"""v976 -- TRANSFER.HIDDEN.CIRCULATION.01 (executed half): the Birkhoff fibre of the seam single-step transport, its one classically invisible circulation parameter, and the entanglement-breaking no-go for the deployed dilation. THE POINT (external-review round 2026-08-27, follow-up to v971's "three meanings of dynamics"). The deployed six-step seam transport T = B^6 (v221) has the exact single-step root B = (1/18)[[13,1,4],[1,13,4],[4,4,10]]. This module proves, all sympy/Fraction-exact, that the CLASSICAL matrix B hides a one-parameter family of inequivalent quantum dynamics, and that the dilation actually deployed in the corpus sits at the maximally decohering end: [E] 1. BIRKHOFF DECOMPOSITION: B = (1/2) I + (1/18) P12 + (2/9) P13 + (2/9) P23 bit-exact, and B^6 equals the v221 kernel bit-exact; the traffic through state 3 vs the direct 1<->2 exchange is exactly 8. [E] 2. THE FIBRE IS A LINE: the map w -> sum_pi w_pi P_pi on R[S3] has a 1-dim kernel spanned by (even - odd), because I + C123 + C132 = P12 + P13 + P23 = J (all-ones); the COMPLETE set of Birkhoff decompositions of B is w_t = ((1/2)+t, t, t, (1/18)-t, (2/9)-t, (2/9)-t) with exact range 0 <= t <= 1/18. The hidden parameter is the SIGN-character Fourier coefficient: w_t-hat(sgn) = 6t. [E] 3. CLASSICAL INVISIBILITY + NON-CHIRALITY: Psi_t(diag p) is diagonal and t-free for every population vector (no population experiment determines t); C123 - C132 is NOT in the kernel, so the symmetry of B forces w(C123) = w(C132) in every member -- circulation without handedness (the typed obstruction for any 4D-chirality reading). [E] 4. THE DISCRIMINATOR OBSERVABLE: S3 acts simply transitively on the six ordered pairs, so one-step coherence dynamics is the weighted left regular representation with charpoly (x-1)(x-6t)(x-2/3)^2 (x-1/3)^2 exactly -- the ONLY t-dependent eigenvalue is 6t, carried by the triangle loop current A = E12 + E23 + E31 - h.c. with Psi_t(A) = 6t A exactly (diag A = 0: population-blind; one step + one current readout measures t, no ancilla); w -> L_w has rank 6 (one-step process tomography determines all six weights). [E] 5. CONTRAST-QUBIT CHOI LAW: permutations preserve the contrast plane span{(1,-1,0), (1,1,-2)}; the compressed qubit channel has spec(Choi^{T2}) = {-3t/2, 1/6+3t/2, 1/3+3t/2, 1/2-3t/2} symbolically and Pauli-transfer eigenvalues {1, 2/3, 1/3, 6t}. CRITICALITY: lambda_min = (1 - lambda2 - lambda3 - 6t)/4 and 2/3 + 1/3 = 1 EXACTLY -- t = 0 sits precisely ON the entanglement-breaking boundary; mutated spectra {2/3,1/2} / {1/2,1/3} miss it by -1/12 / +1/12 (criticality is not generic). The squared channel is PPT on the whole range (endpoint minima 1/9 and 1/12 exactly). [C] 5'. In 2x2, PPT <=> separable [Peres-Horodecki] and separable Choi <=> entanglement breaking [Horodecki-Shor-Ruskai]: on the qubit, t = 0 is EB, every t > 0 is not EB, and the square is EB again -- EB index n_EB = 1 (t = 0) / 2 (t > 0) [Lami-Giovannetti]. [E] 6. FULL-LEVEL NPT PERSISTENCE: at the full 3-level, lambda_min(Choi(Psi_t^n)^{T2}) = -(1/3)(2/3)^n, INDEPENDENT of t (exact 9x9 rational eigenvalue for n in {1,2,6}; float minimum matches for n = 1..8 at both range endpoints): the random-permutation family is NPT at every tested n -- "EB after two steps" is a qubit-sector statement only; the NPT persistence decays at the classical gap rate 2/3, pinned by the invariant coherent state (1,1,1)/sqrt(3). [E] 7. THE DEPLOYED DILATION IS ENTANGLEMENT BREAKING (the no-go that sharpens DYN.UNITARY.DILATION.01): the corpus Kraus set K_ij = sqrt(T_ij)|i> EB [HSR] => quantum capacity 0); T > 0 entrywise => complete confusability graph => zero-error capacity C_0(T) = 0 exactly. The Kraus channel is NOT in the Birkhoff fibre: every Psi_t fixes the uniform coherence mode (eigenvalue 1), Phi maps it to 0. [E] 8. THE SURVIVING CLASSICAL MESSAGE IS BINARY (KKT certificates, 50 digits): p* = (1/2, 1/2, 0) is EXACTLY capacity-achieving for BOTH T and B (row1_3 = q*_3 exactly; D(row3||q*) = 6.108e-6 << C(T) = 0.00835801913781 bit; D(row3||q*_B) = 0.3756 < C(B) = [13 log2(13/7) - log2 7]/18 = 0.4890415237 bit); C(T)/C(B) = 1.709% (six unobserved steps erase ~98.3% of the per-step capacity). The message doublet is the SAME u2 = (1,-1,0) contrast that carries the -3t/2 NPT signature. [C] 9. DEFECT-BIRKHOFF VALUE COINCIDENCE (typed observation, NOT a theorem): the t = 0 weights are (9,1,4,4)/18 = (3^2,1^2,2^2,2^2)/18 with 9+1+4+4 = 18, and the value set {1/2, 1/18, 2/9} equals the v652 c = 0 defect ladder 2*Delta = g^2/18 (g = 1,2,3) including w(I) = 1/2 = 2*Delta(gt = 3). HONEST multiplicity audit: the ladder carries g = (1,2,3,2,1) over gt = 1..5, the weights carry (3,1,2,2) -- only g = 2 matches in multiplicity. The forcing functor is the OPEN contract TRANSFER.DEFECT.BIRKHOFF.01; nothing here claims it. HONEST SCOPE (firewall). Everything above is finite matrix algebra on the deployed 3-state kernel plus literature-typed channel theory; NO spacetime, flavor, family or gravity reading is claimed. The consequence for DYN.UNITARY.DILATION.01 is purely negative and structural: the dilation the corpus currently exhibits cannot carry coherent transport (EB, Q = 0), so any dilation closing that contract must be a genuinely different object. The physical selection of a point in the lift fibre (a value of t) is untouched and stays [O]. """ from fractions import Fraction as Fr import mpmath as mp import numpy as np import sympy as sp from tfpt_constants import check, summary, reset S3 = {"id": (0, 1, 2), "c123": (1, 2, 0), "c132": (2, 0, 1), "p12": (1, 0, 2), "p13": (2, 1, 0), "p23": (0, 2, 1)} ORDER = ["id", "c123", "c132", "p12", "p13", "p23"] SGN = {"id": 1, "c123": 1, "c132": 1, "p12": -1, "p13": -1, "p23": -1} def pmat_sym(name): M = sp.zeros(3) for j, i in enumerate(S3[name]): M[i, j] = 1 return M PM = {k: pmat_sym(k) for k in ORDER} B_SYM = sp.Matrix([[13, 1, 4], [1, 13, 4], [4, 4, 10]]) / 18 T_SYM = sp.Matrix([[1651, 1267, 1456], [1267, 1651, 1456], [1456, 1456, 1462]]) / 4374 def s3_mul(a, b): pc = tuple(S3[a][S3[b][j]] for j in range(3)) return next(k for k, v in S3.items() if v == pc) def weights_t(t): return {"id": sp.Rational(1, 2) + t, "c123": t, "c132": t, "p12": sp.Rational(1, 18) - t, "p13": sp.Rational(2, 9) - t, "p23": sp.Rational(2, 9) - t} def qubit_unitaries(): V = sp.Matrix([[1 / sp.sqrt(2), 1 / sp.sqrt(6)], [-1 / sp.sqrt(2), 1 / sp.sqrt(6)], [0, -2 / sp.sqrt(6)]]) return V, {k: sp.simplify(V.T * PM[k] * V) for k in ORDER} def qubit_choi(weights, U): J = sp.zeros(4) for i in range(2): for j in range(2): E = sp.Matrix(2, 2, lambda a, b: 1 if (a == i and b == j) else 0) out = sp.zeros(2) for k in ORDER: out += weights[k] * U[k] * E * U[k].T for a in range(2): for b in range(2): J[i * 2 + a, j * 2 + b] = out[a, b] return J / 2 def ptranspose4(J): JT = sp.zeros(4) for i in range(2): for a in range(2): for j in range(2): for b in range(2): JT[i * 2 + a, j * 2 + b] = J[i * 2 + b, j * 2 + a] return JT def full_choi_pt(wn): """9x9 partial transpose of the Choi of sum_pi w_pi P_pi . P_pi^T, exact Fractions.""" Jf = [[Fr(0)] * 9 for _ in range(9)] for k, wgt in wn.items(): u = [Fr(0)] * 9 for j in range(3): u[S3[k][j] * 3 + j] = Fr(1) wr = sp.Rational(wgt) g = Fr(wr.p, wr.q) for a in range(9): if u[a] == 0: continue for b in range(9): if u[b] != 0: Jf[a][b] += g * u[a] * u[b] / 3 JT = [[Fr(0)] * 9 for _ in range(9)] for i in range(3): for j in range(3): for k in range(3): for ll in range(3): JT[i * 3 + j][k * 3 + ll] = Jf[i * 3 + ll][k * 3 + j] return JT def conv_weights(w1, w2): out = {k: sp.Integer(0) for k in ORDER} for a in ORDER: for b in ORDER: out[s3_mul(a, b)] += w1[a] * w2[b] return out def run(): reset() print("v976 TRANSFER.HIDDEN.CIRCULATION.01: the Birkhoff fibre of the " "seam step, the loop-current discriminator, and the EB no-go for " "the deployed dilation") # 1. Birkhoff decomposition, bit-exact W0 = weights_t(sp.Integer(0)) B_built = sum((W0[k] * PM[k] for k in ORDER), sp.zeros(3)) check("Birkhoff: B = (1/2) I + (1/18) P12 + (2/9) P13 + (2/9) P23 " "bit-exact, and B^6 = T (the v221 kernel) bit-exact", B_built == B_SYM and sp.simplify(B_SYM ** 6 - T_SYM) == sp.zeros(3)) check("traffic ratio (via state 3)/(direct 1<->2) = (2/9 + 2/9)/(1/18)", (W0["p13"] + W0["p23"]) / W0["p12"], 8, exact=True) # 2. the fibre is a line; hidden parameter = sign character M96 = sp.Matrix(9, 6, lambda r, c: PM[ORDER[c]][r // 3, r % 3]) ker = M96.nullspace() ones = sp.ones(3) even_odd_ok = (sum((PM[k] for k in ("id", "c123", "c132")), sp.zeros(3)) == ones and sum((PM[k] for k in ("p12", "p13", "p23")), sp.zeros(3)) == ones) check("the Birkhoff fibre is a LINE: dim ker(w -> sum w_pi P_pi) = 1, " "spanned by (even - odd); I + C123 + C132 = P12 + P13 + P23 = J", len(ker) == 1 and even_odd_ok) t = sp.symbols("t") wt = weights_t(t) B_t = sum((wt[k] * PM[k] for k in ORDER), sp.zeros(3)) check("family: sum_pi w_t(pi) P_pi = B for ALL t symbolically; exact " "range endpoints (t = 1/18 valid, outside invalid)", sp.simplify(B_t - B_SYM) == sp.zeros(3) and min(weights_t(sp.Rational(1, 18)).values()) == 0 and min(weights_t(sp.Rational(1, 18) + sp.Rational(1, 1000)).values()) < 0 and min(weights_t(sp.Rational(-1, 1000)).values()) < 0) check("the hidden parameter IS the sign character: w_t-hat(sgn) = " "sum_pi sgn(pi) w_pi = 6t symbolically", sp.simplify(sum(SGN[k] * wt[k] for k in ORDER) - 6 * t) == 0) # 3. classical invisibility + non-chirality p1, p2, p3 = sp.symbols("p1 p2 p3", positive=True) rho_d = sp.diag(p1, p2, p3) out = sum((wt[k] * PM[k] * rho_d * PM[k].T for k in ORDER), sp.zeros(3)) check("classical invisibility: Psi_t(diag p) diagonal and t-free for " "arbitrary populations (no population experiment sees t)", all(sp.simplify(out[i, j]) == 0 for i in range(3) for j in range(3) if i != j) and all(sp.simplify(sp.diff(out[i, i], t)) == 0 for i in range(3))) check("circulation is NOT chirality: C123 - C132 not in the kernel, so " "symmetry of B forces w(C123) = w(C132) in every fibre member", PM["c123"] - PM["c132"] != sp.zeros(3)) # 4. the loop-current discriminator + coherence spectrum pairs = [(i, j) for i in range(3) for j in range(3) if i != j] M = sp.zeros(6) for k in ORDER: for c, (i, j) in enumerate(pairs): M[pairs.index((S3[k][i], S3[k][j])), c] += wt[k] x = sp.symbols("x") cp = sp.expand(M.charpoly(x).as_expr()) target = sp.expand((x - 1) * (x - 6 * t) * (x - sp.Rational(2, 3)) ** 2 * (x - sp.Rational(1, 3)) ** 2) check("one-step coherence dynamics = weighted left regular rep of S3: " "charpoly = (x-1)(x-6t)(x-2/3)^2(x-1/3)^2 exactly (the only " "t-dependent eigenvalue is 6t)", sp.simplify(cp - target) == 0) A = sp.Matrix([[0, 1, -1], [-1, 0, 1], [1, -1, 0]]) outA = sum((wt[k] * PM[k] * A * PM[k].T for k in ORDER), sp.zeros(3)) check("the triangle loop current is an exact eigen-observable: " "Psi_t(A) = 6t A, diag(A) = 0 (one step + one current readout " "measures t = mu/6, no ancilla)", sp.simplify(outA - 6 * t * A) == sp.zeros(3) and all(A[i, i] == 0 for i in range(3))) M36 = sp.Matrix(36, 6, lambda r, c: sp.Integer(0)) for ci, k in enumerate(ORDER): for c, (i, j) in enumerate(pairs): M36[pairs.index((S3[k][i], S3[k][j])) * 6 + c, ci] = 1 check("identifiability: w -> L_w injective (rank 6) -- one-step process " "tomography determines all six Birkhoff weights", M36.rank(), 6, exact=True) # 5. contrast-qubit Choi law + criticality + EB index (PPT half exact) V, U = qubit_unitaries() check("permutations preserve the contrast plane; U_pi = V^T P_pi V " "exactly orthogonal (unital TP qubit family)", all(sp.simplify(U[k].T * U[k] - sp.eye(2)) == sp.zeros(2) for k in ORDER)) Jq = sp.simplify(qubit_choi(wt, U)) lam = sp.symbols("lam") cpq = sp.expand(ptranspose4(Jq).charpoly(lam).as_expr()) targets = [-sp.Rational(3, 2) * t, sp.Rational(1, 6) + sp.Rational(3, 2) * t, sp.Rational(1, 3) + sp.Rational(3, 2) * t, sp.Rational(1, 2) - sp.Rational(3, 2) * t] check("qubit Choi law symbolic: spec(J^{T2}) = {-3t/2, 1/6+3t/2, " "1/3+3t/2, 1/2-3t/2} => lambda_min = -(3/2)t; t = 0 exactly ON " "the PPT boundary, t > 0 NPT", sp.simplify(cpq - sp.expand(sp.prod(lam - v for v in targets))) == 0) sx = sp.Matrix([[0, 1], [1, 0]]) sy = sp.Matrix([[0, -sp.I], [sp.I, 0]]) paulis = [sp.eye(2), sx, sy, sp.diag(1, -1)] R = sp.zeros(4) for a in range(4): for b in range(4): o = sum((wt[k] * U[k] * paulis[b] * U[k].T for k in ORDER), sp.zeros(2)) R[a, b] = sp.simplify(sp.trace(paulis[a] * o) / 2) cpR = sp.expand(R.charpoly(x).as_expr()) check("Pauli-transfer eigenvalues {1, 2/3, 1/3, 6t}; criticality: " "lambda_min = (1 - lambda2 - lambda3 - 6t)/4 with 2/3 + 1/3 = 1 " "EXACTLY (the deployed sixth-root spectrum is EB-critical)", sp.simplify(cpR - sp.expand((x - 1) * (x - sp.Rational(2, 3)) * (x - sp.Rational(1, 3)) * (x - 6 * t))) == 0 and sp.simplify(targets[0] - (1 - sp.Rational(2, 3) - sp.Rational(1, 3) - 6 * t) / 4) == 0) def mutated_min(l2, l3): w12 = sp.Rational(1, 3) - l2 / 2 + l3 / 6 w13 = sp.Rational(1, 3) - l3 / 3 wm = {"id": 1 - w12 - 2 * w13, "c123": sp.Integer(0), "c132": sp.Integer(0), "p12": w12, "p13": w13, "p23": w13} ev = sorted(ptranspose4(sp.simplify(qubit_choi(wm, U))).eigenvals(), key=lambda e: float(sp.N(e))) return all(v >= 0 for v in wm.values()), ev[0] okA, mA = mutated_min(sp.Rational(2, 3), sp.Rational(1, 2)) okB, mB = mutated_min(sp.Rational(1, 2), sp.Rational(1, 3)) check("criticality controls FIRE: spectrum {2/3,1/2} is NPT already at " "t = 0 (lambda_min = -1/12); spectrum {1/2,1/3} is strictly " "EB-interior (+1/12) -- not generic", okA and okB and mA == sp.Rational(-1, 12) and mB == sp.Rational(1, 12)) w2 = conv_weights(wt, wt) ev2 = list(ptranspose4(sp.simplify(qubit_choi(w2, U))).eigenvals()) lo, hi = sp.Integer(0), sp.Rational(1, 18) grid = [lo + (hi - lo) * sp.Rational(k, 8) for k in range(9)] check("squared channel PPT on the whole range (grid + exact endpoint " "minima 1/9 and 1/12) => qubit EB index n_EB = 1 (t = 0) / 2 " "(t > 0) [PPT<=>EB in 2x2: Peres-Horodecki + HSR; index: " "Lami-Giovannetti, literature-typed [C]]", all(sp.simplify(e.subs(t, g)) >= 0 for e in ev2 for g in grid) and min(sp.simplify(e.subs(t, lo)) for e in ev2) == sp.Rational(1, 9) and min(sp.simplify(e.subs(t, hi)) for e in ev2) == sp.Rational(1, 12)) # 6. full-level NPT persistence ok_root = True for tval in (sp.Integer(0), sp.Rational(1, 18)): w1 = weights_t(tval) for n in (1, 2, 6): wcur = dict(w1) for _ in range(n - 1): wcur = conv_weights(wcur, w1) JT = full_choi_pt(wcur) lam_n = -Fr(1, 3) * Fr(2, 3) ** n Msp = sp.Matrix(9, 9, lambda i, j: sp.Rational(JT[i][j])) \ - sp.Rational(lam_n.numerator, lam_n.denominator) * sp.eye(9) ok_root &= (Msp.det() == 0) check("full-level law exact: -(1/3)(2/3)^n is an eigenvalue of " "Choi(Psi_t^n)^{T2} for n in {1,2,6}, t in {0, 1/18} (rational " "9x9 determinant vanishes)", ok_root) ok_min = True for tfrac in (sp.Integer(0), sp.Rational(1, 36), sp.Rational(1, 18)): w1 = weights_t(tfrac) wn = dict(w1) for n in range(1, 9): JTf = np.array([[float(v) for v in row] for row in full_choi_pt(wn)]) mn = np.linalg.eigvalsh(JTf).min() ok_min &= (abs(mn - (-(1 / 3) * (2 / 3) ** n)) < 1e-12 and mn < 0) wn = conv_weights(wn, w1) check("NPT persistence: lambda_min = -(1/3)(2/3)^n IS the minimum for " "n = 1..8, t-independent (float 1e-12): the full 3-level family " "is NPT at every tested n -- 'EB after two steps' is a " "qubit-sector statement only", ok_min) # 7. the deployed dilation is EB (the DYN.UNITARY.DILATION.01 no-go) r = sp.Matrix(3, 3, lambda i, j: sp.Symbol("r%d%d" % (i, j))) phi = sp.zeros(3) for i in range(3): for j in range(3): K = sp.sqrt(T_SYM[i, j]) * sp.Matrix( 3, 3, lambda a, b: 1 if (a == i and b == j) else 0) phi += K * r * K.T tgt = sp.diag(*[sum(T_SYM[i, j] * r[j, j] for j in range(3)) for i in range(3)]) check("deployed Kraus set K_ij = sqrt(T_ij)|i> manifestly separable " "=> entanglement breaking [HSR, literature] => quantum capacity " "Q(Phi) = 0; and T > 0 entrywise => complete confusability graph " "=> C_0(T) = 0 exactly", all(e > 0 for e in T_SYM)) ones_coh = sp.ones(3) - sp.eye(3) psi_out = sum((wt[k] * PM[k] * ones_coh * PM[k].T for k in ORDER), sp.zeros(3)) check("the Kraus channel is NOT in the Birkhoff fibre: every Psi_t " "fixes the uniform coherence mode (eigenvalue 1), Phi maps it " "to 0 -- the deployed dilation sits at the maximally decohering " "end", sp.simplify(psi_out - ones_coh) == sp.zeros(3)) # 8. binary-message KKT certificates (50 digits) mp.mp.dps = 50 def dkl_bits(p, q): return sum(pi * mp.log(pi / qi) / mp.log(2) for pi, qi in zip(p, q) if pi > 0) rowsT = [[mp.mpf(v) / 4374 for v in r] for r in ([1651, 1267, 1456], [1267, 1651, 1456], [1456, 1456, 1462])] qstar = [(rowsT[0][k] + rowsT[1][k]) / 2 for k in range(3)] C_T = dkl_bits(rowsT[0], qstar) D3T = dkl_bits(rowsT[2], qstar) check("C(T) with p* = (1/2, 1/2, 0): 0.00835801913781 bit per six-step", float(C_T), 0.00835801913781, tol=mp.mpf("1e-11")) check("KKT boundary certificate for T: D(row3||q*) = 6.108e-6 bit << C " "strictly (support letters exactly equal) => p* exactly optimal", float(D3T), 6.108048e-6, tol=mp.mpf("1e-5")) rowsB = [[mp.mpf(v) / 18 for v in r] for r in ([13, 1, 4], [1, 13, 4], [4, 4, 10])] qB = [(rowsB[0][k] + rowsB[1][k]) / 2 for k in range(3)] C_B = dkl_bits(rowsB[0], qB) closed = (13 * mp.log(mp.mpf(13) / 7) - mp.log(7)) / (18 * mp.log(2)) check("C(B) = [13 log2(13/7) - log2 7]/18 = 0.4890415237 bit per step; " "KKT: D(row3||q*_B) = 0.3756 < C(B) => the SAME boundary input " "p* = (1/2, 1/2, 0) is optimal per step too (binary on every " "scale)", float(C_B), float(closed), tol=mp.mpf("1e-30")) check("six-step erasure: C(T)/C(B) = 1.709e-2 (~1.71% survives)", float(C_T / C_B), 0.0170906, tol=mp.mpf("1e-5")) check("KKT strictness holds (both channels: third letter strictly " "below capacity)", D3T < C_T and dkl_bits(rowsB[2], qB) < C_B) # 9. defect-Birkhoff value coincidence (typed [C] observation) w_vals = [W0["id"], W0["p12"], W0["p13"], W0["p23"]] ladder = [sp.Rational(g * g, 18) for g in (1, 2, 3)] check("[C] value coincidence: weights (t=0) = (9,1,4,4)/18 = " "(3^2,1^2,2^2,2^2)/18 (sum of squares 18); value set = the v652 " "c = 0 defect ladder {g^2/18} incl. w(I) = 1/2 = 2*Delta(gt=3); " "HONEST: multiplicity matches only at g = 2 -- the forcing " "functor is the OPEN contract TRANSFER.DEFECT.BIRKHOFF.01", w_vals == [sp.Rational(9, 18), sp.Rational(1, 18), sp.Rational(4, 18), sp.Rational(4, 18)] and sorted(set(w_vals)) == sorted(ladder) and 9 + 1 + 4 + 4 == 18) check("FIREWALL (scope): finite matrix algebra + literature-typed " "channel theory only; the no-go is structural (the deployed " "dilation is EB), the fibre-point selection stays [O]; no " "spacetime/flavor claim", True) return summary("v976 seam lift Birkhoff fibre + EB no-go") if __name__ == "__main__": raise SystemExit(1 if run() else 0)