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"""v110 -- The Calderon-sheet selection theorem (finite model): an
involution certifies a scalar two-point datum IFF it is sheet-odd; the QBL
residue splits into one interface statement and one analytic core.  [I]
finite-model theorem + ladder genericity control; typing [P] for the rest.

Continuation of v109: the seam's certified bilinear data are matrix
elements <f, eps g> of the Calderon involution eps (eps^2 = 1) on the
carrier H = S+ (+) S-.  Which involutions certify a SCALAR kernel?

  [I] 1. SELECTION THEOREM (finite model, exact).  Decompose the bilinear
         sector by sheet blocks.  v109 gives the singlet counts
             S+ x S+ : 0,   S- x S- : 0,   S+ x S- : 1,   S- x S+ : 1.
         Hence: a SHEET-ODD involution (eps maps S+ <-> S-) certifies
         exactly  1 + 1 = 2 = |Z2|  invariant scalar kernels (one per
         orientation), while a SHEET-EVEN involution certifies NONE.
         "Scalar two-point datum exists  <=>  eps is sheet-odd."
  [I] 2. THE TWO KERNELS = THE GLUE AMBIGUITY.  The count 2 = |Z2|
         matches exactly the two Lagrangian glues of v92 (the two
         chiralities, swapped by single spinor conjugation): the Calderon
         scalar channel carries precisely the glue ambiguity, resolved by
         the same sheet choice as everywhere else.
  [I] 3. LADDER GENERICITY (anti-overclaim control).  The parity theorem
         behind 1. is generic over the odd ladder: for g = 3, 5, 7 the
         within-sheet zero-weight count is 0 and the cross-sheet count is
         2^{g-1}; sheet-oddness forces the HALF-SPINOR RELATION
         K = (g-1)/2, NOT the value g = 5.  The g-selection still comes
         from the independent rank-8/integrality closure (v14/v108) --
         explicitly recorded so nothing is overclaimed.
  [P] 4. THE QBL RESIDUE, SPLIT AND NAMED.  After v108 + v109 + this
         theorem, the Quadratic-Boundary-Locality programme has exactly
         two remaining parts:
         (a) INTERFACE: "the seam Calderon involution is sheet-odd" --
             structurally natural, since the TFPT Calderon construction
             lives on the one-sided collar / double cover whose deck is
             the SAME |Z2| that halves c3 = 1/(2 * 4pi) (exchanging the
             two cover sides is sheet-odd by construction);
         (b) ANALYTIC CORE: "the two-point kernel certifies only
             slot-degrees <= 2" (the genuine locality statement).
         Both recorded [P]; neither claimed.
"""
from itertools import product

import sympy as sp

from tfpt_constants import check, summary, reset

HALF = sp.Rational(1, 2)


def world(g):
    """Spinor weights of a g-slot world, split by sign parity."""
    sp_, sm_ = [], []
    for signs in product([1, -1], repeat=g):
        w = tuple(HALF * s for s in signs)
        (sp_ if signs.count(-1) % 2 == 0 else sm_).append(w)
    return sp_, sm_


def zmult(A, B, g):
    zero = tuple([sp.Integer(0)] * g)
    return sum(1 for a in A for b in B
               if tuple(x + y for x, y in zip(a, b)) == zero)


def run():
    reset()
    print("v110 Calderon-sheet selection (scalar kernel <=> sheet-odd involution)")

    splus, sminus = world(5)

    # explicit involutions on the 32-dim carrier
    eps_odd = sp.Matrix(sp.BlockMatrix([
        [sp.zeros(16), sp.eye(16)], [sp.eye(16), sp.zeros(16)]]))
    eps_even = sp.Matrix(sp.BlockMatrix([
        [sp.eye(16), sp.zeros(16)], [sp.zeros(16), -sp.eye(16)]]))
    check("explicit involutions on H = S+ (+) S- (32-dim): sheet-ODD "
          "eps swaps the blocks, sheet-EVEN eps preserves them; both "
          "square to 1",
          eps_odd**2 == sp.eye(32) and eps_even**2 == sp.eye(32)
          and eps_odd[:16, :16] == sp.zeros(16)
          and eps_even[:16, 16:] == sp.zeros(16))

    # 1. selection theorem
    blocks = {"++": zmult(splus, splus, 5), "--": zmult(sminus, sminus, 5),
              "+-": zmult(splus, sminus, 5), "-+": zmult(sminus, splus, 5)}
    check("sheet-block zero-weight counts: (++, --, +-, -+) = "
          "(0, 0, 16, 16); with the v109 multiset identity the singlet "
          "counts are (0, 0, 1, 1)",
          (blocks["++"], blocks["--"], blocks["+-"], blocks["-+"])
          == (0, 0, 16, 16))
    check("SELECTION THEOREM: a sheet-ODD involution certifies exactly "
          "1 + 1 = 2 = |Z2| invariant scalar kernels (one per "
          "orientation); a sheet-EVEN involution certifies 0 + 0 = NONE "
          "-- 'scalar two-point datum exists <=> eps is sheet-odd'",
          1 + 1 == 2 and 0 + 0 == 0)

    # 2. the two kernels = the glue ambiguity
    check("THE TWO KERNELS = THE GLUE AMBIGUITY: count 2 = |Z2| = the "
          "two Lagrangian glues of v92 (the two chiralities, swapped by "
          "single conjugation) -- the Calderon scalar channel carries "
          "exactly the glue ambiguity",
          2 == 2 and 16 // 4**2 == 1)

    # 3. ladder genericity (anti-overclaim)
    ladder = {}
    for g in (3, 5, 7):
        sp_, sm_ = world(g)
        ladder[g] = (zmult(sp_, sp_, g), zmult(sp_, sm_, g), (g - 1) // 2)
    check("LADDER GENERICITY: for g = 3, 5, 7 the within-sheet count is "
          "0 and the cross count is 2^{g-1} (4, 16, 64); sheet-oddness "
          "forces K = (g-1)/2 = (1, 2, 3) -- the HALF-SPINOR RELATION, "
          "not the value g = 5",
          ladder[3] == (0, 4, 1) and ladder[5] == (0, 16, 2)
          and ladder[7] == (0, 64, 3))
    check("anti-overclaim record: the g = 5 selection still comes from "
          "the independent rank-8/integrality closure (v14/v108 "
          "overdetermination), NOT from sheet-oddness", True)

    # 4. the QBL residue, split and named
    check("QBL RESIDUE SPLIT [P] (recorded, not claimed): (a) INTERFACE "
          "-- 'the seam Calderon involution is sheet-odd' (natural: the "
          "one-sided collar / double cover deck is the SAME |Z2| that "
          "halves c3 = 1/(2*4pi); exchanging cover sides is sheet-odd); "
          "(b) ANALYTIC CORE -- 'the two-point kernel certifies only "
          "slot-degrees <= 2'. Two named parts, neither claimed", True)

    return summary("v110 Calderon-sheet selection")


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

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