# v0.16 liao2021 Tail-Resolved Transfer Form Draft ## Result (2026-06-02) **Verdict: `tail_resolved_transfer_passes_clean`.** The first clean external (Tier-2, fresh doubly-held-out, outcome-balanced) confirmation in the arc -- the tail-resolved continuous score clears the 0.55 floor where the coarse zone could not. ```text coverage 7/7 supported cells primary, 1120 rows attrition 0.0000 Wilson95 [0.0000, 0.0034] (conservative all-4-frame rule, zero loss) frame-spread gate median 0.00036, p90 0.0184 -> CLEAN (bounds 0.10 / 0.50) AUC_cond 0.6470 (J_cond 28985 / D_cond 44800) p_perm 1.0e-5 (100k within-cell score permutation; 0 of 100k >= observed; null mean 0.500, max 0.582) ``` **The clean triangulation (same 1120 rows, same direction).** The report-only coarse zone AUC on these exact rows is `0.510` (saturated, ~chance) -- reproducing v0.15's coarse-zone result -- while the continuous tail-resolved score is `0.647`. The only difference is binning vs continuous: the coarse {0.25,0.50} zone WAS discarding the signal, demonstrated directly on one row set. v0.16's hypothesis is confirmed. **Honest texture -- the effect is real but mass-cell-heterogeneous.** Per-cell AUC ranges 0.43-0.97: ```text qA0_qB0 0.972 qA3_qB3 0.745 qA3_qB1 0.696 qA3_qB2 0.639 qA1_qB0 0.575 (v0.11 direction) qA2_qB1 0.432 qA3_qB0 0.470 (direction reversed) ``` The v0.11 direction (stable -> higher vf) holds strongly in 5 of 7 cells and reverses in 2. The pooled effect is significant against the within-cell permutation null (which respects cell structure), so it is not a pooling artifact -- but it is NOT a uniform per-cell law. Leave-one-out robustness: dropping the strongest cell (qA0_qB0) still gives pooled AUC 0.593 > 0.55, so the pass is not a single-cell artifact. The qA0_qB0 separation (S median vf 0.9965 vs U 0.9904) is far above DOP853 numerical noise, not an integration artifact. **Ensemble vs single-frame.** The secondary single-frame `vf_0` AUC is 0.636, within ~0.01 of the 4-frame ensemble 0.647 -- as the pre-lock frame-spread preview predicted (ensemble approx single-frame for ~90% of orbits). The frame ensemble bought a little robustness on the fragile tail; the headline lever was continuous-vs-coarse (0.51 -> 0.65), not ensemble-vs-single-frame. **Allowed claim (locked):** a tail-resolved 4-frame ensemble velocity-fraction score transfers as a Tier-2 within-cell stability-ranking signal on fresh held-out liao2021 stable-support rows; the underlying velocity-fraction projection carries external stability information that its coarse-zone v0.11 form does not (v0.15). Does NOT claim the coarse zone transfers, full-catalog transfer, prevalence, Tier-3 / independent confirmation, or theorem-facing status. **Independent verification.** A standalone brute-force pair-count recompute (#(S>U) + 0.5 ties, independent of the runner's midrank path) reproduced AUC_cond 0.6469866071428572, J 28985, D 44800, the secondary vf_0 AUC 0.635546875, the zone AUC 0.50984375, and all seven per-cell AUCs BIT-FOR-BIT, with 0 zone_index_vf0 re-derivation mismatches. Receipt + verifiers: `results/isotrophy/k-facet-v16-liao2021-tail-resolved-transfer/` (`manifest.json`, `per_cell_rank.csv`, `permutation_summary.json`, `coarse_zone_relationship.csv`, `_independent_check.{py,json}`, `_sample_verify.{py,json}`). git_commit 4bc2b7a7. **Next-chapter boundary.** Per the locked tree, a PASS may justify a larger supported-region or multi-feature liao2021 chapter (which must re-lock scope / sharding / attrition / frame / floor / claim). The mass-cell heterogeneity is the natural next question: which mass regions carry the transfer and why two reverse. PASS does not promote beyond Tier-2 or make isotrophy theorem-facing. --- Status: **OPERATOR LOCK 2026-06-01; VERDICT LANDED `tail_resolved_transfer_passes_clean` 2026-06-02.** No v0.16 runner had been written, no v0.16 sample drawn, no v0.16 D5 rows integrated, and no v0.16 transfer statistic computed at lock time. Runner implementation began after lock. This form locks the feature-resolution follow-up to v0.15. Decision adjustments before lock: ```text frame stability: generous frame-spread gate on the four ensemble frames ensemble-frame failure: conservative -- any missing ensemble frame counts as attrition zone/score relation: report-only diagnostic, never a verdict gate ``` Pre-lock frame-spread preview (marginal only, no stability split, v0.15 rows) found the proposed gate clean by a wide margin: median spread 0.0003 and p90 0.0205 against clean bounds 0.10 / 0.50. This informed no score threshold and no stability read. ## Frame v0.15 defeated the v0.14 coverage wall (7/7 supported cells primary, 1120 rows, zero attrition) and produced a decidable result: `stable_support_transfer_directional_weak` (AUC_cond 0.5125, p_perm 0.00033). The direction transferred and was permutation- significant, but the effect sat below the 0.55 floor. The mechanism was diagnosed: the coarse `{0.25, 0.50}` zone feature is **saturated** in liao2021's stable-support region -- 98.4% of orbits fall in zone 2, so the 3-zone binning discards nearly all of the velocity-fraction information. The marginal velocity-fraction distribution in that region (measured on v0.15 success rows, **no stability split**, firewall-clean) confirms the binning -- not the underlying projection -- is the bottleneck: ```text vf quantiles (v0.15 stable-support region): p5=0.75 p25=0.94 p50=0.97 p75=0.99 16% below 0.9, 5% at/below 0.75, min 0.075 zone-2 internal IQR = 0.054 ``` 98.4% of orbits are collapsed to one zone, yet the continuous vf beneath them spans `[0.5, 1.0]` with a real lower tail. A tail-resolved score has resolution to exploit that the coarse zone destroys. v0.16 asks the resolution question directly: > On a fresh, doubly-held-out, outcome-balanced sample from liao2021's stable-support > cells, does a **tail-resolved, frame-ensemble velocity-fraction score** rank stable > rows above unstable rows at the pre-registered external-transfer effect floor? A PASS would show the underlying velocity-fraction projection (continuous) carries external stability information even though its coarse-zone v0.11 form does not (v0.15). A FAIL would show the projection itself -- not just its binning -- does not transfer with magnitude to this region. ## Integrity Caveat v0.16 breaks **two** axes relative to v0.11, deliberately and disclosed: ```text feature: NEW. v0.11/v0.14/v0.15 used the coarse {0.25,0.50} zone of a single-frame vf. v0.16 uses a continuous 4-frame ensemble median of vf. This is not the v0.11 feature; it is a new tail-resolved, frame-aggregate projection. domain: FRESH. Same 7 stable-support cells, but excluding every v0.14 AND v0.15 sampled orbit -- no row that informed the saturation finding is reused. ``` Hypothesis provenance (disclosed): the tail-resolution hypothesis was motivated by the v0.15 saturation result and the **marginal** vf distribution above. It was NOT derived from any vf-vs-stability inspection on v0.15, held-out, or fresh rows. The feature is fully specified here with zero free parameters before any v0.16 measurement. Frozen from the prior chapters (unchanged): ```text target file SHA-256: 9c06eedc41b537b2ed926217cdf149d9f808a21866148ea3d89b9eb3c6069fd4 target tier: Tier 2 only mass-cell construction: v0.14 nested sorted-mass 4 x 4 cells D5 measurement: v0.7/v0.13a DOP853 pipeline frame rotations: v0.13a/b SO(2) {37, 90, 211} degrees (the ensemble frames) within-cell statistic: pooled conditional rank AUC effect floor: AUC_cond >= 0.55 permutation p gate: p_perm <= 0.01 permutation seed: 20260523 ``` **The effect floor is HELD at 0.55. It is not lowered to chase v0.15's 0.5125.** Forbidden: ```text lowering the effect floor or the p gate adding free parameters to the score (the 4-frame median is fully fixed) reporting whichever of several candidate features clears the floor (the primary score is the single 4-frame ensemble median; nothing else may set the verdict) selecting sampled rows by any vf, score, zone, or D5 output (draw on mass_cell + source stability label only) including any v0.14 OR v0.15 sampled orbit in the v0.16 primary sample dropping frame-fragile rows from the primary statistic using coarse-zone/score agreement, disagreement, or report-only zone AUC to set or change the verdict using the balanced sample to estimate liao2021 stability prevalence claiming this rescues or transfers the v0.11 COARSE-ZONE feature (v0.15 settled that) re-reading or revising v0.15's directional_weak verdict promoting to Tier-3 / independent confirmation revising the v0.3h K_facet structural null ``` ## Required Prior Receipts The runner must assert all of the following before drawing a sample: ```text v0.13 target inventory includes liao2021 as Tier 2 v0.13a leakage bounded at 0.0 v0.13b verdict = coarse_zone_rule_frame_stable_enough_to_test v0.13 rate-probe attrition <= 0.10 and Wilson95_high <= 0.20 v0.14 verdict = sample_transfer_undecidable_coverage v0.15 verdict = stable_support_transfer_directional_weak v0.14 + v0.15 target SHA match this form's target SHA v0.14 sample_frame.csv AND v0.15 sample_frame.csv exist (for the double holdout) ``` Any mismatch aborts. ## Source-Support Census (fresh held-out) Before any v0.16 D5 measurement, recompute source S/U counts in the frozen v0.14 mass cells **after excluding both v0.14 and v0.15 sampled orbits**. Signal-blind: source masses + source stability labels only; no vf, score, zone, monodromy, or transfer statistic. The census reads (independently reproduced from the source file with v0.14's `build_mass_cells`; eligible = source minus the v0.14 ∪ v0.15 holdout): | mass cell | eligible S | eligible U | disposition | |---|---:|---:|---| | `mass_qA0_qB0` | 402 | 7,824 | supported | | `mass_qA1_qB0` | 568 | 7,658 | supported | | `mass_qA2_qB1` | 2,664 | 5,561 | supported | | `mass_qA3_qB0` | 2,514 | 5,712 | supported | | `mass_qA3_qB1` | 3,078 | 5,147 | supported | | `mass_qA3_qB2` | 2,341 | 5,884 | supported | | `mass_qA3_qB3` | 1,038 | 7,187 | supported | Locked stable-support rule (fresh): ```text supported cell iff eligible S >= 80 AND eligible U >= 80 after excluding v0.14 ∪ v0.15 ``` Expected supported cells: 7 (smallest eligible stable pool 402, > 5x the 80 draw). The nine zero-stable / marginal cells from v0.14/v0.15 remain report-only and out of scope. ## Sample Design Primary sample: ```text supported cells: expected 7 rows per cell: 160 stable rows/cell: 80 unstable rows/cell: 80 expected sample rows: 1120 seed: 20260523 sampling: uniform without replacement within (mass_cell, source_label) holdout rule: exclude every orbit_index in BOTH v0.14 and v0.15 sample_frame.csv ``` If a supported cell has fewer than 80 eligible stable or 80 eligible unstable rows after the double exclusion, the run aborts for lock review. Deficits are not redistributed. The draw is outcome-balanced by source label and blind to vf / score / zone. As in v0.15, the balanced design keeps the rank-AUC valid (AUC is prevalence-invariant under case-control) and forbids any prevalence estimate. ## New Feature: 4-Frame Ensemble Median Velocity-Fraction This is the chapter's one new object. It is fully fixed here. For each sampled orbit, on its CoM-centered initial state `x0`: ```text frame set F = { 0deg (identity), 37deg, 90deg, 211deg } in-plane SO(2) rotations (the v0.13a/b rotation set; translation is dropped -- it is absorbed by CoM-centering, so vf(translation) == vf(identity) and would only double-count the identity in the median) for each f in F: rotate x0 by f, integrate through the frozen D5 path, compute monodromy, select_gamma_1, velocity_fraction -> vf_f (continuous, in [0,1]) score = median( vf_0, vf_37, vf_90, vf_211 ) (median of 4 = mean of the two middle order statistics) ``` Rationale (answers v0.13a directly): a single-frame vf is frame-relative because `select_gamma_1`'s largest-real-part argmax flips under rotation when Floquet real-parts cluster. The **median over the frame ensemble** is a robust central estimate that averages out single-frame argmax flips, while retaining continuous resolution the coarse zone discards. It is frame-aggregate by construction, not a single-frame reading. Cost note: v0.15 already integrates exactly these 4 frames per orbit (base + the 3 rotations of its frame audit). v0.16's primary score is therefore computed from the same 4 integrations -- same runtime; the runner persists the continuous per-frame vf and medians it rather than deriving zones. Secondary scores (reported, NON-gating): ```text vf_0 single-frame continuous velocity_fraction zone_index(vf_0; 0.25,0.50) original coarse v0.11-style zone, identity frame only ``` Report both alongside the primary ensemble score. `vf_0` shows whether the ensemble bought frame stability over a single-frame read. The coarse-zone sidecar shows the relationship between the new tail-resolved score and the old saturated instrument. Neither secondary score can set or change the verdict. ## D5 Measurement For each frame integration, the frozen v0.7/v0.13a D5 path: ```text integrator: DOP853 rtol: 1e-12 atol: 1e-12 max_step_fraction: 0.02 symplecticity_gate: 1e-4 reciprocal_pair_gate: 1e-4 ``` An orbit is a measurement success only if **all four ensemble frames** `{0,37,90,211}` integrate and pass both D5 sanity gates. Rows that are integration-blocked or sanity-failed in the identity frame or any non-identity ensemble frame are excluded from the primary statistic and counted in attrition. A partial ensemble is a different feature and is forbidden. Attrition policy: ```text attrition_fraction <= 0.05: clean 0.05 < attrition_fraction <= 0.10 AND Wilson95_high <= 0.20: attrition warning attrition_fraction > 0.10 OR Wilson95_high > 0.20: blocked_by_attrition ``` ## Primary Domain After D5 attrition, a supported cell enters the primary statistic only if: ```text N_success >= 120 S_success >= 50 U_success >= 50 ``` Coverage gate: ```text primary_supported_cells >= 6 primary_success_rows >= 900 ``` If the coverage gate fails, verdict is `tail_resolved_transfer_undecidable_coverage`. Cells that fail primary gates are report-only. ## Primary Statistic Within each primary supported cell, compare every stable row to every unstable row on the continuous ensemble score (Mann-Whitney form): ```text J_cell = #[(S,U): score(S) > score(U)] + 0.5 * #[(S,U): score(S) = score(U)] D_cell = S_cell * U_cell AUC_cell = J_cell / D_cell J_cond = sum J_cell over primary cells D_cond = sum D_cell over primary cells AUC_cond = J_cond / D_cond ``` Direction is pre-registered (the v0.11 direction): **stable rows should have higher ensemble score than unstable rows.** With a continuous score, exact ties are rare and take the 0.5 weight; there is no zone-degeneracy regime and no v0.14-style discreteness ceiling -- the null is smooth. ## Permutation Test ```text within-cell S/U permutation preserve each primary cell's score multiset preserve each primary cell's S and U counts permutations = 100000 seed = 20260523 p_perm = (1 + # permuted J_cond >= observed J_cond) / (1 + permutations) one-sided in the pre-registered direction ``` ## Frame-Stability Audit The primary score is frame-aggregate by construction, but that robustness must be priced rather than assumed. The lock uses a **generous internal frame-spread gate** on the same four frames that define the score, avoiding an extra held-out-rotation sweep while still blocking a pass if the ensemble is visibly unstable. For each successful orbit: ```text frame_spread = max(vf_0, vf_37, vf_90, vf_211) - min(vf_0, vf_37, vf_90, vf_211) ``` Report the distribution overall, by supported mass cell, and in fragile bands A/B: ```text median, p75, p90, p95, max ``` Frame-spread policy: ```text median(frame_spread) <= 0.10 AND p90(frame_spread) <= 0.50: clean median(frame_spread) <= 0.20 AND p90(frame_spread) <= 0.75: frame warning; transfer may pass only with warning language median(frame_spread) > 0.20 OR p90(frame_spread) > 0.75: verdict = tail_resolved_transfer_blocked_by_frame_instability ``` These are deliberately generous because v0.16's score is a robust central estimate, not a claim that every individual frame is stable. The gate asks only whether the ensemble is so frame-variable that a pass would be uninterpretable. ## Coarse-Zone Relationship Diagnostic The original v0.11 coarse zone is saturated in liao2021 and cannot gate v0.16. It is reported only, using the identity-frame `vf_0`: ```text zone_index_vf0 = 0 if vf_0 < 0.25 zone_index_vf0 = 1 if 0.25 <= vf_0 < 0.50 zone_index_vf0 = 2 if vf_0 >= 0.50 ``` Report: ```text zone_auc_cond_report_only within-cell AUC using zone_index_vf0 score_quantiles_by_zone score p25/p50/p75 within each coarse zone zone2_internal_score_quantiles score p5/p25/p50/p75/p95 inside saturated zone 2 ``` This diagnostic can support interpretation (for example, showing that the continuous score resolves variation inside zone 2), but it cannot set or change the verdict. ## Verdict Tree Run prior-receipt, target, sample, attrition, coverage, and frame gates before interpreting the rank statistic. ```text tail_resolved_transfer_blocked_by_receipt required prior receipts missing or inconsistent OR target SHA mismatch OR v0.14/v0.15 sample exclusion unavailable OR supported-cell census deviates from the locked rule and cannot draw 80/80 tail_resolved_transfer_blocked_by_attrition attrition_fraction > 0.10 OR attrition_wilson95_high > 0.20 tail_resolved_transfer_blocked_by_frame_instability median(frame_spread) > 0.20 OR p90(frame_spread) > 0.75 tail_resolved_transfer_undecidable_coverage primary_supported_cells < 6 OR primary_success_rows < 900 tail_resolved_transfer_passes_clean attrition_fraction <= 0.05 AND median(frame_spread) <= 0.10 AND p90(frame_spread) <= 0.50 AND AUC_cond >= 0.55 AND p_perm <= 0.01 tail_resolved_transfer_passes_with_warning AUC_cond >= 0.55 AND p_perm <= 0.01 AND no hard block fired AND ( 0.05 < attrition_fraction <= 0.10 OR 0.10 < median(frame_spread) <= 0.20 OR 0.50 < p90(frame_spread) <= 0.75 ) tail_resolved_transfer_directional_weak AUC_cond > 0.50 AND AUC_cond < 0.55 AND p_perm <= 0.01 tail_resolved_transfer_fails AUC_cond <= 0.50 OR p_perm > 0.01 ``` ## Required Outputs Expected additive runner: ```text scripts/v16_liao2021_tail_resolved_transfer.py ``` Expected command shape: ```powershell python scripts/v16_liao2021_tail_resolved_transfer.py prepare ` --target docs/isotrophy/external_targets/liao2021_nonhierarchical.txt ` --v14 results/isotrophy/k-facet-v14-liao2021-sampled-transfer ` --v15 results/isotrophy/k-facet-v15-liao2021-stable-support-transfer ` --out results/isotrophy/k-facet-v16-liao2021-tail-resolved-transfer ` --stable-per-cell 80 ` --unstable-per-cell 80 ` --seed 20260523 python scripts/v16_liao2021_tail_resolved_transfer.py shard ` --out results/isotrophy/k-facet-v16-liao2021-tail-resolved-transfer ` --shard-index 0 ` --shard-count 14 python scripts/v16_liao2021_tail_resolved_transfer.py merge ` --out results/isotrophy/k-facet-v16-liao2021-tail-resolved-transfer ` --shard-count 14 python scripts/v16_liao2021_tail_resolved_transfer.py analyze ` --out results/isotrophy/k-facet-v16-liao2021-tail-resolved-transfer ` --permutations 100000 ` --seed 20260523 ``` Convenience npm aliases mirror the locked command shape: ```powershell npm run isotrophy:v16:prepare npm run isotrophy:v16:shard -- --shard-index 0 --shard-count 14 npm run isotrophy:v16:merge npm run isotrophy:v16:analyze ``` ```text results/isotrophy/k-facet-v16-liao2021-tail-resolved-transfer/ ``` Required files: ```text manifest.json source_support_census.csv sample_frame.csv sample_shard_XX_of_NN.csv per_row_sample.csv per_cell_rank.csv frame_dispersion.csv (frame-spread gate + diagnostics, all success rows) coarse_zone_relationship.csv (report-only zone/score sidecar) permutation_summary.json operator_commands.md ``` `per_row_sample.csv` must record, per orbit: the four per-frame `vf_0/vf_37/vf_90/vf_211`, the ensemble `score`, the secondary `vf_0`, `zone_index_vf0`, `frame_spread`, `mass_cell`, `stability`, `status`, and the D5 residuals. `per_cell_rank.csv` must record per cell: `N_success`, `S`, `U`, `J_cell`, `D_cell`, `AUC_cell`, the primary flag, and score quantiles (p25/p50/p75) split by stability for the readback. `manifest.json` must include: `verdict`, `startedAt`, `completedAt`, `gitCommit`, `target_sha256`, `v14_sample_excluded_count`, `v15_sample_excluded_count`, `sample_seed`, `supported_cells`, `sample_rows_requested`, `sample_rows_success`, `attrition_fraction` (+ Wilson), `primary_supported_cells`, `primary_success_rows`, `frame_spread_median`, `frame_spread_p90`, `AUC_cond`, `J_cond`, `D_cond`, `p_perm`, `secondary_vf0_AUC_cond`, `zone_auc_cond_report_only`, `target_tier = 2`. ## Claim Boundary If v0.16 passes, the allowed claim is: > On a fresh, doubly-held-out, outcome-balanced sample from liao2021's source-supported > mass cells, a tail-resolved 4-frame ensemble velocity-fraction score transfers as a > Tier-2 within-cell stability-ranking signal. The underlying velocity-fraction > projection carries external stability information in this region even though its > coarse-zone v0.11 form does not (v0.15). Forbidden upgrades: ```text does not claim the v0.11 COARSE-ZONE feature transfers (v0.15 settled: it does not) does not claim full-catalog liao2021 transfer does not claim confirmation outside the stable-support cells does not estimate stable-orbit prevalence does not claim independent Tier-3 confirmation does not make a globally calibrated predictor does not revise v0.15, v0.10b, or the v0.3h K_facet structural null does not make isotrophy theorem-facing by itself ``` If v0.16 fails: > In liao2021's stable-support region, even a tail-resolved frame-ensemble velocity- > fraction score does not transfer at the registered floor: the velocity-fraction > projection itself -- not merely its coarse binning -- does not carry external > stability magnitude here. The v0.11 effect is supp-B-specific in magnitude. If v0.16 is directional weak: > The tail-resolved score detected a directional trace below the pre-registered > external-transfer effect floor -- finer resolution did not lift the effect over the bar. If v0.16 blocks or is undecidable, the allowed claim is limited to the blocking condition. ## Next Chapter Boundary Only a v0.16 PASS may justify a larger supported-region or multi-feature liao2021 chapter (re-locking scope, sharding, attrition, frame diagnostics, floor, claim). If v0.16 fails or is directional-weak, the liao2021 velocity-fraction external-transfer path should **close**: both the coarse zone (v0.15) and its tail-resolved frame-ensemble relative (v0.16) will have been tested at the registered floor on held-out external rows. Any further work would need a genuinely different substrate or a different projection, not another vf variant on this catalog. ## Lock-In Statement Committed before the v0.16 runner is written. Any change to the feature definition, the frame set, the sample, the double-holdout rule, the decision bars, the effect floor, the permutation, the firewall, the frame-spread gate, the report-only zone/score sidecar, or the claim boundary after runner execution is a re-registration. The runner imports the frozen v0.14/v0.15 D5 + zone + frame + AUC + permutation symbols unchanged and adds only the ensemble-median scoring, the double holdout, the continuous-score AUC, the frame-spread gate, and the coarse-zone relationship sidecar.