# v0.7a' Velocity-Fraction Audit, Restricted-Scope (Pre-Close Confirmation) Status: **VERDICT LANDED 2026-05-24**. Verdict: **`velocity_fraction_restricted_passes_audit`** [PASS]. The restricted-scope audit on the 250 analyzable rows produces `chi^2 = 16.425` vs critical `11.34` (df = 3, `p = 9.28e-4`) with `alignment_tightness = 0.698`, below the 0.8 warning threshold. The velocity-fraction shadow stratifies S/U on the analyzable sub-catalog **independently of the v0.5a branch hash**. The z-fraction sidecar shows the same U-shape direction at `p = 0.021`, below the loud-signal threshold (no fresh circularity audit triggered). This document locks the v0.7a' minimal-scoped chi-squared audit on the restricted catalog (the 250 rows analyzable under v0.7a's R2.A discipline) with **explicit Pass / Partial / Fail criteria tied to the v0.5a branch explanation**. It is the single pre-registered confirmation run that licenses the v0.7 chapter close. ## Verdict (Landed, 2026-05-24) Receipt: `results/isotrophy/k-facet-v07a-prime-restricted-scope/manifest.json`. ```text verdict: velocity_fraction_restricted_passes_audit [PASS] scope: 250 analyzable rows (v0.7a integration_blocked == False) S_count / U_count: 87 / 163 attrition disclosure (permanent on this receipt): v0.7a full catalog rows: 273 v0.7a integration_blocked_count: 23 (8.42%) v0.7a additional sanity_failed_count: 11 v0.7a total data-integrity: 34 / 273 = 12.5% v0.7a' restricted scope: 250 rows ``` Primary vf quartile contingency (re-binned within the 250-row subset): ```text Q_vf N S U S_fraction chi2_contrib 1 63 31 32 0.4921 5.76 <-- gamma_1 mostly positional 2 62 11 51 0.1774 7.95 <-- mixed, slightly velocity-leaning 3 62 18 44 0.2903 0.91 4 63 27 36 0.4286 1.80 <-- gamma_1 mostly velocity ----- chi^2 = 16.43 ``` ```text chi^2: 16.425218 test_branch_taken: chi_squared (min_expected = 21.6, well above the asymptotic floor 5) df: 3 critical (p=0.01): 11.34 p_value (chi-squared(3)): 9.276e-4 alignment_tightness: 0.6984 (<= 0.8 warning threshold) ``` z-fraction sidecar (D1+A; report-only): ```text Q_z N S U S_fraction 1 63 26 37 0.4127 1.16 2 62 12 50 0.1935 6.52 <-- dominant chi^2 contribution 3 62 27 35 0.4355 2.09 4 63 22 41 0.3492 0.00 ----- chi^2 = 9.77 ``` ```text chi^2: 9.772511 p_value (chi-squared(3)): 2.060e-2 alignment_tightness: 0.5000 sidecar_status: report_only indicative_verdict_if_promoted: velocity_fraction_restricted_fails_audit (does NOT meet the loud-signal threshold at p <= 0.01; no fresh circularity audit triggered.) ``` **Structural reading:** > The Floquet velocity-fraction direction shadow stratifies S/U on > the 250 analyzable supp-B rows in a **non-monotone U-shape**: > Q1 (gamma_1 mostly positional, vf <= q25) holds 49.2% S; Q2 > (mixed, slightly velocity-leaning) holds 17.7% S; Q3 holds 29.0% > S; Q4 (gamma_1 mostly velocity, vf > q75) holds 42.9% S. The > "pure" direction ends (Q1 and Q4) both concentrate stable orbits; > the mixed/slightly-velocity-leaning Q2 concentrates unstable > orbits. Alignment-tightness against the v0.5a branch hash is > 0.698, below the 0.8 warning threshold -- the signal is NOT a > branch-shadow re-projection. This is the FIRST low-dimensional > projection in the four-chapter v0.4 + v0.5 + v0.6 + v0.7 audit > chain to produce a stability signal that is statistically > significant AND not bin-locality-aligned with prior chapters' > branch-hash content. **Catalog-wide caveat:** the v0.7a catalog-wide audit was integration-attrited at 8.42% (23 blocked + 11 sanity-failed = 34/273). The restricted-scope PASS holds on the 250 analyzable rows ONLY. The attrited 23 rows cluster at long-period high-m_3 (m_3 = 1.5/1.6/1.7), where the variational equation at rtol = 1e-12 hits the DOP853 step-size floor. Generalizing this result to the full supp-B catalog requires either (a) a substrate-level precision relaxation pre- registration (a separate audit family), or (b) a different orbit representation that avoids the variational-precision wall. Neither is licensed under this form lock. Per the locked outcome interpretation, this verdict closes the v0.7 chapter as: > The Floquet velocity-fraction mechanism produced a positive signal > on the analyzable sub-catalog (`chi^2 = 16.43`, `p = 9.3e-4`, > alignment 0.70), but cannot be evaluated on the full 273-row > catalog at the locked variational precision. The attrited 23 rows > (high-m_3 long-period regime) limit the result's catalog-wide > generalizability. The chapter closes with a **qualified positive** > in the orbit-dynamics shadow family: the first such positive in > the v0.4-v0.5-v0.6-v0.7 chain, and the first to be branch- > independent. Audience: v0.7a' runner; v0.7 chapter-close author; paper-side reviewer of the integration-attrited close. Companions: - `kfacet_v07a_velocity_fraction_audit_form.md` -- v0.7a parent form + verdict `velocity_fraction_blocked_integration_attrition`. The R1 + R2.A amendments and the locked non-circularity sentence carry forward unchanged. - `kfacet_v05a_branch_map_form.md` -- v0.5a branch hash (the alignment-tightness comparison axis). - `kfacet_v06a_energy_quartile_audit_form.md` -- precedent for the alignment-tightness threshold tree (0.8 warning, 0.95 severe). Frame: v0.7a's 273-row audit was integration-attrited at 8.42% (`integration_blocked_count = 23 > 14`). The catalog cannot be honestly evaluated at variational precision 1e-12, but the 250 analyzable rows DO carry computed velocity_fraction values with sd = 0.144 (well above the constant-feature retirement floor). v0.7a' is the single minimum-scope pre-registered confirmation that asks: **on the analyzable subset, does the velocity-fraction shadow stratify S/U at the registered floor, and does it do so independently of the v0.5a branch hash?** The Pass / Partial / Fail outcome tree below is locked **before** the chi-squared, alignment-tightness, or sidecar p-value are read. All three outcomes close the v0.7 chapter; the verdict's specific shape determines how the chapter close is worded. ## What v0.7a' Is v0.7a' is a **catalog-side audit on a restricted scope**, NOT a predictor: ```text Scope: the 250 analyzable rows from v0.7a's per_row_table.csv (rows with integration_blocked == False). Excluded: 23 integration-blocked rows + 11 R1-sanity-failed rows (still in per_row_table but flagged). The attrition is RECORDED on every receipt; it is NOT re-evaluated against the 5% gate (v0.7a's gate already fired and is the reason this scope is restricted). Feature: velocity_fraction(row), already computed in v0.7a per the D5 + B operational definition. No new variational compute. Existing per_row_table.csv is the input. Binning: RE-COMPUTED quartiles within the 250-row subset using numpy.quantile(method='linear'). This is necessary because v0.7a's quartile assignments were never written (the original runner blocked at the verdict step). The re-binning is mechanical and uses the same quartile method. Contingency: 4 x 2 over (Q_vf, S/U), df = 3. Critical: chi-squared(3) at p = 0.01 = 11.34. Alignment guard: max over Q_vf of (fraction-in-any-single-v0.5a-branch_ label-bucket). Thresholds inherited from v0.6a/v0.7a: 0.8 (warning) and 0.95 (severe). Sparse-cell fallback: v0.6b inheritance verbatim (seed = 20260523, n_permutations = 10000). For quartile binning on 250 rows with 87 S / 163 U, the chi-squared branch is overwhelmingly likely; fallback documented for completeness. Sidecar: z_fraction (D1+A direction-projection feature), quartile-binned within the 250-row subset. REPORT-ONLY (no verdict claim). ``` ## Pass / Partial / Fail Outcome Tree (Locked Pre-Data) The verdict is determined by the joint state of `chi^2` and the alignment-tightness scalar against the v0.5a branch_label. The tree is locked before either quantity is computed: ```text verdict tree: if min_occupied_bin_count < 2: verdict = velocity_fraction_restricted_inconclusive_sparse (sparse contingency on the 250-row subset; structurally improbable for quartile binning but recorded for completeness) elif chi^2_vf > 11.34: if alignment_tightness <= 0.8: verdict = velocity_fraction_restricted_passes_audit [PASS] elif alignment_tightness <= 0.95: verdict = velocity_fraction_restricted_passes_audit_alignment_warning [PARTIAL] else: verdict = velocity_fraction_restricted_passes_audit_severe_alignment [PARTIAL-severe] else: verdict = velocity_fraction_restricted_fails_audit [FAIL] ``` ### Outcome interpretation (locked) ```text [PASS] velocity_fraction_restricted_passes_audit: On the 250-row analyzable subset, vf chi-squared exceeds 11.34 AND vf is not bin-aligned with the v0.5a branch hash (alignment <= 0.8). The velocity-fraction shadow carries stability information independent of the v0.5a branch hash on the analyzable subset. The v0.7 chapter closes as: "the Floquet velocity-fraction mechanism produced a positive signal on the analyzable sub-catalog, but cannot be evaluated on the full 273-row catalog at the locked variational precision. The attrited 23 rows (high-m_3 long-period regime) limit the result's catalog-wide generalizability." [PARTIAL] velocity_fraction_restricted_passes_audit_alignment_warning (alignment in (0.8, 0.95]) or [PARTIAL-severe] velocity_fraction_restricted_passes_audit_severe_alignment (alignment > 0.95): vf stratifies in-sample on the analyzable subset, but the stratification tracks the v0.5a branch hash tightly. This is the same bin-locality phenomenon v0.6a exposed for the conserved- quantity shadow. The v0.7 chapter closes as: "the Floquet velocity-fraction shadow does not carry stability information beyond labeling the v0.5a branch on the analyzable sub-catalog; the in-sample positive is branch-shadow content re-projected onto orbit-dynamics coordinates." [FAIL] velocity_fraction_restricted_fails_audit: vf chi-squared does not exceed 11.34 on the analyzable sub-catalog. The v0.7 chapter closes as: "the Floquet velocity-fraction shadow does not stratify stability on the analyzable sub-catalog; the joint v0.4 + v0.5 + v0.6 + v0.7 envelope is sealed against catalog-coordinate and orbit-dynamics low-dimensional shadows on this catalog." ``` **All three outcomes close the v0.7 chapter and the isotrophy program.** The Pass outcome adds the qualifier that the result is restricted to the analyzable subset; the Partial and Fail outcomes are sharper methodological statements. ## Attrition Disclosure (Recorded Permanently) Every v0.7a' receipt records: ```text v0.7a_full_catalog_row_count: 273 v0.7a_analyzable_row_count: 250 v0.7a_integration_blocked_count: 23 v0.7a_sanity_failed_count: 11 v0.7a_attrition_fraction: 0.0842 v0.7a_total_data_integrity_fraction: 0.125 v0.7a' restricted scope used: 250 rows (analyzable subset only) v0.7a' S_count / U_count: 87 / 163 ``` The attrition is permanent on the v0.7a' verdict. No amount of chi-squared signal on the restricted subset can recover the catalog-wide claim that v0.7a was registered to make. v0.7a' is explicitly the **confirmation read** before chapter close, not a re-attempt at the original catalog-wide audit. ## Why a Single Restricted-Scope Audit, Not v0.7a'' or v0.7b The isotrophy program's pre-mortem discipline says: when an audit attrites, the chapter closes; the program does not chase incremental re-registrations down a sequence of progressively-relaxed scopes. v0.7a' is registered as a SINGLE confirmation audit, not a sequence. The verdict goes to one of {PASS, PARTIAL, PARTIAL-severe, FAIL, INCONCLUSIVE_SPARSE}. Each verdict closes the v0.7 chapter. The isotrophy program retires at end-of-v0.7 with the four-chapter envelope (v0.4 / v0.5 / v0.6 / v0.7) as the publishable methodology contribution. No v0.7b held-out predictor will be registered under this scope. No further v0.7 sub-registrations will be authored. ## Non-Circularity Audit ```text The 250-row analyzable subset is the v0.7a output, filtered by integration_blocked == False. The filter uses ONLY the integrator's runtime status (Did compute_monodromy_vectorized raise?), which is a function of the row's variational equation, not of the stability label or any feature. The filter is information-disjoint from S/U. The velocity_fraction values are the v0.7a output, computed under the locked D5 + B operational definition + R1-amended sanity gates. The non-circularity sentence and firewall from v0.7a carry forward unchanged. The re-binning to quartiles within the 250-row subset uses numpy.quantile on the velocity_fraction column only. The S/U label is the held-back test column, joined at the chi-squared step. ``` The v0.7a' alignment-tightness scalar against v0.5a branch_label is the same comparison the v0.6a / v0.7a form locks used. No new circularity risks. ## Free Parameter Count ```text Number of free parameters fitted to the data: 0 ``` The scope (analyzable subset), the binning rule (quartiles within subset), the quantile method (numpy linear interpolation), the bin assignment convention (right-closed lower), the df formula (occupied_bins - 1), the critical value (11.34), the alignment- tightness thresholds (0.8, 0.95), the sparse-cell fallback parameters, and the Pass / Partial / Fail outcome tree are all locked pre-data. ## Implementation Plan ```text Suggested script: scripts/v07a_prime_restricted_scope_audit.py Inputs: results/isotrophy/k-facet-v07a-velocity-fraction-audit/per_row_table.csv results/isotrophy/k-facet-v05a-branch-map/per_row_table.csv (the v0.5a per_row_table is already joined into v0.7a's per_row_table via the branch_label column; reading v0.5a directly is optional.) Output: results/isotrophy/k-facet-v07a-prime-restricted-scope/manifest.json results/isotrophy/k-facet-v07a-prime-restricted-scope/per_row_table.csv results/isotrophy/k-facet-v07a-prime-restricted-scope/contingency_table_vf.csv results/isotrophy/k-facet-v07a-prime-restricted-scope/contingency_table_z.csv Expected runtime: seconds. No new variational compute. ``` ## Lock-In Statement This audit form is committed before the chi-squared, alignment- tightness, sidecar chi-squared, or sidecar alignment-tightness has been read for the 250-row analyzable subset. Any change to the scope, feature, binning, df, critical value, alignment-tightness thresholds, Pass/Partial/Fail outcome tree, or sparse-cell fallback parameters after the v0.7a' runner is executed is a re-registration, not a refinement. This form lock is the LAST audit registration in the isotrophy program. Any outcome closes the v0.7 chapter and licenses the program retirement document. ## Doc Trail - `kfacet_v07a_velocity_fraction_audit_form.md` -- v0.7a parent audit with attrition verdict. - `kfacet_v07_mechanism_preregistration.md` -- v0.7 parent registration. - `kfacet_v05a_branch_map_form.md` -- v0.5a branch hash (alignment comparison axis). - `kfacet_v06a_energy_quartile_audit_form.md` -- alignment-tightness pattern. - `kfacet_v07_writeup.md` -- v0.7 chapter close (qualified-positive on restricted domain; landed 2026-05-24). - `kfacet_v08_mechanism_preregistration.md` -- v0.8 parent registration on Floquet direction-purity (opens v0.8). --- Restricted-scope confirmation audit. Pass / Partial / Fail tree locked. PASS landed at chi^2 = 16.43, p = 9.3e-4, alignment 0.698. v0.7 chapter closed as qualified-positive on restricted domain; v0.8 opened on Floquet direction-purity.