"""v113 -- One kernel is the whole net: the QBL input merges with the R2 (holomorphy) premise. [I] exact Pfaffian/Wick + polarization theorems in the finite model + tower bookkeeping; gate typing [P]/[A] unchanged. The remaining QBL input (v112) was: "the seam certifies through a single scalar 2-point kernel." This module anchors that input to the established Gate-A objects (v83/v87/v89/v92): the carrier net (D5)_1 x (A3)_1 IS a free-fermion (quasi-free) theory, and for quasi-free theories the single 2-point kernel determines EVERYTHING. [I] 1. MAJORANA BOOKKEEPING (exact arithmetic). Level-1 central charges: c(D5)_1 = 45/9 = 5 = g_car, c(A3)_1 = 15/5 = 3 = N_fam, c(SO(16))_1 = 120/15 = 8, c(E8)_1 = 248/31 = 8. The carrier net is SO(10)_1 x SO(6)_1 = 10 + 6 = 16 free Majorana fermions, c = 16/2 = 8 -- and the WHOLE extension tower carrier(mu=16) -> SO(16)_1(mu=4) -> E8_1(mu=1) carries the SAME 16 free fermions; only the certified glue grows (index steps 2 x 2 = 4 = |mu_4|, v89). [I] 2. WICK / PFAFFIAN (the one kernel determines all correlations). In the exact 10-Majorana model (5 slots, integer Jordan-Wigner): ALL 210 vacuum 4-point functions and ALL 210 6-point functions equal the Pfaffian of the single 2-point kernel. "One scalar 2-point kernel" is not a truncation for this net -- it is the complete specification. [I] 3. THE KERNEL IS A CALDERON INVOLUTION OF RANK g. The 2-point matrix is M = I + iA with A integer, antisymmetric, A^2 = -I; hence P = M/2 is a projection with rank P = 5 = g_car and eps = iA is an involution (eps^2 = 1) whose +/- eigenspaces are the annihilation/creation sides. At seam level (16 Majoranas) the same construction has rank 8 = rank E8: THE CENTRAL CHARGE IS THE RANK OF THE ONE KERNEL (c = 5 for the carrier block, c = 8 for the seam hull). [I] 4. ONE KERNEL <=> ONE STATE. The joint kernel of the five annihilators is exactly 1-dimensional: the polarization determines the vacuum uniquely -- kernel => state => (Wick) all correlations. [P] 5. PREMISE MERGE (recorded, not claimed). QBL's remaining input "single scalar 2-point kernel" = "the seam state is quasi-free" = the defining property of the free-fermion net that the R2 gate already posits (Gate-A theorem (H') 'seam net = index-4 mu_4 extension of the carrier net', v89 -- same 16-fermion content at every tower level by 1.). So QBL adds NO assumption beyond R2; upgrade contract: when R2 closes, QBL closes [L] along free net => quasi-free vacuum => one kernel => sheet-odd (v110) => self-counting tower (v109/v112) => complete transport (v111) => carrier. The gate itself stays [P]/[A]. """ from itertools import combinations import sympy as sp from tfpt_constants import check, summary, reset def jw(g): """Exact Jordan-Wigner annihilators on the 2^g Fock space.""" eye2, zee = sp.eye(2), sp.diag(1, -1) amat = sp.Matrix([[0, 1], [0, 0]]) ops = [] for i in range(g): mats = [zee] * i + [amat] + [eye2] * (g - 1 - i) full = mats[0] for m in mats[1:]: full = sp.Matrix(sp.kronecker_product(full, m)) ops.append(full) return ops def pfaffian(kmat, idx): """Pfaffian of the antisymmetric submatrix indexed by idx (recursive).""" if not idx: return sp.Integer(1) head = idx[0] total = sp.Integer(0) for t, j in enumerate(idx[1:]): rest = [x for x in idx[1:] if x != j] total += (-1) ** t * kmat[head][j] * pfaffian(kmat, rest) return total def run(): reset() print("v113 quasi-free kernel (one kernel is the whole net)") # 1. bookkeeping check("MAJORANA BOOKKEEPING: c(D5)_1 = 45/9 = 5 = g_car, c(A3)_1 = " "15/5 = 3 = N_fam, c(SO16)_1 = 120/15 = 8, c(E8)_1 = 248/31 " "= 8; carrier = 10 + 6 = 16 free Majoranas, c = 16/2 = 8 -- " "the whole tower carrier -> SO(16)_1 -> E8_1 carries the SAME " "16 fermions, index steps 2 x 2 = 4 = |mu_4| (v89)", sp.Rational(45, 9) == 5 and sp.Rational(15, 5) == 3 and sp.Rational(120, 15) == 8 and sp.Rational(248, 31) == 8 and 10 + 6 == 16 and sp.Rational(16, 2) == 8 and sp.sqrt(sp.Rational(16, 4)) * sp.sqrt(sp.Rational(4, 1)) == 4) # build the exact 10-Majorana model g = 5 a = jw(g) ad = [x.T for x in a] cs = [] for i in range(g): cs.append(a[i] + ad[i]) cs.append(sp.I * (ad[i] - a[i])) vac = sp.zeros(2 ** g, 1) vac[0] = 1 def vev(ops): v = vac for o in reversed(ops): v = o * v return (vac.T * v)[0] m2 = sp.Matrix(10, 10, lambda j, k: vev([cs[j], cs[k]])) amat = (m2 - sp.eye(10)) / sp.I pol = (sp.eye(10) + sp.I * amat) / 2 # 3. Calderon involution of rank g check("THE KERNEL IS A CALDERON INVOLUTION OF RANK g: M = I + iA " "with A integer antisymmetric, A^2 = -I; P = M/2 projection " "with rank P = 5 = g_car = c(D5)_1; eps = iA involution " "(eps^2 = 1) -- the certified kernel IS the polarization", sp.simplify(amat + amat.T) == sp.zeros(10) and all(x.is_integer for x in amat) and sp.simplify(amat * amat + sp.eye(10)) == sp.zeros(10) and sp.simplify(pol * pol - pol) == sp.zeros(10) and pol.rank() == 5 and sp.simplify((sp.I * amat) ** 2) == sp.eye(10)) # 2. Wick / Pfaffian k10 = [[vev([cs[x], cs[y]]) if x < y else 0 for y in range(10)] for x in range(10)] ok4 = all(sp.simplify(vev([cs[i] for i in q]) - pfaffian(k10, list(q))) == 0 for q in combinations(range(10), 4)) ok6 = all(sp.simplify(vev([cs[i] for i in s]) - pfaffian(k10, list(s))) == 0 for s in combinations(range(10), 6)) check("WICK/PFAFFIAN: ALL 210 vacuum 4-point functions AND all 210 " "6-point functions equal the Pfaffian of the single 2-point " "kernel -- one kernel determines all correlations (exact)", ok4 and ok6) # 4. one kernel <=> one state check("ONE KERNEL <=> ONE STATE: the joint kernel of the five " "annihilators is exactly 1-dimensional -- the polarization " "determines the vacuum uniquely", 32 - sp.Matrix.vstack(*a).rank() == 1) # 3b. seam level a16 = sp.zeros(16) for i in range(8): a16[2 * i, 2 * i + 1] = 1 a16[2 * i + 1, 2 * i] = -1 p16 = (sp.eye(16) + sp.I * a16) / 2 check("SEAM LEVEL: the 16-Majorana kernel has rank 8 = rank E8 = c " "-- THE CENTRAL CHARGE IS THE RANK OF THE ONE KERNEL (5 for " "the carrier block, 8 for the seam hull)", sp.simplify(p16 * p16 - p16) == sp.zeros(16) and p16.rank() == 8) # 5. premise merge check("PREMISE MERGE [P] (recorded, not claimed): QBL's input " "'single scalar 2-point kernel' = 'the seam state is " "quasi-free' = the defining property of the free-fermion net " "the R2 gate already posits (Gate-A (H') index-4 extension, " "v89; same 16-fermion content at every tower level). QBL adds " "NO assumption beyond R2; when R2 closes, QBL closes [L] " "along free net => one kernel => sheet-odd => self-counting " "tower => complete transport => carrier. Gate typing [P]/[A] " "unchanged", True) return summary("v113 quasi-free kernel") if __name__ == "__main__": raise SystemExit(1 if run() else 0)