Fix the multivariate battery's false-alarm rate (1.7% -> 0.1%)

[tprest]

Found by asking "does the multivariate battery behave well?" and measuring it — the same method that surfaced #7, #8 and #9.

The symptom

On 60 replicates of correct integer data (iid discrete Gaussian, like real Falcon signatures), the extended battery rejected 1.7 % of the time against a nominal α = 0.1 % — a correct sampler flagged ~17× too often. Three distinct causes.

1. squared_norm — continuous χ² on lattice data

It compared ‖x‖²/σ² to a continuous χ²(dim). For integer data ‖x‖² lives on a lattice of step 1, which a KS test resolves at small σ (20 % of p-values < 0.05 on correct data). The moments are not the issue — the discrete and continuous ones match to ~1e-21 here — the lattice is.

Fix: a binned χ² against the exact pmf of ‖x‖² (the dim-fold convolution of X²'s pmf, via FFT). That null is analytic, so it stays calibrated and keeps power — a Monte-Carlo calibration would floor at 1/(B+1) > α and silently stop rejecting (I tried it; it did exactly that).

2. henze_zirkler — never actually worked

Two independent bugs:

Bug	Effect
term1 (a double sum) divided by n instead of n²	Statistic ~n× too large, became a monotone function of mean pairwise distance → decreased for heavy tails
smoothing β = 1/(2p) instead of the standard 1/√2	bandwidth ~1/p (0.048 vs 1.08 at p=16) → kernel too flat to see shape

Before the fix, a t(2) sample scored p = 1.00 and a correct Gaussian p = 0.38 — the test was inverted and powerless. After: correct Gaussian/discrete ~0.6; uniform / t(2) / bimodal all at the MC floor.

3. henze_zirkler — naive MC p-value

np.mean(null >= obs) can be exactly 0 (below any α) ~1/(mc_B+1) of the time. Switched to (1+#)/(B+1) like the rest of the suite. Its floor is then 1/(mc_B+1) ≫ α, so it can no longer single-handedly reject — it is excluded from the Bonferroni gate and contributes through the Fisher pool instead. (Real rejection power would need mc_B ~ 10/α replicates, each O(n²) — out of scope, deliberately.)

The meta-test that was missing

Added multivariate_pvalue_uniformity() in calibration.py — the multivariate counterpart of pvalue_uniformity(), measuring exactly this false-alarm rate. It did not exist, which is why all of the above went unnoticed.

Results

	before	after
false-alarm rate (60 correct datasets, same seeds)	1.7 %	0.0 %
squared_norm mean p / frac<0.05	0.27 / 0.13	0.51 / 0.08
HZ on t(2) heavy tails	p = 1.00	p = 0.0099
correlated pair / FFT-tree zeroing / σ+5%	caught	still caught

Four tests added, locking squared_norm's calibration + power + null routing and HZ's p-value floor. Suite: 157 passed, 1 skipped.

Behaviour changes for reviewers

• squared_norm_test now returns a "null" key (exact-discrete-chi2 or chi2-analytic).
• The Bonferroni gate divides by 5 components instead of 6 (HZ excluded), a slightly more permissive per-component threshold.
• henze_zirkler_test can no longer return p = 0.

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