# v0.9 Mechanism Preregistration: Signed Floquet Direction-Composition Status: **REGISTERED 2026-05-24. v0.9a child verdict landed 2026-05-24: `signed_vf_three_zone_fails_audit_chi2`** at chi^2 = 7.42 vs critical 9.21 (df=2, p = 0.0245). The U-shape pattern check also fails (mixed > positional). Substantive finding: the v0.7a' U-shape was a quartile-boundary artifact at vf ~ 0.297; under physical cutpoint at vf = 0.25 the pattern is monotone-increasing (positional 11% S < mixed 34% S < velocity-heavy 44% S), but the 3 x 2 chi^2 does not reach the p = 0.01 floor. The v0.7a' chi^2 = 16.43 positive is NOT invalidated; the U-shape MECHANISM hypothesis is. See `kfacet_v09a_signed_vf_three_zone_form.md` for the full verdict. v0.9b (held-out predictor) is NOT licensed under the fails verdict. This document locks the v0.9 body / projection / observable and the **anti-circular framing discipline** for any future child form lock that tests vf-ordering hypotheses after v0.7a' already produced a chi^2 = 16.43 positive on a vf-ordering chi-squared of independence. Audience: v0.9a form-lock author; paper-side reviewer of the v0.8 -> v0.9 transition (purity-shadow structural-negative + signed direction-composition open). Companions: - `kfacet_v08_writeup.md` -- v0.8 chapter close (structural-negative on unsigned direction-purity) that opened v0.9. - `kfacet_v08a_purity_quartile_audit_form.md` -- v0.8a fails verdict + the density-asymmetry diagnostic that licensed signed-direction as the next hypothesis. - `kfacet_v07a_prime_restricted_scope_form.md` -- v0.7a' PASS at chi^2 = 16.43; the source observation v0.9 must NOT silently re-test. - `kfacet_v07a_velocity_fraction_audit_form.md` -- v0.7a parent audit (vf operational definition; non-circularity sentence carries forward). Frame: v0.7a' produced a chi^2 = 16.43 (p = 9.3e-4) signal on quartile-binned velocity-fraction vs S/U with alignment 0.698 against the v0.5a branch hash. v0.8a tested whether this signal is captured by unsigned direction-purity `abs(vf - 0.5)` and failed (chi^2 = 4.94, p = 0.176). The v0.8a diagnostic purity_signed contingency reproduced v0.7a' chi^2 = 16.43 exactly, confirming the signal is real but lives in the **signed** vf direction. v0.9 promotes signed direction-composition to the primary mechanism and asks: **what specific pattern in vf vs S/U is real, beyond the already-known fact that there is some pattern?** ## What v0.9 Is Not v0.9 is NOT: - **A re-test of the v0.7a' positive under a different framing.** Any chi^2 of independence test on vf-ordering (raw vf quartiles, signed quartiles, binary vf-side, log-vf, any monotone transform) reproduces v0.7a's chi^2 = 16.43 by linear-transform invariance. Such a test would silently re-confirm v0.7a' under a new name -- the kind of mechanism-laundering the pre-mortem discipline was built to prevent. - **A v0.7c held-out vf predictor.** v0.7a' was attrition-blocked on the catalog-wide audit; the chapter closed on a qualified- positive on restricted domain. Promoting v0.7a' to a held-out predictor under a new name (v0.9b vf-quartile predictor) would not be honest -- v0.7b was explicitly NOT licensed under the attrition verdict. - **A re-projection of vf onto a different feature axis.** vf itself is the load-bearing scalar; v0.9 asks what SPECIFIC STRUCTURE in vf vs S/U is testable beyond chi^2 of independence. ## Body / Projection / Observable ```text Body: the 250-row analyzable supp-B subset from v0.7a (rows where compute_monodromy_vectorized completed at R1-amended sanity gates). Inherited from v0.8 verbatim. Projection: orbit-dynamics signed direction-composition shadow row -> vf(row) = velocity-fraction of gamma_1 Same operational definition as v0.7a' (locked selection rule, tie-break cascade, CoM reduction, mass-weighted norm). Observable: per-row (vf, stability) on 250 rows joined with v0.5a branch_label for alignment-tightness diagnostics. Per-row computational cost: zero. vf values are derived from v0.7a per_row_table.csv. No new variational integration is needed. ``` The projection axis is **vf as a signed scalar in [0, 1]**, not folded. v0.9 inherits v0.7a/v0.7a's vf with no transform. ## Anti-Circular Framing Discipline (Load-Bearing) The central methodological challenge: v0.7a' has already established chi^2 of independence = 16.43 on quartile-binned vf vs S/U on this same 250-row subset. Any v0.9 test that: - Uses vf-ordering as the feature, AND - Uses chi^2 of independence as the test statistic, AND - Tests on the same 250-row subset, **reproduces v0.7a' by construction** (linear-transform invariance of chi^2 of independence under monotone reordering of feature bins). Such a test is not a fresh hypothesis test; it is a re-statement of v0.7a'. For v0.9 to be honestly non-circular, **at least one of** the three above conditions must be broken: ```text 1. Different feature derivation: Use a non-monotone transform of vf (e.g., partition into three physically-motivated zones with pre-registered cutpoints from physical reasoning, NOT from v0.7a' quartile cutpoints). 2. Different test statistic: Use a hypothesis-specific test that is NOT chi^2 of independence. Examples: Jonckheere-Terpstra monotone-trend test (asks "is the relationship monotone?"); piecewise-binomial U-shape goodness-of- fit test (asks "does the U-shape fit a specific predicted pattern?"); contingency-shape Cochran-Armitage trend test ("is there a monotone trend?"). 3. Different evaluation domain: Test on a HELD-OUT subset of the 250-row analyzable rows where the held-out partition is pre-registered (e.g., leave-one-m_3- bin-out, with the v0.7a' quartile cutpoints frozen as features from training folds). Combinations are allowed. The v0.9a form lock must SPECIFY which condition(s) it breaks and argue the non-circularity explicitly. ``` This anti-circular discipline is **load-bearing** for v0.9. A v0.9a form lock that does not break at least one of the three conditions is **rejected pre-compute** as a re-test of v0.7a'. ## Operational Definition ```text vf is inherited unchanged from v0.7a: vf(row) = ||delta_v||^2 / (||delta_q||^2 + ||delta_v||^2) where (delta_q, delta_v) is the largest-real-part Floquet eigenvector under CoM reduction + mass-weighted norm. Locked selection rule, tie-break cascade, and reference frame from v0.7a apply. vf is in [0, 1] for all 250 analyzable rows. No new transform is applied at the parent level; the v0.9a form lock will specify how vf is used (zones, trend test, held-out, etc.). ``` ## Non-Circularity Audit The v0.7a locked non-circularity sentence is **re-asserted verbatim** for v0.9: > v0.9 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. v0.9-specific anti-circular sentence (load-bearing): > v0.9 does not test chi^2 of independence on vf-ordering against > S/U on the v0.7a analyzable subset; that test has already been > performed in v0.7a' and any v0.9 repetition under different > binning labels reproduces v0.7a' by linear-transform invariance. > v0.9 tests a specific pattern hypothesis (e.g., monotone-trend > rejection, three-zone S-fraction prediction, held-out fold > classification) that v0.7a' did NOT pre-register. The v0.9a form lock must include both sentences explicitly. ## Disallowed Features (Inheritance + v0.9-Specific) The v0.9 form locks may NOT use: ```text v0.4b / v0.5 / v0.6 / v0.7 / v0.8 inheritance: - stability label S/U, - any function of M_i eigenvalue magnitudes (spectral radius, unit-circle count, max |lambda|, Krein signature, etc.), - v0.4a Pass classification, - v0.5a branch hash output (as feature; alignment-tightness diagnostic only), - Q_E / Q_|L| / any v0.6a quartile, - E / |L| as scalars (retired in v0.6), - unsigned purity = abs(vf - 0.5) as primary (v0.8a closed it). v0.9-specific: - chi^2 of independence on vf-ordering vs S/U as the PRIMARY test statistic on the 250-row subset (would re-test v0.7a'; see anti-circular discipline above). - v0.7a' quartile cutpoints as v0.9 binning (would inherit v0.7a's data-driven partition). - the v0.7a' Q_vf labels as features (would silently re-test). - any hypothesis registered AFTER reading the v0.7a' contingency that "predicts" the shape v0.7a' already observed (would be post-hoc, not pre-registered). ``` ## Inheritance Discipline From v0.5 / v0.6 / v0.7 / v0.8 v0.9 inherits all prior register-shaping disciplines AND adds the anti-circular framing as a new permanent discipline: 1. **Audit-then-predictor separation.** v0.9a registers an audit under one of the three anti-circular forms; v0.9b (if licensed by v0.9a passing) registers a held-out predictor. 2. **Asymmetric falsifier for the predictor.** v0.9b requires BOTH `McNemar p <= 0.01` AND positive accuracy delta against the always-U baseline. 3. **Held-out partition discipline.** v0.9b defaults to leave-one-m_3-bin-out on the 250-row analyzable subset. 4. **Tie rule + absent-bin fallback.** Inherited from v0.5b. 5. **Alignment-tightness guard.** v0.9a MUST pre-register an alignment-tightness scalar against the v0.5a branch hash (thresholds 0.8 warning / 0.95 severe). 6. **Sparse-cell fallback tree.** Inherited from v0.6b (seed 20260523, n_permutations 10000). 7. **Constant-feature retirement.** vf_sd on the 250 rows was 0.144 in v0.7a; retirement at sd < 0.01 is structurally unlikely. 8. **Anti-circular framing (v0.9-new).** Any v0.9a form lock must break at least one of (feature derivation, test statistic, evaluation domain) relative to v0.7a's chi^2-of-independence- on-vf-quartiles test. ## Candidate v0.9a Audit Forms (Sketches) The user-specified candidates: ```text A. Ordered quartile audit (Q1/Q2/Q3/Q4 labels preserved). STATUS: REJECTED under anti-circular discipline. Reason: chi^2 of independence on ordered vf quartiles vs S/U reproduces v0.7a's chi^2 = 16.43 by construction. The "ordered" label adds no new test; the statistic does not change under re-labeling. B. Binary threshold audit at vf < 0.5 vs vf >= 0.5. STATUS: REJECTED under anti-circular discipline. Reason: collapses the v0.7a' Q1+Q2 vs Q3+Q4 contingency. The binary chi^2 (df = 1) is determined by the same sufficient statistics that produced v0.7a's chi^2 = 16.43. Re-test under coarser binning. C. Three-zone audit (positional-pure / mixed / velocity-heavy). STATUS: LIVE CANDIDATE if zone boundaries are pre-registered with PHYSICAL MOTIVATION, NOT data-driven from v0.7a's quartile cutpoints. Example: vf in [0, 0.2) (positional-dominant), [0.2, 0.6) (mixed), [0.6, 1] (velocity-heavy). Cutpoints (0.2, 0.6) would need physical justification (e.g., a Hamiltonian-flow-specific threshold or a Floquet-theory-derived boundary). 3 x 2 contingency, chi^2(2) at p = 0.01, critical 9.21. STILL TESTS CHI^2 OF INDEPENDENCE; the non-circularity rests ENTIRELY on the cutpoints being non-data-driven. D. Ordinal trend + nonmonotonicity test. STATUS: LIVE CANDIDATE (cleanest non-circular shape). Use a Jonckheere-Terpstra test of "is vf vs S/U monotone?" -- the v0.7a' U-shape predicts FAIL-TO-FIT-MONOTONE. This is a hypothesis-specific test, NOT chi^2 of independence. Verdict: pass if Jonckheere-Terpstra fails to reject H_0 (vf and S/U are not monotonically related); fail if monotone fit succeeds (would contradict the U-shape). df / critical: pre-registered from the test's null distribution. Recommendation: D is the cleanest anti-circular path. C is allowed ONLY if the zone boundaries are explicitly justified physically. A and B are rejected. ``` ## Open Design Questions (Resolved in v0.9a Form Lock) 1. **v0.9a form choice (C/D).** D is the cleanest anti-circular form. C is allowed with explicit physical motivation for zone boundaries. 2. **If D: which monotone-trend test?** Candidates: Jonckheere- Terpstra, Cochran-Armitage trend test on ordered categories, Mann-Kendall on the per-row (vf, S) pairs. Each has different power characteristics. 3. **If C: physical motivation for zone boundaries?** Must be pre-registered with explicit physical/Hamiltonian rationale, NOT data-driven from v0.7a' quartile cutpoints. 4. **Predictor licensing.** A pass at v0.9a licenses v0.9b (held-out predictor on the 250-row subset). The held-out discipline is straightforward (mirrors v0.5b); the audit shape is what makes v0.9 honest. ## Pre-Mortem (Recorded Now) ```text Risk 1: v0.9a passes by repeating v0.7a' under a different name. Mitigation: anti-circular discipline rejects forms A, B outright. The v0.9a form lock must include both non-circularity sentences explicitly. Receipts must record which of (feature, statistic, domain) was broken. Risk 2: v0.9a fails because the U-shape is real but our specific pattern hypothesis (e.g., "vf < 0.2 has 50% S") doesn't match the actual data distribution. Mitigation: this would be a clean structural-negative result; the chapter closes on the hypothesis-specific test, not on the U-shape itself. v0.9c could register a different specific hypothesis. Risk 3: The U-shape is real but is bin-locality-driven (some m_3 sub-bin produces the high-S Q1 + high-S Q4 pattern by chance). v0.7a' alignment was 0.698 against branch_label (below 0.8), but the alignment was on (m_3, z_0) buckets, not on m_3 alone. Mitigation: v0.9a must report a per-m_3 vf-pattern diagnostic (S_fraction by vf-zone, stratified by m_3) before any verdict is licensed. Risk 4: The 250-row subset is biased toward shorter-period orbits (attrition concentrated at high-m_3 long-period). The U- shape may be specific to the short-period regime. Mitigation: per-m_3 analyzable counts are on the v0.8a receipt; v0.9b's held-out partition will see thin folds at high m_3. ``` ## Stop Conditions 1. **v0.9a form lock fails anti-circular audit.** Form rejected pre-compute; chapter closes immediately on the v0.9 hypothesis. 2. **v0.9a constant-feature retirement.** Structurally unlikely given vf_sd = 0.144 on the 250-row subset. 3. **v0.9a alignment-tightness > 0.8.** v0.9b cannot inherit the default leave-one-m_3-bin-out partition. 4. **v0.9a fails the hypothesis-specific test.** Chapter closes as a structural-negative on the specific pattern hypothesis. v0.9c (a different specific hypothesis) may be registered separately, but the user should weigh bandwidth tradeoff against Phase 15+. 5. **v0.9b fails the held-out gate.** Chapter closes as a projection-limit (mirroring v0.5) on the signed direction- composition mechanism. ## Lock-In Statement This parent registration is committed before: - any v0.9a form lock has been authored, - any specific pattern hypothesis has been pre-registered, - the per-m_3 vf-pattern diagnostics have been examined. The body / projection / observable, the operational definition of vf (inherited from v0.7a), the anti-circular framing discipline, the disallowed-feature list, and the inheritance discipline are locked. Any change after the v0.9a form lock is a re-registration of the parent, not a refinement of a child form. The anti-circular framing is **permanent discipline** for any future v0.9-family registration. ## Doc Trail - `kfacet_v08_writeup.md` -- v0.8 structural-negative chapter close (opens v0.9). - `kfacet_v08a_purity_quartile_audit_form.md` -- v0.8a fails verdict + density-asymmetry diagnostic. - `kfacet_v07a_prime_restricted_scope_form.md` -- v0.7a' PASS (the load-bearing observation v0.9 must NOT silently re-test). - `kfacet_v07a_velocity_fraction_audit_form.md` -- v0.7a parent (vf operational definition). - `kfacet_v05a_branch_map_form.md` -- v0.5a branch hash (alignment- tightness comparison axis). --- Parent registration locked. v0.9a form lock pending. The anti-circular framing is the load-bearing discipline: any v0.9a form that does NOT break at least one of (feature derivation, test statistic, evaluation domain) relative to v0.7a's chi^2-of-independence-on-vf-quartiles is rejected pre-compute.