# v0.7 Mechanism Preregistration: gamma_1 Direction-of-Instability Status: **REGISTERED 2026-05-24. v0.7 CHAPTER CLOSED 2026-05-24 as qualified-positive on restricted domain.** v0.7a child verdict landed 2026-05-24: `velocity_fraction_blocked_integration_attrition` at integration_blocked_count = 23 (8.42% of catalog). v0.7a' restricted-scope confirmation audit landed PASS at chi^2 = 16.43 (p = 9.3e-4) with alignment 0.698 against the v0.5a branch hash on the 250 analyzable rows. The Floquet velocity-fraction direction shadow is the first low-dimensional projection in the v0.4/v0.5/v0.6/v0.7 envelope to produce a non-branch-aligned positive signal. Chapter close at `kfacet_v07_writeup.md`. **v0.8 opens** on Floquet direction-purity at `kfacet_v08_mechanism_preregistration.md`. This document locks the v0.7 body / projection / observable and the non-circularity discipline. Audience: v0.7a/v0.7b form-lock author; paper-side reviewer of the v0.6 -> v0.7 transition (catalog-coordinate close + orbit-dynamics open). Companions: - `kfacet_v06_writeup.md` -- v0.6 conditional-independence chapter close (the joint v0.4 + v0.5 + v0.6 catalog-coordinate envelope that opened v0.7). - `kfacet_v04b_gamma3_form.md` (retired) and `kfacet_v04b_gamma3prime_form.md` (falsified) -- v0.4b disallowed- feature provenance. - `kfacet_v04a_domain_map_preregistration.md` -- supp-B monodromy computation source; per-row M_i and eigenstructure are derivable from this receipt. - v0.3 gamma_1 sidecar in the v0.3h audit chain -- the (sidecar) origin of the gamma_1 quantity, now promoted to primary registration. Frame: v0.4 ruled out the Z_2 tangent/orbit-gauge-rigidity shadow as a stability projector. v0.5 ruled out the 2-bit catalog branch shadow as a held-out predictor. v0.6 ruled out the continuous catalog-coordinate (E, |L|) shadow beyond branch labeling. **v0.7 leaves catalog-coordinate space**: it asks whether the row's per-orbit monodromy operator has a geometric structure — specifically the DIRECTION of its largest-real-part eigenvector — that stratifies stability. ## What v0.7 Is Not v0.7 is NOT: - **A Floquet-magnitude-based feature.** The unstable eigenvalue's modulus IS the stability indicator (or trivially equivalent to it). v0.4b's disallowed-feature list explicitly retired such features for stability circularity. v0.7 uses the eigenvector DIRECTION as a geometric quantity in phase space, not the eigenvalue magnitude or the count of off-unit-circle eigenvalues. - **A within-U-rows analysis.** Conditioning the audit on stability = U is direct circularity. v0.7's gamma_1 quantity must be defined for ALL 273 rows (S and U) under a single operational rule, and the audit asks whether the geometric feature stratifies S vs U across the full catalog (or a stratum thereof). - **A re-use of v0.4b's K_fib tangent-isotypic decomposition.** v0.4b retired the tangent isotypic split because `F_beta` does not preserve K_fib on supp-B. v0.7 must NOT use the K_fib-isotypic-projection step. The gamma_1 direction is computed from the raw monodromy M_i, not from the K_fib slice. - **A re-projection onto the v0.5a (m_3, z_0) branch shadow or the v0.6 (E, |L|) shadow.** v0.7's projection axis is the per-row eigenvector DIRECTION in phase space, not the catalog-coordinate features that v0.4/v0.5/v0.6 already exhausted. ## Body / Projection / Observable ```text Body: supplementary-B piano-trio orbit as primary orbit-dynamics object. domain: 273 rows, already verified Z2_clean by v0.4a; per-row M_i and Floquet eigenstructure are derivable from the v0.4a receipt. Projection: orbit-dynamics direction shadow row -> gamma_1(row), the eigenvector DIRECTION in phase space of the row's largest-real-part Floquet eigenvalue, projected onto a pre-registered geometric reference frame. Observable: per-row gamma_1 direction feature (a scalar or a small-cardinality bin label derived from the direction) joined with the supp-B stability label S/U. Per-row computational cost: monodromy M_i and its eigenstructure are already computed in the v0.4a two-pass classifier; v0.7 may reuse those numerics under the disallowed-feature audit. No new orbit integration. Per-row feature extraction is seconds. ``` The projection family is **orbit-dynamics**: the monodromy M_i carries information about how the orbit's tangent space evolves under one period of the dynamics. The largest-real-part Floquet eigenvector captures the orbit's most-dynamically-active direction in tangent space — for U rows this is the unstable direction; for S rows it is the direction along which neutral perturbations are largest. ## Operational Definition Challenges gamma_1's operational definition is the load-bearing design problem of this parent registration. Three concerns: 1. **Floquet eigenvalue choice for S rows.** Stable orbits have all Floquet eigenvalues on the unit circle, paired by conjugation and typically degenerate at +1 (translations, time direction, energy-conservation Jordan block, etc.). Picking ONE eigenvalue among the degenerate ones is arbitrary unless a deterministic tie-break is locked. 2. **Eigenvector direction ambiguity.** Eigenvectors are defined up to sign (or up to a complex phase for complex eigenvalues). The form lock must specify a deterministic normalization. 3. **Reference frame.** gamma_1 lives in 6D phase space (delta-r, delta-v) at the orbit anchor. Projecting onto a pre-registered geometric axis requires choosing the reference frame (anchor body frame, center-of-mass frame, Z_2-symmetry frame, etc.). These concerns are NOT resolved by the parent registration. They are flagged so the v0.7a form-lock author can take direction and lock a specific operational definition before any compute. ## Candidate Operational Definitions of gamma_1 (Sketches for v0.7a) The v0.7a form lock will pick one of the following (or a registered variant), with explicit non-circularity provenance: ```text D1. Largest-real-part eigenvalue: gamma_1(row) = (unit-normalized) eigenvector of the Floquet eigenvalue whose REAL PART is largest. - For S rows: picks the eigenvalue closest to +1 from the right (real part = 1 if all eigenvalues are unit modulus; tie-break below). - For U rows: picks the unstable eigenvalue (real > 1 by definition). - tie-break: among eigenvalues with equal real part, pick the one with smallest absolute imaginary part. Further ties go to smallest argument. D2. Largest-magnitude eigenvalue: gamma_1(row) = eigenvector of the eigenvalue with largest |lambda|. - For S rows: |lambda| = 1 for all; arbitrary among unit-circle eigenvalues. - For U rows: unique unstable eigenvalue (largest |lambda| > 1). - tie-break: argument closest to 0, then smallest absolute imaginary part. D3. Restricted to "well-defined" rows; companion sidecar for ill-defined rows: gamma_1(row) defined only when the eigenvalue choice is unambiguous (e.g., a unique largest-|lambda| eigenvalue exists above a pre-registered floor). Other rows go to a "gamma_1_undefined" bin which is recorded but report-only. - Risk: the "well-defined" subset may correlate with stability (most U rows are well-defined; many S rows are not), which reintroduces circularity through the back door. - Pre-registered guard: alignment-tightness scalar between "gamma_1_defined" and the v0.5a branch label, with a stop condition at alignment > 0.8. D4. Z_2-isotypic phase-space split of gamma_1: Decompose gamma_1 into its Z_2-even and Z_2-odd components under the supp-B (12)-swap symmetry. Use the EVEN/ODD FRACTION as the feature (continuous in [0, 1]). - Domain-non-circular: uses the catalog's known Z_2 symmetry (verified by v0.4a's domain map), not the stability label. - Risk: re-uses the Z_2 projection that v0.4 already ruled out. Promotion requires explicit paper-side justification that the DIRECTION-level Z_2 information differs from the tangent-isotypic-DIMENSION information v0.4 tested. D5. Phase-space angle decomposition (spatial vs velocity): gamma_1 in 6D space splits naturally into a 3D delta-r component and a 3D delta-v component. Use the ratio |gamma_1_velocity| / (|gamma_1_velocity| + |gamma_1_spatial|) as the feature (continuous in [0, 1]). - Non-circular: ratio is a geometric scalar of the eigenvector. - Risk: may be degenerate (a Hamiltonian flow's tangent dynamics naturally mixes position and velocity); the feature may not vary across rows. ``` The v0.7a form lock must lock ONE of D1-D5 (or a registered variant) with EXPLICIT non-circularity provenance. Multiple parallel definitions are NOT permitted under a single v0.7a registration. ## Non-Circularity Audit (Load-Bearing) ```text The Floquet eigenstructure of M_i is computed from the row's monodromy operator, which in turn is computed from the orbit's tangent dynamics under one period. This computation: - uses the row's published initial conditions (m_i, r_i, v_i), - uses the row's published period T, - uses the three-body Hamiltonian dynamics, - does NOT use the supp-B stability label S/U. Therefore the EIGENVECTOR of M_i is a function of the orbit and its anchor-period only; it does not inspect S/U. The CIRCULARITY risk is in three places, and the v0.7a form lock must address all three: 1. EIGENVALUE-CHOICE CIRCULARITY: - "Pick the eigenvalue WITH MAGNITUDE > 1" picks only on U rows. This IS the stability label. DISALLOWED. - "Pick the largest-magnitude eigenvalue regardless of whether it's > 1" is allowed IF a deterministic tie-break is locked pre-data for the unit-circle case. - "Pick the eigenvalue with the largest real part" is allowed under the same tie-break discipline. 2. WELL-DEFINEDNESS CIRCULARITY: - "gamma_1 is defined only when..." where the condition correlates with stability (e.g., "only when there's an unstable eigenvalue") reintroduces circularity through the back door. - If the form lock uses a definedness-condition, an alignment- tightness scalar between "gamma_1_defined" and the stability label MUST be pre-registered with a stop condition. 3. FEATURE-EXTRACTION CIRCULARITY: - "Use the unstable Lyapunov rate as the feature" IS the stability label. DISALLOWED. - "Use the eigenvector DIRECTION's projection onto a pre-registered geometric axis" is allowed. - "Use the eigenvalue's MAGNITUDE or any function of it" is DISALLOWED (this is the v0.4b inheritance). The non-circularity provenance must be argued explicitly in the v0.7a form lock document. A v0.7a registration whose operational definition does not pass this audit is rejected pre-compute. ``` ## Disallowed Features (Inheritance from v0.4b + v0.7-specific) The v0.7 form locks may NOT use: ```text v0.4b inheritance: - the stability label S/U (or any function of it), - spectral radius, unit-circle count, max |lambda|, or any function of M_i eigenvalues' MAGNITUDES, - Krein signature, - "has off-unit-circle Floquet pair" (this IS S/U), - any feature derived from the K_fib tangent-isotypic decomposition retired in v0.4b's gamma_3 form lock. v0.5/v0.6 inheritance: - the v0.5a branch hash output (any function of (m_3, z_0) indicator bits), - Q_E or Q_|L| or any function of the v0.6a quartile assignments, - E, |L|, or any conserved-quantity scalar -- v0.6 closed on these and they are retired for v0.7. v0.7-specific: - The MAGNITUDE of any Floquet eigenvalue (re-asserted). - Any operational definition where gamma_1's definedness condition correlates with stability at alignment > 0.8 (this is a back-door circularity). - Any feature that requires conditioning on S or U separately (within-S or within-U analyses are NOT permitted at the primary audit level; only post-hoc diagnostics). ``` ## Inheritance Discipline From v0.5 / v0.6 The v0.5 and v0.6 chapter closes locked six register-shaping lessons. v0.7 inherits all six: 1. **Audit-then-predictor separation.** v0.7a registers a chi-squared independence audit on a binned gamma_1 feature; v0.7b 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.** v0.7b requires BOTH `McNemar p <= 0.01` AND positive accuracy delta against the always-U baseline. Inherited from v0.5b. 3. **Held-out partition discipline.** v0.7b defaults to leave-one-m_3-bin-out (12 bins with N >= 5 on supp-B), matching v0.5b. Alternative partitions must be motivated paper-side and pre-registered. 4. **Tie rule and absent-bin fallback.** v0.7b inherits the conservative tie rule (predict U) and the global-majority fallback for absent bins, with `absent_bin_fallback_used` flagged on the receipt. 5. **Alignment-tightness guard (v0.6 inheritance).** v0.7a MUST pre-register an alignment-tightness scalar between the gamma_1 feature's bins and the v0.5a branch label, with a stop condition at alignment > 0.8 (alignment-warning verdict) or alignment > 0.95 (re-registration mandate). v0.6a's value of 0.956 demonstrated this guard is load-bearing for any catalog-correlated feature. 6. **Sparse-cell fallback tree (v0.6 inheritance).** v0.7a and v0.7b MUST pre-register the three-way fallback (chi-squared / exact permutation / sparse-inconclusive) with locked thresholds and locked permutation seed. This is permanent discipline. ## Candidate Audit Forms (Sketches for v0.7a Registration) The v0.7a form lock will pick one of the following audit forms with specific binning and df: ```text A. Direction-projection quartile audit: Project gamma_1 onto a pre-registered geometric axis (e.g., body-3 spatial axis, center-of-mass frame z-axis, or Z_2-even projection). Quartile-bin the projection magnitude over supp-B's 273-row distribution. 4-bin x 2-class contingency. df = 3 (or occupied_bins - 1 if sparse-cell fallback fires). B. Velocity-fraction quartile audit: Use the gamma_1_velocity / (gamma_1_velocity + gamma_1_spatial) ratio (continuous in [0, 1]). Quartile-bin over supp-B. 4-bin x 2-class contingency, df = 3. C. Z_2 isotypic-fraction quartile audit: Use the gamma_1_even / (gamma_1_even + gamma_1_odd) Z_2 fraction. Quartile-bin. 4-bin x 2-class. df = 3. Requires explicit non-circularity vs v0.4b's retired isotypic dimension feature. D. Direction-angle binary audit: "Predict S iff gamma_1 lies in pre-registered angular sector X." 2-bin x 2-class. df = 1, critical 6.63 at p = 0.01. Coarser test; more power if the angular structure is sharp. ``` All four forms must include the alignment-tightness guard against the v0.5a branch label and the sparse-cell fallback tree. ## Open Design Questions (Resolved in v0.7a/v0.7b Form Locks) 1. **Operational definition (D1-D5).** Which definition of gamma_1? D1 (largest real part) and D2 (largest magnitude) are most-natural; D4 (Z_2 isotypic) and D5 (velocity fraction) are derived features. 2. **Tie-break discipline.** Which specific deterministic tie-break handles the S-row degeneracy? Pre-registered, not data-driven. 3. **Reference frame.** Anchor frame, center-of-mass frame, or Z_2-symmetry frame for the geometric projection? 4. **Audit form (A-D).** Which projection of gamma_1 is the primary audit feature? 5. **Sparse-cell threshold inheritance.** v0.6b used min_expected = 5 and min_occupied_bin_count = 2 with permutation seed 20260523. v0.7a inherits these unless explicitly re-registered. 6. **|L|-loud-signal threshold from v0.6.** A residual question from v0.6: the |L| sidecar at p = 0.074 was below the loud-signal threshold for fresh registration. v0.7 explicitly does NOT re-register |L|; conserved quantities are retired. ## Pre-Mortem (Recorded Now) ```text Risk 1: gamma_1's operational definition is ill-defined for S rows in a way that DOES align with stability. Mitigation: alignment-tightness scalar pre-registered with stop condition at alignment > 0.8 against the stability label (NOT just against the v0.5a branch). Risk 2: gamma_1's eigenvector direction depends on numerical precision of the v0.4a monodromy computation. Stability of the eigenvector direction across the v0.4a Pass 1 vs Pass 2 tolerances must be checked on the 24 Pass 2-rescued rows as a sanity gate. Risk 3: The non-circularity audit may fail. If the v0.7a operational definition cannot be made non-circular under reasonable framings, the v0.7 chapter closes as "operational_definition_failed_non_circularity_audit" without any compute, and the program either: - opens v0.7' with a fresh operational definition, - opens v0.8 with a different mechanism family (action-angle / KAM, m_3 = 0.4 sub-catalog targeted, or close-without-v0.8). Risk 4: gamma_1 may be near-constant on supp-B. If the per-row gamma_1 distribution under the locked operational definition has insufficient spread, the constant-feature retirement rule (v0.5/v0.6 inheritance) fires and the audit cannot run. Mitigation: the v0.7a runner emits a per-row gamma_1 distribution as a pre-audit diagnostic; the form lock must check distribution spread before claiming a verdict. ``` ## Stop Conditions 1. **Non-circularity audit failure.** If the v0.7a operational definition cannot be argued non-circular, the form lock is rejected pre-compute. The v0.7 chapter closes immediately. 2. **Constant gamma_1 on supp-B.** If the per-row gamma_1 feature has insufficient variation (e.g., all rows project to the same bin), constant-feature retirement fires. The chapter closes as "gamma_1_constant_on_supp_b". 3. **Alignment-tightness firing.** If the v0.7a binning has alignment > 0.8 against the v0.5a branch label, v0.7b cannot inherit the default leave-one-m_3-bin-out partition. v0.7b must re-register with an alignment-breaking partition (within-branch audit, mirroring v0.6b's pattern). 4. **v0.7a fails the chi-squared / permutation test.** The chapter closes as a structural-negative on the gamma_1 direction shadow. v0.7c (alternative gamma_1 formulation) may be registered separately or the program closes. ## Lock-In Statement This parent registration is committed before: - any per-row gamma_1 has been computed under any operational definition, - any specific binning has been locked, - the non-circularity audit has been argued in writing for any specific operational definition. The body / projection / observable, the non-circularity audit framing, the disallowed-feature list, the inheritance discipline from v0.5/v0.6, and the stop conditions are locked. Any change to those after the v0.7a form lock is a re-registration of the parent, not a refinement of a child form. Implementation may proceed to: 1. Add a per-row monodromy + Floquet eigenstructure computation against the v0.4a manifest. The v0.4a Pass 1/Pass 2 receipts already carry M_i numerically; v0.7 may reuse those numerics under the disallowed-feature audit (no K_fib decomposition is used). 2. Sanity-check the eigenvector direction stability across Pass 1 vs Pass 2 tolerances on the 24 Pass 2-rescued rows. 3. Emit a per-row gamma_1 distribution as a pre-audit diagnostic. No audit verdict is licensed until a v0.7a form lock is separately registered with: - an explicit operational definition of gamma_1 (D1-D5 or a registered variant), - an explicit non-circularity argument that addresses all three circularity risks above, - a deterministic tie-break for the S-row degeneracy, - an alignment-tightness guard against the v0.5a branch label, - a sparse-cell fallback tree. ## Doc Trail - `kfacet_v07a_velocity_fraction_audit_form.md` -- v0.7a child form lock (D5 + B; velocity-fraction quartile audit; D1 + A direction- projection sidecar report-only). - `kfacet_v06_writeup.md` -- v0.6 conditional-independence chapter close (opens v0.7). - `kfacet_v06a_energy_quartile_audit_form.md` -- v0.6a catalog audit (alignment-tightness guard provenance). - `kfacet_v06b_within_branch_energy_audit_form.md` -- v0.6b within-branch audit (sparse-cell fallback tree provenance). - `kfacet_v05a_branch_map_form.md`, `kfacet_v05b_branch_predictor_form.md` -- v0.5 audit+predictor pattern (asymmetric falsifier inheritance). - `kfacet_v04b_gamma3_form.md` (retired) and `kfacet_v04b_gamma3prime_form.md` (falsified) -- v0.4b disallowed- feature provenance for Floquet-magnitude features. - `kfacet_v04a_domain_map_preregistration.md` -- supp-B monodromy source (per-row M_i is derivable from this receipt). --- Parent registration locked. v0.7a child form lock + verdict landed 2026-05-24 (`kfacet_v07a_velocity_fraction_audit_form.md`): D5 + B (velocity-fraction quartile audit) was locked with the non-circularity sentence; D1 + A (direction-projection quartile) was the named report-only sidecar; R1 amendment relaxed symplecticity gate to 1e-4; R2.A amendment added per-row integration-failure fallback with 5% attrition gate. **Verdict: `velocity_fraction_blocked_integration_attrition`** at `integration_blocked_count = 23` (8.42% > 5% threshold). The audit is integration-attrited at variational precision 1e-12, not feature-falsified. v0.7b is not licensed; no chi-squared verdict.