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n int64 6 48 | N int64 216 111k | theta float64 0.79 3.14 | s_theta float64 0.59 4 | lambda0 float64 0 0.01 | n3lam0 float64 0.18 2.07 | slope_data float64 -3.36 -2.73 ⌀ | Rn_lambda0 float64 0.93 2.19 | t float64 0 0.01 |
|---|---|---|---|---|---|---|---|---|
6 | 216 | 3.141593 | 4 | 0.009601 | 2.073914 | null | 0.92872 | 0.000558 |
8 | 512 | 3.141593 | 4 | 0.00365 | 1.868929 | -3.36176 | 1.140263 | 0.000454 |
10 | 1,000 | 3.141593 | 4 | 0.001754 | 1.75433 | -3.283578 | 1.280073 | 0.000485 |
12 | 1,728 | 3.141593 | 4 | 0.000974 | 1.683114 | -3.227298 | 1.376547 | 0.000466 |
16 | 4,096 | 3.141593 | 4 | 0.000391 | 1.600565 | -3.174809 | 1.499118 | 0.000535 |
20 | 8,000 | 3.141593 | 4 | 0.000194 | 1.554693 | -3.130312 | 1.572855 | 0.000872 |
24 | 13,824 | 3.141593 | 4 | 0.00011 | 1.52572 | -3.103177 | 1.621712 | 0.000885 |
32 | 32,768 | 3.141593 | 4 | 0.000046 | 1.491426 | -3.079025 | 1.681997 | 0.001818 |
48 | 110,592 | 3.141593 | 4 | 0.000013 | 1.459475 | -3.05341 | 1.740712 | 0.005895 |
6 | 216 | 0.785398 | 0.585786 | 0.000849 | 0.18343 | null | 2.193518 | 0.000441 |
8 | 512 | 0.785398 | 0.585786 | 0.000387 | 0.198167 | -2.731369 | 1.956017 | 0.000532 |
10 | 1,000 | 0.785398 | 0.585786 | 0.000204 | 0.203851 | -2.873279 | 1.873601 | 0.000611 |
12 | 1,728 | 0.785398 | 0.585786 | 0.00012 | 0.207157 | -2.91176 | 1.82774 | 0.00065 |
16 | 4,096 | 0.785398 | 0.585786 | 0.000052 | 0.210958 | -2.936806 | 1.776797 | 0.000881 |
20 | 8,000 | 0.785398 | 0.585786 | 0.000027 | 0.213103 | -2.954655 | 1.748842 | 0.001223 |
24 | 13,824 | 0.785398 | 0.585786 | 0.000016 | 0.214487 | -2.964483 | 1.731099 | 0.00157 |
32 | 32,768 | 0.785398 | 0.585786 | 0.000007 | 0.216175 | -2.972761 | 1.709781 | 0.002917 |
48 | 110,592 | 0.785398 | 0.585786 | 0.000002 | 0.217825 | -2.981245 | 1.689252 | 0.009744 |
U(1) defect phase-crossover experiment
Numerical study of the smallest eigenvalue of the unnormalized scalar connection Laplacian of an n×n open square grid with exactly one phased edge (plus 2D torus and 3D box extensions). Part of a multi-agent project on local frustration vs global spectral visibility.
Start here: REPORT.md (v3, post-audit) — results with [T]/[A]/[N]/[C]/[O] evidence labels and answers to the eight central questions. Pre-correction report: provenance/REPORT_v2.md. Validation: VALIDATION_REPORT.md.
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 chain [T]: 1/(Nλ₀) = G′_uv(n) + (1+R_eff′)/s_θ (up to the O(λ₀) shift) — phase enters only additively; the slope is phase-independent as a consequence.
- Exact collapse: y⁻¹ = 1 + R_eff′ + s_θ·G′_uv(n). The 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 extrapolation: c = 0.63660(5) ≈ 2/π (O(1/log n) correction fit, corner θ=π).
- 3D box: λ₀ = Θ(n⁻³) (n⁻² refuted); matched-n phase ratios cross the s_θ-ratio — phase visible in the leading constant (transient geometry).
- Corner-zero explained [T]: U = diag(g)·T commutes with L_π iff θ=π; +1 sector forces ψ(corner)=0 exactly; ground state in +1 sector for all tested n=3…128 [N].
- Refuge bound holds 18/18; torus limit unresolved at n≤192.
Contents
u1_experiment_v2.py— main experiment script (validate|sweep2d|green|refuge|torus|box3d|plots|upload)audit/— audit-fix scripts (derived | sector | reorg | report stages)measurements.csv,green.csv,torus.csv,box3d.csv,refuge.csv,groundstates.npz(the*_rows.jsonfiles 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 metricssummary.json,validation.json,env.jsonplots/fig1…fig8— figuresv1/— round-1 run kept for provenance (superseded);provenance/REPORT_v2.md— pre-audit report
Reproduce: U1_OUT=out python u1_experiment_v2.py validate,sweep2d,green,refuge,plots,upload then validate,torus,box3d,upload; audit pass: python audit/audit_main.py derived,sector,reorg (deps: numpy 2.5.3, scipy 1.18.1, pyamg 5.3.0, matplotlib; all CPU).
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