# v0.7a Velocity-Fraction Audit Form Lock (D5 + B) Status: **VERDICT LANDED 2026-05-24 (R1 + R2.A amendments applied)**. Verdict: **`velocity_fraction_blocked_integration_attrition`** at `integration_blocked_count = 23` (8.42% of catalog, above the pre-registered 5%/14-row threshold). Of the 250 analyzable rows, an additional 11 failed the R1-amended symplecticity / reciprocal-pair gates. The catalog cannot be evaluated at the locked variational precision (`rtol = atol = 1e-12`); the audit is integration-attrited, not feature-falsified. See the "Verdict (Landed)" section below. This document locks the v0.7a univariate-velocity-fraction chi-squared audit form under the parent registration's candidate slots **D5** (velocity-fraction direction property) and **B** (velocity-fraction quartile audit). The D1 + A audit (largest-real-part eigenvector direction-projection quartile) is registered as a **report-only named sidecar** and emits no independent verdict (also not licensed under the attrition verdict). **Amendment R1 (2026-05-24): symplecticity sanity-gate threshold relaxed from 1e-6 to 1e-4** based on the 7-row vectorized smoke evidence. See the "Amendment R1: Sanity-Gate Threshold" section below. **Amendment R2.A (2026-05-24): per-row integration-failure fallback discipline** added after the full-catalog runner crashed at row 76 (O_194 at m_3=0.5). See the "Amendment R2.A: Per-Row Integration- Failure Fallback" section below. ## Verdict (Landed, 2026-05-24) Receipt: `results/isotrophy/k-facet-v07a-velocity-fraction-audit/manifest.json`. The R1-amended runner ran through all 273 supp-B rows over ~4.5 hours under the R2.A try-catch and R2.C append-per-row disciplines. Result: ```text verdict: velocity_fraction_blocked_integration_attrition total catalog rows: 273 S / U (catalog): 97 / 176 analyzable rows: 250 integration_blocked_count: 23 attrition_threshold_count: 14 (5% of 273, ceiling) attrition_fired: True (23 > 14) sanity_gate_failures (analyzable rows with symp > 1e-4 or recip > 1e-4): 11 total data-integrity issues: 34 / 273 = 12.5% constant_feature_retirement check: vf_sd (over analyzable rows): 0.1444 threshold: 0.01 retired: False (vf has real variation, but the audit is not licensed to read it) total runtime: 266.9 min (~4.5 h) ``` Integration-blocked rows by m_3: ```text m_3 blocked count labels 0.5 1 O_{194} 0.9 4 O_{796}, O_{1008}, O_{1079}, O_{1194} 1.0 3 O_{1153}, O_{1200}, O_{1362} 1.5 4 O_{152}, O_{166}, O_{175}, O_{187} 1.6 6 O_{255}, O_{258}, O_{315}, O_{317}, O_{324}, O_{327} 1.7 5 O_{178}, O_{185}, O_{188}, O_{231}, O_{246} --- 23 ``` All 23 blocks fired in `compute_monodromy_vectorized` with `Required step size is less than spacing between numbers`. The base `integrate_orbit` succeeded for every row; the failure is the matrix variational equation `dY/dt = J(t) Y` along the integrated orbit at `rtol = atol = 1e-12`. Sanity-gate failures (the 11 analyzable rows where symp or recip exceeded the R1 1e-4 threshold) cluster at m_3 in {0.6, 0.7, 1.0, 1.1, 1.4, 1.6}. Worst symplecticity residual was O_321(1.6) at 1.12e-3 (11x the gate). Worst reciprocal-pair residual was O_642(1.0) at 9.17e-3 (90x the gate). **Structural reading:** > The orbits where the variational equation is most numerically > challenging — long-period high-m_3 rows in the 1.4..1.7 cluster > and a tail at m_3 = 0.9..1.0 — cluster precisely in the regime > v0.4a's two-pass classifier was built to handle (Pass 2 tight > tolerances at identity_rotation_tolerance = 1e-9, phase_grid = > 361). At v0.7a's variational precision rtol = atol = 1e-12, the > DOP853 step adapter cannot find a feasible step for the > 324-dimensional matrix variational equation along these orbits. > The audit is NOT feature-falsified; it is INTEGRATION-ATTRITED at > the locked precision. The catalog cannot be evaluated honestly > under this precision configuration. **Joint v0.4 + v0.5 + v0.6 + v0.7 envelope (the publishable multi-chapter statement):** > Four sequential pre-registered projections of the supp-B body — > the Z_2 symmetry shadow (tangent-isotypic and orbit-gauge-rigidity, > v0.4), the 2-bit catalog branch shadow (v0.5), the continuous > orbit-level conserved-quantity shadow E and |L| (v0.6), and the > orbit-dynamics gamma_1 direction shadow (v0.7a velocity-fraction) -- > have produced three distinct chapter-close types: structural- > negative (v0.4), projection-limit (v0.5), conditional-independence > (v0.6), AND a fourth methodological close type, integration- > attrition (v0.7), where the audit cannot honestly run on this > catalog at the registered precision because the variational > equation along the most-challenging orbits fails DOP853 step > control. **Receipts:** ```text docs/isotrophy/kfacet/kfacet_v07a_velocity_fraction_audit_form.md (this document) results/isotrophy/k-facet-v07a-velocity-fraction-audit/manifest.json results/isotrophy/k-facet-v07a-velocity-fraction-audit/per_row_table.csv results/isotrophy/k-facet-v07a-velocity-fraction-audit/smoke/smoke_manifest.json results/isotrophy/k-facet-v07a-velocity-fraction-audit/smoke/smoke_per_row_table.csv scripts/v07a_velocity_fraction_smoke.py scripts/v07a_velocity_fraction_audit.py ``` No chi-squared verdict is licensed under this audit run. The D1+A sidecar (z-fraction quartile) is also not evaluated; the same catalog-integrity issue applies. Audience: v0.7a runner; v0.7b form-lock author (conditional on v0.7a passing); paper-side reviewer of the v0.6 -> v0.7 transition. Companions: - `kfacet_v07_mechanism_preregistration.md` -- v0.7 parent registration (operational definitions D1-D5, audit forms A-D, three circularity risks, disallowed-feature list, inheritance discipline). - `kfacet_v06_writeup.md` -- v0.6 conditional-independence chapter close that opened v0.7. - `kfacet_v06b_within_branch_energy_audit_form.md` -- v0.6b sparse-cell fallback tree pattern (inheritance source). - `kfacet_v06a_energy_quartile_audit_form.md` -- v0.6a alignment- tightness guard pattern (inheritance source). - `kfacet_v04a_domain_map_preregistration.md` -- supp-B monodromy computation source; per-row M_i is derivable from this receipt. Frame: v0.6 closed as a conditional-independence result — the continuous catalog-coordinate (E, |L|) shadow does not carry stability information beyond labeling the v0.5a branch hash. v0.7 leaves catalog-coordinate space entirely; v0.7a is the first orbit-dynamics audit. Among the parent's five candidate operational definitions (D1-D5), this form lock picks **D5** because it minimizes the tie-break surface and stays furthest from eigenvalue-ordering as the feature-selection mechanism. ## Why D5 + B Over Alternatives (Codex Rationale) > v0.7 is already walking close to the circularity cliff by touching > Floquet eigenstructure at all. D1 "largest-real-part" is > mathematically natural, but it still chooses the feature through > eigenvalue ordering, and eigenvalue ordering is too close to the > S/U definition. We'd save D1 for a sidecar after D5 establishes > the clean floor. The locked picks therefore prioritize **non-circularity over discriminative power**. D5 is a direction property that exists for any 6D phase-space vector and is independent of eigenvalue ordering once a direction is fixed. The S-row eigenvalue degeneracy tie-break is documented but minimized: D5's feature does NOT depend on resolving the degeneracy because the velocity-fraction within any eigenspace is well-defined. D1 + A is preserved as a **named sidecar** (report-only): the natural "physicist's instinct" version. If D5 + B passes, D1 + A is an interpretable companion; if D5 + B fails but D1 + A lights up, the sidecar signal may be selection-rule contaminated and would require a fresh circularity audit, not a refinement of v0.7a. ## Amendment R1: Sanity-Gate Threshold (Locked 2026-05-24) The v0.7a form lock pre-registered a symplecticity sanity gate at `1e-6`. A 7-row vectorized smoke (sentinels O_50, O_62, O_67, O_434 at m_3=0.4 and O_242, O_282, O_284 at m_3=1.0) at `rtol = atol = 1e-12` showed symplecticity residuals in the range `7.9e-8 .. 3.84e-5` -- 5 of 7 rows above the pre-locked 1e-6 threshold. The reciprocal-pair gate at `1e-4` passed for all 7 rows (worst-case `8.35e-5` on O_242). The residuals scale with period and Floquet amplification, not with implementation breakage: strongly unstable rows (max_re = 39 on O_242, max_re = 25 on O_284) have larger absolute residuals because the unstable Floquet mode quadratically amplifies integration error in the `M_i^T omega M_i - omega` quantity. Long-period S rows (O_434 at period 200) show similar amplification via accumulated step error. The pre-locked 1e-6 was a pre-run guess. The smoke is the appropriate empirical evidence to amend the threshold. Blocking the chapter on 1e-6 would say "our sanity gate was too tight," not "the mechanism failed" — and the reciprocal-pair gate (the load-bearing test that the monodromy preserves the Floquet pairing structure) passes uniformly. ```text v0.7a-r1 sanity-gate amendment: rtol = 1e-12 (unchanged) atol = 1e-12 (unchanged) max_step_fraction = 0.02 (unchanged) symplecticity_gate: old: 1e-6 new: 1e-4 reciprocal_pair_gate: 1e-4 (unchanged) justification: 7-row vectorized smoke (rtol = atol = 1e-12, max_step = 0.02*T) showed symplecticity residuals in 7.9e-8 .. 3.84e-5, while reciprocal-pair residuals all passed at 1e-4. Residuals scale with period and Floquet amplification and are treated as the variational-integration precision floor, not as feature evidence. firewall (locked permanently): symplecticity residual, reciprocal-pair residual, max_re, and the selected eigenvalue are QC/provenance only. They do NOT enter the velocity-fraction feature, the binning, the chi-squared statistic, the alignment-tightness scalar, or the verdict. Any future amendment that promotes a sanity diagnostic into the feature path is a fresh circularity audit, not a refinement. ``` This amendment is locked before the amended-gate smoke is re-run. Any subsequent change to either gate after the audit verdict lands is a re-registration, not a refinement. ## Amendment R2.A: Per-Row Integration-Failure Fallback (Locked 2026-05-24) The v0.7a R1-amended full-catalog runner crashed at row 76 of 273 (O_194 at m_3=0.5) with: ```text RuntimeError: vectorized variational integration failed for O_{194}(0.5): Required step size is less than spacing between numbers. ``` This is a pre-sanity numerical-precision failure inside scipy's DOP853 step adapter on the matrix variational equation. The base `integrate_orbit` succeeded; `compute_monodromy_vectorized` could not find a feasible step at `rtol = atol = 1e-12`. It is NOT a symplecticity or reciprocal-pair sanity-gate violation -- the sanity gates never get to evaluate because M_i is never produced. The pre-locked form lock anticipated symplecticity / reciprocal-pair failures but did not anticipate pre-sanity integrator failures. R2.A pre-registers the row-level fallback discipline. ```text v0.7a-r2.a per-row integration-failure fallback: trigger: any RuntimeError raised from compute_monodromy_vectorized for a given row (typically "Required step size is less than spacing between numbers", but any IVP failure mode). per-row marking: the row is recorded on per_row_table.csv with: - row identity preserved (label, index, m3, z0, period, stability, branch_label), - symplecticity_status = "integration_blocked", - reciprocal_pair_status = "integration_blocked", - velocity_fraction = null / NaN, - z_fraction = null / NaN, - Q_vf, Q_z = null, - all gamma_1 fields = null, - integration_error_message = the exception message, - integration_blocked = True. chi-squared denominator: blocked rows are EXCLUDED from the chi-squared catalog (the 273 reduces to (273 - integration_blocked_count)) and from the alignment-tightness denominator. Quartile cutpoints are computed on the non-blocked subset only. attrition threshold (locked): if integration_blocked_count > 5% of catalog (= ceil(0.05 * 273) = 14 rows for supp-B): verdict = velocity_fraction_blocked_integration_attrition. No chi-squared verdict is licensed; the catalog is too integration-attrited for the audit to run honestly. else: audit proceeds on (273 - integration_blocked_count) rows. firewall (re-asserted permanently): integrator error messages and blocked-row status are QC/provenance only. They do NOT enter the velocity-fraction feature, the binning, the chi-squared statistic, the alignment-tightness scalar, or the verdict (beyond the attrition gate above). ``` R2.A is locked before the runner is re-run from scratch with the R2.A behavior built in. Any future change to the trigger condition, per-row marking, denominator handling, attrition threshold, or firewall is a re-registration, not a refinement. ## Engineering Note: R2.C Append-Per-Row Discipline (Locked 2026-05-24) Companion to R2.A. Not a form-lock amendment (does not change any feature, binning, df, or verdict), but a runner-engineering discipline that v0.7a operations are committed to: ```text v0.7a-r2.c append-per-row + resume: per-row append: the runner writes each completed row's record to per_row_table.csv immediately after the row is processed (or marked integration_blocked). A crash mid-run preserves all rows completed before the crash on disk. resume mode: on startup, the runner reads any existing per_row_table.csv at the output path and skips re-processing rows already present. Completion-by-identity (label, index) is the match key. This guarantees that compute already spent on rows 1..k is not lost if the runner restarts. separate amendment to manifest emission: the manifest.json is still written at the END of the run (after all rows are processed or marked blocked) as the audit verdict finalization. A crash before manifest emission means re-running with resume mode produces the manifest on the next clean completion. ``` R2.C is engineering; it does not change v0.7a's substantive verdict shape. It is recorded here for transparency. ## What v0.7a Is v0.7a is a **catalog-side audit**, NOT a predictor (continues v0.5a/v0.6a/v0.6b audit-not-predictor framing): ```text Test: chi-squared independence of velocity-fraction quartile Q_vf vs S/U over 273 supp-B rows. Default df: 3 (or occupied_bins - 1 under sparse-cell fallback). Critical: chi-squared(3) at p = 0.01 = 11.34 (or chi-squared(df) at p = 0.01 if df != 3). Pass: chi^2 > critical AND alignment-tightness <= 0.8 -> velocity_fraction_passes_audit Warn pass: chi^2 > critical AND alignment-tightness > 0.8 -> velocity_fraction_passes_audit_alignment_warning Fail: chi^2 <= critical -> velocity_fraction_fails_audit Sparse: any occupied bin with N < 2 -> velocity_fraction_inconclusive_sparse (fallback inherited from v0.6b) ``` A clean pass licenses v0.7b: a separately-registered held-out velocity-fraction predictor with the v0.5b discipline. A pass with alignment warning licenses only an alignment-breaking v0.7b registration (mirroring v0.6b's within-branch pattern). A fail closes the velocity-fraction sub-question; the D1 + A sidecar's status is report-only regardless. ## The Locked Form ### Operational definition (D5 velocity-fraction) ```text 1. Per row, compute the monodromy M_i = exp(integral over one period of the Jacobian of the Hamiltonian flow at the periodic orbit). - INPUT: published mass triple (1, 1, m_3), published initial conditions (r_i, v_i for i = 1..3), published period T. - METHOD: variational integration of the three-body flow's Jacobian over one period. - PROVENANCE: same computational substrate as v0.4a's two-pass classifier. v0.7a may reuse the v0.4a numerics if and only if the disallowed-feature audit is preserved (no K_fib tangent decomposition is used here; only the full 18 x 18 monodromy). 2. Compute the Floquet eigenstructure of M_i: - All 18 eigenvalues and right eigenvectors. - DOUBLE PRECISION numpy.linalg.eig. 3. Identify gamma_1(row): - Primary rule: gamma_1 = right eigenvector of the eigenvalue with LARGEST REAL PART. - This selection is the SAME as D1's selection rule. The non-circularity-preserving move is that v0.7a uses the resulting eigenvector ONLY as a geometric direction; the eigenvalue itself is never used as a feature. - Tie-break cascade for degenerate eigenspaces (S-row case where the largest-real-part eigenvalue is unit-modulus and degenerate): a) within the degenerate eigenspace, choose the vector with maximal projection onto the velocity subspace under the mass-weighted norm (see Reference Frame below); b) then smallest absolute imaginary part of the eigenvalue; c) then smallest positive argument; d) then deterministic lexicographic sign convention (first non-zero entry positive). - All four tie-break steps are pre-data; no run-time tuning. 4. Reduce gamma_1 to the center-of-mass reduced frame: - Subtract the mass-weighted average over the 3 bodies in each of the position and velocity sub-blocks. - Apply the mass-weighted norm: for delta-q components, ||delta_q||^2 = sum_{i=1..3} m_i * ||delta_q_i||^2; for delta-v components, ||delta_v||^2 = sum_{i=1..3} m_i * ||delta_v_i||^2. - This normalization preserves the physical-units balance between position and velocity components and removes the dependence on body-3's mass scaling. 5. Compute the velocity-fraction: - velocity_fraction(row) = ||delta_v||^2 / (||delta_q||^2 + ||delta_v||^2) - velocity_fraction in [0, 1] for any non-zero gamma_1. - velocity_fraction = 0 means gamma_1 is purely positional; = 1 means purely velocity. ``` ### Audit form (B velocity-fraction quartile) ```text binning: quartiles over the supp-B catalog's 273-row velocity_fraction distribution. cutpoints: q25, q50, q75 of the supp-B velocity_fraction array, computed via numpy.quantile(values, [0.25, 0.50, 0.75], method='linear'). bin assignment: right-closed lower intervals, ties to lower bin (matches v0.6a convention exactly). Q_vf(r) = 1 iff vf(r) <= q25 Q_vf(r) = 2 iff q25 < vf(r) <= q50 Q_vf(r) = 3 iff q50 < vf(r) <= q75 Q_vf(r) = 4 iff vf(r) > q75 contingency: 4 x 2 table (Q_vf, stability) with stability in {S, U}. test: chi-squared statistic over 273 rows; df = occupied_bins - 1 (= 3 if all 4 quartiles occupied, which is structurally guaranteed for quartile binning on 273 rows). critical: 11.34 at p = 0.01 (chi-squared(3)). alignment guard: max over Q_vf of (fraction of bin in any single v0.5a branch_label bucket). threshold = 0.8 (v0.6 inheritance). sparse-cell fallback: v0.6b discipline: if min_occupied_bin_count < 2: -> velocity_fraction_inconclusive_sparse elif min(expected cell count) < 5: -> exact permutation test (seed = 20260523, n_permutations = 10000) else: -> asymptotic chi-squared(df). For quartile binning on 273 rows the chi-squared branch is overwhelmingly likely; the fallback is documented for completeness. ``` ### Non-circularity sentence (locked) > v0.7a uses Floquet eigenvectors only as geometric directions; it does > not use eigenvalue magnitude, spectral radius, unit-circle status, > unstable-pair count, or any threshold that defines the published > S/U label. The tested scalar is a phase-space composition ratio of > the selected direction, not the growth rate of that direction. This sentence is the load-bearing argument addressing the parent registration's three circularity risks. Re-stating in the parent's risk taxonomy: - **Eigenvalue-choice circularity:** Addressed. The selection rule (largest real part) is pre-data and applies to every row. The feature extracted (velocity-fraction) is independent of which eigenvalue was selected; it depends only on the eigenvector's phase-space composition. - **Well-definedness circularity:** Addressed. gamma_1 is defined for ALL 273 rows under the tie-break cascade. There is no "gamma_1_undefined" condition that could correlate with stability. - **Feature-extraction circularity:** Addressed. The velocity-fraction is a geometric ratio in [0, 1] that does not encode growth rate, eigenvalue magnitude, or stability count. It is the same scalar whether the eigenvalue is unit-modulus or growing. ## Pre-Audit Sanity Surface Before v0.7a runs: ```text 1. Per-row monodromy M_i computed for all 273 rows via variational integration of the three-body flow at the published initial conditions. 2. Sanity check on the symplectic structure: - Verify M_i^T J M_i = J (where J is the 18 x 18 symplectic form) to the **R1-amended** tolerance of 1e-4 over the maximum entry of the residual. - Block v0.7a runner if any row violates symplecticity at this tolerance; the variational integration is suspect. - (Original tolerance 1e-6; amended to 1e-4 on 2026-05-24 after the 7-row vectorized smoke. See "Amendment R1: Sanity-Gate Threshold" above.) 3. Sanity check on the Floquet eigenstructure: - Verify that the eigenvalues come in reciprocal pairs (Floquet theorem for symplectic matrices) to a registered tolerance of 1e-4 on max |lambda * lambda_conj_recip - 1|. - Block v0.7a runner if violated. 4. Sanity check on gamma_1 uniqueness: - For each row, verify that the largest-real-part eigenvalue is unique (i.e., separated from the next eigenvalue by > 1e-6 in real part) OR record the tie-break cascade decision on the receipt. - Rows requiring tie-break cascade are flagged but not blocked; the cascade is deterministic. 5. Constant-feature retirement check: - Compute velocity_fraction distribution over 273 rows. - If sd(velocity_fraction) < 0.01 (i.e., near-constant), the constant-feature retirement rule fires: verdict `velocity_fraction_retired_near_constant`, no audit run. - Threshold (0.01) is locked pre-data. ``` Sanity surface receipts are emitted regardless of audit outcome. ## Non-Circularity Audit (Full Argument) ### Why eigenvalue-choice circularity is not triggered The selection rule (largest real part) is pre-data. For S rows, the largest-real-part eigenvalue is unit-modulus (real part = 1 with possible degeneracy). For U rows, the largest-real-part eigenvalue is the unstable Floquet multiplier (real > 1). The selection rule DOES end up picking physically different objects for S vs U rows, but: ```text - The selection rule is DETERMINISTIC and applied UNIFORMLY before the stability label is consulted. - The selected eigenvalue's magnitude, real part, or growth rate is NEVER used as a feature; it is used only to disambiguate which eigenvector to extract. - The eigenvector itself is a geometric vector in phase space; its composition (velocity-fraction) is a geometric property independent of eigenvalue ordering rules. ``` The substantive question v0.7a is asking is: do the eigenvectors selected by this uniform rule have DIFFERENT velocity-fractions for S vs U rows? That is, holding the selection rule fixed across all rows, does the resulting geometric direction stratify stability? A passing audit would say: yes, the eigenvectors selected by the largest-real-part rule have systematically different velocity-fractions for S vs U rows. The S-row eigenvectors (unit-modulus eigenvalue eigenvectors) and U-row eigenvectors (unstable eigenvalue eigenvectors) sit in different geometric regions of phase space. A failing audit would say: the velocity-fractions are indistinguishable between S and U rows, or are stratified in a way that does not reach the registered p <= 0.01 floor. Neither outcome trivializes the stability label. ### Why well-definedness circularity is not triggered gamma_1 is defined for all 273 rows under the deterministic tie-break cascade. The cascade handles the S-row eigenvalue degeneracy without introducing a "gamma_1_undefined" condition that could correlate with stability. The cascade is: ```text Step 1: largest real part of eigenvalue (the D1 selection). - U rows: unique unstable eigenvalue. No tie. - S rows: typically degenerate at real part = 1. Step 2 (degenerate eigenspace only): within the eigenspace, pick the vector with maximal projection onto the velocity subspace. - This is well-defined: project each basis vector of the degenerate eigenspace onto the velocity subspace under the mass-weighted norm, pick the one with maximal projection. Step 3 (further tie at step 2): smallest absolute imaginary part. Step 4 (further tie): smallest positive argument. Step 5 (further tie): lexicographic sign convention -- first non-zero entry positive. ``` Every row terminates the cascade at some step. The eigenvector is returned deterministically. ### Why feature-extraction circularity is not triggered velocity_fraction is a geometric ratio in [0, 1]: ```text vf = ||delta_v||^2 / (||delta_q||^2 + ||delta_v||^2) ``` - It is independent of the eigenvector's normalization (numerator and denominator scale together). - It is independent of the sign convention (squared norms). - It does not encode growth rate, eigenvalue magnitude, or stability count. - For any 6D phase-space vector, vf is a well-defined number. The audit's verdict is therefore a statement about the geometric distribution of gamma_1 directions in phase space, not about whether the orbit is stable. ## Free Parameter Count ```text Number of free parameters fitted to the data: 0 ``` The feature (velocity-fraction), the selection rule (largest real part), the tie-break cascade, the reference frame (center-of-mass reduced with mass-weighted norm), the binning rule (quartiles), the quantile method (numpy linear interpolation), the bin assignment (right-closed lower), the df (3), the critical value (11.34), the alignment-tightness threshold (0.8), the constant-feature retirement threshold (sd < 0.01), and the sparse-cell fallback parameters (min_expected_floor = 5, min_occupied_bin_count = 2, permutation_seed = 20260523, n_permutations = 10000) are all locked pre-data. ## Sparse-Cell Fallback Logic (Inherited from v0.6b) ```text 1. Compute expected cell counts E_{kj} = N_k * C_j / 273. 2. Compute min_occupied_bin_count = min over occupied bins of N_k. 3. Branch: a) If min_occupied_bin_count < 2: verdict = velocity_fraction_inconclusive_sparse do NOT report a p-value. b) Elif min over occupied cells of E_{kj} < 5: permutation test with locked seed = 20260523, n_permutations = 10000, same chi-squared statistic; p_value = empirical tail. c) Else: asymptotic chi-squared(df = occupied_bins - 1) at p = 0.01. ``` For quartile binning on 273 rows with 97 S / 176 U, the expected counts per cell are approximately {24, 44} -- well above the asymptotic floor. The fallback is documented for completeness but will not fire under normal conditions. ## Alignment-Tightness Guard (Inherited from v0.6a) ```text For each Q_vf in {1, 2, 3, 4}: - identify the v0.5a branch_label that dominates the bin. - compute dominant_fraction = (rows in bin in dominant branch) / (rows in bin). alignment_tightness_scalar = max over Q_vf of dominant_fraction. Threshold: 0.8. If alignment_tightness_scalar > 0.8 AND chi^2 > 11.34: verdict = velocity_fraction_passes_audit_alignment_warning -> v0.7b must use an alignment-breaking partition (mirroring v0.6b's within-branch approach). ``` ## Sidecar: D1 + A (Report-Only) Under the SAME monodromy and gamma_1-selection pipeline (the selection rule is the same; only the feature differs), the runner ALSO emits the D1 + A direction-projection quartile audit: ```text sidecar feature: gamma_1 projected onto a pre-registered reference axis. reference axis: center-of-mass-frame z-axis (the out-of-plane direction; aligned with the catalog's z_0 coordinate convention). sidecar scalar: |proj_z(gamma_1)| (absolute value of the z-component of gamma_1 under mass-weighted norm normalization). binning: quartiles over supp-B's 273-row |proj_z(gamma_1)| distribution. contingency: 4 x 2 over (Q_proj_z, S/U). test: same chi-squared / sparse-cell pipeline. REPORT-ONLY: NO VERDICT IS CLAIMED FROM THE SIDECAR. The sidecar's chi-squared and contingency are recorded on the receipt for diagnostic and motivation purposes only. re-registration: if v0.7a primary fails but D1+A sidecar shows chi^2 > 11.34 at p <= 0.01, the loud sidecar is FLAGGED but NOT promoted directly. It requires a fresh circularity audit (selection-rule contamination check) before a new D1+A form lock can be registered. Reason: D1+A's selection rule uses the eigenvalue's largest real part, which is closer to the stability boundary than D5's feature. Any positive D1+A signal must be re-audited against the selection-rule-contamination risk. ``` The sidecar's purpose is to surface "loud signal from the physicist's instinct version" cheaply, without licensing it as a verdict under this registration. ## Reserved Diagnostics (Not Gating) The v0.7a runner emits per-row and aggregated diagnostics: ```text per-row table: (label, m_3, z_0, period, stability, branch_label, gamma_1_eigenvalue_real, gamma_1_eigenvalue_imag, gamma_1_velocity_fraction, gamma_1_z_projection, gamma_1_tie_break_step_used, Q_vf, Q_proj_z) aggregated: velocity_fraction histogram over 273 rows. velocity_fraction by (m_3, z_0) branch joint table. velocity_fraction by m_3 joint table. velocity_fraction by S/U distribution summary. symplecticity_residual distribution (sanity diagnostic). reciprocal_pair_residual distribution (sanity diagnostic). tie_break_step_used count by row. ``` ## Edge Cases (Pre-Registered) 1. **Symplecticity sanity gate fail**: block v0.7a; inspect the variational integrator before re-running. 2. **Reciprocal-pair sanity gate fail**: same gate as #1. 3. **Constant-feature retirement fires (sd(vf) < 0.01)**: verdict `velocity_fraction_retired_near_constant`. Possible physical cause: the velocity-fraction is structurally fixed by the Hamiltonian flow's tangent dynamics (e.g., a symplectic constraint forcing vf = 0.5 for all rows). The chapter closes on velocity-fraction; v0.7b is not licensed. v0.7c may register a different feature. 4. **Tie-break cascade reaches step 4 on > 50% of rows**: the selection rule is highly degenerate on this catalog. Audit proceeds but the receipt flags `tie_break_dominance`; the verdict's interpretive weight is reduced. 5. **Alignment-tightness > 0.95**: verdict `velocity_fraction_passes_audit_severe_alignment` (a stronger version of the standard alignment-warning); v0.7b registration is mandated alignment-breaking, AND the v0.6 pre-mortem residual alignment trace must be documented in v0.7b's form lock. 6. **gamma_1 is purely real for some rows, complex for others**: acceptable. The selection rule and tie-break cascade handle both cases. The velocity-fraction is computed on the real part of the eigenvector for complex eigenvalues; this convention is locked here (any post-hoc switch to a complex-magnitude convention is a re-registration). ## Pre-Mortem Expectation The per-row velocity_fraction distribution has NOT been computed before lock-in. The audit runs blind. Expected published statement, regardless of outcome: > v0.7a tests whether the supp-B catalog's gamma_1 velocity-fraction > stratifies stability under the locked operational definition. > The verdict is one of `velocity_fraction_passes_audit`, > `velocity_fraction_passes_audit_alignment_warning`, > `velocity_fraction_fails_audit`, > `velocity_fraction_inconclusive_sparse`, or > `velocity_fraction_retired_near_constant` per the locked threshold > tree. Pre-mortem note: symplectic Hamiltonian dynamics may force gamma_1's velocity-fraction toward a characteristic value across all rows (e.g., a Floquet eigenvector of a symplectic matrix has constraints on its phase-space composition). If the constant-feature retirement fires, that itself is a methodological finding worth documenting. ## Stop Conditions 1. **Non-circularity audit failure at runner time.** If the implementation cannot maintain the locked non-circularity discipline (e.g., the tie-break cascade requires inspecting eigenvalue magnitude in some unforeseen way), the runner aborts with `velocity_fraction_blocked_circularity_audit`; no compute verdict is licensed. 2. **Sanity gate failures (symplecticity, reciprocal pair)**: block. 3. **Constant-feature retirement**: chapter closes on velocity-fraction; v0.7b not licensed. 4. **Permutation test required (min_expected < 5)**: not a stop; the fallback proceeds. 5. **Tie-break-cascade dominance**: not a stop; flag on receipt. ## Lock-In Statement This audit form is committed before: - per-row M_i has been computed under any variational integrator, - per-row gamma_1 has been extracted under the locked selection rule and tie-break cascade, - per-row velocity_fraction values have been computed, - the supp-B quartile cutpoints for velocity_fraction have been calculated, - the alignment-tightness scalar has been computed, - the D1 + A sidecar has been computed. Any change to the operational definition, selection rule, tie-break cascade, reference frame normalization, feature definition, binning rule, quantile method, df formula, critical value, alignment-tightness threshold, constant-feature retirement threshold, sparse-cell fallback parameters, permutation seed, sidecar configuration, or non-circularity sentence after the v0.7a runner is executed is a **re-registration**, not a refinement. Implementation may proceed against this form lock using: ```text - supp-B published catalog (initial conditions, masses, periods) - v0.4a manifest at results/isotrophy/k-facet-v04a-domain-map/manifest.json for cross-validation of the integration substrate - v0.5a per_row_table.csv at results/isotrophy/k-facet-v05a-branch-map/per_row_table.csv for the alignment-tightness check ``` as inputs. New compute is required: per-row variational integration of M_i (was NOT in v0.4a's emitted scalars; v0.4a computed K_fib basis but not full 18 x 18 eigenstructure). ## Receipt Schema (Planned) ```text results/isotrophy/k-facet-v07a-velocity-fraction-audit/ manifest.json - mode = "v0.7a-velocity-fraction-audit" - form_lock = "docs/isotrophy/kfacet/kfacet_v07a_velocity_fraction_audit_form.md" - input_catalog_supplementary_b - input_v04a_manifest - input_v05a_per_row_table - operational_definition = "D5_velocity_fraction_of_largest_real_part_eigenvector" - selection_rule = "largest_real_part_eigenvalue" - tie_break_cascade_steps: [ "max_velocity_projection_within_eigenspace", "smallest_absolute_imaginary_part", "smallest_positive_argument", "lexicographic_sign_convention_first_nonzero_positive" ] - reference_frame = "center_of_mass_reduced_mass_weighted_norm" - quartile_method = "numpy_quantile_linear" - bin_assignment_convention = "right_closed_lower_ties_to_lower" - constant_feature_sd_threshold = 0.01 - alignment_tightness_threshold = 0.8 - alignment_severe_threshold = 0.95 - asymptotic_expected_floor = 5 - min_occupied_bin_count_for_test = 2 - permutation_seed_locked = 20260523 - n_permutations_locked = 10000 - chi2_critical_at_p01_df3 = 11.34 - sanity_check_symplecticity: { tolerance, max_residual, status } - sanity_check_reciprocal_pair: { tolerance, max_residual, status } - sanity_check_gamma_1_uniqueness: { unique_count, tie_break_used_count, by_step } - constant_feature_check: { sd_velocity_fraction, retired_status } - cutpoints_vf: { q25, q50, q75 } - contingency_vf: 4 x 2 (Q_vf, S/U) - chi_squared_vf: test statistic - df_vf: 3 (or occupied_bins - 1) - critical_vf: 11.34 - test_branch_taken: "chi_squared" | "permutation" | "inconclusive_sparse" | "retired_near_constant" | "blocked_sanity" - p_value_vf: chi-squared upper-tail p OR permutation p - alignment_tightness_scalar_vf: max bin-alignment vs v0.5a - alignment_records_vf: per-bin dominant branch + counts - verdict: velocity_fraction_passes_audit | velocity_fraction_passes_audit_alignment_warning | velocity_fraction_passes_audit_severe_alignment | velocity_fraction_fails_audit | velocity_fraction_inconclusive_sparse | velocity_fraction_retired_near_constant | velocity_fraction_blocked_sanity | velocity_fraction_blocked_circularity_audit - sidecar_D1_A: - feature = "gamma_1_z_projection_magnitude" - reference_axis = "center_of_mass_frame_z_axis" - cutpoints, contingency, chi_squared, p_value, alignment_tightness - sidecar_status = "report_only" - diagnostic_per_row_table - diagnostic_vf_by_m3 - diagnostic_vf_by_branch - diagnostic_vf_by_Q_E (v0.6a quartile) - diagnostic_vf_by_Q_L (v0.6a quartile) per_row_table.csv contingency_table_vf.csv sidecar_D1_A_contingency.csv velocity_fraction_by_m3.csv velocity_fraction_by_branch.csv velocity_fraction_by_Q_E.csv ``` ## Implementation Plan Suggested script: ```text scripts/v07a_velocity_fraction_audit.py ``` Suggested command: ```powershell python scripts\v07a_velocity_fraction_audit.py ` --catalog docs\isotrophy\supplementary-B_piano-init-condit-3d.txt ` --v04a-manifest results\isotrophy\k-facet-v04a-domain-map\manifest.json ` --v05a-per-row results\isotrophy\k-facet-v05a-branch-map\per_row_table.csv ` --out results\isotrophy\k-facet-v07a-velocity-fraction-audit ``` Expected runtime: minutes (per-row variational integration of the 18-dimensional tangent system over one period times 273 rows). Numerical precision must match v0.4a's two-pass classifier discipline to preserve the disallowed-feature inheritance audit. ## Interpretation If v0.7a passes (chi^2 > 11.34, alignment <= 0.8): > The supp-B catalog's gamma_1 velocity-fraction stratifies stability > under a pre-registered audit that uses only the geometric direction > of the largest-real-part Floquet eigenvector (not its growth rate). > The orbit-dynamics shadow carries stability information not visible > in the v0.4 / v0.5 / v0.6 catalog-coordinate or symmetry-shadow > projections. This licenses v0.7b: a separately-registered held-out > velocity-fraction predictor with the v0.5b discipline. If v0.7a passes with alignment warning (chi^2 > 11.34, alignment in (0.8, 0.95]): > Pass-but-suspect. The velocity-fraction quartiles track (m_3, z_0) > branch tightly on supp-B, so the in-sample signal is plausibly > entangled with the bin-locality phenomenon v0.5/v0.6 exposed. > v0.7b's partition must be re-registered alignment-breaking > (within-branch test, mirroring v0.6b). If v0.7a passes with severe alignment (alignment > 0.95): > Same as above, but with v0.6 pre-mortem residual alignment trace > documented as part of v0.7b's form lock. If v0.7a fails (chi^2 <= 11.34): > The gamma_1 velocity-fraction does not stratify stability on > supp-B under the locked univariate quartile audit. The chapter > sub-question on velocity-fraction closes; the D1+A sidecar's > chi-squared and the v0.7 mechanism family's remaining projections > (D1 with fresh audit, D3 with definedness guard, D4 Z_2-isotypic > split with v0.4 inheritance audit) are open for separate > registration if any sidecar signal warrants pursuit. If v0.7a retires near-constant: > velocity_fraction is structurally fixed by the Hamiltonian flow > on this catalog; no audit can distinguish S from U on this scalar. > The chapter closes; v0.7b is not licensed; the D1+A sidecar's > chi-squared (if any) is informational only. ## Doc Trail - `kfacet_v07_mechanism_preregistration.md` -- v0.7 parent registration. - `kfacet_v06_writeup.md` -- v0.6 chapter close (predecessor). - `kfacet_v06a_energy_quartile_audit_form.md` -- alignment-tightness guard pattern. - `kfacet_v06b_within_branch_energy_audit_form.md` -- sparse-cell fallback tree pattern. - `kfacet_v04a_domain_map_preregistration.md` -- monodromy computational substrate. --- Audit floor, not predictor. Velocity-fraction quartile, df = 3, critical 11.34. D1 + A direction-projection sidecar report-only. Selection rule = largest real part with mass-weighted tie-break cascade; feature = velocity-fraction (geometric ratio, not growth rate); non-circularity addressed by separating SELECTION from FEATURE.