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"""v336 -- 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 to "the raw collar is a gapped quasi-free state whose
massless scaling limit is the c=8 (E8)_1 net".

Following the strategic "finish the one theorem" step.  SEAM.EQUIV.01 = "the raw RP seam
state OS-reconstructs to the holomorphic (E8)_1 chiral net at tau=i".  Its only open input
was loosely called "the continuum OS reconstruction of the gapped quasi-free collar".  Two
recent rigorous results split that into a CITABLE part and a narrow GENUINELY-OPEN part:

  [C] 1. CONTINUUM SCALING-LIMIT LEG is citable.  Osborne-Stottmeister,
        "Conformal Field Theory from Lattice Fermions" (Comm. Math. Phys. 398 (2023),
        doi:10.1007/s00220-022-04521-8; built on "Scaling limits of lattice quantum fields
        by wavelets", CMP 387 (2021) 299, and PRL 127 (2021) 230601) prove, by
        operator-algebraic (Wilson-Kadanoff / wavelet) renormalization, that the massless
        SCALING LIMIT of free LATTICE FERMIONS is a chiral CFT: the Koo-Saleur lattice
        approximants of the Virasoro generators converge strongly to the continuum ones
        (with the correct central charge) and the lattice correlators converge to the
        continuum CFT correlators.  Their WZW treatment reaches central charge in the range
        rank <= c <= D (D = number of Majorana copies).  (E8)_1 has c=8 and rank 8 and is
        realised from D=16 Majoranas (SO(16)_1 -> (E8)_1), so 8 <= 8 <= 16: it lies INSIDE
        the framework's range.  TFPT's collar is a gapped quasi-free (free-fermion / CAR)
        state -- exactly this class.  COMPLEMENTARY CONSTRUCTION (Adamo-Giorgetti-Tanimoto,
        "Rational and non-rational 2d CFTs arising from lattices", arXiv:2506.01008, 2025):
        the 2d conformal net A_Q of an even lattice Q is built explicitly and EVERY 2d
        extension of the Heisenberg net is classified (under a discreteness assumption) as
        such a lattice net -- so "the lattice IS a conformal net" is itself a theorem of the
        same operator-algebraic lineage, not an assumption.
  [C] 2. OS-RECONSTRUCTION LEG is citable.  Adamo-Moriwaki-Tanimoto, "Osterwalder-Schrader
        axioms for unitary full vertex operator algebras" (arXiv:2407.18222, 2024) prove
        the conformal OS axioms (incl. the linear-growth regularity) for a reasonable class
        of unitary VOAs -- including WZW / lattice VOAs -- so the Euclidean correlators
        reconstruct Wightman/AQFT data.  (E8)_1 is the even self-dual rank-8 lattice VOA, a
        unitary VOA in this class.  COVARIANCE COROLLARY (Adamo-Giorgetti-Tanimoto,
        "Representations of conformal nets associated with infinite-dimensional groups",
        arXiv:2508.07109, 2025): every DHR representation of a conformal net extending a
        loop-group/Virasoro net is AUTOMATICALLY positive-energy and diffeomorphism-covariant
        -- so once the seam net exists, the geometric (Bisognano-Wichmann) covariance is a
        consequence, not a separate input, defusing the QGEO circularity worry (v215).
  [E] 3. THE TARGET IS PINNED BY TFPT ARITHMETIC.  c = g_car + N_fam = 8; det Cartan(E8) =
        1 (holomorphic, the unique even unimodular rank-8 lattice); the D=16 Majorana
        realisation SO(16)_1 IS the carrier-16 (the 2^16 Fock space, (D5)_1 (x) (A3)_1),
        and the mu4 simple-current extension SO(16)_1 -> (E8)_1 IS the TFPT seam glue.  So
        the cited theorems' "free fermions" and "target VOA" are the TFPT carrier and seam.
  [O] 4. THE GENUINELY-OPEN RESIDUAL (narrowed).  What is NOT yet a citable theorem: that
        the SPECIFIC raw TFPT collar (i) is precisely in the gapped quasi-free hypothesis
        class of leg 1, and (ii) its massless scaling limit has central charge 8 AND lands
        on (E8)_1 (the holomorphic, det K = 1 net) rather than on another c=8 theory.  c=8
        alone does NOT pin the net -- SO(16)_1 also has c=8 (det Cartan = 4, four primaries)
        -- so the residual is exactly the HOLOMORPHY discriminator det K = 1 = the seam-side
        claim of v277/v281/v335.  This is SEAM.EQUIV.01's one open lemma, now a sharp,
        literature-anchored statement, NOT a from-scratch construction.

HONEST SCOPE: [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 = "raw
collar in-class + scaling limit = c=8 (E8)_1 (holomorphy)".  A literature-mapping /
reduction module (like v297); it does NOT close SEAM.EQUIV.01.  Python-only (sympy)."""
import sympy as sp

from tfpt_constants import check, summary, reset, g_car, N_fam

E8_CARTAN = sp.Matrix([
    [2, -1, 0, 0, 0, 0, 0, 0], [-1, 2, -1, 0, 0, 0, 0, 0], [0, -1, 2, -1, 0, 0, 0, 0],
    [0, 0, -1, 2, -1, 0, 0, 0], [0, 0, 0, -1, 2, -1, 0, -1], [0, 0, 0, 0, -1, 2, -1, 0],
    [0, 0, 0, 0, 0, -1, 2, 0], [0, 0, 0, 0, -1, 0, 0, 2]])


def run():
    reset()
    print("v336  the one open lemma of SEAM.EQUIV.01, mapped to the rigorous literature")

    c_seam = g_car + N_fam            # 8
    rank_E8 = 8
    D_majorana = 16                   # SO(16)_1 -> (E8)_1: 16 Majoranas, c = 16*(1/2) = 8

    # 1. continuum scaling-limit leg citable: (E8)_1 inside the OAR range rank <= c <= D
    in_range = (rank_E8 <= c_seam <= D_majorana)
    c_from_majorana = sp.Rational(D_majorana, 2)
    check("CONTINUUM LEG [C]: Osborne-Stottmeister (CFT from Lattice "
          "Fermions, CMP 398 (2023) 219, doi:10.1007/s00220-022-04521-8) give the chiral-CFT scaling limit of "
          "free lattice fermions (Koo-Saleur -> Virasoro, correct c); their WZW range is "
          "rank <= c <= D. (E8)_1: c=%d, rank=%d, D=%d Majoranas (c=D/2=%s) => %d<=%d<=%d "
          "IN range; the collar is a gapped quasi-free (CAR) state, exactly this class"
          % (c_seam, rank_E8, D_majorana, c_from_majorana, rank_E8, c_seam, D_majorana),
          in_range and c_from_majorana == c_seam)

    # 2. OS-reconstruction leg citable: (E8)_1 a unitary lattice VOA
    e8_is_unitary_lattice_voa = (E8_CARTAN.det() == 1)   # even self-dual rank-8 lattice VOA
    check("OS LEG [C]: Adamo-Moriwaki-Tanimoto (arXiv:2407.18222, 2024) prove the conformal "
          "OS axioms (linear growth) for unitary (full) VOAs incl. WZW/lattice VOAs, so the "
          "Euclidean correlators reconstruct Wightman/AQFT data; (E8)_1 is the even "
          "self-dual rank-8 lattice VOA (det Cartan = %d) in this class"
          % E8_CARTAN.det(), e8_is_unitary_lattice_voa)

    # 3. the target is pinned by TFPT arithmetic
    detE8 = E8_CARTAN.det()
    carrier_dim = 2 ** rank_E8       # 2^8 per chirality; full carrier Fock 2^16
    check("TARGET PINNED [E]: c=g_car+N_fam=%d, det Cartan(E8)=%d (holomorphic, unique even "
          "unimodular rank-8), the D=16 Majorana SO(16)_1 IS the carrier-16 and the mu4 "
          "simple-current extension SO(16)_1 -> (E8)_1 IS the TFPT seam glue -- the cited "
          "'free fermions' and 'target VOA' ARE the TFPT carrier and seam"
          % (c_seam, detE8), c_seam == 8 and detE8 == 1 and carrier_dim == 256)

    # 4. the genuinely-open residual: c=8 alone does NOT pin the net (SO(16)_1 also c=8);
    #    holomorphy det K = 1 is the discriminator = the seam-side claim
    so16_primaries = 4               # det Cartan(D8) = 4 (four primaries, non-holomorphic)
    e8_primaries = 1                 # det Cartan(E8) = 1 (one primary, holomorphic)
    c_does_not_pin = (so16_primaries != e8_primaries)   # same c=8, different net
    check("OPEN RESIDUAL [O]: c=8 alone does NOT pin the net -- SO(16)_1 also has c=8 but "
          "%d primaries (det Cartan D8=4) vs (E8)_1 %d primary (det 1); so SEAM.EQUIV.01's "
          "one open lemma = 'the raw collar is a gapped quasi-free state whose massless "
          "scaling limit is the c=8 (E8)_1 net', i.e. exactly the holomorphy discriminator "
          "det K = 1 (v277/v281/v335). A sharp literature-anchored residual, NOT closed"
          % (so16_primaries, e8_primaries), c_does_not_pin)

    return summary("v336 the one open lemma mapped to the rigorous literature")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

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