# v0.8 Mechanism Preregistration: Floquet Direction-Purity Status: **REGISTERED 2026-05-24. v0.8a child verdict landed 2026-05-24: `purity_quartile_fails_audit`** at chi^2 = 4.94 (p = 0.176, alignment 0.587, sd(purity) = 0.096). The unsigned purity transform `abs(vf - 0.5)` does NOT capture the v0.7a' positive at the registered floor; the v0.7a' signal lives in the **signed** direction (diagnostic purity_signed contingency reproduces chi^2 = 16.43 exactly). The failure is explained by the vf distribution's density asymmetry (184/250 rows below vf = 0.5; max vf = 0.87, no rows near vf = 1) -- the purity symmetry assumption is violated by the catalog's actual distribution. v0.8b (held-out predictor) is NOT licensed under the fails verdict. See `kfacet_v08a_purity_quartile_audit_form.md`. This document locks the v0.8 body / projection / observable and the non-circularity discipline. Form locks for v0.8a (audit: purity-binned audit or monotone purity predictor) and v0.8b (held-out predictor with attrition carried as domain restriction) are separately registered children. Audience: v0.8a/v0.8b form-lock author; paper-side reviewer of the v0.7 -> v0.8 transition (qualified-positive close + direction-purity mechanism open). Companions: - `kfacet_v07_writeup.md` -- v0.7 chapter close (qualified-positive on restricted domain) that opened v0.8. - `kfacet_v07a_velocity_fraction_audit_form.md` -- v0.7a parent audit; source of the velocity-fraction operational definition and the locked non-circularity sentence (carries forward unchanged). - `kfacet_v07a_prime_restricted_scope_form.md` -- v0.7a' restricted- scope confirmation; source of the analyzable subset (250 rows) and the U-shaped direction-purity signature. - `kfacet_v05b_branch_predictor_form.md` -- v0.5b held-out predictor pattern; inheritance source for v0.8b's discipline. Frame: v0.7a' produced a statistically-significant (chi^2 = 16.43, p = 9.3e-4) branch-independent (alignment 0.698) **non-monotone U-shaped** signal on the velocity-fraction quartile contingency: Q1 (gamma_1 mostly positional) 49% S, Q2 (mixed, slightly velocity) 18% S, Q3 29% S, Q4 (gamma_1 mostly velocity) 43% S. The U-shape suggests the load-bearing axis is **direction-purity**, not raw velocity-fraction. v0.8 promotes purity to the primary operational quantity and asks whether it predicts held-out stability on the analyzable subset. ## What v0.8 Is Not v0.8 is NOT: - **A retreat from the v0.7a non-circularity sentence.** The non-circularity sentence carries forward verbatim: Floquet eigenvectors are used as geometric directions only; eigenvalue magnitude and stability thresholds never enter the feature. The purity scalar `abs(vf - 0.5)` is a deterministic function of vf, inheriting vf's non-circularity provenance. - **A v0.7c that promotes the sidecar.** The v0.7a' z-fraction sidecar (D1 + A) did NOT pass the loud-signal threshold and is not promoted under v0.8. If the v0.8 chapter wants to extend to z-direction structure, that is a separate fresh registration with its own circularity audit. - **A catalog-wide claim.** The domain is fixed at the 250 analyzable rows from v0.7a (rows with `integration_blocked == False`). The 23 integration-blocked rows + 11 R1-sanity-failed rows are **not** in the v0.8 domain. Attrition is recorded permanently on every v0.8 receipt; v0.8 makes no claims about rows outside the analyzable subset. - **A re-binning of the v0.7a' Q_vf quartiles.** v0.8 uses purity as a fresh feature, with its own binning rule locked at v0.8a. Borrowing v0.7a' quartile cutpoints would conflate the registered- positive feature (vf quartile) with the proposed mechanism (purity). v0.8 needs its own binning to evaluate purity on its own terms. ## Body / Projection / Observable ```text Body: the 250-row analyzable supp-B subset from v0.7a (catalog rows where compute_monodromy_vectorized completed successfully at rtol = atol = 1e-12, max_step_fraction = 0.02). Attrition carried as a permanent domain restriction. Projection: orbit-dynamics direction-purity shadow row -> purity(row) = abs(vf(row) - 0.5) where vf is the v0.7a velocity-fraction (gamma_1's mass-weighted velocity-norm divided by total mass-weighted norm under CoM reduction), inheriting v0.7a's selection rule, tie-break cascade, and reference frame. Equivalently: purity = min(vf, 1 - vf) reversed and shifted (purity high <-> direction near a pure end {gamma_1 mostly positional or gamma_1 mostly velocity}; purity low <-> direction maximally mixed at vf = 0.5). Observable: per-row pair (purity, stability) on 250 analyzable rows, joined with the v0.5a branch_label for alignment-tightness diagnostics. Per-row computational cost: zero. purity is derived from existing v0.7a per_row_table.csv; no new variational integration is needed. ``` The projection axis is **direction-purity**, a monotone transform of the v0.7a U-shape signature. Where v0.7a' tested whether the vf quartile distribution stratifies S/U (non-monotone signal), v0.8 tests whether purity stratifies S/U (expected-monotone signal: high purity -> high S, low purity -> low S). ## Operational Definition of Purity ```text For each analyzable row r: purity(r) = abs(vf(r) - 0.5) range: [0, 0.5] purity(r) = 0 iff vf(r) = 0.5 (maximally mixed direction) purity(r) = 0.5 iff vf(r) in {0, 1} (perfectly pure direction; numerically rare for any real Floquet eigenvector). ``` The definition is **deterministic, monotone in `abs(vf - 0.5)`, and inherits all of v0.7a's non-circularity provenance**. No new operational quantities are introduced. Equivalent forms (recorded but not used as the primary feature): ```text purity_alt(r) = 1 - 2 * min(vf(r), 1 - vf(r)) = 2 * abs(vf(r) - 0.5) ∈ [0, 1] (scaled version; same ordering) purity_signed(r) = vf(r) - 0.5 ∈ [-0.5, 0.5] (signed; preserves which "pure end") ``` `purity_signed` is **recorded as a diagnostic** for the U-shape asymmetry check (do Q1-pure orbits behave the same as Q4-pure orbits?), but is **not the primary feature**. Using the signed version as primary would silently re-test the v0.7a' Q1-vs-Q4 split, which is the conditioning the v0.7a' result already used. ## Non-Circularity Audit ```text purity is a monotone deterministic function of vf, so its non-circularity provenance is exactly v0.7a's: vf is computed from gamma_1, the eigenvector of the largest- real-part Floquet eigenvalue of M_i, under the locked tie-break cascade. The eigenvalue ORDERING is used to disambiguate which eigenvector to extract; the eigenvalue MAGNITUDE never enters the feature. vf is a geometric ratio of mass-weighted-norm components after CoM reduction; purity is the distance from the mid-point of this ratio. Neither vf nor purity uses the stability label, the Floquet spectral radius, the unstable-pair count, the Krein signature, the v0.4a Pass classification, the v0.5a branch hash output, or the v0.6 conserved-quantity quartile. The v0.5a branch_label is joined ONLY to compute the alignment- tightness scalar (a diagnostic guard against branch-shadow re-projection), not to construct purity. S/U is the held-back test column, joined at the chi-squared step. ``` The non-circularity sentence from v0.7a is **re-asserted** for v0.8: > v0.8 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 distance-from-mixed > phase-space composition ratio of the selected direction, not the > growth rate of that direction. ## Disallowed Features (Inheritance + v0.8-Specific) The v0.8 form locks may NOT use: ```text v0.4b / v0.5 / v0.6 / v0.7 inheritance: - the stability label S/U or any function of it, - any function of M_i eigenvalue magnitudes (spectral radius, unit-circle count, max |lambda|, Krein signature, etc.), - the row's v0.4a Pass 1 / Pass 2 provenance, - the v0.5a branch hash output as a feature (it remains available as the alignment-tightness comparison axis only), - Q_E / Q_|L| / any v0.6a quartile, - E or |L| as scalars (retired in v0.6 close). v0.8-specific: - the SIGNED form purity_signed as the primary feature (re-tests the v0.7a' Q1-vs-Q4 split; available as diagnostic only). - vf itself as the primary feature (v0.7a/v0.7a' already used this; v0.8 tests purity, not raw vf). - the v0.7a' Q_vf quartile assignments as features (would conflate v0.7a' positive with v0.8 mechanism test). - any quantity derived from the 23 integration-blocked rows or the 11 R1-sanity-failed rows (they are outside the v0.8 domain). ``` ## Inheritance Discipline From v0.5 / v0.6 / v0.7 v0.8 inherits all prior register-shaping disciplines: 1. **Audit-then-predictor separation.** v0.8a registers an audit on a binned (or monotone) purity feature; v0.8b separately registers a held-out predictor. An audit pass does NOT license a predictor; the predictor must pass its own pre-registered held-out gate. 2. **Asymmetric falsifier for the predictor.** v0.8b requires BOTH `McNemar p <= 0.01` AND positive accuracy delta against the always-U baseline. Inherited from v0.5b. 3. **Held-out partition discipline.** v0.8b defaults to leave-one-m_3-bin-out (over m_3 bins with N >= 5 on the 250-row analyzable subset). Alternatives must be motivated paper-side and pre-registered. Per the user's direction, **attrition is carried as part of the domain, not hidden**: the held-out folds are the m_3 bins of the analyzable subset, and the receipt explicitly records that the catalog-wide audit was attrited at 8.4%. 4. **Tie rule and absent-bin fallback.** v0.8b inherits the conservative tie rule (predict U) and the global-majority fallback for absent bins. 5. **Alignment-tightness guard.** v0.8a MUST pre-register an alignment-tightness scalar against the v0.5a branch hash with the locked 0.8 warning / 0.95 severe thresholds. The v0.7a' alignment was 0.698 -- below the 0.8 warning floor -- but the purity feature is a transform, not the same as vf; the alignment must be re-evaluated on purity quartiles (or whatever binning v0.8a locks). 6. **Sparse-cell fallback tree.** v0.8a and v0.8b inherit the v0.6b discipline (chi-squared / exact permutation / sparse-inconclusive) with locked thresholds and locked permutation seed (20260523, n_permutations = 10000). 7. **Constant-feature retirement.** If purity has sd < 0.01 on the 250-row subset, constant-feature retirement fires. Pre-mortem estimate from v0.7a': vf_sd was 0.144, so |vf - 0.5| should have sd > 0.05 (well above the floor). ## Candidate Audit Forms (Sketches for v0.8a Registration) The v0.8a form lock will pick ONE of the following (or a registered variant). All forms inherit the alignment-tightness guard, the sparse-cell fallback tree, and the constant-feature retirement threshold. ```text A. Purity-quartile audit (analogue of v0.7a's B): Quartile-bin purity over the 250-row subset. 4 x 2 contingency over (Q_purity, S/U). df = 3, critical 11.34 at p = 0.01. Expected shape if v0.7a' signature is monotone in purity: Q1 (low purity, mixed direction) -> low S_fraction Q4 (high purity, near a pure direction) -> high S_fraction Cleanest extension of the v0.7a' pattern. B. Monotone purity-threshold predictor (single-cut audit): "predict S iff purity > purity_threshold" threshold = pre-registered quantile (e.g., median, q60, q75). 2 x 2 contingency vs S/U, df = 1, critical 6.63 at p = 0.01. Coarser test but a direct monotone-direction claim. C. Continuous correlation audit: Spearman rank correlation between purity and S (treating S as binary 1, U as 0). One-sample test of H_0: rho = 0 vs H_1: rho > 0 (one-sided since v0.7a' predicts positive correlation). Closest to the "distance from mixed" intuition; no binning choice. D. Two-bin median-split audit (cleanest minimal test): Median-split purity over the 250-row subset. 2 x 2 contingency, df = 1, critical 6.63. Pre-registered direction: above-median purity has higher S_fraction. The form lock can include a binomial-tail test as a power-conservative alternative. ``` The v0.8a form lock must lock ONE of A-D (or a registered variant) BEFORE running the audit. Multiple parallel audits are NOT permitted under a single v0.8a registration. ## Candidate Predictor Forms (Sketches for v0.8b Registration) If v0.8a passes, v0.8b registers a held-out predictor. Candidate shapes: ```text E. Bin-majority on the v0.8a purity binning (mirrors v0.5b): train per-bin majority on training folds (leave-one-m_3-bin-out); predict per-bin majority on test fold. F. Threshold rule on purity (mirrors a fixed v0.8a-style cut): "predict S iff purity > purity_threshold" threshold pre-registered as a quantile of the training fold's purity distribution. G. Logistic regression on purity (1 free parameter): single-feature logistic predictor; held-out McNemar against always-U. Free parameter must be counted explicitly; df adjustments recorded. ``` v0.8b's form lock must pre-register exactly one shape and one partition (default leave-one-m_3-bin-out on the 250-row analyzable subset). Free parameters must be counted explicitly. ## Open Design Questions (Resolved in v0.8a/v0.8b Form Locks) 1. **v0.8a audit form (A/B/C/D).** Most natural extension of v0.7a' is A (purity-quartile audit), mirroring v0.7a's B audit form. D (median-split) is a power-conservative two-bin alternative. C (Spearman correlation) avoids the binning choice entirely but introduces a different test statistic. 2. **Purity binning method.** Quartile (4 bins) vs median split (2 bins) vs continuous. Pre-registered before any chi-squared computation. 3. **v0.8b partition.** Default leave-one-m_3-bin-out on the 250 analyzable rows. The m_3 bin distribution after attrition needs to be checked: some bins may have < 5 rows after attrition. v0.8a's receipt MUST report the per-m_3-bin analyzable row counts before v0.8b registration. 4. **U-shape vs monotone diagnostic.** v0.7a' showed a U-shape; v0.8 tests monotone-in-purity. v0.8a's receipt MUST emit the diagnostic purity_signed quartile contingency to check whether Q1-pure and Q4-pure orbits behave the same in S-fraction (if not, the U-shape is asymmetric and the purity transform may not be the cleanest mechanism). ## Pre-Mortem (Recorded Now) ```text Risk 1: purity could be a near-constant on the 250-row analyzable subset if vf is concentrated near 0 or 1. Mitigation: constant-feature retirement at sd < 0.01. v0.7a' vf_sd was 0.144; purity sd should be similar order (vf is roughly uniform over the subset, so purity is roughly uniform over [0, 0.5]). Risk 2: the v0.7a' U-shape may be asymmetric -- Q1-pure orbits and Q4-pure orbits may have different S-fractions for physical reasons. If asymmetric, the purity transform conflates two different physical mechanisms. Mitigation: v0.8a receipt MUST emit the purity_signed contingency as a diagnostic. If the asymmetry is large (e.g., one tail has 60% S and the other 30%), v0.8b may need re-registration with a non-purity feature. Risk 3: the attrition (23 blocked + 11 sanity-failed) is concentrated at long-period high-m_3. The held-out leave-one-m_3-bin-out partition for v0.8b on the analyzable subset will have very thin folds at high m_3 (m_3 = 1.7 has only ~5 analyzable rows). Low statistical power on those folds. Mitigation: v0.8a's receipt reports per-m_3 analyzable row counts. v0.8b may need to drop high-m_3 bins from the gating partition (recording them as report-only) if N < 5 after attrition. Risk 4: v0.8a passes but v0.8b fails (the v0.5/v0.6 pattern). This would close the chapter as another projection-limit, but with the difference that the v0.8a positive is the FIRST non-branch-aligned positive in the chain. A v0.8b fail does not undo v0.7's qualified-positive close. Mitigation: pre-mortem expectation recorded; the chapter close framing handles all outcomes cleanly. ``` ## Stop Conditions 1. **Constant-feature retirement on purity.** Chapter closes on purity; the v0.8 mechanism cannot be evaluated under this feature. 2. **v0.8a fails the chi-squared / permutation test.** Chapter closes as a structural-negative on the purity shadow. 3. **v0.8a alignment-tightness > 0.8.** v0.8b cannot inherit the default leave-one-m_3-bin-out partition; an alignment-breaking re-registration is required. 4. **v0.8b fails the held-out predictor gate.** Chapter closes as a projection-limit on the purity shadow (mirroring v0.5). 5. **U-shape asymmetry diagnostic large (signed-quartile spread > 0.2).** v0.8a's receipt flags the issue; v0.8b may register a non-purity feature instead. ## Lock-In Statement This parent registration is committed before: - any v0.8a binning has been chosen, - the purity distribution over the 250-row subset has been computed, - the per-m_3 analyzable row counts have been examined, - any v0.8b partition has been locked. The body / projection / observable, operational definition of purity, non-circularity audit framing, disallowed-feature list, and inheritance discipline are locked. Any change after the v0.8a form lock is a re-registration of the parent, not a refinement of a child form. Implementation may proceed to: 1. Compute purity = abs(vf - 0.5) for each of the 250 analyzable rows in v0.7a per_row_table.csv. Catalog-only; seconds of compute total. 2. Emit a per-row purity diagnostic table (label, m_3, z_0, vf, purity, purity_signed, stability, branch_label). 3. Emit per-m_3-bin analyzable row counts on the v0.8a receipt before any predictor partition is locked. No audit verdict is licensed until a v0.8a form lock is separately registered. ## Doc Trail - `kfacet_v07_writeup.md` -- v0.7 qualified-positive chapter close (opens v0.8). - `kfacet_v07a_velocity_fraction_audit_form.md` -- v0.7a parent audit (non-circularity sentence + amendments). - `kfacet_v07a_prime_restricted_scope_form.md` -- v0.7a' restricted- scope confirmation (source of the U-shape signature). - `kfacet_v05b_branch_predictor_form.md` -- v0.5b held-out predictor pattern (inheritance source for v0.8b). - `kfacet_v06b_within_branch_energy_audit_form.md` -- sparse-cell fallback tree (inheritance source). - `kfacet_v06a_energy_quartile_audit_form.md` -- alignment-tightness guard (inheritance source). --- Parent registration locked. v0.8a (audit) and v0.8b (held-out predictor) form locks pending. The non-circularity sentence carries forward unchanged; the domain is fixed at the 250 analyzable rows; attrition is permanent on the receipt and is carried as part of the domain, not hidden.