🔗 #11 Conditional MI I(f;Y|S) GROUNDED

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.

8 / 8

gate-groups pass

0.00

AR(1) null FPR (both seeds)

1.0

power @ dim(S) 1·3·5

2 seeds

both 8/8

Preflight (resource-only): load1 0.17 · 40 GiB available · si/so ~0 · ClickHouse active, readonly=2. 5c/5G/no-swap capped; single-thread BLAS. Slice staged read-only at /tmp/i11_slice.parquet.

The complete battery

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 AR(1) hardening — the load-bearing new result

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.

High-dimensional-S (curse-of-dim FN). With the interaction conditioned on a 3- and 5-dimensional S (7-dim joint at N=1800), power stays 1.0 — the admit-strength interaction survives moderate-dim conditioning. Published caveat: measured for the strong exemplar; a weaker interaction at high dim degrades on the N-vs-dim frontier → route to #12/#13.

The published blind spot — imperfect conditioning

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).

Conditional axis opened. #11 is the instrument that catches the interaction-only signals the marginal instruments (#4/#5/#7/#19) flagged FNR=1 and routed here — grounded on a genuine sign-XOR archetype (marginal MI 0.04, CMI 0.31). Next: #12 knockoffs → #13 DML. 13 / 20 instruments grounded.