#!/usr/bin/env python3 """v1005 -- relative pencil bridges: exact Hankel/winding identities plus the circularity-flagged relative pencil (NOT a P2 selector). Provenance: experiments/tfpt-discovery/relative_pencil_bridges_probe.py (review wave 6, 2026-08-30; 42/42). THE POINT. Exact corpus identities, honestly typed. [E-finite] EXACT: anchor Hankel identity p_n p_{n+2} - p_{n+1}^2 = 2^{n+1} symbolic + rank-2 moment structure + Jacobi matrix spec {1,2} diag (4/3, 5/3) (provenance v106/v119); winding spectrum {5/2, 1, 1} + fixed plane (provenance v4). [C] EXACT BUT CIRCULARITY-FLAGGED: relative pencil -- unique t = 2 with lambda = 7/3, factorization (3lambda-7)(lambda^2-2lambda+2), CM block J0^2 = -I, dual number mod 2, saturated lattice index 25 = g_car^2 SNF (1,1,25) (provenance v568/v95/v410/v135) -- BUT carrier 5 already sits in the column sums of Q_2 and C: NOT a P2 selector until Q, C, epsilon are derived from seam geometry independently. [C] INTERPRETIVE: Klein-four SNF (2,2) only after denominator clearing D = 2d -- consistent with side-blind nulls (v521), not a theorem. MUST-FAIL: t = 0 fails the augmentation eigencondition; perturbed C breaks the factorization; p_n = 2+3^n breaks the Hankel law. HONEST SCOPE (firewall): no import of verification modules (provenance is source-text only); circularity flag is BINDING; AX.P2.01 typing UNCHANGED. Python-only / Wolfram mirror deferred. """ from __future__ import annotations from pathlib import Path import sympy as sp from sympy.matrices.normalforms import smith_normal_form from tfpt_constants import check as suite_check, summary, reset HERE = Path(__file__).resolve().parent ROOT = HERE.parent EXACT_VERIFIED = "EXACT_VERIFIED" EXACT_VERIFIED_PROVENANCE_UNVERIFIED = ( "EXACT_VERIFIED_PROVENANCE_UNVERIFIED" ) INTERPRETIVE_C = "INTERPRETIVE[C]" REFUTED = "REFUTED" VERDICT_ENUM = { EXACT_VERIFIED, EXACT_VERIFIED_PROVENANCE_UNVERIFIED, INTERPRETIVE_C, REFUTED, } def check(label: str, condition: bool) -> None: """Record one deterministic check against tfpt_constants.check.""" suite_check(label, bool(condition)) def positive_snf_diagonal(matrix: sp.Matrix) -> tuple[int, ...]: """Return positive Smith invariant factors, including no zero padding.""" smith = smith_normal_form(matrix, domain=sp.ZZ) return tuple( abs(int(smith[index, index])) for index in range(min(smith.shape)) if smith[index, index] != 0 ) def primitive_integer_vector(vector: sp.Matrix) -> sp.Matrix: """Clear denominators and divide by the gcd, fixing the first sign.""" denominator = sp.ilcm(*(sp.denom(entry) for entry in vector)) integral = sp.Matrix([int(entry * denominator) for entry in vector]) divisor = sp.igcd(*(abs(entry) for entry in integral if entry)) primitive = integral.applyfunc(lambda entry: entry // divisor) first_nonzero = next(entry for entry in primitive if entry) return -primitive if first_nonzero < 0 else primitive def source_contains(relative_path: str, *needles: str) -> bool: """Check provenance without executing or importing a corpus module.""" source = (ROOT / relative_path).read_text(encoding="utf-8") return all(needle in source for needle in needles) def claim_1_relative_pencil() -> str: print("\nCLAIM 1 — RELATIVE PENCIL") t, lam = sp.symbols("t lam") epsilon = sp.ones(3, 1) q_t = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, t, 1]]) center = sp.Matrix([[4, 3, 0], [4, 5, 2], [5, 5, 3]]) transfer_t = sp.simplify(q_t.inv() * center) augmentation = sp.simplify(epsilon.T * transfer_t) expected_augmentation = sp.Matrix( [[sp.Rational(7, 3), sp.Rational(19, 3) - 2 * t, sp.Rational(19, 3) - 2 * t]] ) check("epsilon^T Q_t^-1 C has the claimed symbolic form", augmentation == expected_augmentation) eigen_equations = [ sp.expand(entry) for entry in augmentation - lam * epsilon.T ] eigen_solutions = sp.solve(eigen_equations, (t, lam), dict=True) check("augmentation eigencondition uniquely gives t=2, lambda=7/3", eigen_solutions == [{lam: sp.Rational(7, 3), t: 2}]) check("the genuine t=0 twin fails the augmentation eigencondition", len(set(augmentation.subs(t, 0))) != 1) q_2 = q_t.subs(t, 2) transfer = transfer_t.subs(t, 2) pencil = sp.expand((lam * q_2 - center).det()) expected_pencil = 3 * lam**3 - 13 * lam**2 + 20 * lam - 14 check("det(lambda Q_2-C)=3 lambda^3-13 lambda^2+20 lambda-14", pencil == expected_pencil) check("pencil factors as (3 lambda-7)(lambda^2-2 lambda+2)", sp.factor(pencil) == (3 * lam - 7) * (lam**2 - 2 * lam + 2)) check("Spec(Q_2^-1 C)={7/3,1+i,1-i}", transfer.eigenvals() == { sp.Rational(7, 3): 1, 1 + sp.I: 1, 1 - sp.I: 1, }) residue = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) winding = residue + 6 * sp.ones(3, 1) * sp.Matrix([[1, 0, 0]]) sheet = q_2 * sp.diag(0, 1, 1) delta_q = (epsilon.T * sheet**2 * epsilon)[0] check("coefficient tuple equals (N_fam,Delta_Q,det L,det C)", (3, delta_q, winding.det(), center.det()) == (3, 13, 20, 14)) radial = primitive_integer_vector( (3 * transfer - 7 * sp.eye(3)).nullspace()[0] ) cm_operator = sp.simplify( transfer**2 - 2 * transfer + 2 * sp.eye(3) ) cm_basis_1 = sp.Matrix([-1, 1, 0]) cm_basis_2 = sp.Matrix([0, -1, 1]) cm_basis = sp.Matrix.hstack(cm_basis_1, cm_basis_2) block = (cm_basis.T * cm_basis).inv() * cm_basis.T * transfer * cm_basis expected_block = sp.Matrix([[1, 1], [-1, 1]]) check("primitive radial lattice is Z(4,18,3)", radial == sp.Matrix([4, 18, 3])) check("CM plane is exactly x+y+z=0", cm_operator.rref()[0] == sp.Matrix([[1, 1, 1], [0, 0, 0], [0, 0, 0]])) check("chosen saturated CM basis gives A=[[1,1],[-1,1]]", block == expected_block) j_zero = expected_block - sp.eye(2) block_mod_2 = expected_block.applyfunc(lambda entry: entry % 2) check("J0=A-I satisfies J0^2=-I", j_zero**2 == -sp.eye(2)) check("A mod 2 is the all-one block and squares to zero", block_mod_2 == sp.ones(2) and (block_mod_2**2).applyfunc(lambda entry: entry % 2) == sp.zeros(2)) basis_change = sp.Matrix.hstack(radial, cm_basis_1, cm_basis_2) saturated_index = abs(int(basis_change.det())) check("radial plus saturated CM lattice has index 25", saturated_index == 25 and positive_snf_diagonal(basis_change) == (1, 1, 25)) alternate_saturated = sp.Matrix.hstack( radial, cm_basis_1 + cm_basis_2, cm_basis_2 ) nonsaturated = sp.Matrix.hstack( radial, 2 * cm_basis_1, cm_basis_2 ) check("saturated basis changes preserve 25; arbitrary choices do not", abs(int(alternate_saturated.det())) == 25 and abs(int(nonsaturated.det())) == 50 and positive_snf_diagonal(nonsaturated) == (1, 1, 50)) check("Q_2 provenance: corpus Q(c) family and 0/2 twins", source_contains( "verification/v568_bit_selector_ladder.py", "def Qc(c):", "return sp.Matrix([[3, 1, 0], [3, 2, 0], [3, c, 1]])", "order_coords(Qc(0))", "order_coords(Qc(2))", )) check("C provenance: corpus centered flavor operator", source_contains( "verification/v95_centered_diamond.py", "C = R + Q * sp.diag(1, 0, 0)", "C == sp.Matrix([[4, 3, 0], [4, 5, 2], [5, 5, 3]])", )) check("Delta_Q=13 and det L=20 are genuine corpus quantities", source_contains( "verification/v410_sheet_generator_binary.py", "DELTA_Q = 13", "(ONE.T * V**2 * ONE)[0] == DELTA_Q", ) and source_contains( "verification/v135_det_surface.py", "(det R, det C, det L) = (8, 14,", "20)", )) q_snf = positive_snf_diagonal(q_2) center_snf = positive_snf_diagonal(center) check("circularity audit invariants are exact", tuple(epsilon.T * q_2) == (9, 5, 1) and tuple(epsilon.T * center) == (13, 13, 5) and q_2.det() == 3 and q_snf == (1, 1, 3) and center.det() == 14 and center_snf == (1, 1, 14)) perturbed_center = center + sp.diag(0, 0, 1) perturbed_pencil = sp.factor((lam * q_2 - perturbed_center).det()) check("mutant C+E33 breaks the claimed factorization", perturbed_pencil != (3 * lam - 7) * (lam**2 - 2 * lam + 2)) print(" PROVENANCE: Q_t/Q_0/Q_2 are corpus objects at " "verification/v568_bit_selector_ladder.py:91-92,175-184; " "C is corpus at verification/v95_centered_diamond.py:45-56,80-82.") print(" INDEX: 25 is intrinsic only for the primitive radial line plus " "the saturated CM-plane lattice; nonsaturated generators give " "multiples such as 50.") print(" CIRCULARITY: carrier 5 is already visible in the column sums of " "both Q_2 and C. Their determinants/SNFs do not recover 5. A " "P2 replacement therefore requires independent proofs that Q, C, " "Q^-1 C, and epsilon are canonically forced by seam geometry " "without using the carrier selector.") return INTERPRETIVE_C def claim_2_anchor_moments() -> str: print("\nCLAIM 2 — ANCHOR MOMENTS") n = sp.symbols("n", integer=True, nonnegative=True) def moment(index: sp.Expr) -> sp.Expr: return 2 + 2**index check("p_n=2*1^n+1*2^n is the measure 2 delta_1+delta_2", sp.simplify(moment(n) - (2 * 1**n + 2**n)) == 0) hankel_2 = sp.simplify( moment(n) * moment(n + 2) - moment(n + 1)**2 ) check("symbolic 2x2 Hankel identity is 2^(n+1)", sp.simplify(hankel_2 - 2**(n + 1)) == 0) check("2x2 Hankel identity holds numerically for n=0..12", all( moment(index) * moment(index + 2) - moment(index + 1)**2 == 2**(index + 1) for index in range(13) )) hankel_3 = sp.Matrix( 3, 3, lambda row, column: moment(n + row + column) ) check("all shifted 3x3 Hankel determinants vanish symbolically", sp.simplify(hankel_3.det()) == 0) check("moments satisfy p_(n+2)=3p_(n+1)-2p_n", sp.simplify( moment(n + 2) - 3 * moment(n + 1) + 2 * moment(n) ) == 0) mass = sp.Integer(3) alpha_0 = sp.Rational(1, mass) * (2 * 1 + 1 * 2) variance_numerator = ( 2 * (1 - alpha_0)**2 + (2 - alpha_0)**2 ) beta_1 = sp.sqrt(variance_numerator / mass) phi_0 = {1: 1 / sp.sqrt(mass), 2: 1 / sp.sqrt(mass)} phi_1 = { point: (point - alpha_0) / sp.sqrt(variance_numerator) for point in (1, 2) } alpha_1 = sp.simplify( sum( weight * point * phi_1[point]**2 for point, weight in ((1, 2), (2, 1)) ) ) beta_from_inner_product = sp.simplify( sum( weight * point * phi_0[point] * phi_1[point] for point, weight in ((1, 2), (2, 1)) ) ) jacobi = sp.Matrix([[alpha_0, beta_1], [beta_1, alpha_1]]) expected_jacobi = sp.Matrix([ [sp.Rational(4, 3), sp.sqrt(2) / 3], [sp.sqrt(2) / 3, sp.Rational(5, 3)], ]) check("orthonormalization derives the claimed Jacobi matrix", beta_1 == beta_from_inner_product and jacobi == expected_jacobi) check("Jacobi spectrum={1,2}, trace=3, determinant=2", jacobi.eigenvals() == {1: 1, 2: 1} and jacobi.trace() == 3 and jacobi.det() == 2) anchor = (1, 1, 2) e_1 = sum(anchor) e_2 = sum( anchor[left] * anchor[right] for left, right in ((0, 1), (0, 2), (1, 2)) ) check("Jacobi diagonal is (e1(a)/p0,e2(a)/p0)=(4/3,5/3)", (jacobi[0, 0], jacobi[1, 1]) == (sp.Rational(e_1, 3), sp.Rational(e_2, 3))) check("anchor-moment and normalized-ratio provenance is in corpus", source_contains( "verification/v106_review_validation.py", "def p_sum(n):", "p_n(a) = 2 + 2^n", ) and source_contains( "verification/v119_review_validation_2.py", "ANCHOR RATIO TRIAD", "sp.Rational(4, 3)", "sp.Rational(5, 3)", )) mutant = lambda index: 2 + 3**index check("mutant p_n=2+3^n breaks the 2^(n+1) Hankel law", mutant(0) * mutant(2) - mutant(1)**2 != 2) print(" PROVENANCE: p_n=2+2^n is corpus at " "verification/v106_review_validation.py:70-94; the normalized " "anchor ratios (e1,e2)/p0=(4/3,5/3) are corpus at " "verification/v119_review_validation_2.py:60-92.") return EXACT_VERIFIED def claim_3_winding_spectrum() -> str: print("\nCLAIM 3 — WINDING SPECTRUM") residue = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) one = sp.ones(3, 1) e_1_row = sp.Matrix([[1, 0, 0]]) winding = residue + 6 * one * e_1_row relative = residue.inv() * winding expected_relative = sp.Matrix([ [sp.Rational(5, 2), 0, 0], [sp.Rational(3, 2), 1, 0], [sp.Rational(-3, 2), 0, 1], ]) check("true corpus R and L give the claimed R^-1 L", relative == expected_relative) check("Spec(R^-1 L)={5/2,1,1}", relative.eigenvals() == {sp.Rational(5, 2): 1, 1: 2}) x_1, x_2, x_3 = sp.symbols("x_1 x_2 x_3") vector = sp.Matrix([x_1, x_2, x_3]) fixed_residual = sp.simplify((relative - sp.eye(3)) * vector) check("fixed plane is exactly x_1=0", fixed_residual == sp.Matrix([ sp.Rational(3, 2) * x_1, sp.Rational(3, 2) * x_1, sp.Rational(-3, 2) * x_1, ])) check("5/2 equals det L / det R", sp.Rational(winding.det(), residue.det()) == sp.Rational(20, 8) == sp.Rational(5, 2)) check("R and winding-shifted L provenance is genuine", source_contains( "verification/v4_flavor_matrix.py", "R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]])", "L = R + 6 * W", "anchor = first column of R = (1,1,2)", )) print(" PROVENANCE: R and L=R+6*1*e1^T are corpus objects at " "verification/v4_flavor_matrix.py:22-24; R's first column is " "the anchor a=(1,1,2) at lines 57-59.") return EXACT_VERIFIED def claim_4_dual_frame_klein_four() -> str: print("\nCLAIM 4 — DUAL FRAME KLEIN FOUR") residue = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) anchor = sp.Matrix([1, 1, 2]) one = sp.ones(3, 1) d = anchor.T * residue.inv() n = sp.Matrix([[5, -9, 6]]) evaluation_frame = sp.Matrix.hstack(one, anchor) literal_block = sp.Matrix.vstack(d, n) * evaluation_frame integral_d = 2 * d integral_block = sp.Matrix.vstack(integral_d, n) * evaluation_frame expected_block = sp.Matrix([[0, 2], [2, 8]]) check("d=a^T R^-1=(-1/2,-1/2,1) and n=(5,-9,6)", d == sp.Matrix([[ sp.Rational(-1, 2), sp.Rational(-1, 2), 1 ]]) and n == sp.Matrix([[5, -9, 6]])) check("literal evaluations with d give [[0,1],[2,8]], not B", literal_block == sp.Matrix([[0, 1], [2, 8]]) and literal_block != expected_block) check("primitive integral covector D=2d reconstructs reviewer B", integral_d == sp.Matrix([[-1, -1, 2]]) and integral_block == expected_block) check("det B=-4 and SNF(B)=diag(2,2)", integral_block.det() == -4 and positive_snf_diagonal(integral_block) == (2, 2)) check("cokernel is Z2 x Z2, not Z4", positive_snf_diagonal(integral_block) == (2, 2) and integral_block.applyfunc(lambda entry: entry % 2) == sp.zeros(2)) check("d and n provenance is genuine", source_contains( "verification/v225_dual_normal_frame.py", "d = (A.T * R.inv())", "n = sp.Matrix([[5, -9, 6]])", "list(n * L) == [L.det(), 0, 0] == [20, 0, 0]", )) check("a corpus side-blind null exists", source_contains( "verification/v521_seam_bit_rp_blind.py", "RP/Theta battery is side-blind", "EIGHTH", )) print(" PROVENANCE: d and n are corpus objects at " "verification/v225_dual_normal_frame.py:32-39,48-62; an explicit " "side-blind null is verification/v521_seam_bit_rp_blind.py:1-10.") print(" NORMALIZATION AUDIT: the advertised B is not the literal " "(d,n)-evaluation matrix. It uses the denominator-cleared " "primitive integral normal D=2d; hence determinant 4 and the " "Klein-four cokernel are exact only after that extra normalization.") print(" CONCLUSION: determinant-4/Smith data cannot select cyclic mu4 " "gluing; its cokernel is Z2 x Z2, consistent with the corpus " "side-blind nulls.") return INTERPRETIVE_C def run(): reset() print("v1005 -- relative pencil bridges (circularity-flagged)") verdicts = { "relative_pencil": claim_1_relative_pencil(), "anchor_moments": claim_2_anchor_moments(), "winding_spectrum": claim_3_winding_spectrum(), "dual_frame_klein_four": claim_4_dual_frame_klein_four(), } check("all claim verdicts use the frozen enum", set(verdicts.values()) <= VERDICT_ENUM) print("\nCLAIM VERDICTS") for claim, verdict in verdicts.items(): print(f" {claim}: {verdict}") check( "FIREWALL (scope): no verification-module import (provenance is " "source-text only); circularity flag BINDING -- carrier 5 already " "sits in column sums of Q_2 and C, not a P2 selector; AX.P2.01 " "typing UNCHANGED; no status-marker move", True, ) return summary( "v1005 relative pencil: Hankel/winding [E-finite]; pencil " "circularity-flagged [C]; Klein-four after D=2d [C]" ) if __name__ == "__main__": raise SystemExit(1 if run() else 0)