Character Saturation and Spectral Factorization of Lattice Dirac Operators

Computational Repository Companion code and data for: Spectral Factorization of Lattice Dirac Operators Companion Computations: Holonomic Structure, Monodromy, and Sin-Cos Duality Requirements Python 3.8+ numpy, scipy, mpmath Optional: matplotlib (for figure generation) Install: pip install numpy scipy mpmath matplotlib Repository Structure Each computation has its own directory containing src/ (scripts) and results/ (output).Scripts are self-contained and can be run independently. Directory Paper Reference Description 01_moment_factorization/ Main, Thm 2 + App B Exact rational moments via beta-integral factorization 02_hadamard_embedding/ Main, Thm 3 W_BCC(k) = W_4D(M^T k) verification 03_spectral_entropy/ Main, Table 2 Spectral entropy comparison: SC, BCC, FCC 04_log_X_pslq/ Main, Sec V; Comp, Sec IV 87-digit computation + PSLQ identification 05_holonomic_ode/ Comp, Sec VIII.A Holonomic recurrence search + 4th-order ODE extraction 06_monodromy_transcendence/ Main, Thm 7; Comp, Sec VIII.C Monodromy computation, transcendence proof, irreducibility 07_sin_cos_duality/ Comp, Sec VII Sin-cos involution, isospectrality, correlation 08_diamond_resolvent/ Comp, Sec VIII.D-E Diamond ODE, monodromy, scalar comparison 09_converse_characterization/ Comp, Sec I Lattice catalog: saturation vs non-saturation 10_quantum_walk_isotropy/ Comp, Sec II Quartic anisotropy ratio for BCC, SC, FCC 11_large_d_asymptotics/ Comp, Sec III + V Large-d spectral limit + anti-concentration 12_mahler_measure/ Comp, Sec IV Mahler measure characterization of 13_subgroup_hierarchy/ Comp, Sec VI Subgroup factorization hierarchy in (Z/2Z)^d 14_frobenius_analysis/ Comp, Sec VIII.B Frobenius compatibility at singular points data/ Both papers Exact rational moments (601 terms, JSON) figures/ Comp, Fig 1 Generated figures Shared Data data/ contains exact rational moments as JSON files: dirac_moments.json -- for n = 0, ..., 600 (Dirac dispersion) scalar_moments.json -- for n = 0, ..., 600 (gauge Laplacian) diamond_moments.json -- for n = 0, ..., 600 (diamond operator) Moments are stored as exact fractions "numerator/denominator". Key Results Quantity Value Script 6 (exact) 01_moment_factorization/ 99/2 (exact) 01_moment_factorization/ 1.41371520...55 (87 digits) 04_log_X_pslq/ Dirac ODE order 4 (minimal) 05_holonomic_ode/ tr(M_1 M_2) 16 (exact integer) 06_monodromy_transcendence/ Corr(X, R) -5/9 07_sin_cos_duality/ Diamond ODE order 3 08_diamond_resolvent/ Computation Dependency Graph 01_moment_factorization (exact moments) | +-> 05_holonomic_ode (recurrence + ODE) | | | +-> 06_monodromy_transcendence (Theorem 7) | | | +-> 14_frobenius_analysis (local structure) | +-> 04_log_X_pslq (87-digit constant) | | | +-> 12_mahler_measure (Laurent polynomial) | +-> 08_diamond_resolvent (diamond ODE + comparison) | +-> 07_sin_cos_duality (involution theorem) 02_hadamard_embedding (independent) 03_spectral_entropy (independent) 09_converse_characterization (independent) 10_quantum_walk_isotropy (independent) 11_large_d_asymptotics (independent) 13_subgroup_hierarchy (independent) Reproducing Results Each script is self-contained. To reproduce a specific computation: cd 06_monodromy_transcendence/ python src/monodromy_multiprecision.py Most scripts accept --Nk to control grid resolution (default 64 or 128).High-precision computations use mpmath with configurable mp.dps. Notation X(k) = |F(k)|^2: Dirac dispersion, spectrum [0, 16] W(k) = 8[1 - cos k_1 cos k_2 cos k_3]: gauge Laplacian, spectrum [0, 16] X_D(k) = X(k) + R(k): diamond operator R(k): sin-cos dual of X(k), same spectral moments F_mu(k) = 4 sin(k_mu) prod_{nu != mu} cos(k_nu): BCC Dirac hopping n = 8: BCC coordination number d = 3: spatial dimension