Prospectively fixed moderate-imbalance extension, version 1
Status: local pre-execution specification after all v0.6 results and critiques were known. This is not external preregistration or a claim that the full paper is confirmatory. The geometries were used earlier; only these independently seeded new catalogues and procedures are prospectively evaluated. Timing and reference-equivalence checks, and population-truth integrations, precede freezing; no extension estimator outcomes have been generated before freeze.

Questions: How do ordering probability and comparison error change under moderate sample-size imbalance? Does matching at the smaller size help, and what is the absolute-estimation cost? Does subtracting the permutation mean materially change either outcome?

Scope: L1 (1,1 Gaussian pair) and Q1x (4,3 blocks with one quadratic pair and independent extra coordinates), using the identical validated generators from the unchanged archive. Total population information targets are 0.4, 0.5 and 0.6 nats. New 0.5 settings and existing settings pass independent FFT and adaptive real-space integration at tolerance 1e-8 nats. The numerical table is recorded separately. These two deliberately contrasting geometries do not establish generalisation to arbitrary dependence structures.

Sample sizes: A has 100 or 200 observations. B has ratio 1,1.25,1.5,2 or3 times the A size (100,125,150,200,300; and200,250,300,400,600). Stream A and B use independent 600-row parents; each evaluated size is a nested prefix. For each design,target and stream there are601 independent parents. Target comparisons (IA,IB) are (.4,.4),(.5,.5),(.6,.6),(.4,.5),(.5,.4),(.4,.6),(.6,.4). Thus nonzero differences .1/.2 are tested with the stronger dependence in either size position, along with equal-information controls. Reusing a bank across comparisons does not create additional independent replicates. No selective cell reporting.

Estimation: unchanged exact validated fast wrapper, k=4, B=100 whole-row permutations per dataset, standardisation and strict maximum-norm neighbour counts inherited unchanged. Return KSG raw/centred, Gaussian-copula raw/centred/analytic. Preserve negative scores. The analytic GC correction remains only exactly unbiased for ordinary Gaussian covariance, not necessarily after empirical rank normalisation. Preprocessing and permutations are recalculated independently for each dataset or subset.

Min-N matching: at equal sizes reuse baseline estimates. At each unequal comparison retain A in full. Independently draw five row subsets of B of size NA, without replacement within a subset; subsets may overlap. Average the five subset scores before subtracting A. Identical subsets and permutations supply all estimators. A five-subset average is one parent-level result, not five independent replicates. Subset identities, scores and permutation MCSE are retained.

Outcomes: for every design, sample-size cell, truth comparison and estimator report mean estimated difference IB-IA, signed bias, comparison RMSE, absolute RMSE separately for each side, and directional preference P(estimated difference>0). At nonzero truth differences report correct ordering with pointwise Wilson95% intervals and paired matching-minus-baseline improvement in probability; provide paired parent-level percentile-bootstrap95% interval and exact McNemar test from improved/worsened replicate counts. At zero true difference no correct ordering is defined; report directional preference and difference error. Absolute calibration includes GC model approximation error. Report raw-minus-centred corrections and replicate-level spread as well as outcome differences. Paired interval resamples independent replicate IDs, never raw observations or individual overlapping subsets.

Analysis bootstrap: 2000 bootstrap draws, fixed independent analysis streams keyed by geometry,truth comparison,anchor,ratio,estimator,metric. No intervals have a simultaneous-coverage guarantee.601 replicates are fixed from ~4percentage-point pointwise normal-approximation95% precision at probability0.5 (Wilson convention used in reporting). Do not treat Monte Carlo replication as observational empirical validation.

Randomness: NumPy SeedSequence entropy202610153741129, PCG64. Spawn key(version1,design index,target index,replicate0..600,stream0/1,role0=data/1=permutation/2=subset/3=analysis,n,anchor,subset). Parent generation uses n=anchor=subset=0. All new seeds are separate from v0.6. No worker-dependent streams. Source and configuration SHA256 hashes and UTC timestamp are frozen before first main task.

Stopping: complete all7,212 independent parents and180,300 estimator calls (including17,? no, counts are determined in config and runner). Explicit authoritative count: per geometry,target,replicate there are2 calls for stream A,48 for stream B, hence 50*2*3*601=180300 calls and18,030,000 permutations shared across returned estimators. There is no result-based early stop, replication increase, re-seeding, geometry deletion or estimator selection. Resource cap8 workers,1 numerical-library thread each. Interrupted tasks resume from verified configuration-identical parent checkpoints. Record numerical failures; any implementation fix must be logged with original outputs preserved and same seeds rerun. Unresolved failures are disclosed, not dropped.

Reporting: all numeric results require complete frozen bank and analysis integrity checks. Retain original v0.6 evidence and distinguish this extension. No Zenodo deposit, DOI, independent statistician approval or real-data validation is implied by this work.
