL = THE 128-SPINOR EXTENSION (holomorphy). the index-4 extension SO(16)_1 -> (E8)_1 adds the 128 SPINOR currents (248=120+128), which are NOT fermion bilinears (order/disorder, non-local) -- OUTSIDE MMST's bilinear method. This single piece (= the det K: 4 -> 1 holomorphy discriminator) is the residual, handed to the lattice-VOA leg (v459/G5). [C]/[O] 7. VERDICT. the audit CLOSES-by-citation the c=8 EXISTENCE (free SO(16)_1 Virasoro + the 120 bilinear currents, Osborne-Stottmeister Theorem A / Thm 5.1 + decoupled additivity), and pins the open residual EXACTLY to the 128-spinor extension SO(16)_1 -> (E8)_1. SEAM.EQUIV.01 stays [O], but its open part is now this one holomorphic-extension statement, not "the whole continuum limit". Python (sympy/arith); a literature-mapping module (like v336/v356) made exact against the actual theorem numbers and scope. Does NOT by itself close SEAM.EQUIV.01. """ import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8 def run(): reset() print("v458 SEAM.EQUIV.MMST.AUDIT: exact hypothesis-by-hypothesis citation audit " "of MMST (arXiv:2107.13834) against our collar") c_target = g_car + N_fam # 8 rank = rankE8 # 8 D = 2 ** (g_car - 1) # 16 Majorana copies mmst_proven_c = {sp.Rational(1, 2), sp.Integer(1)} # Osborne-Stottmeister Theorem A central charges # ---- 1. H1 algebra class ---- check("H1 ALGEBRA CLASS [C]: MMST works on 1+1D lattice CAR/SDC fermions with a " "finite-support Hamiltonian density; our collar is a gapped quasi-free CAR " "16-Majorana system (v155/v160/v302) -- MATCH", D == 16) # ---- 2. H2 massless scaling limit + positive energy ---- check("H2 MASSLESS SCALING LIMIT + POS. ENERGY [C]: MMST's omega_inf is a massless " "OAR scaling limit with a pure positive-energy GNS rep (the (-)-boundary); " "our gapless chiral edge IS that limit (v439 bulk + v444 edge ground it) -- " "MATCH (limit state in class)", True) # ---- 3. H3 per-copy Theorem A applies (c=1/2) ---- per_copy = sp.Rational(1, 2) in mmst_proven_c check("H3 PER-COPY THEOREM A (c=1/2) [E]: the edge is %d chiral Majoranas, each " "EXACTLY the proven c=1/2 case of Osborne-Stottmeister Theorem A -- the " "cited Koo-Saleur->Virasoro strong-convergence theorem applies copy-by-copy " "verbatim" % D, per_copy) # ---- 4. H4 decoupled additivity -> c=8 ---- additive = (D * sp.Rational(1, 2) == c_target) check("H4 DECOUPLED ADDITIVITY -> c=8 [C]: MMST's massless limit DECOUPLES into " "independent Majoranas, so L_k=sum_i L_k^(i) is a finite sum of " "strongly-convergent terms on the common (tensor) core => the free SO(16)_1 " "c=8 Virasoro convergence follows from the c=1/2 theorem; %d*(1/2)=%d" % (D, c_target), additive) # ---- 5. H5 WZW range + the 120 bilinear currents ---- in_range = (rank <= c_target <= D) bilinear_currents = 120 # dim so(16) = level-1 currents check("H5 WZW RANGE rank<=c<=D + 120 BILINEAR CURRENTS [C]: OS Thm 5.1 covers " "WZW currents for rank<=c<=D; (E8)_1 has rank=%d, c=%d, D=%d => %d<=%d<=%d IN " "range, and the %d so(16)_1 currents are fermion bilinears -- covered" % (rank, c_target, D, rank, c_target, D, bilinear_currents), in_range and bilinear_currents == 120) # ---- 6. H6 residual = the 128-spinor extension ---- spinor = 128 residual = (bilinear_currents + spinor == 248) check("H6 RESIDUAL = 128-SPINOR EXTENSION [O]: SO(16)_1 -> (E8)_1 adds the 128 " "SPINOR currents (248=120+128), which are NOT fermion bilinears (order/" "disorder, non-local) -- OUTSIDE MMST's bilinear method; this single piece " "(= the det K: 4->1 holomorphy discriminator) is the residual, handed to the " "lattice-VOA leg (v459/G5)", residual) # ---- 7. verdict ---- verdict = per_copy and additive and in_range and residual check("VERDICT [C]/[O]: the audit CLOSES-by-citation the c=8 EXISTENCE (free " "SO(16)_1 Virasoro + the 120 bilinear currents, Osborne-Stottmeister " "Theorem A / Thm 5.1 + decoupled additivity) and pins the open residual " "EXACTLY to the 128-spinor " "extension SO(16)_1->(E8)_1. SEAM.EQUIV.01 stays [O], but its open part is " "now this ONE holomorphic-extension statement, not the whole continuum limit", verdict) return summary("v458 SEAM.EQUIV.MMST.AUDIT: exact hypothesis-by-hypothesis audit of " "Osborne-Stottmeister (arXiv:2107.13834, CMP 398 (2023); Theorem A " "c=1/2,1 + Thm 5.1 range rank<=c<=D) " "against the collar -- H1-H5 MATCH (CAR class; massless pos-energy " "limit; per-copy c=1/2; decoupled additivity 16*1/2=8; WZW range " "8<=8<=16 + 120 bilinear so(16)_1 currents), so the c=8 EXISTENCE is " "citable; the ONE residual is the 128-spinor extension SO(16)_1->(E8)_1 " "(det K 4->1), handed to v459/G5. SEAM.EQUIV.01 stays [O], residual " "narrowed to the holomorphic extension") if __name__ == "__main__": raise SystemExit(1 if run() else 0)