Summary
This page records the actual numbers. Every framework experiment writes a JSONL metadata file with full solver parameters and three brake-exponent estimators (polyfit, GLS-AR(1), Bayesian-AR(1)), plus a TXT human-summary file. The tables below are direct extracts from validation/catalogue.md.
KPZ universality (1+1D) — the cleanest cross-shadow call
| Quantity | Median across 3 shadows | \(\sigma_{\text{cross}}\) | Target | Error |
|---|
| \(\beta\) (growth exponent) | 0.316 | 0.004 | 1/3 ≈ 0.333 | 5.1% |
| \(\alpha\) (roughness) | 0.505 | 0.025 | 1/2 = 0.500 | 1.0% |
| \(z = \alpha/\beta\) | 1.64 | 0.13 | 3/2 = 1.500 | 9.3% |
3 microscopic models (Ballistic Deposition, Restricted Solid-on-Solid, Corner Growth/discrete-TASEP), \(L \in \{256, 1024, 2048\}\), 64 realisations per cell. Tracy–Widom GOE skewness sign matches. scripts.
Wigner–Dyson RMT universality — per-shadow brakes
| Entry distribution | \(\beta\) |
|---|
| Gaussian | 0.891 |
| Bernoulli ±1 | 0.892 |
| Uniform | 0.901 |
| Exponential-centred | 0.899 |
| Student-t (df=5) | 0.884 |
| Median | 0.892 |
| \(\sigma_{\text{cross}}\) | 0.006 |
| \(\sigma_{\text{within}}\) (any single distribution) | 0.05–0.12 |
Cross-shadow KS distance to Wigner-GOE surmise = 0.0064. \(\sigma_{\text{cross}}\) is 5% of within-shadow noise — universality call.
Ricci-flow Type-I/Type-II classification
| Profile | \(\beta\) | Class |
|---|
| A — symmetric dumbbell | 1.083 | Type-I (target 1.0) |
| B — asymmetric dumbbell | 1.086 | Type-I (target 1.0) |
| C — cap-flat degenerate | 1.164 | Type-II (β > 1) |
Within-Type-I \(\sigma = 0.0015\); Type-II gap = +0.080 — gap exceeds within-class noise by 50×. Recovers Hamilton–Perelman Type-I/Type-II classification.
3D Ising critical exponents (Wolff cluster MC)
| Quantity | Framework | Target | Error |
|---|
| \(T_c\) | 4.51235 ± 0.0066 | 4.51153 | 0.02% |
| \(\beta_{\text{mag}}\) | 0.318 ± 0.021 | 0.327 | 2.8% |
| \(\gamma/\nu\) | 1.93 ± 0.04 | 1.96 | 1.7% |
Fluid turbulence — cross-shadow brake-principle (~190 cells)
| Shadow class | Cells | \(\beta\) summary |
|---|
| 3D NS-Galerkin DNS | 30 | 28/30 with \(\beta < 1\); 2 outliers traced to under-resolution |
| Burgers 1D | 21 | 21/21 (100%); brake \(1-\beta\) saturates 0.15 for \(\mathrm{Re} \ge 10^3\) |
| MHD 1D | 24 | 24/24 (100%) |
| Klein–Gordon nonlinear | 9 | 9/9 (100%) |
| NLS-1D (integrable) | 9 | \(\beta \approx 1.03\); legitimately no brake |
| Maxwell linear | 9 | \(\beta \approx 1.02\); legitimately no brake |
| Sabra shell | 24 | \(\mathrm{Re} = 10^3\)–\(10^{10}\); brake \(\approx 0.45 \pm 0.05\) across 8 orders of magnitude |
| EDQNM closure | 21 | 21/21 (100%) |
| LES Smagorinsky | 12 | \(\beta \approx 1.02\); engineering tool lacks brake (structural defect) |
Climate — CO\(_2\) multi-cadence (instance 23)
| Cadence | \(\beta\) | \(R^2\) | Notes |
|---|
| Modern (Mauna Loa, 1958–2024) | +4.04 | 0.68 | Anthropogenic, strong fit |
| Holocene (200 yr – 12 ka) | −2.54 | 0.03 | Fails Theorem 3 gate (log-range 0.15) |
| Vostok (0–420 ka) | +3.00 | 0.09 | Natural paleo |
| EPICA (0–800 ka) | +2.96 | 0.08 | Natural paleo |
Headline finding
Within-paleo cross-cadence \(\sigma = 0.024\) (natural CO\(_2\) scale-invariant millennial → 100-millennial). \(|\beta_{\text{anth}} - \beta_{\text{nat}}| = 2.13\), separation 5.7σ. Once Milankovitch cycles peeled by EMD, natural-CO\(_2\) trend brake \(\beta = -0.24 \pm 0.45\) (below threshold). Framework reads anthropogenic CO\(_2\) as a distinct cascade class.
Climate — CMIP6 tipping cross-shadow (instance 16)
| Element | Models | \(\beta\) median | \(\sigma_{\text{cross}}\) | Direction | Verdict |
|---|
| Arctic sea-ice | 21/21 | −0.341 | 0.694 | 1.000 (decline) | robust |
| AMOC | 19/19 | +0.309 | 1.545 | 1.000 (decline) | partially-robust |
| Amazon NPP | 25/25 | +1.717 | 1.874 | 0.800 | inconclusive |
Gravitational waves
| Event | Significance | Time error | Method |
|---|
| GW150914 | +23.6σ combined | 37 ms | cross-detector consensus |
| GW151226 | +3.7σ combined | 25 ms | cross-detector consensus |
| GW190521 | +5.76σ | — | consensus_signal (real heavy BBH) |
| GW190426_190642 | — | — | consensus_noise (matches LIGO demotion) |
NANOGrav 15-yr: \(\gamma = 4.11 \pm 0.39\), within 1σ of SMBH-binary expectation 13/3 ≈ 4.33; alternatives (cosmic strings γ ≈ 2, primordial γ ≈ 5, phase transition γ ≈ 7) excluded at > 2σ.
Seismology — Tohoku 2011 aftershocks
USGS catalog, 1716 \(M \ge 4.5\) events over 14 days post-mainshock. Omori–Utsu fit \(n(t) = K/(t + c)^p\) gives \(p = 1.184\), \(R^2 = 0.916\). Literature range 1.0–1.3 (well-matched). Theorem 1 reading: \(\beta > 1\) is Type-II slow asymptotic decay — matches the physical reality of long aftershock tails.
Nuclear — \(E = mc^2\) across AME2020
| Metric | Value |
|---|
| Isotopes evaluated | 2,545 (\(A \ge 4\)) |
| Median residual | \(3 \times 10^{-9}\) |
| Maximum residual | \(1 \times 10^{-7}\) |
Caveat reported: AME tabulates BE from mass via \(E = mc^2\); the test is internally consistent at \(10^{-8}\), not independent confirmation. Honest-scope statement publicised through \(\mathscr{A}\).
Fast Radio Bursts
| Quantity | Value |
|---|
| Repeaters analysed (CHIME 2023, \(N \ge 5\) bursts) | 21 |
| Total bursts | 326 |
| FRB 20180916B period recovered | 16.33 d (0.01% error) |
| Monte-Carlo \(p\)-value | 0.005 |
| Repeaters clearing both gates | 1/21 (the only published periodic CHIME repeater) |