T4: regulation rates from spontaneous fluctuations
Synthetic, dimensionless two-sensor neuron (n = 3, k = 2) with an Ornstein-Uhlenbeck input current; no organism result. Proposal T4 in spec/13-proposals.md.
Gates: 15/15 passed.
256 replicas × 1500 time units observed per rate, sampled every τ_mix = 1.0; rates from lagged covariances at lags (5, 15).
| Rate scale c | ε | Var(z) on the slice (trace) ± SE | Linear theory | True loop-gain eigenvalues | From fluctuations | From fluctuations + observation noise | From recovery |
|---|---|---|---|---|---|---|---|
| 0.02 | 0.0339 | 0.000376 ± 3e-06 | 0.000379 | (0.00639, 0.0339) | (0.00673, 0.0329) | (0.00746, 0.033) | (0.00629, 0.0336) |
| 0.04 | 0.0677 | 0.000742 ± 7e-06 | 0.000737 | (0.0128, 0.0677) | (0.0125, 0.0674) | (0.0121, 0.0684) | (0.0126, 0.0672) |
| 0.08 | 0.135 | 0.00139 ± 1.04e-05 | 0.0014 | (0.0256, 0.135) | (0.0261, 0.131) | (0.0265, 0.127) | (0.0251, 0.135) |
| Measurement | Result |
|---|---|
| Non-scalar C: mean difference / largest covariance difference (SE units) | 1.72e-11 / 54.9 |
| Hidden cascade (τ_c = 78.2): lag dependence of the rate estimate, direct / cascade | 0.0591 / 0.768 |
| Gate | Passed |
|---|---|
| regulation slow relative to activity (epsilon < 0.15 at every rate) | yes |
| stationary mean independent of rate (< 0.1 of spread) | yes |
| spread grows with rate: 4x rate gives > 3x variance | yes |
| variance ratio matches linear theory (within 4 SE) | yes |
| stationary covariance matches Lyapunov prediction (< 10%) | yes |
| conserved quantities do not fluctuate | yes |
| fluctuation rates match true rates (< 10%) | yes |
| fluctuation rates within 3 SE of true rates | yes |
| recovery rates match true rates (< 3%) | yes |
| Onsager: fluctuation and recovery rates agree (< 10%) | yes |
| observation noise as large as the fluctuations: rates within 3 SE of true | yes |
| non-scalar C: equal means | yes |
| non-scalar C: covariances differ (> 5 SE) | yes |
| direct observation: estimate independent of lag (< 10%) | yes |
| hidden cascade: estimate depends on lag (> 2x the direct dependence) | yes |
Seed 5. Full values: report.json.