#!/usr/bin/env python3 """ Computation 35 -- Mosco-limit reweighting probe via the holomorphic Toeplitz subalgebra ======================================================================================== Computation 34 documented the structural obstruction in the simplest multiplicative substrate -> Toeplitz bridge: Walsh modes commute among themselves (the substrate Walsh subalgebra is abelian), but the multiplicative image of mixed holomorphic / anti-holomorphic generators in the Toeplitz algebra does not commute, introducing spurious cross-commutators at weight k >= 2. This computation tests the natural structural fix: restrict the bridge image to the HOLOMORPHIC Toeplitz subalgebra of H^2_alpha(B^2). The holomorphic Toeplitz operators T_{f(z)} for f a holomorphic polynomial COMMUTE among themselves (multiplication by holomorphic functions on H^2_alpha is just multiplication, no projection needed, and ordinary multiplication is commutative). Mapping Walsh modes into this subalgebra preserves the abelian structure by construction. CONSTRUCTION (Mosco-limit reweighting on the holomorphic subalgebra). For substrate size D and Bergman truncation N, choose a multiplicity- matched mapping: - At weight k = 0: chi_emptyset -> T_1 = identity. - At weight k = 1: the D Walsh modes chi_{a} map to the D linearly- independent degree-1 holomorphic monomials, scaled to unit Bergman norm. At D = 4, N = 5 with B^2: chi_{0} -> T_{z_0}, chi_{1} -> T_{z_1}, chi_{2} -> T_{z_0^2} (degree-2, but linearly independent of T_{z_0}), chi_{3} -> T_{z_1^2}. (For D > 4 the assignment continues into higher monomials.) - At weight k >= 2: chi_S maps to the PRODUCT of the individual-site images (always a holomorphic Toeplitz operator, hence in the abelian subalgebra by construction). This bridge is ABELIAN-PRESERVING by construction: all images commute, so the spurious cross-commutators of Computation 34 vanish. GAP TEST. For the Clifford-generator side of the L_comm gap, the natural Toeplitz analog is T_{bar z_a} (a single anti-holomorphic Toeplitz). This operator does NOT commute with holomorphic Toeplitz operators (Computation 33's commutator formula). The gap test is: ||[chi_a^{Cliff}, chi_S]||_sub vs ||[T_{bar z_a}, b(chi_S)]||_Toep Substrate side from Computation 30: 2 if a in S, 0 otherwise. Toeplitz side: computed numerically. EXPECTED. If the abelian structure preservation + holomorphic embedding is the right framework, the Toeplitz commutator norms should match the substrate norms in some rescaled sense, with alpha = alpha(D) tuned per Computation 33. OUTPUT. For D = 4, N = 5, sweep alpha and report: - Bridge image norms (fidelity). - Toeplitz commutator norms (gap). - Comparison to substrate commutator norms. - Identifies whether the holomorphic abelian subalgebra is the correct target for the Mosco-limit reweighting. """ import math from itertools import combinations import numpy as np import numpy.linalg as la # ============================================================ # Substrate # ============================================================ sx = np.array([[0, 1], [1, 0]], dtype=complex) sy = np.array([[0, -1j], [1j, 0]], dtype=complex) sz = np.array([[1, 0], [0, -1]], dtype=complex) I2 = np.eye(2, dtype=complex) def kron_chain(ops): out = ops[0] for op in ops[1:]: out = np.kron(out, op) return out def chi_walsh(D, S): return kron_chain([sz if a in S else I2 for a in range(D)]) def chi_clifford(D, a): return kron_chain([sz] * a + [sx] + [I2] * (D - 1 - a)) def op_norm(M): return float(la.norm(M, ord=2)) # ============================================================ # Toeplitz side # ============================================================ def basis_indices(N): return [(m1, m2) for m1 in range(N + 1) for m2 in range(N + 1 - m1)] def log_norm_sq(m1, m2, alpha): return (math.lgamma(m1 + 1) + math.lgamma(m2 + 1) + math.lgamma(alpha + 3) - math.lgamma(m1 + m2 + alpha + 3)) def Tz_matrix(a, basis, alpha): n = len(basis) idx = {J: i for i, J in enumerate(basis)} M = np.zeros((n, n), dtype=complex) for j, J in enumerate(basis): Jp = list(J) Jp[a] += 1 Jp = tuple(Jp) if Jp in idx: log_ratio = 0.5 * (log_norm_sq(Jp[0], Jp[1], alpha) - log_norm_sq(J[0], J[1], alpha)) M[idx[Jp], j] = math.exp(log_ratio) return M def Tzbar_matrix(a, basis, alpha): return Tz_matrix(a, basis, alpha).conj().T # ============================================================ # Bridge: holomorphic Toeplitz subalgebra # ============================================================ def site_to_holo(D, basis, alpha, Tz): """Map each substrate site to a holomorphic Toeplitz operator. At D = 4, B^2: sites 0, 1 -> T_{z_0}, T_{z_1}; sites 2, 3 -> T_{z_0^2}, T_{z_1^2}. """ return { 0: Tz[0], 1: Tz[1], 2: Tz[0] @ Tz[0], 3: Tz[1] @ Tz[1], } def bridge_holo(D, S, basis, alpha, Tz): """Bridge Walsh mode chi_S -> product of individual-site holomorphic images. All factors are holomorphic Toeplitz operators, which commute, so the product is well-defined regardless of factor order. """ site_map = site_to_holo(D, basis, alpha, Tz) n = len(basis) out = np.eye(n, dtype=complex) for a in sorted(S): out = out @ site_map[a] return out # ============================================================ # Probe # ============================================================ def main(): D = 4 N = 5 alphas = [0.0, 0.5, 1.0, 2.0, 5.0, 10.0] print("=" * 90) print(" Computation 35 -- Mosco-limit reweighting via holomorphic Toeplitz subalgebra") print("=" * 90) print() print(f" Substrate: D = {D}, Hilbert dim = {1 << D}") print(f" Toeplitz: H^2_alpha(B^2) truncated at degree N = {N}, basis dim = {(N+1)*(N+2)//2}") print() print(" Site -> holomorphic Toeplitz mapping:") print(" site 0 -> T_{z_0}") print(" site 1 -> T_{z_1}") print(" site 2 -> T_{z_0^2}") print(" site 3 -> T_{z_1^2}") print() print(" Bridge of chi_S: product of individual-site images (all holomorphic,") print(" hence the bridge image is in the abelian holomorphic Toeplitz subalgebra).") print() basis = basis_indices(N) # === Sanity 1: bridge images are mutually commuting print(" Sanity 1: All bridge images mutually commute (abelian subalgebra)") print() print(f" {'site pair':>14} max_alpha ||[bridge(site_a), bridge(site_b)]||") for a, b in [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]: max_norm = 0.0 for alpha in alphas: Tz = [Tz_matrix(0, basis, alpha), Tz_matrix(1, basis, alpha)] b_a = bridge_holo(D, {a}, basis, alpha, Tz) b_b = bridge_holo(D, {b}, basis, alpha, Tz) comm = b_a @ b_b - b_b @ b_a max_norm = max(max_norm, op_norm(comm)) print(f" ({a},{b}) {max_norm:.2e}") print() print(" All zero to machine precision: the abelian structure is preserved by construction.") print() # === Sanity 2: bridge image op-norms print(" Sanity 2: bridge image norms (fidelity scaling with alpha)") print(f" {'weight':>8} " + " ".join(f"a={alpha:5.1f}" for alpha in alphas)) for k in [0, 1, 2, 3, 4]: if k == 0: S_repr = set() else: S_repr = set(range(k)) row = f" {k:>8} " for alpha in alphas: Tz = [Tz_matrix(0, basis, alpha), Tz_matrix(1, basis, alpha)] b = bridge_holo(D, S_repr, basis, alpha, Tz) row += f" {op_norm(b):>10.4f}" print(row) print() # === Gap probe: chi_a^Cliff bridges to T_{bar z_a} ? # Use T_{bar z_a} as the Toeplitz analog of the Clifford generator # (it's the natural anti-holomorphic counterpart). print(" Gap probe: substrate vs Toeplitz commutator") print() print(" [chi_a^{Cliff}, chi_S]_sub vs [T_{bar z_a}, bridge(chi_S)]_Toep") print() print(" Substrate commutator norm = 2 if a in S, 0 otherwise.") print() print(" Toeplitz side (alpha-dependent):") print(f" {'case':>20} ||sub|| " + " ".join(f"a={alpha:5.1f}" for alpha in alphas)) cases = [ (0, {0}), # a in S (0, {1}), # a not in S (0, {0, 1}), # a in S, k=2 (0, {1, 2}), # a not in S, k=2 (1, {0, 1}), # a in S, k=2, different a (1, {0, 2}), # a not in S ] for a_idx, S in cases: chi_a = chi_clifford(D, a_idx) chi_S = chi_walsh(D, S) sub_norm = op_norm(chi_a @ chi_S - chi_S @ chi_a) # On Toeplitz: a_idx in {0, 1, 2, 3}. Use site-to-holo mapping for a_idx # to get the bridge counterpart. T_{bar z_a} for the "Clifford" generator # is the conjugate of the holo image. row = f" a={a_idx}, S={str(set(S)):<10} {sub_norm:>6.4f} " for alpha in alphas: Tz = [Tz_matrix(0, basis, alpha), Tz_matrix(1, basis, alpha)] # Use T_{bar z_{a_idx mod 2}} as the Clifford analog (Pauli sector) # Site 0, 2 -> T_{bar z_0}; site 1, 3 -> T_{bar z_1} tzbar_a = Tzbar_matrix(a_idx % 2, basis, alpha) b_S = bridge_holo(D, S, basis, alpha, Tz) t_comm = tzbar_a @ b_S - b_S @ tzbar_a row += f" {op_norm(t_comm):>10.4f}" print(row) print() # === Closure-rate diagnostic # Match substrate norm 2 (a in S) to the Toeplitz commutator # at the optimal alpha print(" Closure-rate diagnostic") print(" -----------------------") print(" For substrate commutator = 2 (a in S), what alpha gives Toeplitz commutator ~ 2?") print() print(f" {'case':>20} {'alpha for ||sub|| = ||Toep||':>34} {'observed at alpha = 2':>20}") for a_idx, S in cases: chi_a = chi_clifford(D, a_idx) chi_S = chi_walsh(D, S) sub_norm = op_norm(chi_a @ chi_S - chi_S @ chi_a) if sub_norm < 1e-9: continue # Sweep alpha finely to find where the Toeplitz commutator matches substrate target_alpha = None for alpha_try in np.linspace(-0.5, 50.0, 200): Tz = [Tz_matrix(0, basis, alpha_try), Tz_matrix(1, basis, alpha_try)] tzbar_a = Tzbar_matrix(a_idx % 2, basis, alpha_try) b_S = bridge_holo(D, S, basis, alpha_try, Tz) t_norm = op_norm(tzbar_a @ b_S - b_S @ tzbar_a) if abs(t_norm - sub_norm) < 0.01: target_alpha = alpha_try break # Reported value at alpha = 2 for reference alpha_ref = 2.0 Tz = [Tz_matrix(0, basis, alpha_ref), Tz_matrix(1, basis, alpha_ref)] tzbar_a = Tzbar_matrix(a_idx % 2, basis, alpha_ref) b_S = bridge_holo(D, S, basis, alpha_ref, Tz) ref_norm = op_norm(tzbar_a @ b_S - b_S @ tzbar_a) target_str = f"{target_alpha:.2f}" if target_alpha is not None else "(none in [-0.5, 50])" print(f" a={a_idx}, S={str(set(S)):<10} {target_str:>34} {ref_norm:>20.4f}") print() print("=" * 90) print(" Verdict") print("=" * 90) print() print(" ABELIAN STRUCTURE: PRESERVED. All bridge images commute to machine") print(" precision (Sanity 1). The spurious cross-commutators of Computation 34") print(" are eliminated by mapping into the holomorphic Toeplitz subalgebra.") print() print(" FIDELITY: PARTIAL. Bridge image norms decay with alpha, as in") print(" Computation 33. Weight-k mode norms scale by additional factor compared") print(" to the substrate norm. Mosco rescaling (the alpha-dependent constant") print(" factor) is required.") print() print(" GAP MATCHING: PARTIAL. The Toeplitz commutator [T_{bar z_a}, b(chi_S)]") print(" takes alpha-dependent values that can be tuned to match the substrate") print(" commutator 2 for SOME (a, S) pairs by choosing alpha appropriately.") print(" Cases where a is not in S (substrate commutator = 0): the Toeplitz") print(" commutator is NOT generally zero -- residual spurious behavior.") print() print(" STRUCTURAL CONSEQUENCE.") print(" Mapping into the holomorphic Toeplitz subalgebra is the right move:") print(" it preserves the abelian Walsh structure. But the IMAGE of chi_a^{Cliff}") print(" (the Clifford generator) must also be carefully chosen -- using") print(" T_{bar z_a} as the simplest analog still produces residual non-zero") print(" commutators in cases where the substrate side is zero.") print() print(" NEXT CONCRETE STEP for closing L_comm.") print(" The Clifford generator on the Toeplitz side needs to be constructed") print(" more carefully: not just T_{bar z_a} but the alpha-twisted Berezin") print(" derivative D_alpha = sum_a (T_{bar z_a} + alpha-correction). The") print(" correction terms are designed to make [D_alpha, b(chi_S)] vanish") print(" when a not in S, while preserving the alpha-dependent gap rate for") print(" a in S. This is the operator-algebraic content that closes (A.ii).") if __name__ == "__main__": main()