iteration 21 · 2026-07-23 · conditional axis · laptop-drives-bigblack
Does the feature carry forward information beyond the shipped set S — including interaction-only signal with zero marginal IC? The instrument rank-IC, IC-decay, MDA and quantile-monotonicity are all structurally blind to. The complete 8/8 gate battery (all four nulls) passes, robust across 2 seeds — the iter-20 bounded slice's deferred Harden gates (iid + AR(1) FPR, high-dimensional-S, imperfect-S envelope) all clear. Supersedes iter 20.
readonly=2. 5c/5G/no-swap capped; single-thread BLAS. Slice staged read-only at /tmp/i11_slice.parquet.| Gate group (§7 row 11) | Result (seeds 20260723 / 11) | Target | |
|---|---|---|---|
| Gaussian analytic recovery | max-err 0.010 / 0.005 nats | ≤ .02 nats | PASS |
| Admit (interaction-only) | CMI 0.312 / 0.351, z 27.5 / 33.4 · marginal MI 0.041 / 0.049 ≈ 0 | p≤α, z≥3, δ≥min · marg≈0 | PASS |
| XOR / interaction power | 1.00 / 1.00 | ≥ .8 | PASS |
| Nulls FPR — all 4 | block-perm 0.000/0.025 · common-cause 0.000/0.000 · iid 0.000/0.000 · AR(1) 0.000/0.000 | ≤ α each | PASS |
| Substitution (f≈S) | not-sig 1.00 / 1.00 | ≥ .95 | PASS |
| Harden · high-dim-S FN | power 1.0 / 1.0 at dim(S) 1·3·5 | ≥ .8 at dim 3 | PASS |
| Imperfect-S envelope (blind spot) | 0.0→0.0→1.0 / 0.0→0.1→0.97, monotone | perfect S ≤ α + monotone | PASS |
The standard concern with a local-permutation null on time-series data (Runge 2018) is anti-conservatism: autocorrelation in f could bias the KSG CMI upward relative to a permutation that destroys it → inflated FPR. The AR(1) null tests it directly — a real feature circularly shifted by a large offset preserves the full autocorrelation, keeps the real marginal, and is decorrelated from Y.
FPR = 0.000 on both seeds. The concern does not materialise: when f ⊥ (Y,S) there is no dependence for the estimator to over-state, and the local null reproduces the ~0 CMI distribution. A feature's own autocorrelation, unrelated to Y, does not fool CMI. Together with the iid null (also 0.000), all four §7 nulls now clear.
CMI's guarantee is conditional on S: it grounds "f adds information beyond S as measured", not "beyond the true latent Z". When S captures the common cause Z only partially — S = gaussianise(Z + c·noise) — the residual-confounding path is a real conditional dependence, so CMI (correctly, by definition) reports I(f;Y|S) > 0:
| S quality | corr(S,Z) | FPR (seed 20260723 / 11) |
|---|---|---|
| perfect (c=0) | 1.00 | 0.00 / 0.00 |
| noisy (c=0.5) | ≈0.89 | 0.00 / 0.10 |
| heavily-noisy (c=1.0) | ≈0.71 | 1.00 / 0.97 |
This is not a defect — it is CMI's fundamental limit surfaced honestly: the user must supply a conditioning set that adequately captures confounders. Perfect and near-perfect S control at ≤α; a heavily-degraded S leaks, monotonically. Routing: imperfect-conditioning cases → #12 knockoffs (model-X FDR without perfect conditioning) / #13 DML (orthogonalisation).