Spectral Fidelity of the Discrete Witten Laplacian on S¹: Discretization Artifacts, Structure-Preserving Schemes, and Bayesian Parameter Recovery from Spectral Data

# Spectral Fidelity of the Discrete Witten Laplacian on S¹ — v2 **Paper:** Spectral Fidelity of the Discrete Witten Laplacian on S¹: Discretization Artifacts, Structure-Preserving Schemes, and Bayesian Parameter Recovery from Spectral Data **Authors:** Frédéric David Blum, Claude, Catalyst AI **Version:** v2 (with Phase III corrections from multi-AI peer review) ## Key Results ### Semi-Positivity (Phase I) - Standard 3-pt stencil: λ₀ < 0 in 100% of 96 configurations - Factored d_W†·d_W: λ₀ ≥ 0 in 98% (2% at machine ε) - FEEC: λ₀ ≈ 10⁻¹⁶ at all resolutions ### Phase-Shift Isospectrality (Phase III) - At N=64, δ=0.37·dx: matrix changes 0.48%, spectrum stable at 3.7×10⁻¹⁵ - Explained by DFT fractional shift conjugation (unitary, exact for all δ) - **Negative controls:** Dirichlet BC → spectrum shifts 10⁻⁴; non-uniform grid → shifts 15-130% - Same perturbation, 15 orders of magnitude different response ### Bayesian E_J,c Inference - Linear model: r = 0.9996, RMSE = 0.07 MHz (proof-of-concept) - Nonlinear model: fails (r = NaN, 219% error) — honest assessment included ### Phase III Fixes (from GPT-5.2 Think review) - Fix 1: Dense vs sparse solver comparison — gap behavior confirmed as physics - Fix 2: Three-metric verification protocol — isospectrality confirmed genuine - Fix 3: Nonlinear Bayesian — failure honestly documented ## Files - `spectral_fidelity_paper_v2.docx` — Full paper - `code/` — All Python scripts (Phase I, II, III, negative controls) - `figures/` — All figures including negative controls - `data/` — Phase III results (JSON) ## Reproducibility All results reproducible in < 120 seconds on standard hardware. ```bash pip install numpy scipy matplotlib python code/witten_s1_test.py # Phase I python code/witten_s1_phase2.py # Phase II python code/witten_s1_phase3.py # Phase III python code/negative_control.py # Negative controls ```