nachenberg/u1-defect-crossover

U(1) defect phase-crossover experiment

Numerical study of the smallest eigenvalue of the unnormalized scalar connection Laplacian of graphs carrying a single phased edge (open square grids, 2D torus, 3D box). Part of a multi-agent project on local frustration vs global spectral visibility.

New reader? Start with GUIDE.md — a plain-language walkthrough (no spectral-graph-theory background needed). Then REPORT.md (v3.1) for the full results with [T]/[A]/[N]/[C]/[O] evidence labels. Validation: VALIDATION_REPORT.md. History: CHANGELOG.md; immutable prior states under provenance/ and v1/.

Version: v3.1 (2026-10-05). Cite this state via CITATION.cff; the exact immutable commit is the one introducing CITATION.cff: cc214cd6ac3a7fb373a525d5620fa892e2704fc6.

TL;DR

• Green slopes G′_uu ~ c_x log n confirmed to ~1e-4: corner 2/π, flat 1/π, interior 1/(2π); ratio 4.002 : 2.000 : 1.
• Exact identity [T]: y⁻¹ = 1 + R_n(λ₀); the Green/effective-resistance form y⁻¹ ≈ 1 + R_eff′ + s_θ·G′_uv(n) is the identity plus a spectral-shift remainder (measured +0.0868 at n=128, θ=π) — phase enters only additively; the slope is phase-independent as a consequence.
• Fitted r* = R_eff′ + s_θ·β₀ is phase-dependent — not an identification of R_eff′ (plateau check: 0.3023473 vs 0.3023443).
• Z_n tail extrapolation strongly consistent with 2/π: c = 0.63660(5) (extrapolation, not direct asymptotic confirmation).
• 3D box: λ₀ = Θ(n⁻³) (n⁻² refuted); matched-n phase ratios cross the s_θ-ratio — phase visible in the leading constant; the Green-like quantity remains small over tested sizes, consistent with bounded Green behavior.
• Corner-zero explained [T]: U = diag(g)·T commutes with L_π iff θ=π; the +1 sector forces ψ(corner)=0 exactly; ground state in the +1 sector for all tested n=3…128 [N].
• Refuge bound holds 18/18; torus limit unresolved at n≤192.
• Next experiment (PREREGISTERED_EXPERIMENT.md): matched-environment sweep — registered, not run; the pure-tree arm was replaced by a cycle-containing tree-like environment (no cycle ⇒ no holonomy ⇒ nothing to test).

Contents

• GUIDE.md, REPORT.md, PREREGISTERED_EXPERIMENT.md, CHANGELOG.md, CITATION.cff
• u1_experiment_v2.py — main experiment script (validate|sweep2d|green|refuge|torus|box3d|plots|upload); audit/ — audit scripts
• measurements.csv, green.csv, torus.csv, box3d.csv, refuge.csv, groundstates.npz (the *_rows.json files are the raw row dumps behind each CSV)
• sector.csv — θ=π symmetry/sector check; zn_recomputed.csv, box3d_matched.csv, constants.json — audit-derived metrics; preflight_environment_check.json — structural preflight for the registered experiment
• summary.json, validation.json, env.json
• plots/fig1…fig8 — figures
• v1/ — round-1 run (superseded); provenance/REPORT_v2.md, provenance/REPORT_v3.md — verbatim report history

Reproduce: pip install numpy scipy pyamg matplotlib huggingface_hub then U1_OUT=out python u1_experiment_v2.py validate,sweep2d,green,refuge,plots,upload and validate,torus,box3d,upload; audit pass: python audit/audit_main.py derived,sector,reorg. All CPU; ≈$0.05 total compute so far.