"""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 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)