Focused simulation protocol version 0.6
Comparability of permutation centred mutual information estimates

Status and provenance
This is a prospectively specified amendment for the new analyses supporting a revised manuscript. It replaces the unexecuted expanded grid in reconstructed protocol v0.5. It is not a public preregistration. A timestamped local manifest records the exact configuration and executable files before any new main-study estimator outputs are examined. The earlier manuscript results, its 5,000-catalogue quadratic follow-up, the truth-validation results, estimator-equivalence checks and timing measurements were already known. They are retained as prior evidence. The main study may produce favourable, unfavourable or inconclusive results; all prespecified cells and replicates are reported.

Reason for amendment
The expanded v0.5 proposal implied approximately 7,789 serial run-hours under provisional benchmark assumptions. Before generating main outcomes, the scope is restricted to the mechanisms necessary to evaluate comparability across sample size and dependence geometry. The amendment is not a selective replication cut under R9. It replaces that proposed grid and its uniform-reduction rule explicitly. The original proposal remains in the archive. Replication is 200 independent parent catalogues per design, target and stream throughout this focused study. N above 500 is excluded from the new centred study; earlier raw-score experiments through N = 2000 remain historical supplemental results.

Question and estimands
The primary question is whether subtracting the mean of whole-row permutation scores makes estimates of mutual information comparable across samples. The quantities evaluated separately are absolute mean calibration error, error in the difference between two independent samples, and distortion of a cumulative profile whose generating distribution and true information remain constant. All information and errors are in nats, using natural logarithms. A permutation reference under an alternative is the expectation of the statistic after permuting the observed sample; it is not assumed to equal an independent sample from the exact product of population marginals.

Generating designs and truth
C and epsilon are independent standard-normal vectors. Active coordinate pairs use D_j = sqrt(s) g(C_j) + sqrt(1-s) epsilon_j. Linear g(c)=c; quadratic g(c)=(c^2-1)/sqrt(2); absolute-value g(c)=(abs(c)-sqrt(2/pi))/sqrt(1-2/pi). Inactive coordinates are independent standard normals. L1 and Q1 have dimensions (1,1) with one active pair. L3 and Q3 have dimensions (4,3) with three active pairs. Q1x has dimensions (4,3) with one active pair. T3 starts from L3 and replaces D_2 by D_2 + D_1^2 - 1; this invertible within-D shear preserves information. H1 and H3 use the absolute-value signal with dimensions (1,1) and (4,3), with one and three active pairs respectively.
L1, L3, T3, Q1, Q1x and Q3 are development designs. H1 and H3 are reserved geometry validation. True total information is matched at 0.10, 0.15, 0.40 and 0.60 nats. Linear and T3 truths are analytic. Quadratic and absolute-value truths are obtained from independent FFT and adaptive real-space integration, agreeing within 1e-8 nats. The 32 required matched settings and their residuals are recorded in focused_matched_information.csv. Target-specific reference checks cover each distinct Q/H signal setting with ten million draws and all four T3 targets with ten million draws, with a three-Monte-Carlo-SE diagnostic. Fixed seeds and every check are retained. Q1x shares the Q1 scalar truth reference and has an independent generator check at s=0.64 in validation package v0.2. L1/L3 have analytic truths and representative independent generator checks. Zero-information implementation controls cover all eight designs.

Independent parent bank
For every design, stream A has true information 0.10 and 0.40 and 200 independent parent catalogues of 500 rows per target. Evaluate prefixes N = 78, 156, 300 and 500 from each parent. Stream B independently has targets 0.10, 0.15, 0.40 and 0.60, 200 parent catalogues of 500 rows per target, and prefixes N = 78 and 500. A and B use independent random streams. Their equally numbered replicates form independent sample pairs; independence applies across replicate numbers. Prefixes within one parent remain nested and dependent. No extra evidence is claimed by reusing a parent across several reported comparisons.
Additional null controls use designs L1, L3 and T3 at s=0, stream A, 200 parents and prefixes 78 and 500. They have no P1 subsampling. Null controls are implementation and calibration diagnostics, not an expanded power experiment.

Estimators
Use KSG estimator 1 with the maximum norm, within-sample standardisation by sample standard deviation with ddof=1, and strict marginal neighbour counts. Evaluate k = 3, 4, 5 and 8. KSG4 centred is the primary procedure; other k values are prespecified sensitivities and are not selected by their outcomes. Preserve negative estimates. Whole kinematic rows are permuted, retaining within-D dependence. Baseline B=100 permutations are shared by all estimators on a dataset. The raw estimate and permutation mean are reported separately, together with the centred estimate and its conditional permutation Monte Carlo SE.
GC uses average ranks, normal scores Phi^{-1}((rank-0.5)/N) and sample covariance. Report raw GC, permutation-centred GC and analytically corrected GC. The correction is delta(N,p,q)=[b_p(N)+b_q(N)-b_(p+q)(N)]/2, where b_m(N)=sum_(j=1)^m psi((N-j)/2)+m log(2)-m log(N-1). Subtract delta. This correction is exact for log-determinants of ordinary Gaussian sample covariance; exact unbiasedness is not asserted after empirical rank transformation. Analytically corrected-and-centred GC equals centred GC and is not counted twice. GC errors relative to true information include population approximation error outside its model.
The optimized implementation shares exact distance calculations and neighbour queries. It must reproduce the original raw and per-permutation values on the fixed equivalence tests before main execution. The archived v0.2 implementation remains unchanged and is packaged alongside it.

Fixed common-size procedure P1
The common size is 78. Reuse the baseline estimate at N=78. At each larger prefix, independently draw five row subsets of size 78, without replacement within a subset. Subsets may overlap. Apply the same row indices to C and D and recalculate all preprocessing and permutations for every subset. Average the five estimates separately for each procedure. Each parent/prefix/subset is generated once and may be reused in the prespecified E2 and E3 summaries. Subsets at different prefixes use distinct streams conditional on their shared parent.
For E2, apply P1 to both sides when either side has N=500; this includes equal-size 500/500 comparisons. This is a common-78 procedure, not a procedure that retains the larger minimum size for every comparison. For E3, it applies at every prefix above 78. Mean-bias reduction is not sufficient to establish superiority: RMSE and the loss of precision are also reported. Equal information alone does not guarantee cancellation of bias. Cancellation in expectation follows for equally sized iid samples from the same full joint distribution.

Permutation sensitivity
Before observing main outcomes, select exactly 20 of the 200 parent replicate IDs for each design, target and stream using an independent selection stream. Extend every baseline prefix belonging to those parents to B=500, retaining the identical first 100 permutations. A single B=500 call may supply both summaries. The extension covers baseline calls only; P1 subset estimates remain at B=100. This restriction is an explicit amendment to v0.5. It is not evidence that P1 is insensitive to its permutation count. Compare B100 and B500 estimates within the same selected parents and report signed changes, absolute changes and their Monte Carlo uncertainty. Do not use the sensitivity results to choose B or drop cells.

Experiments and comparisons
E1 uses stream A only at targets 0.10 and 0.40, for all four prefixes and all designs, plus the separately labelled null controls. Report raw and centred calibration, analytic-GC calibration, SD and RMSE. Do not pool stream B selectively to increase replication.
E2 pairs A at I_A=0.10 or 0.40 with B at I_B=r I_A, r in {1,1.5}. Within each generating design, evaluate (N_A,N_B)=(78,78),(500,500),(78,500),(500,78). Both directions are retained. In addition, compare L3(A) with T3(B) and Q1x(A) with Q3(B), at equal true information, both targets and the same four size pairs. These comparisons isolate geometry-dependent error at matched block dimensions. Baseline and common-78 versions use the preassigned parent estimates. No unrelated design rankings are inferred from these controls.
E3 uses stream A only, with the four nested prefixes and constant truth within each parent. Report the mean error at each prefix, endpoint difference and range of the mean profile. All estimator variants are retained. KSG4 centred is primary. Per-parent trends are descriptive and are not tested against an arbitrary rank-correlation threshold.
Reserved H1/H3 data are generated only after the development results have been summarised and the interpretation or guidance to be evaluated is recorded with a timestamp and hash. The fixed estimator set, P1 rule and criteria do not change. No learned k-selection rule is fitted. The reserved families test the extent to which observations about the fixed procedures extend to a different dependence geometry; they do not validate every possible nonlinear relation.

Uncertainty and reporting thresholds
For E1, the replicate error is estimate minus true information. For E2 it is (estimate_B-estimate_A)-(I_B-I_A). Report the mean error, sample SD, RMSE and a two-sided 95% Student-t Monte Carlo interval for the mean over the 200 independent replicate records. This replaces the proposed bootstrap mean interval for these scalar outcomes. For P1 effects, retain baseline and P1 estimates together and report paired uncertainty in their changes. E2 correct-ordering probabilities, with Wilson intervals, are descriptive; no ordering pass threshold is imposed. Ratio summaries use only positive estimates on both sides and report the excluded fraction. No logarithms of estimates are used.
Use tau=max(0.01,0.10 I_ref) as a reporting threshold, not a universal accuracy guarantee. I_ref is the target for E1 and E3 and I_A for E2. A mean-error interval entirely within [-tau,tau] is within tolerance; an interval wholly outside is outside tolerance; otherwise it is indeterminate. This assesses the Monte Carlo mean, not the reliability of a single observed comparison.
For E3 let mu_j be the mean error at prefix j, Delta=mu_last-mu_first, and H=max(mu_j)-min(mu_j). Use 9,999 bootstrap resamples of independent whole parent catalogues, keeping all prefixes together. For each resample compute the maximum absolute centred error in pairwise prefix contrasts. Its 95th percentile is c. Report [Delta-c,Delta+c] and [max(0,H-c),H+c]. The profile is within tolerance only if the endpoint interval lies inside [-tau,tau] and the upper range limit is at most tau. It is outside tolerance if the endpoint interval lies wholly outside that band or the lower range limit exceeds tau; otherwise indeterminate. These are approximate Monte Carlo uncertainty intervals. H>=abs(Delta), so the two criteria are not independent confirmations. There is no simultaneous confidence guarantee over the full experiment grid.
There is no automatic publication-format decision or claim of success based on a number of passing cells. Statistical calibration, usefulness, limitations and novelty determine the interpretation. Every prespecified outcome remains available regardless of direction.

Execution and reproducibility
Use separate SeedSequence streams for data, permutations, subset selection, sensitivity selection and analysis bootstrap. The original study seed bases 202611010000 for development and 202612010000 for reserved validation are retained. The complete new schema, role map and task coordinates are stored in the configuration; worker count and scheduling never determine random streams. Benchmarks and truth checks use distinct seed bases. Preserve full input settings, file hashes, seed provenance, numerical-library versions, timing records and failures.
The proposed focused bank contains 10,200 independent parent catalogues, 26,800 baseline calls and 80,000 P1 calls, for 106,800 calls in total. Exactly 2,680 baseline calls have 400 additional permutations. The total is 11,752,000 permutation evaluations shared by the eleven returned scores. This is a call-count statement, not a count of independent observations or separate tests. The full configuration is authoritative. The executor uses bounded parallel workers and one numerical-library thread per worker.
No cell is dropped after outcomes are seen. Failed numerical tasks are recorded, diagnosed and rerun only after a documented implementation correction, with the original seed and all prior failure records retained. An interrupted valid task can resume unchanged. A numerical failure that cannot be resolved makes the affected unit indeterminate and is disclosed. No result-dependent replication increase or reseeding is permitted.

Reporting of the existing manuscript
Legacy raw-score calibration, unmatched power comparisons, group-preserving tests, selection quadrature and observational sensitivities remain labelled as earlier analyses. The 78-star application is an illustration, not the basis for general simulation conclusions. Its original KSG choice is k=5 and is not silently relabelled k=4. No claim is made to replicate the full APOGEE/Gaia application of Pandey and Mondal or to infer a new stellar population. Both earlier quadratic signal allocations remain documented, with their different total information made explicit.

Authorship and AI assistance
ChatGPT/Codex and Claude assisted with design critique, drafting, code, computation, figure preparation and revision. This is disclosed accurately. A local pre-execution record is not described as public preregistration. Human authors must review and approve the final scientific content and submission. Preparation of this package does not submit or publicly deposit the manuscript.
