iter 1 — ξ pinned to Chatterjee's own implementation FIXED

Attack item R-3 · 2026-07-29 · harness findings/evolution/audits/2026-07-29-probe-hardening-loop/harness/xi_oracle_parity.py · gate mise run xi:oracle / /xi:oracle

Why this went first

Every other kernel in this repo is pinned to a canonical FOSS oracle at ≤1e-9 — bartels, cecp, coxstuart, hoeffding, lkurt, recurr, signflux. chatterjee_xi, the rotation cascade's now-binding stage, was the only unpinned statistic. Nothing downstream can be trusted while the estimand itself is unpinned, so this had to precede every measurement.

Prediction, written before running

Expected a real divergence under ties. The prior session had measured 2e-3 between the repo and xicorpy and hypothesised a tie-formula disagreement — the repo always using the general l_i denominator, the reference switching to the simplified n²−1 form when it detects no ties.

That prediction was wrong, and the reason matters.

Method — two legs, because ξ is randomized

XICOR breaks X-ties at random (rank(xvec, ties.method="random")) and exposes no seed argument. A pointwise comparison is therefore only meaningful when there are no X-ties.

Y-tie fractions are swept 0→50% deliberately: that band is precisely where the simplified and general forms diverge, and the hypothesised 2e-3 gap was attributed to ~10% Y-ties.

Result — leg 1: exact

case                                y_ties          repo         XICOR     |diff|
L1_independent_y000                  0.000  -0.011568753  -0.011568753   2.44e-15
L1_independent_y100                  0.100  -0.022417659  -0.022417659   4.44e-16
L1_independent_y309                  0.309  -0.010649721  -0.010649721   8.88e-16
L1_independent_y500                  0.500   0.022020022   0.022020022   1.55e-15
L1_quadratic_y309                    0.309   0.601991216   0.601991216   2.22e-16
L1_monotone_y309                     0.309   0.951658716   0.951658716   0.00e+00
...  15 cases, max |diff| = 2.44e-15

Floating-point exact — six orders of magnitude tighter than the 1e-9 gate, at every Y-tie fraction including the 30.9% that matches the worst real cell.

Result — leg 2: consistent

case                          x_ties        repo  XICOR mean       sd       z    p_two  verdict
L2_independent_x100            0.100   -0.039642   -0.040459  0.00459    0.18   0.8486  ok
L2_independent_x500            0.500    0.003532   -0.009294  0.00883    1.45   0.1619  ok
L2_quadratic_x100              0.100    0.644543    0.649344  0.00176   -2.73   0.0060  ok
L2_quadratic_x500              0.500    0.626789    0.626989  0.00329   -0.06   0.9475  ok

What this establishes

1. The repo's ξ is canonical. It agrees with Chatterjee's own package to floating-point precision across the whole tie range. The rank conventions match exactly: rank(x, ties.method="random") for X, ties.method="max" on ±Y.
2. The prior 2e-3 divergence is refuted as diagnosed. It was measured against xicorpy, a third-party package — not against canon. XICOR never branches on ties, so the tie-formula hypothesis had nothing to attach to. xicorpy is additionally unsafe here: its tie detector tests duplicate (x,y) row pairs, not tied values within a column, so a 30.9%-tied column with distinct pairs reports "no ties" and proceeds silently.
3. The binding stage is now gated — /xi:oracle, /xi:doctor, /xi:check-full, following the repo's established oracle-namespace pattern.

The harness produced a false positive on itself, and that is recorded

The first version of leg 2 asked whether the repo value fell inside XICOR's observed [min, max] over 200 draws. It flagged L2_quadratic_x100 for sitting 3.1e-5 below the observed minimum at z = −2.63. That is a support test built on an order statistic, and min/max over 200 draws is a poor support estimator — the "failure" was an ordinary draw.

Replaced with the two-sided smoothed empirical p-value, which is the construction the research flagged XICOR's own permutation branch for omitting (its mean(rp > xi) is unsmoothed, so not conservative). Recorded rather than quietly corrected, because a gate that has produced a false positive should carry that history.

One non-central case — evidence for R-2, not a parity defect

L2_quadratic_x100 clears the adjusted threshold (p=0.0060 vs 0.00167) but is the one case that is not comfortably central, and z held at −2.73 when B went from 200 to 2000. The honest reading: the repo's deterministic hash tie-break is one specific rearrangement, not a random draw, so there is no reason for it to be distributed as a uniform draw from the rearrangement distribution. A systematically slightly-low value is expected. That is exactly the estimand mismatch Chatterjee's own remedy addresses:

"Alternatively, one can consider taking the average of ξₙ over all possible increasing rearrangements of the Xᵢ's."
— arXiv:1909.10140, remark (8)

So this case is evidence for R-2 (de-randomisation), and it is carried forward as such.

Carried forward

Evidence: harness/xi_oracle_parity_evidence.json (21 cases, full numbers). Reproduce: mise run xi:oracle.