"""v987 -- DYN.UNITARY.DILATION.01 [O update: the OS/dilation LADDER is executed at kernel + free-field + interacting-kernel + quantum-coupled band level]; v971 / v984 / v977 (read-only anchors). THE POINT (promote round 2026-08-28). v984 executed the DISCRETE collision/QCA leg. This module re-derives the next four executed legs from the seam single-step kernel B (exact sixth root of the v221 transfer; B = [[13,1,4],[1,13,4],[4,4,10]]/18): [E] A. KERNEL OS CONTINUATION (probe dilation_os_continuation): B = P1 + (2/3) P2 + (1/3) P3 > 0 with exact orthogonal projectors; H = -log B = log(3/2) P2 + log 3 P3 is PSD with gap log(3/2); T(t) = e^{-tH} is a stochastic semigroup inside the cone; U(t) = e^{-itH} is a unitary group with Wick anchor U(-i n) = B^n for n = 1..6. MUST-FAIL: the v971 negative kernel (second eigenvalue -1/10) has no real half-step. [E] B. SIZE-UNIFORM FREE BAND (probe dilation_field_os_uniform): T_L = B^(x L) has H_L = sum_x h_x commuting local, gap log(3/2) UNIFORM in L (exact at L = 2, 3 via the projector algebra); D2: the per-cell reduced channel of the collision band is B at L = 1, 2; LR radius 1 on the L = 2 band (far-cell commutator machine-zero; near-cell strictly nonzero). [E/N] C. INTERACTING CONGRUENCE FAMILY (probe dilation_interacting_os): T_eps = K^{1/2} B^(x L) K^{1/2} is SPD for EVERY tested diagonal coupling (congruence of a PD matrix); the one-sided mutant K B^(x L) is not symmetric. Gap plateau measured (~0.17 at L = 2..4, eps = 0.3); quasi-locality of -log T measured (2-site coefficients decay with distance). [N] D. QUANTUM SWAP COUPLING (probe dilation_quantum_coupled_band): W_theta = U_int(theta) (x)_x V_x with h = SWAP is unitary, CPTP at the reduced cell, d(theta) ~ theta^2 (leading even power), and an even-bond matching layer of the same SWAP family makes the far cell vanish at machine precision. HONEST SCOPE (firewall): finite, discrete-time, 1+1D kernel / band. The CONTINUOUS field-level OS reconstruction, thermodynamic-limit branch selection, and any 3+1D reading stay [O]. No marker move. Python-only / Wolfram mirror deferred (GATE.WOLFRAM.02 convention of the v974-v986 wave). Provenance: experiments/tfpt-discovery/dilation_os_continuation_probe.py, dilation_field_os_uniform_probe.py, dilation_interacting_os_probe.py, dilation_quantum_coupled_band_probe.py (ALL PASS; not imported). """ from itertools import product as iproduct import numpy as np import sympy as sp from tfpt_constants import check, summary, reset B_SYM = sp.Matrix([[13, 1, 4], [1, 13, 4], [4, 4, 10]]) / 18 BF = np.array([[13.0, 1.0, 4.0], [1.0, 13.0, 4.0], [4.0, 4.0, 10.0]]) / 18.0 def projectors(): u1 = sp.Matrix([1, 1, 1]) / sp.sqrt(3) u2 = sp.Matrix([1, -1, 0]) / sp.sqrt(2) u3 = sp.Matrix([1, 1, -2]) / sp.sqrt(6) return u1 * u1.T, u2 * u2.T, u3 * u3.T def collision_V(): """v977/v984 collision V = C (U_B (x) I) as a numeric 9x9.""" s13sq = sp.Rational(2, 9) c13sq = 1 - s13sq s12sq = sp.Rational(1, 14) s23sq = sp.Rational(2, 7) c12sq, c23sq = 1 - s12sq, 1 - s23sq s12, c12 = sp.sqrt(s12sq), sp.sqrt(c12sq) s23, c23 = sp.sqrt(s23sq), sp.sqrt(c23sq) s13, c13 = sp.sqrt(s13sq), sp.sqrt(c13sq) eid = -4 / sp.sqrt(65) + sp.I * 7 / sp.sqrt(65) U = sp.Matrix([ [c12 * c13, s12 * c13, s13 / eid], [-s12 * c23 - c12 * s23 * s13 * eid, c12 * c23 - s12 * s23 * s13 * eid, s23 * c13], [s12 * s23 - c12 * c23 * s13 * eid, -c12 * s23 - s12 * c23 * s13 * eid, c23 * c13]]) Un = np.array([[complex(sp.N(U[i, j], 20)) for j in range(3)] for i in range(3)]) Cn = np.zeros((9, 9)) for j in range(3): for k in range(3): Cn[j * 3 + ((j + k) % 3), j * 3 + k] = 1.0 return Cn @ np.kron(Un, np.eye(3)) def kron_power(M, L): out = np.array([[1.0]]) for _ in range(L): out = np.kron(out, M) return out def coupling_diag(L, eps): D = 3 ** L v = np.zeros(D) for s in range(D): digits = [] r = s for _ in range(L): digits.append(r % 3) r //= 3 v[s] = eps * sum(1.0 for x in range(L) if digits[x] == digits[(x + 1) % L]) return np.exp(v) def transfer_int(L, eps): BL = kron_power(BF, L) k = coupling_diag(L, eps) K12 = np.sqrt(k) return (K12[:, None] * BL) * K12[None, :] def swap_even_H(L): """even-bond matching layer: SWAP on (0,1), (2,3), ... (no wrap).""" D = 3 ** L H = np.zeros((D, D), dtype=float) for x in range(0, L - 1, 2): a, b = x, x + 1 for s in range(D): digits = [] r = s for _ in range(L): digits.append(r % 3) r //= 3 digits = digits[::-1] nd = digits[:] nd[a], nd[b] = nd[b], nd[a] t = 0 for d in nd: t = t * 3 + d H[t, s] += 1.0 return H def run(): reset() print("v987 DYN.UNITARY.DILATION.01: OS/dilation ladder " "(kernel + free band + interacting congruence + quantum SWAP)") P1, P2, P3 = projectors() check("A SPECTRAL [E]: B = P1 + (2/3) P2 + (1/3) P3 with orthogonal " "projectors summing to I", sp.simplify(B_SYM - (P1 + sp.Rational(2, 3) * P2 + sp.Rational(1, 3) * P3)) == sp.zeros(3) and sp.simplify(P1 + P2 + P3 - sp.eye(3)) == sp.zeros(3) and all(sp.simplify(P * Q) == sp.zeros(3) for P, Q in ((P1, P2), (P1, P3), (P2, P3)))) H = sp.log(sp.Rational(3, 2)) * P2 + sp.log(3) * P3 u1 = sp.Matrix([1, 1, 1]) / sp.sqrt(3) eigH = sorted(sp.simplify(x) for x in H.eigenvals()) check("A OS HAMILTONIAN [E]: H = -log B PSD, spectrum " "{0, log(3/2), log 3}, gap log(3/2), ground = uniform", sp.simplify(H - H.T) == sp.zeros(3) and sp.simplify(H * u1) == sp.zeros(3, 1) and eigH == [0, sp.log(sp.Rational(3, 2)), sp.log(3)]) t, s = sp.symbols("t s", real=True) Tt = P1 + sp.exp(-t * sp.log(sp.Rational(3, 2))) * P2 \ + sp.exp(-t * sp.log(3)) * P3 row_sums = sp.simplify(Tt * sp.Matrix([1, 1, 1])) prod_diff = (Tt * Tt.subs(t, s) - Tt.subs(t, t + s)).applyfunc( lambda x: sp.simplify(sp.powsimp(sp.expand(x), force=True))) check("A SEMIGROUP [E]: T(t) rows sum to 1 and T(t)T(s) = T(t+s) " "symbolically (inside the stochastic cone)", sp.simplify(row_sums - sp.Matrix([1, 1, 1])) == sp.zeros(3, 1) and prod_diff == sp.zeros(3)) Thalf = Tt.subs(t, sp.Rational(1, 2)) check("A HALF STEP [E]: B^{1/2} = T(1/2) is real and entrywise " "positive (the chain interpolates inside the cone)", all(sp.simplify(sp.im(x)) == 0 for x in Thalf) and all(sp.N(x) > 0 for x in Thalf)) Ut = P1 + sp.Rational(2, 3) ** (sp.I * t) * P2 \ + sp.Rational(1, 3) ** (sp.I * t) * P3 check("A UNITARY GROUP [E]: U(t)U(s) = U(t+s) and U(t)^dag = U(-t)", sp.simplify(Ut * Ut.subs(t, s) - Ut.subs(t, t + s)) == sp.zeros(3) and sp.simplify(Ut.H - Ut.subs(t, -t)) == sp.zeros(3)) wick = all(sp.simplify(Ut.subs(t, -sp.I * n) - B_SYM ** n) == sp.zeros(3) for n in range(1, 7)) check("A WICK ANCHOR [E]: U(-i n) = B^n for n = 1..6 " "(six-step macro-gate T = B^6 = e^{-6H} included)", wick) Bbad = P1 - sp.Rational(1, 10) * P2 + sp.Rational(1, 3) * P3 half_bad = P1 + sp.sqrt(sp.Rational(-1, 10)) * P2 \ + sp.sqrt(sp.Rational(1, 3)) * P3 check("A MUST-FAIL [E]: v971 negative kernel (lam2 = -1/10) is " "doubly stochastic with det < 0 and its half-step is NON-REAL " "-- no self-adjoint OS Hamiltonian", sp.simplify(Bbad * sp.Matrix([1, 1, 1]) - sp.Matrix([1, 1, 1])) == sp.zeros(3, 1) and sp.simplify(Bbad.det()) < 0 and any(sp.simplify(sp.im(x)) != 0 for x in half_bad)) h_eig = [0, sp.log(sp.Rational(3, 2)), sp.log(3)] ok_gap = True for L in (2, 3): spec = sorted(set(sum(h_eig[i] for i in tup) for tup in iproduct(range(3), repeat=L)), key=lambda e: sp.N(e)) gap = sp.simplify(spec[1] - spec[0]) ok_gap &= spec[0] == 0 and sp.simplify( gap - sp.log(sp.Rational(3, 2))) == 0 check("B UNIFORM GAP [E]: H_L = sum_x h_x has ground 0 and gap " "log(3/2) EXACTLY at L = 2, 3 (uniform in L, no closing)", ok_gap) P = [P1, P2, P3] lam = [sp.Integer(1), sp.Rational(2, 3), sp.Rational(1, 3)] B2 = sp.Matrix(sp.kronecker_product(B_SYM, B_SYM)) recon = sp.zeros(9, 9) for i in range(3): for j in range(3): recon += lam[i] * lam[j] * sp.Matrix( sp.kronecker_product(P[i], P[j])) check("B WICK L=2 [E]: B^(x2) = sum_{ij} e^{-(h_i+h_j)} P_i (x) P_j " "(product OS Hamiltonian generates the product chain)", sp.simplify(B2 - recon) == sp.zeros(9)) Vn = collision_V() def per_cell_channel(L): outs = [] for basis in range(3): p = np.zeros(3) p[basis] = 1.0 rho = np.array([1.0]) for _ in range(L): rho = np.kron(rho, np.diag(p)) rho = np.kron(rho, np.diag([1.0, 0, 0])) W = np.array([1.0]) for _ in range(L): W = np.kron(W, Vn) big = W @ rho.astype(complex) @ W.conj().T dims = [3] * (2 * L) red = big.reshape(dims + dims) for f in range(2 * L - 1, 0, -1): n = red.ndim // 2 red = np.trace(red, axis1=f, axis2=f + n) outs.append(np.real(np.diag(red))) return np.array(outs).T ok_d2 = True ref = None for L in (1, 2): M = per_cell_channel(L) if ref is None: ref = M ok_d2 &= np.allclose(M, BF, atol=1e-10) and np.allclose( M, ref, atol=1e-10) check("B D2 [E/N]: per-cell reduced channel is B at L = 1, 2 " "identically -- one fresh ancilla per cell per step, linear " "environment, size-consistent", ok_d2) L = 2 Wcoll = np.kron(Vn, Vn) def op_on(factor, M, nfac=4): out = np.array([[1.0]], dtype=complex) for f in range(nfac): out = np.kron(out, M if f == factor else np.eye(3)) return out lam1 = np.zeros((3, 3), dtype=complex) lam1[0, 1] = lam1[1, 0] = 1.0 zgen = np.diag([1.0, -1.0, 0.0]).astype(complex) O0 = op_on(0, lam1) O1 = Wcoll.conj().T @ O0 @ Wcoll far = max(np.max(np.abs(O1 @ op_on(2, g) - op_on(2, g) @ O1)) for g in (lam1, zgen)) near = max(np.max(np.abs(O1 @ op_on(0, g) - op_on(0, g) @ O1)) for g in (lam1, zgen)) check("B LR RADIUS 1 [N]: after one collision step an S0-local " "operator commutes with the far cell S1 of the L = 2 band " "(dev %.1e < 1e-10) and the near-cell commutator is STRICTLY " "nonzero (velocity = 1, not 0)" % far, far < 1e-10 and near > 1e-6) ok_pd = True for L in (2, 3): for eps in (-1.0, -0.3, 0.3, 1.0): w = np.linalg.eigvalsh(transfer_int(L, eps)) ok_pd &= w[0] > 0 check("C CONGRUENCE [E]: T_eps = K^{1/2} B^(xL) K^{1/2} is SPD for " "every tested (L, eps) including NEGATIVE couplings -- " "positivity is a congruence fact", ok_pd) T_one = coupling_diag(3, 0.7)[:, None] * kron_power(BF, 3) asym = float(np.max(np.abs(T_one - T_one.T))) check("C MUST-FAIL [E]: one-sided coupling K B^(xL) is NOT " "symmetric (max asym %.2f) -- no self-adjoint OS Hamiltonian" % asym, asym > 1e-3) gaps = [] for L in range(2, 5): w = np.linalg.eigvalsh(transfer_int(L, 0.3)) gaps.append(float(np.log(w[-1] / w[-2]))) print(" gap(L=2..4):", ["%.4f" % g for g in gaps]) check("C GAP PLATEAU [N]: gap(L) stays bounded away from zero " "(min %.4f > 0.1) with last relative step < 0.15 at eps = 0.3 " "-- no collapse with size" % min(gaps), min(gaps) > 0.1 and abs(gaps[-1] - gaps[-2]) / gaps[-1] < 0.15) L = 4 T4 = transfer_int(L, 0.3) w4, v4 = np.linalg.eigh(T4) H4 = -(v4 @ np.diag(np.log(np.maximum(w4, 1e-300))) @ v4.T) b1 = np.diag([1, -1, 0]) / np.sqrt(2) b2 = np.diag([1, 1, -2]) / np.sqrt(6) b3 = np.zeros((3, 3)) b3[0, 1] = b3[1, 0] = 1 / np.sqrt(2) BASIS = [b1, b2, b3] def two_site_coeff(Hm, d): best = 0.0 for A in BASIS: for C in BASIS: op = np.array([[1.0]]) for site in range(L): if site == 0: op = np.kron(op, A) elif site == d: op = np.kron(op, C) else: op = np.kron(op, np.eye(3)) best = max(best, abs(float(np.sum(Hm * op.T))) / 3 ** (L - 2)) return best c1, c2 = two_site_coeff(H4, 1), two_site_coeff(H4, 2) print(" 2-site coeffs d=1,2:", "%.2e" % c1, "%.2e" % c2) check("C QUASI-LOCALITY [N]: 2-site coefficients of H = -log T " "decay with distance (ratio %.3f < 0.8) -- D1 interacting " "witness at kernel level" % (c2 / c1 if c1 else 1), c1 > 0 and c2 / c1 < 0.8) Lq = 2 Hsw = swap_even_H(Lq) wsw, vsw = np.linalg.eigh(Hsw) Wcoll2 = np.kron(Vn, Vn) def embed_sys_unitary(Usys): """Usys acts on (S0, S1); embed as U (x) I on ancillas of the factor order (S0, A0, S1, A1).""" Ufull = np.zeros((81, 81), dtype=complex) for s in range(81): digits = [] r = s for _ in range(4): digits.append(r % 3) r //= 3 digits = digits[::-1] sys = digits[0] * 3 + digits[2] for t_sys in range(9): amp = Usys[t_sys, sys] if abs(amp) < 1e-16: continue nd = digits[:] nd[0] = t_sys // 3 nd[2] = t_sys % 3 t = 0 for d in nd: t = t * 3 + d Ufull[t, s] += amp return Ufull def keep_S0(rho): red = rho.reshape([3] * 8) for f in (3, 2, 1): n = red.ndim // 2 red = np.trace(red, axis1=f, axis2=f + n) return red theta = 0.2 Uint_sys = vsw @ np.diag(np.exp(-1j * theta * wsw)) @ vsw.T Wth = embed_sys_unitary(Uint_sys) @ Wcoll2 check("D UNITARY [N]: W_theta = U_int(theta) (x)_x V_x is unitary " "at L = 2, theta = 0.2 (max |W^dag W - I| < 1e-9)", np.max(np.abs(Wth.conj().T @ Wth - np.eye(81))) < 1e-9) mix = np.eye(3) / 3.0 def reduced_diag(W): cols = [] for j in range(3): rho = np.array([1.0], dtype=complex) rho = np.kron(rho, np.diag([float(k == j) for k in range(3)])) rho = np.kron(rho, np.diag([1.0, 0.0, 0.0])) rho = np.kron(rho, mix) rho = np.kron(rho, np.diag([1.0, 0.0, 0.0])) cols.append(np.real(np.diag(keep_S0(W @ rho @ W.conj().T)))) return np.array(cols).T dths = [] for th in (0.0, 0.05, 0.1, 0.2): Uth = vsw @ np.diag(np.exp(-1j * th * wsw)) @ vsw.T W = embed_sys_unitary(Uth) @ Wcoll2 dths.append(float(np.max(np.abs(reduced_diag(W) - BF)))) print(" d(theta) at 0, 0.05, 0.1, 0.2:", ["%.3e" % d for d in dths]) xs = np.log([0.05, 0.1, 0.2]) ys = np.log(np.maximum(dths[1:], 1e-30)) power = float(np.polyfit(xs, ys, 1)[0]) print(" fitted power d ~ theta^{%.3f}" % power) check("D d(0) = 0 [N]: reduced diagonal kernel recovers B exactly " "at theta = 0 (max |M-B| = %.1e)" % dths[0], dths[0] < 1e-10) check("D d(theta) ~ theta^2 [N]: leading even power in {1.5, 2.5} " "(fit %.3f; probe ~1.98) and d -> 0 continuously" % power, 1.5 < power < 2.5 and dths[1] <= dths[2] + 1e-15 and dths[2] <= dths[3] + 1e-15) choi = np.zeros((9, 9), dtype=complex) for i in range(3): for j in range(3): ein = np.zeros((3, 3), dtype=complex) ein[i, j] = 1.0 rho = np.array([1.0], dtype=complex) rho = np.kron(rho, ein) rho = np.kron(rho, np.diag([1.0, 0.0, 0.0])) rho = np.kron(rho, mix) rho = np.kron(rho, np.diag([1.0, 0.0, 0.0])) red = keep_S0(Wth @ rho @ Wth.conj().T) choi[3 * i:3 * i + 3, 3 * j:3 * j + 3] = red w_choi = np.linalg.eigvalsh(0.5 * (choi + choi.conj().T)) check("D CPTP [N]: reduced per-cell channel at theta = 0.2 has " "Choi PSD (min eig %.1e) -- openness survives SWAP coupling" % w_choi[0], w_choi[0] > -1e-9) sw2 = Hsw @ Hsw check("D LR MATCHING [E]: even-bond SWAP is hermitian, SWAP^2 = I " "on the paired cells (range 1) -- far unmatched cells are " "machine-zero by construction (probe L = 4 far-cell zero is " "this skeleton; Tr SWAP = 3 on C^3 x C^3)", float(np.max(np.abs(Hsw - Hsw.T))) < 1e-15 and np.allclose(sw2, np.eye(9)) and abs(float(np.trace(Hsw)) - 3.0) < 1e-12) check("FIREWALL (scope): kernel + free band + diagonal-coupling " "interacting + one SWAP family, finite L, discrete time; " "continuous field-level OS, thermodynamic-limit branch " "selection and any 3+1D reading stay [O]; no marker move", True) return summary("v987 OS/dilation ladder: kernel exact + free band " "+ interacting congruence + quantum SWAP") if __name__ == "__main__": raise SystemExit(1 if run() else 0)