"""v981 -- PRIME.LSTAR.DUAL_HOLE.01 [E] (consolidation wave 14, round 356, note DCCVII, 2026-08-27): THE BORODIN PARTICLE-HOLE DUALITY -- the exact equivalence L* <=> R > (1/2) I for the dual hole kernel at half filling, with the reciprocal dual weight and the by-design anti-correlation of the two phi blocks, all re-derived from scratch in exact arithmetic (pure Fractions on a discrete orthogonal-polynomial model + sympy closed forms on the cosine lattice, no probe imports), plus the sealed round verdict re-run as exact decision logic on the frozen record aggregates. This is the wave's strongest single theorem: the r354 anti-correlation of the weight block against the pair-geometry block is DUALITY ALGEBRA BY DESIGN, and the honest main verdict is DUALITY_REPARAM_ONLY -- the L* lane goes FINAL to the specialist memo. THE EXACT LAYER (theorem grade, module-own): [E] 1. BORODIN COMPLEMENTATION AT HALF FILLING: on a discrete support X of S = 2N - 1 atoms with positive weight u, the rank-N projection kernel Pi_N^u (built from monic orthogonal polynomials in exact Fractions) and the rank-(N-1) kernel of the DUAL weight u_vee_j = 1/(u_j P'(x_j)^2) (P the monic node polynomial of the FULL support) satisfy Pi_N^u + G Pi_{N-1}^{u_vee} G^{-1} == I with the RATIONAL conjugator G = diag(1/(u_j P'(x_j))) -- no square roots; bit-exact on the 5-atom rational model; the dropped-P'^2 mutant (m1) and the rank-N mutant (m2) each break the complementation by an exact nonzero amount. [E] 2. THE L-ENSEMBLE TRANSFORMATION: Q = (Pi_N^u)_{Y,Y} on any pair Y and E := Q (I - Q)^{-1} satisfy the exact roundtrip Q == E (I + E)^{-1}; via the complementation, I - Q == G_Y R G_Y^{-1} with R = (Pi_{N-1}^{u_vee})_{Y,Y}, hence E == G_Y (R^{-1} - I) G_Y^{-1} bit-exact. [E] 3. THE SPECTRAL MAP AND THE EQUIVALENCE: charpoly(E) == charpoly(R^{-1} - I) exactly (diagonal conjugation), so the eigenvalues satisfy lam(E) = 1/lam(R) - 1 and L* <=> lam_max(E) < 1 <=> R > (1/2) I, margin = 1 - lam_max(E) == 2 - 1/lam_min(R) -- gated in exact arithmetic on a passing AND a failing model (both directions of the equivalence realized). [E] 4. THE PAIR IDENTITIES: the diagonal entries are conjugation- invariant, so p = 1 - E_11 == 2 - (R^{-1})_11 and q = 1 - E_22 == 2 - (R^{-1})_22 exactly; trace and determinant give p + q == tr(2I - R^{-1}) and p q - c^2 == det(2I - R^{-1}) exactly -- the r342 pair block IS the (1,2) principal minor of 2I - R^{-1} up to the rational conjugation (entry identity gated with the explicit G factors). [E] 5. THE COSINE-LATTICE CLOSED FORMS (S = 5, symbolic): the document support x_j = cos(pi j / S) has P(x) = 2^{1-S}(x+1) U_{S-1}(x) exactly; |P'(x_j)| = 2^{1-S} S kappa_j/(1 - x_j) with kappa_j = 1 (j < S), 2 (j = S); with the document weight u_j = (1/S) c_j (1 - x_j) F_j (F = |f| generic positive) the dual weight is EXACTLY u_vee_j PROPTO c_j (1 - x_j) / F_j (one j-independent constant, gated symbolically): the original weight carries +log F, the dual weight -log F. [E] 6. ANTI-CORRELATION BY DESIGN: u_j u_vee_j == 1/P'(x_j)^2 definitionally, so d log u_vee / d log F == -1 while d log u / d log F == +1 (symbolic): the weight block W = log(u_1 u_2) and the dual block DW = log(u_vee_1 u_vee_2) are ALGEBRAIC MIRRORS up to the F-free geometry factor -- the measured r354 anti-correlation (corr -0.999998) is duality algebra, not a new mechanism. THE FROZEN CENSUS LAYER (sealed record aggregates re-run as decision logic; census values are GATES): r356 borodin_dual_hole_probe (34/34, SPEC 5d277d57 freeze / final record sealed; ~115 s; zero amendments): the support gate holds BITWISE on 85/85 windows (S == L/2 == 2 N_w - 1: the r228 half-filling law N_w = ceil(S/2) IS Borodin's complementary rank condition -- an integer lemma, re-proved here for every odd S); the spectral map margin == 2 - 1/lam_min(R) at <= 9.9e-10; the dual weight three ways with route B == route C bitwise-tight (4.4e-16); corr(psi57 DW, psi57 W) = -0.999998 <= ANTI_BAR -0.99 (ANTI_DESIGN_GATED); L* == R > (1/2) I live to lam_min(R) - 1/2 = +5.9e-11 on EXT6; ALL FOUR ladderable dead worlds violate R > (1/2) I structurally (lam_min(R) - 1/2 in [-0.500, -0.083] vs MAIN +4e-5 .. +6e-11); the r227/r228 obstruction is structurally absent (positive lift: min u >= 6.1e-12 > 0 on every union node vs the signed measure's zero-weight kill) -- BUT the sealed carrier clauses FAIL (best dual-block leave-out 1.021 > 0.5, no compression against the r354 four-block basis): the duality is a REPARAMETRIZATION -- verdict DUALITY_REPARAM_ONLY, the lane goes final to the memo; census-positive: RESERVE_LOCALIZED (the LOCAL dual 2x2 block predicts the log reserve without p/q/c readback: corr +0.9982 / leave-out 0.068 on the 57; EXT puretest +0.9828 / 0.165) and AC_CLASS_EXCLUDED (the rescaled pair positions are family-constant against pi^2 f^2/4 AND a_rhoK = 1.4222 where Lubinsky-type AC universality demands rho_K -> const: the AC hard-edge class is measurably excluded -- Fable's falsifier, banked for the memo). HONEST SCOPE (firewall): the exact layer is finite discrete-OP algebra -- theorem grade on rational models and symbolic closed forms on the cosine lattice; the 85-window census is a FROZEN RECORD AGGREGATE (the probe is sealed experiments-side, pinned in rh/INVENTORY.json, re-verified by run_rh.py); the equivalence L* <=> R > (1/2) I is a REPARAMETRIZATION of the open condition, NOT progress on its truth -- PRIME.LSTAR.SUBORDINATION.01 stays [O] and the lane is FROZEN at rh/problem/lstar_problem.tex; the remaining one-object question is the ~500x smallness of the difference of the by-design-coupled blocks (now a question about local dual two-point data). No zeros, no prime oracles. NOT evidence for or against the Riemann Hypothesis. NO L* CLAIM. NO RH CLAIM. Python-only per GATE.WOLFRAM.02. PROVENANCE: discovery probe borodin_dual_hole_probe.py (34/34, SPEC freeze 5d277d576df75d3a, two-commit protocol 58bb09bb / 5d351e90, zero post-freeze amendments); round 356, note DCCVII, 2026-08-27; literature: Borodin 2002 (duality of discrete OP ensembles, Thm 5), Lubinsky IMRP 2008 (AC universality, used as the excluded prediction); the memo addendum is rh/problem/lstar_problem.tex (round-356 addendum, commit ddf3d94b). """ from fractions import Fraction as Fr import sympy as sp from tfpt_constants import check, summary, reset # ---------------- exact matrix helpers (Fractions) ---------------- def mat_mul(A, B): n, k, m2 = len(A), len(B), len(B[0]) return [[sum(A[i][t] * B[t][j] for t in range(k)) for j in range(m2)] for i in range(n)] def mat_inv(A): n = len(A) M = [row[:] + [Fr(1) if i == j else Fr(0) for j in range(n)] for i, row in enumerate(A)] for col in range(n): piv = next(r for r in range(col, n) if M[r][col] != 0) M[col], M[piv] = M[piv], M[col] pv = M[col][col] M[col] = [v / pv for v in M[col]] for r in range(n): if r != col and M[r][col] != 0: f = M[r][col] M[r] = [a - f * b for a, b in zip(M[r], M[col])] return [row[n:] for row in M] def eye(n): return [[Fr(1) if i == j else Fr(0) for j in range(n)] for i in range(n)] def mat_eq(A, B): return all(a == b for ra, rb in zip(A, B) for a, b in zip(ra, rb)) # ---------------- discrete OP kernel (exact) ---------------- def monic_ops(xs, ws, nmax): """monic orthogonal polynomials (coefficient lists, ascending) for the measure sum w_j delta_{x_j}, by Gram-Schmidt in exact Fractions; returns (polys, norms).""" def peval(c, x): v = Fr(0) for a in reversed(c): v = v * x + a return v polys, norms = [], [] for k in range(nmax): c = [Fr(0)] * k + [Fr(1)] # x^k, monic for pj, hj in zip(polys, norms): ip = sum(w * peval(c, x) * peval(pj, x) for x, w in zip(xs, ws)) f = ip / hj c = [a - f * b for a, b in zip(c, pj + [Fr(0)] * (len(c) - len(pj)))] h = sum(w * peval(c, x) ** 2 for x, w in zip(xs, ws)) polys.append(c) norms.append(h) return polys, norms def projection_kernel(xs, ws, N): """Pi_N(i, j) = w_j sum_{k R > (1/2) I at half filling, the " "reciprocal dual weight and the by-design anti-correlation " "(round 356 frozen)") # the 5-atom rational model, S = 2N - 1 with N = 3 xs = [Fr(-2), Fr(-1), Fr(0), Fr(1), Fr(3)] ws = [Fr(1, 3), Fr(1, 2), Fr(1), Fr(2, 5), Fr(1, 7)] S, N = len(xs), 3 check("model setup: S == 2N - 1 (half filling, S = 5, N = 3), " "u > 0 on every node (the r227/r228 zero-weight obstruction " "is structurally absent on the positive lift eta = mu + nu)", S == 2 * N - 1 and all(w > 0 for w in ws)) Pi = projection_kernel(xs, ws, N) check("Pi_N^u is an exact rank-N projection: Pi^2 == Pi and " "tr Pi == N, pure Fractions", mat_eq(mat_mul(Pi, Pi), Pi) and sum(Pi[i][i] for i in range(S)) == N) Pp = node_poly_derivs(xs) wv = [1 / (w * pp ** 2) for w, pp in zip(ws, Pp)] PiV = projection_kernel(xs, wv, N - 1) G = [1 / (w * pp) for w, pp in zip(ws, Pp)] conj = [[G[i] * PiV[i][j] / G[j] for j in range(S)] for i in range(S)] comp = [[Pi[i][j] + conj[i][j] for j in range(S)] for i in range(S)] check("BORODIN COMPLEMENTATION bit-exact: Pi_N^u + G Pi_{N-1}" "^{u_vee} G^{-1} == I with the reciprocal dual weight " "u_vee = 1/(u P'^2) and the RATIONAL conjugator G = " "diag(1/(u P')) -- no square roots", mat_eq(comp, eye(S))) # m1: dropped P'^2 ; m2: rank N instead of N-1 wv_m1 = [1 / (w * pp) for w, pp in zip(ws, Pp)] PiV_m1 = projection_kernel(xs, wv_m1, N - 1) comp_m1 = [[Pi[i][j] + G[i] * PiV_m1[i][j] / G[j] for j in range(S)] for i in range(S)] PiV_m2 = projection_kernel(xs, wv, N) comp_m2 = [[Pi[i][j] + G[i] * PiV_m2[i][j] / G[j] for j in range(S)] for i in range(S)] check("mutants CAUGHT exactly: m1 (dual weight without P'^2) and " "m2 (rank N instead of N - 1) each break the " "complementation by an exact nonzero amount", not mat_eq(comp_m1, eye(S)) and not mat_eq(comp_m2, eye(S))) # ---- the L-ensemble transformation on the pair Y Y = [1, 3] Q = sub2(Pi, Y) ImQ = [[(Fr(1) if i == j else Fr(0)) - Q[i][j] for j in range(2)] for i in range(2)] E = mat_mul(Q, mat_inv(ImQ)) IpE = [[(Fr(1) if i == j else Fr(0)) + E[i][j] for j in range(2)] for i in range(2)] check("L-ENSEMBLE ROUNDTRIP bit-exact: E = Q(I - Q)^{-1} and " "Q == E(I + E)^{-1} on the pair Y", mat_eq(mat_mul(E, mat_inv(IpE)), Q)) R = sub2(PiV, Y) Rinv = mat_inv(R) GY = [G[Y[0]], G[Y[1]]] E_from_R = [[GY[i] * (Rinv[i][j] - (Fr(1) if i == j else Fr(0))) / GY[j] for j in range(2)] for i in range(2)] check("E == G_Y (R^{-1} - I) G_Y^{-1} bit-exact (restriction of " "the complementation to the pair; G diagonal, so the " "restriction commutes)", mat_eq(E, E_from_R)) check("PAIR IDENTITIES exact: p = 1 - E_11 == 2 - (R^{-1})_11, " "q = 1 - E_22 == 2 - (R^{-1})_22 (diagonal conjugation-" "invariant); pq - c^2 == det(2I - R^{-1}) and p + q == " "tr(2I - R^{-1}); the off-diagonal carries the explicit " "rational conjugator (c-entry identity with G factors)", 1 - E[0][0] == 2 - Rinv[0][0] and 1 - E[1][1] == 2 - Rinv[1][1] and ((1 - E[0][0]) * (1 - E[1][1]) - E[0][1] * E[1][0] == (2 - Rinv[0][0]) * (2 - Rinv[1][1]) - Rinv[0][1] * Rinv[1][0]) and E[0][1] == GY[0] * Rinv[0][1] / GY[1]) # ---- the spectral map + the equivalence (symbolic + both signs) r11, r12, r21, r22, lam = sp.symbols("r11 r12 r21 r22 lam") g1, g2 = sp.symbols("g1 g2", nonzero=True) Rs = sp.Matrix([[r11, r12], [r21, r22]]) Gs = sp.diag(g1, g2) Es = Gs * (Rs.inv() - sp.eye(2)) * Gs.inv() cp_E = sp.expand(Es.charpoly(lam).as_expr()) cp_R = sp.expand((Rs.inv() - sp.eye(2)).charpoly(lam).as_expr()) check("SPECTRAL MAP symbolic: charpoly(E) == charpoly(R^{-1} - I) " "for generic R and diagonal conjugator -- lam(E) = " "1/lam(R) - 1, hence lam_max(E) < 1 <=> lam_min(R) > 1/2 " "and margin = 1 - lam_max(E) == 2 - 1/lam_min(R)", sp.simplify(cp_E - cp_R) == 0) lamsE = sp.Matrix([[sp.Rational(E[0][0]), sp.Rational(E[0][1])], [sp.Rational(E[1][0]), sp.Rational(E[1][1])]] ).eigenvals() lamsR = sp.Matrix([[sp.Rational(R[0][0]), sp.Rational(R[0][1])], [sp.Rational(R[1][0]), sp.Rational(R[1][1])]] ).eigenvals() lmaxE = sp.Max(*lamsE.keys()) lminR = sp.Min(*lamsR.keys()) check("EQUIVALENCE realized on the model, exact: lam_max(E) < 1 " "AND lam_min(R) > 1/2 AND margin identity 1 - lam_max(E) " "== 2 - 1/lam_min(R) (sympy exact surds)", bool(sp.simplify(lmaxE - 1) < 0) and bool(sp.simplify(lminR - sp.Rational(1, 2)) > 0) and sp.simplify((1 - lmaxE) - (2 - 1 / lminR)) == 0) # a failing model: push one weight so hard the pair block crosses ws_bad = [Fr(1, 3), Fr(50), Fr(1), Fr(50), Fr(1, 7)] Pi_b = projection_kernel(xs, ws_bad, N) wv_b = [1 / (w * pp ** 2) for w, pp in zip(ws_bad, Pp)] PiV_b = projection_kernel(xs, wv_b, N - 1) R_b = sub2(PiV_b, Y) lminRb = sp.Min(*sp.Matrix( [[sp.Rational(R_b[0][0]), sp.Rational(R_b[0][1])], [sp.Rational(R_b[1][0]), sp.Rational(R_b[1][1])]] ).eigenvals().keys()) Q_b = sub2(Pi_b, Y) E_b = mat_mul(Q_b, mat_inv( [[(Fr(1) if i == j else Fr(0)) - Q_b[i][j] for j in range(2)] for i in range(2)])) lmaxEb = sp.Max(*sp.Matrix( [[sp.Rational(E_b[0][0]), sp.Rational(E_b[0][1])], [sp.Rational(E_b[1][0]), sp.Rational(E_b[1][1])]] ).eigenvals().keys()) check("BOTH DIRECTIONS realized: the loaded mutant model violates " "R > (1/2) I exactly where lam_max(E) >= 1 -- the " "equivalence has teeth on the failing side too", bool(sp.simplify(lminRb - sp.Rational(1, 2)) < 0) == bool(sp.simplify(lmaxEb - 1) >= 0) and bool(sp.simplify(lminRb - sp.Rational(1, 2)) < 0)) # ---- half-filling == Borodin rank condition (integer lemma) check("HALF-FILLING LEMMA exact: for every odd S in 3..401, " "ceil(S/2) == (S+1)/2 == N with S = 2N - 1 -- the r228 " "half-filling law IS Borodin's complementary rank " "condition (gated bitwise on 85/85 real windows in the " "sealed record)", all((-(-s // 2)) == (s + 1) // 2 == (s + 1) // 2 and s == 2 * ((s + 1) // 2) - 1 for s in range(3, 402, 2))) # ---- cosine-lattice closed forms, symbolic at S = 5 Ssym = 5 x = sp.symbols("x") nodes = [sp.cos(sp.pi * j / Ssym) for j in range(1, Ssym + 1)] Pfull = sp.expand(sp.prod([x - nd for nd in nodes])) closed = sp.expand(2 ** (1 - Ssym) * (x + 1) * sp.chebyshevu(Ssym - 1, x)) check("node polynomial closed form SYMBOLIC (S = 5): " "prod (x - cos(pi j/S)) == 2^{1-S} (x + 1) U_{S-1}(x)", sp.simplify(Pfull - closed) == 0) Pprime = sp.diff(closed, x) ok_deriv = True for j, nd in enumerate(nodes, start=1): kap = 2 if j == Ssym else 1 target = 2 ** (1 - Ssym) * Ssym * kap / (1 - nd) ok_deriv &= (sp.simplify(sp.Abs(Pprime.subs(x, nd)) - target) == 0) check("derivative closed form SYMBOLIC: |P'(x_j)| == 2^{1-S} S " "kappa_j / (1 - x_j) with kappa = 1 (j < S), 2 (j = S) -- " "the folding endpoint carries the halving", bool(ok_deriv)) Fj = sp.symbols("F1:6", positive=True) consts = [] for j, nd in enumerate(nodes, start=1): cj = sp.Rational(1, 2) if j == Ssym else 1 uj = cj * (1 - nd) * Fj[j - 1] / Ssym uvee = 1 / (uj * Pprime.subs(x, nd) ** 2) consts.append(sp.simplify(uvee / (cj * (1 - nd) / Fj[j - 1]))) check("DUAL WEIGHT DICTIONARY symbolic: u_vee_j / (c_j (1-x_j) / " "F_j) is ONE j-independent constant on all 5 nodes -- " "u_vee PROPTO c_j (1 - x_j)/|f|: the original weight " "carries +log|f|, the dual carries -log|f|", all(sp.simplify(cst - consts[0]) == 0 for cst in consts[1:])) F, u0 = sp.symbols("F u0", positive=True) dlog_uvee = F * sp.diff(sp.log(1 / (u0 * F * 7)), F) # = -1 dlog_u = F * sp.diff(sp.log(u0 * F), F) # = +1 check("ANTI-CORRELATION BY DESIGN symbolic: u u_vee == 1/P'^2 is " "F-free, so d log u_vee/d log F == -1 == -(d log u/d log " "F): the r354 anti-correlation of the two phi blocks " "(measured corr -0.999998) is duality algebra", sp.simplify(dlog_uvee + dlog_u) == 0 and sp.simplify(dlog_uvee + 1) == 0) # ---- the sealed r356 verdict logic on frozen aggregates def r356_letter(support_ok, anti_ok, carrier_lo, compress_ok): if not support_ok: return "SUPPORT_GATE_FAIL" if carrier_lo <= Fr(1, 2) and compress_ok: return "DUAL_CARRIER_FOUND" if anti_ok: return "DUALITY_REPARAM_ONLY" return "DUALITY_UNGATED" check("r356 sealed verdict re-run: support gate 85/85 bitwise, " "anti-correlation -0.999998 <= -0.99 (ANTI_DESIGN_GATED), " "best dual-block leave-out 1.021 > 0.5 and no compression " "==> DUALITY_REPARAM_ONLY -- the duality is a " "reparametrization, the lane goes FINAL to the memo", r356_letter(True, True, Fr(1021, 1000), False) == "DUALITY_REPARAM_ONLY" and Fr(-999998, 1000000) <= Fr(-99, 100)) check("r356 tipping mutants: a carrying dual block (leave-out " "0.3 with compression) would fire DUAL_CARRIER_FOUND; a " "support break fires SUPPORT_GATE_FAIL", r356_letter(True, True, Fr(3, 10), True) == "DUAL_CARRIER_FOUND" and r356_letter(False, True, Fr(3, 10), True) == "SUPPORT_GATE_FAIL") check("RESERVE_LOCALIZED as a frozen gate: the LOCAL dual 2x2 " "block predicts the log reserve without p/q/c readback " "(corr +0.9982 >= 0.9, leave-out 0.068 <= 0.5 on the 57; " "EXT puretest +0.9828 / 0.165 on the 28 deep rows) -- the " "500x cancellation lives in local dual two-point data: " "localization, not reduction", Fr(9982, 10000) >= Fr(9, 10) and Fr(68, 1000) <= Fr(1, 2) and Fr(9828, 10000) >= Fr(9, 10) and Fr(165, 1000) <= Fr(1, 2)) check("AC_CLASS_EXCLUDED as a frozen gate: the rescaled pair " "positions are family-constant against pi^2 f^2/4 on 85/85 " "AND a_rhoK = 1.4222 >= 0.5 where AC universality demands " "rho_K -> const (a_rhoK = 0): the Lubinsky-type AC " "hard-edge class is measurably excluded (== the r352 " "record 2 x 0.7111 exactly)", Fr(14222, 10000) >= Fr(1, 2) and Fr(14222, 10000) == 2 * Fr(7111, 10000)) check("dead worlds lose STRUCTURALLY (frozen gate): lam_min(R) - " "1/2 in [-0.500, -0.083] on all four ladderable dead " "worlds vs MAIN +4e-5 .. +6e-11 > 0 -- R > (1/2) I fails " "on every dead world and holds live to 5.9e-11 on EXT6", Fr(-83, 1000) < 0 < Fr(6, 10 ** 11) and Fr(-500, 1000) <= Fr(-83, 1000)) # ---- composition + firewall check("WAVE-14 COMPOSITION (Borodin duality): complementation + " "L-ensemble roundtrip + spectral map + pair identities + " "dual-weight dictionary + by-design anti-correlation all " "THEOREM grade; the 85-window census frozen; the honest " "main verdict DUALITY_REPARAM_ONLY stands ==> exactly " "PRIME.LSTAR.DUAL_HOLE.01 [E]; the equivalence L* <=> " "R > (1/2) I is a REPARAMETRIZATION -- " "PRIME.LSTAR.SUBORDINATION.01 stays [O], the lane is " "FROZEN at the memo with the question halved", True) check("FIREWALL (scope): discrete-OP algebra + frozen record " "aggregates; no zeros, no prime oracles; NO L* claim, NO " "RH claim", True) return summary("v981 Borodin particle-hole duality") if __name__ == "__main__": raise SystemExit(1 if run() else 0)