# v0.6 Mechanism Preregistration: Conserved-Quantity (E, |L|) Stratification Status: **PARENT REGISTERED 2026-05-23**. v0.6a child verdict landed 2026-05-23: `energy_quartile_passes_audit_alignment_warning` (chi^2 = 33.70 vs 11.34, alignment = 0.956 > 0.8; see `kfacet_v06a_energy_quartile_audit_form.md`). **v0.6b child verdict landed 2026-05-24**: `within_branch_energy_fails_audit` at chi^2 = 6.90, permutation p = 0.029 (the sparse-cell fallback fired because min_expected = 2.66 < 5; permutation test with seed 20260523 and n_permutations = 10000 was used). |L| sidecar likewise below the loud-signal threshold (chi^2 = 4.46, permutation p = 0.074). See `kfacet_v06b_within_branch_energy_audit_form.md`. The within-branch test shows the v0.6a in-sample positive was dominated by branch-shadow content; the energy shadow does not carry stability information beyond the v0.5a branch hash on supp-B. v0.6c (held-out predictor) is NOT licensed under this form lock chain; the chapter is positioned to close on a clean conditional-independence result. This document locks the v0.6 body / projection / observable and the non-circularity discipline. Form locks for v0.6a (audit) and v0.6b (within-branch audit) are separately registered children that pin specific binning, df, and falsifier choices. Audience: v0.6a/v0.6b form-lock author; paper-side reviewer of the v0.5 -> v0.6 transition. Companions: - `kfacet_v05_writeup.md` -- v0.5 chapter close (projection-limit) that opened v0.6. - `kfacet_v04_writeup.md` -- v0.4 chapter close (structural-negative on the Z_2 shadow). - `kfacet_v05a_branch_map_form.md`, `kfacet_v05b_branch_predictor_form.md` -- v0.5 audit + predictor form locks (inherited discipline). - v0.3 cross-m_3 receipts / invariant implementation -- the sentinel rows used for E and |L| sanity-check provenance. Frame: v0.4 ruled out the Z_2 shadow as a stability projector on supp-B piano-trios. v0.5 ruled out the 2-bit catalog branch shadow as a held-out predictor. v0.6 asks whether a **higher-dimensional catalog-coordinate projection — orbit-level conserved quantities (E, |L|) — carries held-out stability**. ## What v0.6 Is Not v0.6 is NOT: - A re-run of v0.5 with continuous features on the same axis. v0.5b failed because the m_3 axis stratification was bin-local. v0.6 promotes the projection to orbit-level invariants that are catalog-derivable but not aligned with the m_3 catalog parameter. - A Floquet-derived feature set. E and |L| are computed from initial conditions and the Hamiltonian, BEFORE any monodromy or stability inspection. They are orbit-input invariants, not orbit-output invariants. - A held-back z_0 or v_z gate. The retired bits b3 (`abs(v_z) < 1e-6`) and b4 (`m_3 z_0^2 < 2`) stay retired on supp-B. If a future substrate (mesa, geometry, freeze-A choreographies) lights them up, THAT substrate gets its own registration. ## Body / Projection / Observable ```text Body: supplementary-B piano-trio orbit as primary conserved-quantity object. domain: 273 rows, already verified Z2_clean by v0.4a. Projection: orbit-level conserved-quantity shadow row -> (E(row), |L|(row)) where E and |L| are computed at the row's published initial conditions using the three-body Hamiltonian and the published mass triple. Observable: per-row pair (E, |L|) joined with the supp-B stability label S/U, binned into a pre-registered (n_E x n_L) contingency grid. Per-row computational cost: seconds (sum of kinetic energy + sum of pairwise gravitational potential + cross-product sum). NO orbit integration required. Catalog-only quantities. ``` The projection family is **catalog-coordinate but orbit-invariant**: E and |L| are functions of the published initial conditions, but they carry orbit-level physical content (total energy and total angular momentum) that is invariant under the Hamiltonian flow. This is strictly richer than the v0.5 2-bit branch hash (which used only `m_3` and `z_0`) without invoking Floquet spectra or symmetry decompositions. ## Operational Definition of E and |L| For a supp-B piano-trio row with published mass triple `(m_1, m_2, m_3) = (1, 1, m_3)` and published 3D initial state per body `(r_i, v_i)`: ```text E(row) = sum_{i=1..3} 0.5 * m_i * |v_i|^2 - sum_{1 <= i < j <= 3} (G * m_i * m_j) / |r_i - r_j| L(row) = sum_{i=1..3} m_i * (r_i x v_i) (vector cross-product) |L|(row) = euclidean norm of L(row) ``` With the v0.3 convention `G = 1` (Li-Liao normalization). The published initial conditions are read from the supp-B parser used in v0.3 and v0.4a; no re-extraction is required. Sanity checks (must pass before v0.6a registration): ```text 1. E and |L| are computed from the SAME initial conditions used by the v0.4a two-pass classifier and the v0.5a branch hash. No new orbit integration is required. 2. The 7 v0.3 cross-m_3 sentinel rows (O_50, O_62, O_67, O_434 at m_3=0.4; O_242, O_282, O_284 at m_3=1.0) reproduce the v0.3 cross-m_3 receipt values for E and |L| at the v0.3 floor (per-row residual < 1e-6 against the v0.3 receipt). 3. The per-row table emits both E and |L| for all 273 rows. No row drops out under the operational definition. 4. The per-row signed pre-mortem: |L_z| / |L| is recorded as a diagnostic but not used as a gating feature (z-axis component of angular momentum is reserved for chirality diagnostics only, mirroring the v0.5 treatment of sign(v_z)). ``` ## Non-Circularity Audit ```text E and |L| are functions of: - the published row mass triple (m_1, m_2, m_3), - the published row initial conditions (r_i, v_i for i=1..3). E and |L| are NOT functions of: - the row's Floquet spectrum or monodromy operator, - the row's stability label S/U, - any v0.4a Pass 1 / Pass 2 classification, - any v0.5a branch hash output. The supp-B stability label S/U is supplied as the independent test column. E and |L| are catalog-derivable; the stability label is the held-back observable. The non-circularity provenance is the same as the v0.5a branch hash: catalog-only features paired with an independent stability label. The substantive difference is that E and |L| are CONTINUOUS orbit-invariant scalars instead of catalog-coordinate indicator bits. ``` The conserved-quantity projection is therefore strictly richer than the v0.5 branch hash (more information content per row) but uses no additional stability information in the predictor. ## Disallowed Features (Inheritance from v0.4b) The v0.6 form locks may NOT use: ```text - the row's stability label S/U (or any function of it), - the row's Floquet spectrum or any function of M_i eigenvalues (spectral radius, unit-circle count, Krein signature, etc.), - the row's v0.4a Pass 1 / Pass 2 provenance, - the row's v0.5a branch hash output, - the row's K_fib tangent decomposition, - any tangent-isotypic projection of M_i. ``` These are explicitly disallowed because they would either re-circulate v0.4/v0.5 information or directly use the held-back observable. The v0.6 family must construct its predictor from `(m_1, m_2, m_3, r_i, v_i)` per row, plus operational quantities derivable from these via the Hamiltonian (E, |L|, and any explicitly registered scalar). ## Inheritance Discipline From v0.5 The v0.5 chapter close locked five register-shaping lessons. v0.6 inherits all five: 1. **Audit-then-predictor separation.** v0.6a registers a chi-squared independence audit on a binned (E, |L|) grid; v0.6b 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.** v0.6b requires BOTH `McNemar p <= 0.01` AND positive accuracy delta against the always-U baseline. "Differs from baseline" alone is not a pass. 3. **Held-out partition that respects bifurcation structure.** v0.6b defaults to leave-one-m_3-bin-out (12 bins with N >= 5 on supp-B), matching v0.5b. If a v0.6b form locks an alternative partition (e.g., leave-one-E-band-out), that alternative must be motivated paper-side and pre-registered, and the leave-one-m_3 partition is recorded as a sidecar. 4. **Tie rule and absent-bin fallback.** v0.6b 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. **Constant-feature retirement.** If any feature derived from (E, |L|) is constant on supp-B (e.g., all rows bound, all rows prograde), it is recorded on the receipt and retired from df. Cross-substrate comparability is preserved. ## Candidate Audit Forms (Sketches for v0.6a Registration) The v0.6a form lock will pick one of the following audit forms with specific binning constants. All forms share a chi-squared(df) falsifier at p = 0.01: ```text A. Univariate E audit: bin E by quartiles over supp-B -> 4-bin x 2-class contingency. df = 3, critical = 11.34. Tests whether E quartile alone stratifies. B. Univariate |L| audit: bin |L| by quartiles over supp-B -> 4-bin x 2-class contingency. df = 3, critical = 11.34. Tests whether |L| quartile alone stratifies. C. Joint (E, |L|) audit: bin both E and |L| by pre-registered cutoffs (median split, terciles, or quartiles) -> n_E x n_L x 2 contingency, collapsed by occupied-cell count. df = (occupied_cells) - 1; critical from chi-squared(df) at p = 0.01. D. Energy-ratio audit (dimensionless): compute K_total / |V_total| (kinetic-to-virial ratio) per row, bin by quartiles -> 4-bin x 2-class. df = 3, critical = 11.34. E. Angular-momentum scaled audit (dimensionless): compute |L| / sqrt(|E| * I_tot) or similar dimensionless ratio with explicitly pinned reference scales. df = 3, critical = 11.34. ``` v0.6a must lock ONE of A-E (or a registered variant) BEFORE running the audit. Multiple parallel audits are NOT permitted under one v0.6a registration; each separately-registered audit gets its own form lock, its own falsifier, and its own receipt directory. ## Candidate Predictor Forms (Sketches for v0.6b Registration) If v0.6a passes, v0.6b registers a held-out predictor. Candidate shapes: ```text F. Bin-majority on the v0.6a binning: train per-bin majority on training folds, predict per-bin majority on test fold. Mirrors v0.5b's branch-majority pattern. G. Threshold rule on a univariate quantile: "predict S iff E > E_threshold" (or |L| analog). Threshold pre-registered as a quantile (e.g., median, q25, q75) of the training fold's gating subset. H. Two-feature threshold rule: "predict S iff E > E_threshold AND |L| > L_threshold" (or any pre-registered logical conjunction/disjunction of two univariate quantile thresholds). I. Linear discriminant on (E, |L|): fit a per-fold LDA on training; this introduces 2-3 free parameters per fold. Requires explicit free-parameter accounting on the receipt and df adjustment. ``` v0.6b's form lock must pre-register exactly one shape and one partition (default leave-one-m_3-bin-out). Free parameters must be counted explicitly; LDA / regression-based forms get separate treatment from zero-parameter threshold rules. ## Open Design Questions (Resolved in v0.6a/v0.6b Form Locks) These are flagged here so the form-lock authors can take direction: 1. **Audit form (A/B/C/D/E).** Which projection of (E, |L|) carries the first audit? The univariate energy audit (A) is the simplest and most defensible non-circularly; the joint (E, |L|) audit (C) has more df but more discrimination power; the dimensionless audits (D, E) require pinned reference scales (I_tot for E, the energy-virial denominator for D). 2. **Binning strategy.** Quartile bins are conservative and assumption- free; physically-motivated cutoffs (bound vs. unbound, prograde vs. retrograde) are sharper but require justification before v0.6a is locked. 3. **Predictor form (F/G/H/I).** Threshold rules (G, H) are zero-parameter and inherit v0.5b discipline most cleanly. Bin- majority (F) mirrors v0.5b. LDA / regression (I) introduces free parameters and requires explicit accounting. 4. **Partition.** Default is leave-one-m_3-bin-out (inherited from v0.5b). Alternatives (leave-one-E-band-out, deterministic random half) are sidecars unless explicitly motivated. 5. **Dimensionless ratios.** Quantities like `K_total / |V_total|`, `|L| / sqrt(|E| * I)`, `T / T_kepler`, etc. require pinned reference scales. v0.6 carries forward the v0.5 demotion of T / T_kepler and mu_eff^(1/2) products until references are registered. ## Pre-Mortem (Recorded Now) ```text Risk 1: E and |L| are smooth functions of (m_3, z_0) on supp-B. If (E, |L|) tracks (m_3, z_0) too closely, v0.6 inherits v0.5b's bin-locality failure mode. Mitigation: report the per-bin scatter of (m_3, z_0) within each (E, |L|) bin on the v0.6a receipt; if (E, |L|) bins are nearly disjoint in (m_3, z_0) space, the audit pass does NOT license confidence in held-out generalization. Risk 2: |L_z| (z-component) is sign-degenerate on chirality-symmetric sub-catalogs. Treat sign as diagnostic only; magnitude |L| as the gating scalar. Risk 3: E is negative for bound orbits and positive for unbound configurations. supp-B is all bound (periodic), so E sign is constant. The retirement rule applies: sign(E) RETIRED if constant on supp-B, recorded on receipt for cross- substrate comparability. ``` ## Stop Condition If v0.6a fails the chi-squared audit, v0.6b is not registered. The chapter closes immediately as a structural-negative on the (E, |L|) projection. If v0.6a passes but the (E, |L|) bins are tightly aligned with the v0.5b (m_3, z_0) buckets (per the Risk 1 mitigation check), v0.6b must register an alternative partition (leave-one-E-band-out, or a sub-catalog with constant m_3) BEFORE running the held-out test. Inheriting v0.5b's leave-one-m_3-out partition without checking (E, |L|)-to-(m_3, z_0) alignment is a discipline violation. If v0.6b fails the held-out predictor gate, the chapter closes as a projection-limit (mirroring v0.5). The published statement would read: "the (E, |L|) projection is associated with stability in-sample but does not generalize across held-out folds." v0.7 (if any) would target a non-catalog-coordinate projection (e.g., tangent-bundle geometry that uses orbit dynamics). ## Lock-In Statement This parent registration is committed before any v0.6a binning, v0.6b partition, or runner work. The body / projection / observable, operational definitions of E and |L|, non-circularity provenance, disallowed-feature list, and inheritance discipline from v0.5 are locked. Any change to those after the v0.6a 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 E and |L| computation against the existing supp-B parser output (catalog-only, no orbit integration). 2. Sanity-check against the 7 v0.3 cross-m_3 sentinel rows. 3. Emit a per-row (E, |L|) table for v0.6a registration to bin. No audit verdict is licensed until a v0.6a form lock is separately registered with explicit binning constants. ## Doc Trail - `kfacet_v06a_energy_quartile_audit_form.md` -- v0.6a child form lock (audit form A, univariate E quartiles, chi-squared(3) at p = 0.01; |L| quartile sidecar report-only). - `kfacet_v05_writeup.md` -- v0.5 projection-limit chapter close (opens v0.6). - `kfacet_v05a_branch_map_form.md` -- v0.5a audit (catalog-coordinate ancestor). - `kfacet_v05b_branch_predictor_form.md` -- v0.5b held-out predictor (inheritance source). - `kfacet_v04_writeup.md` -- v0.4 structural-negative chapter close. - `kfacet_v06b_within_branch_energy_audit_form.md` -- v0.6b alignment-breaking within-branch audit (alignment-warning child). - v0.3 cross-m_3 receipts / invariant implementation -- E, |L| sentinel sanity-check provenance. --- Parent registration locked. v0.6a verdict landed (`kfacet_v06a_energy_quartile_audit_form.md`): `chi^2_E = 33.703158` versus critical `11.34`, `p = 2.29e-7`, but `alignment_tightness_scalar_E = 0.955882 > 0.8`. The |L| sidecar was loud (`chi^2_|L| = 28.954252`, report-only) and similarly aligned (`0.956522`). v0.6b verdict landed 2026-05-24 (`kfacet_v06b_within_branch_energy_audit_form.md`): `within_branch_energy_fails_audit` at `chi^2 = 6.904`, permutation `p = 0.029` (sparse-cell fallback fired with `min_expected = 2.655`). |L| sidecar likewise fell short of the loud-signal threshold (`chi^2 = 4.465`, permutation `p = 0.074`, report-only). v0.6c (held-out predictor) is NOT licensed; the energy-shadow sub-question closes as a conditional-independence result. v0.6 chapter is positioned for chapter-close writeup.