#!/usr/bin/env python3 """ Computation 34 -- Explicit substrate <-> Toeplitz bridge probe (D = 4) ======================================================================= Implements a CONCRETE candidate bridge map between the PST substrate algebra at size D = 4 and the truncated weighted Bergman / Toeplitz algebra on H^2_alpha(B^2) at degree N = 5. Builds on Computation 33, which verified the BS 2022 framework has the required non-proportional alpha-scaling at the operator level, by attempting to write down the explicit substrate <-> Toeplitz identification that is the load-bearing remaining task for closing L_comm. CONSTRUCTION. Substrate side (D = 4): - Hilbert space (C^2)^{otimes 4}, dim 16. - Abelian Walsh modes chi_S = product of sigma_z over S subset {0,1,2,3}, C(4, k) modes at weight k: 1, 4, 6, 4, 1 for k = 0, 1, 2, 3, 4. Toeplitz side (H^2_alpha(B^2) at degree N = 5): - Monomial basis z_1^{m_1} z_2^{m_2}, m_1 + m_2 <= 5, dim 21. - Toeplitz operators T_{z_a}, T_{bar z_a} with explicit matrix elements (Computation 33). - alpha treated as a free parameter, tuned per the BS 2022 closure criterion. BRIDGE MAP (candidate). For weight k Walsh modes chi_S, |S| = k: - k = 0: chi_emptyset -> identity (trivially). - k = 1: chi_{a} -> T_{z_a} (a in {0, 1}: site labels match Toeplitz coords). For a in {2, 3}: chi_{a} -> T_{bar z_{a-2}} (parity-flipped). - k = 2: chi_{a, b} -> T_{z_a} T_{z_b} (multiplicative on the Toeplitz side). - k >= 3: chi_S -> product T_{z_a}^* T_{z_b} ... with parity-tracked indices. This is the SIMPLEST possible multiplicative bridge using only T_{z_a} and T_{bar z_a} generators. It is a NECESSARY scaffolding for the BS 2022 framework but not yet sufficient: the Mosco-limit reweighting and the spinor structure are not yet incorporated. WHAT THIS PROBE MEASURES. For each pair (chi_a^{Cliff}, chi_S) appearing in Computation 30: - The discrete commutator op-norm: 2 if a in S, 0 otherwise. - The Toeplitz analog under the candidate bridge. - Their difference under the bridge identification. The output establishes (a) which aspects of the bridge are consistent with BS 2022's framework, (b) which specific terms still need a Mosco-limit averaging or spinor extension to match. EXPECTED RESULT. The simplest multiplicative bridge will MATCH at weight 1 (where it reduces to Computation 33's measurement) but will MISMATCH at higher weights, because: - Substrate Walsh modes commute among themselves (abelian); - Toeplitz products do not generally commute. This documents the structural reason that the bridge cannot be purely multiplicative; it must average over Mosco-limit groupings to match the abelian Walsh structure to the non-abelian Toeplitz output. This computation is therefore a SCOPING ANALYSIS, not a closure: it confirms what works (operator norms, single-mode commutators) and exposes the precise mismatch (abelian vs non-abelian) that the next step in (A.ii) closure has to bridge. """ import math from itertools import combinations import numpy as np import numpy.linalg as la # ============================================================ # Substrate side # ============================================================ 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): """Abelian Walsh mode = sigma_z at sites in S, I elsewhere.""" return kron_chain([sz if a in S else I2 for a in range(D)]) def chi_clifford(D, a): """JW Clifford chi_a^{Cliff} = sigma_z^{a}.""" return kron_chain([sz] * a + [sx] + [I2] * (D - 1 - a)) def op_norm(M): return float(la.norm(M, ord=2)) # ============================================================ # Toeplitz side: weighted Bergman H^2_alpha(B^2) # ============================================================ 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): """log ||z_1^m1 z_2^m2||^2_alpha = log[m1! m2! Gamma(alpha+3) / Gamma(m1+m2+alpha+3)].""" 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): """Matrix of T_{z_a} on ONB basis: T_{z_a} e_J = sqrt(...) e_{J + e_a}.""" 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): """T_{bar z_a} = (T_{z_a})^* on H^2_alpha.""" return Tz_matrix(a, basis, alpha).conj().T # ============================================================ # Bridge map # ============================================================ def bridge_walsh_mode(D, S, basis, alpha, Tz, Tzbar): """Map abelian Walsh mode chi_S to a Toeplitz operator. Convention used here: - sites 0, 1 of substrate <-> z_0, z_1 on B^2 (holomorphic) - sites 2, 3 of substrate <-> bar z_0, bar z_1 (anti-holomorphic) chi_S is mapped to the PRODUCT of the corresponding Toeplitz generators, in lexicographic site order. For k = 0 (S empty), the bridge image is I. """ n = len(basis) out = np.eye(n, dtype=complex) for a in sorted(S): if a < 2: out = out @ Tz[a] else: out = out @ Tzbar[a - 2] return out # ============================================================ # Probe # ============================================================ def main(): D = 4 N = 5 alphas = [0.0, 1.0, 2.0, 5.0] print("=" * 90) print(" Computation 34 -- explicit substrate <-> Toeplitz bridge probe (D = 4)") 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(f" Bridge map: site a in 0,1 -> T_{{z_a}}; site a in 2,3 -> T_{{bar z_{{a-2}}}}") print() basis = basis_indices(N) # Single-mode op-norms (sanity) print(" Sanity: op-norm of bridge image of single-mode chi_{a}") print(f" {'a':>3} ||chi_{{a}}||_sub " + " ".join(f"a={alpha:5.1f}" for alpha in alphas)) for site in range(D): sub_norm = op_norm(chi_walsh(D, {site})) row = f" {site:>3} {sub_norm:>14.6f} " for alpha in alphas: Tz = [Tz_matrix(0, basis, alpha), Tz_matrix(1, basis, alpha)] Tzbar = [Tzbar_matrix(0, basis, alpha), Tzbar_matrix(1, basis, alpha)] b = bridge_walsh_mode(D, {site}, basis, alpha, Tz, Tzbar) row += f" {op_norm(b):>10.4f}" print(row) print() # Commutator gap test: [chi_a^Cliff, chi_S] vs bridge of same print(" Commutator gap: ||[chi_a^{Cliff}, chi_S]||_sub vs ||[bridge(chi_a^Cliff), bridge(chi_S)]||_Toep") print() print(" case ||sub_comm|| " + " ".join(f"alpha={alpha:5.1f}" for alpha in alphas)) cases = [ (0, {0}), # a in S, single site (0, {0, 1}), # a in S, k=2 (0, {1}), # a not in S, single site (0, {1, 2}), # a not in S, k=2 (1, {0, 1}), # a in S, k=2 (2, {0, 2}), # a in S, mixed holo/antiholo ] for a_cliff, S in cases: chi_a = chi_clifford(D, a_cliff) chi_S = chi_walsh(D, S) sub_comm = chi_a @ chi_S - chi_S @ chi_a sub_norm = op_norm(sub_comm) # Bridge each side # chi_a^Cliff has weight 1 in the JW sense (sigma_x at site a, sigma_z below). # For the bridge, treat the Clifford generator as map of site a. # Simplest: bridge maps chi_a^Cliff -> T_{z_a} (or T_{bar z_{a-2}}). row = f" a={a_cliff}, S={str(set(S)):<10} {sub_norm:>14.6f} " for alpha in alphas: Tz = [Tz_matrix(0, basis, alpha), Tz_matrix(1, basis, alpha)] Tzbar = [Tzbar_matrix(0, basis, alpha), Tzbar_matrix(1, basis, alpha)] b_a = bridge_walsh_mode(D, {a_cliff}, basis, alpha, Tz, Tzbar) b_S = bridge_walsh_mode(D, S, basis, alpha, Tz, Tzbar) t_comm = b_a @ b_S - b_S @ b_a row += f" {op_norm(t_comm):>11.4f}" print(row) print() print(" Interpretation.") print() print(" Cases where a is in S (substrate commutator is exactly 2):") print(" The Toeplitz commutator under the simplest multiplicative bridge") print(" does NOT generally equal 2. The discrepancy is the bridge gap") print(" that the Mosco-limit averaging or spinor extension has to absorb.") print() print(" Cases where a is not in S (substrate commutator is 0):") print(" The Toeplitz commutator under the bridge may be NON-ZERO.") print(" These are spurious commutators that the bridge introduces because") print(" Toeplitz products do not commute among themselves.") print() print(" Both phenomena document the same structural fact: the substrate") print(" Walsh subalgebra is ABELIAN, but the multiplicative bridge maps") print(" it into a NON-ABELIAN Toeplitz subalgebra. The Mosco-limit") print(" reweighting at fixed weight k must restore the abelian structure") print(" on the bridge image, while preserving the alpha-dependent") print(" gap-rate scaling of Computation 33.") print() # The structural prediction: at large alpha (deep into the ball), # all commutator norms shrink. Check. print(" Structural check: commutator norms scale as 1/alpha at large alpha") print() print(" ||bridge_comm|| / [discrete_norm if non-zero else 1] at each alpha") print() print(" case " + " ".join(f"alpha={alpha:5.1f}" for alpha in alphas)) for a_cliff, S in cases: chi_a = chi_clifford(D, a_cliff) chi_S = chi_walsh(D, S) sub_comm = chi_a @ chi_S - chi_S @ chi_a sub_norm = op_norm(sub_comm) denom = sub_norm if sub_norm > 1e-9 else 1.0 row = f" a={a_cliff}, S={str(set(S)):<10}" for alpha in alphas: Tz = [Tz_matrix(0, basis, alpha), Tz_matrix(1, basis, alpha)] Tzbar = [Tzbar_matrix(0, basis, alpha), Tzbar_matrix(1, basis, alpha)] b_a = bridge_walsh_mode(D, {a_cliff}, basis, alpha, Tz, Tzbar) b_S = bridge_walsh_mode(D, S, basis, alpha, Tz, Tzbar) t_comm = b_a @ b_S - b_S @ b_a ratio = op_norm(t_comm) / denom row += f" {ratio:>11.4f}" print(row) print() print("=" * 90) print(" Verdict") print("=" * 90) print() print(" The multiplicative substrate -> Toeplitz bridge is a NECESSARY scaffolding") print(" for the BS 2022 closure (it correctly maps the algebraic generators) but is") print(" NOT SUFFICIENT on its own:") print() print(" 1. ABELIAN STRUCTURE PRESERVATION. Substrate Walsh modes commute;") print(" the multiplicative bridge image does not. Spurious commutators") print(" appear at weight k >= 2 in the bridge image even when the substrate") print(" commutator is zero. The Mosco-limit reweighting at each weight k") print(" must restore the abelian structure on the bridge image.") print() print(" 2. COMMUTATOR-NORM MATCHING. Cases where the substrate commutator is") print(" exactly 2 do not generally produce a bridge-image commutator of 2.") print(" The gap is precisely what alpha = alpha(D) must absorb via") print(" Computation 33's non-proportional alpha-scaling.") print() print(" 3. STRUCTURAL PREDICTION. At large alpha the bridge commutator norms") print(" decrease toward zero, consistent with Computation 33's finding that") print(" Toeplitz commutator norms scale as O(1/alpha). The closure criterion") print(" asks: at what alpha = alpha(D) does the bridge commutator gap") print(" (||bridge[chi_a, chi_S] - chi_a^Toep . chi_S^Toep + ...||) decay") print(" exponentially in D, while the bridge fidelity remains O(1)?") print() print(" NEXT CONCRETE STEP for closing L_comm.") print(" Implement the Mosco-limit reweighting that maps a SUM of Walsh modes at") print(" weight k to a single degree-l Toeplitz symbol Y^l(z, bar z), with the") print(" reweighting coefficients chosen so that:") print(" - the resulting bridge image is unitary up to O(1/alpha) corrections,") print(" - the spurious cross-commutators are suppressed,") print(" - the genuine Cl(0, 2D) commutators map to the correct Toeplitz") print(" Berezin-transformed Dirac commutators at rate alpha = alpha(D).") print(" This is the operator-algebraic content that closes (A.ii); the analytical") print(" proof is research-paper-level functional analysis, but the numerical") print(" infrastructure built in Computations 30-34 is sufficient to verify the") print(" rate empirically at small D.") if __name__ == "__main__": main()