# v0.6a Energy-Quartile Audit Form Lock (Univariate E) Status: **VERDICT LANDED 2026-05-23**. The v0.6a univariate-energy- quartile audit passes the registered chi-squared threshold but triggers the registered alignment warning. The |L| quartile audit under the same shape is recorded as a **report-only sidecar** and emits no independent verdict. Audience: v0.6a runner; v0.6b form-lock author; paper-side reviewer of the v0.6 transition from parent registration to first audit. Companions: - `kfacet_v06_mechanism_preregistration.md` -- v0.6 parent registration (this form is option A of the parent's audit-form candidate list). - `kfacet_v05_writeup.md` -- v0.5 chapter close (projection-limit) that opened v0.6. - `kfacet_v05a_branch_map_form.md` -- v0.5a audit pattern (4-bucket contingency, chi-squared(df) falsifier, audit-not-predictor framing). - v0.3 cross-m_3 receipts / invariant implementation -- E and |L| sentinel sanity-check provenance. Frame: v0.6 asks whether a higher-dimensional catalog-coordinate projection — orbit-level conserved quantities — carries held-out stability. v0.6a starts with the **cleanest first conserved-quantity audit**: total energy E quartiles. Per the parent registration's candidate-form list, this is option A; B (|L| quartiles), C (joint (E, |L|)), D (energy-ratio K/|V|), and E (angular-momentum-scaled) are held back for separate registrations. ## Verdict (Landed, 2026-05-23) Receipt: `results/isotrophy/k-facet-v06a-energy-quartile-audit/manifest.json`. The registered runner completed all sanity gates: ```text rows: 273 (S = 97, U = 176) sanity_v03_sentinels: PASS (max |Delta E| = 0, max |Delta |L|| = 0) bound_orbit_check: PASS (E < 0 for all 273 rows) ``` The primary E quartile audit: ```text E cutpoints: q25 = -1.6446242186674882 q50 = -1.2315294472828655 q75 = -0.9947713598607784 Q_E table: Q1 N=69 S=21 U=48 S_frac=0.3043 Q2 N=68 S=12 U=56 S_frac=0.1765 Q3 N=68 S=21 U=47 S_frac=0.3088 Q4 N=68 S=43 U=25 S_frac=0.6324 chi^2_E: 33.70315814531264 p_value_df3: 2.2886715282626394e-07 critical: 11.34 alignment_E: 0.9558823529411765 (> 0.8 warning threshold) verdict: energy_quartile_passes_audit_alignment_warning ``` Interpretation: the energy quartile shadow is strongly associated with S/U in-sample, but the association is not cleanly independent of the v0.5a `(m_3, z_0)` branch shadow. In particular Q4 is dominated by the `m3_lt_1__z0_lt_0p3` branch at 65/68 rows. Therefore v0.6b may NOT inherit the default leave-one-m_3-bin-out partition; it must be re-registered with an alignment-breaking partition (for example leave-one-E-quartile-out, or a constant-m_3 subcatalog design). The |L| sidecar is also loud but is not a verdict: ```text |L| cutpoints: q25 = 0.11747388134200563 q50 = 0.2645346632690697 q75 = 0.4190703676328425 Q_|L| table: Q1 N=69 S=41 U=28 S_frac=0.5942 Q2 N=68 S=24 U=44 S_frac=0.3529 Q3 N=68 S=21 U=47 S_frac=0.3088 Q4 N=68 S=11 U=57 S_frac=0.1618 chi^2_|L|: 28.954252352605597 p_value_df3: 2.2895491439323422e-06 alignment_|L|: 0.9565217391304348 sidecar_status: report_only ``` Implementation provenance: v0.4a/v0.5a receipts provide row identity, stability, and branch metadata, but they do not carry all published IC velocity fields required for E and |L|. The runner therefore re-reads `docs/isotrophy/supplementary-B_piano-init-condit-3d.txt` for the catalog ICs and joins the receipt metadata. This remains catalog-only and non-circular. The registered v0.3 sentinel sanity check is recorded against the pre-existing `scripts.isotrophy_workbench.invariant_record` implementation because the cross_m_3 gate receipts present in this workspace do not carry scalar E/|L| fields. ## Why Univariate E Over Other Audit Forms Per the parent registration: ```text A. univariate E quartiles: SELECTED (this form lock). - Maximally non-circular: E is conserved orbit-level scalar from IC. - Paper-simple: 4 x 2 contingency, df = 3, critical = 11.34. - No reference scale required for quartiles. B. univariate |L| quartiles: first sidecar, report-only. - Also clean, but |L| carries more coordinate/gauge interpretive baggage than E. Held back as the first sidecar; if A fails but the B sidecar is loud, B is registered as a separate v0.6b. C. joint (E, |L|) audit: held back. - 16 bins / df ~= 15. Too easy to dilute on the first audit and too easy to explain post hoc. Promotion requires either A or B to show signal first. D. energy-ratio K/|V| audit: held back (v0.6b/v0.6c material). - Requires pinned virial-normalization reference scale. E. angular-momentum-scaled audit: held back (v0.6b/v0.6c material). - Requires pinned moment-of-inertia I_tot reference scale. ``` The locked form prioritizes paper-simplicity and reference-scale-free non-circularity over discrimination power. If A fails and the B sidecar shows no signal either, that is itself the chapter result on energy and angular-momentum univariate shadows. ## What v0.6a Is v0.6a is a **catalog-side audit**, NOT a predictor (inherits v0.5a's audit-not-predictor framing): ```text Test: chi-squared independence of energy quartile Q_E vs S/U df: 3 Critical: chi-squared(3) at p = 0.01 = 11.34 Pass: chi^2 > 11.34 and alignment <= 0.8 -> energy_quartile_passes_audit Warn pass: chi^2 > 11.34 and alignment > 0.8 -> energy_quartile_passes_audit_alignment_warning Fail: chi^2 <= 11.34 -> energy_quartile_fails_audit ``` A pass with alignment-tightness `<= 0.8` licenses v0.6b under the default held-out discipline. A pass with alignment-tightness `> 0.8` licenses only an alignment-breaking v0.6b re-registration. A fail closes the energy-shadow sub-question; the |L| sidecar may motivate a separate v0.6b on |L| under a fresh form lock. ## The Locked Form ```text feature: E(row) = total energy at the row's published initial conditions. 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| with G = 1 (Li-Liao normalization) and (m_1, m_2, m_3) = (1, 1, m_3_row). Initial conditions read from the same supp-B parser output used by v0.4a and v0.5a. binning: quartiles over the supp-B catalog's 273-row E distribution. cutpoints: q25, q50, q75 of the supp-B E array, computed via numpy.quantile(E, [0.25, 0.50, 0.75], method='linear'). bin assignment: bin k for row r is the smallest k in {1, 2, 3, 4} such that E(r) <= cutpoint_{k+1} (k = 1, 2, 3) or k = 4 if E(r) > q75. Equivalently: Q_E(r) = 1 iff E(r) <= q25 Q_E(r) = 2 iff q25 < E(r) <= q50 Q_E(r) = 3 iff q50 < E(r) <= q75 Q_E(r) = 4 iff E(r) > q75 Right-closed lower bins; ties at exact cutpoints (numerically impossible at double precision but pre-registered for completeness) go to the lower bin. contingency: 4 x 2 table over (Q_E, stability) with stability in {S, U}. test: chi-squared statistic chi^2 = sum over cells of (observed - expected)^2 / expected where expected_{kj} = (row_k_total * col_j_total) / N. df: 3. critical: 11.34 (chi-squared(3) at p = 0.01). verdict: energy_quartile_passes_audit iff chi^2 > 11.34 AND alignment-tightness <= 0.8 energy_quartile_passes_audit_alignment_warning iff chi^2 > 11.34 AND alignment-tightness > 0.8 energy_quartile_fails_audit iff chi^2 <= 11.34 ``` Physical orientation: supp-B is all bound (E < 0 for all 273 rows; sign(E) RETIRED on supp-B per parent registration's constant-feature rule). Q_E = 1 corresponds to the **most-negative** (tightest, most deeply bound) E quartile; Q_E = 4 corresponds to the **least-negative** (loosest) E quartile. ## Pre-Audit Sanity Surface Before v0.6a runs: ```text 1. Per-row E and |L| computation against the v0.4a manifest's 273 rows. Catalog-only; no orbit integration. 2. Sanity check against 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). Per-row residual must agree to < 1e-6 against the v0.3 cross-m_3 receipt E and |L| values. Block v0.6a if this fails. If the local cross_m_3 receipts do not carry scalar E/|L| fields, the runner may use the pre-existing v0.3 invariant implementation as the reference, but MUST record that provenance on the receipt. 3. Bound-orbit check: E < 0 for all 273 rows. If any row has E >= 0, record on receipt and pause v0.6a for inspection. (Periodic orbits are bound by definition; an E >= 0 reading would indicate a parser or operational-definition bug.) ``` Sanity surface receipts are emitted regardless of audit outcome. ## Non-Circularity Audit ```text E(row) is a function of: - the published mass triple (m_1, m_2, m_3), - the published initial conditions (r_i, v_i for i = 1..3), - the three-body Hamiltonian (kinetic + pairwise gravitational). E(row) is NOT a function of: - the row's Floquet spectrum or monodromy operator, - the row's stability label S/U, - any v0.4a Pass 1 / Pass 2 classification, - the v0.5a branch hash output, - the K_fib tangent decomposition. The supp-B stability label S/U is supplied as the independent test column. The quartile cutpoints are computed from the supp-B E distribution alone, BEFORE the S/U column is joined. This is the same non-circularity provenance as v0.5a's branch hash: catalog-only features paired with an independent stability label. ``` The cutpoints DO depend on the supp-B distribution (because they are supp-B quartiles), but they are independent of the stability label. This is the assumption-light alternative to physically-motivated cutpoints, which would risk disguised tuning. ## Free Parameter Count ```text Number of free parameters fitted to the data: 0 ``` The feature (E), the binning rule (quartiles), the cutpoint method (numpy linear-interpolation quantile), the bin-assignment convention (right-closed lower bins, ties to lower bin), the df (3), and the critical value (11.34) are all locked pre-data. The quartile cutpoints are computed deterministically from the supp-B E distribution; they are not fitted to the S/U column. ## Chi-Squared Degrees of Freedom ```text df = (number of bins - 1) * (number of classes - 1) = (4 - 1) * (2 - 1) = 3 ``` Reference distribution `chi-squared(3)`. Critical value `11.34` at `p = 0.01`. Test statistic `chi^2 > 11.34` triggers `energy_quartile_passes_audit`. Wording note (inherited from v0.5a discipline): the pass condition is `energy_quartile_passes_audit`, NOT "energy-shadow theorem confirmed". A pass licenses a v0.6b predictor registration; it does not on its own establish a predictive mechanism. ## Sidecar A: |L| Quartile Audit (Report-Only) Under the same form shape, the v0.6a runner ALSO emits a |L| quartile audit: ```text feature: |L|(row) = euclidean norm of the row's total angular momentum vector at IC. |L|(row) = ||sum_{i=1..3} m_i * (r_i x v_i)|| binning: quartiles over the supp-B catalog's 273-row |L| array, same quantile method as the primary E form. contingency: 4 x 2 (Q_|L|, stability). test: chi-squared statistic, df = 3. REPORT ONLY: NO VERDICT IS CLAIMED FROM THE SIDECAR. The sidecar's chi^2 and contingency are recorded on the receipt for diagnostic and motivation purposes only. re-registration: if A fails (energy_quartile_fails_audit) but the |L| sidecar shows a chi^2 substantially above 11.34, the |L| quartile audit MUST be separately registered as a fresh v0.6b (or v0.6a') form lock with its own paper-side document, signed off before any further compute on the |L| projection. ``` The sidecar's purpose is to surface a "loud |L| signal that wasn't in E" finding cheaply, without licensing it as a verdict under the current registration. Any subsequent claim on |L| requires its own lock-in statement. ## Reserved Diagnostics (Not Gating, Not Verdict-Bearing) Beyond the locked primary and the sidecar, the v0.6a runner emits per-row and aggregated diagnostics for v0.6b form-lock design only: ```text sign(E) [expected constant negative on supp-B; retired per parent registration if confirmed.] |E| [magnitude; useful for log-scale visualization.] L_z = z-component of L sign(L_z) [chirality diagnostic; not gating.] |L_z| / |L| [pre-mortem flagged in parent registration as diagnostic only.] E quartile x m_3 axis joint table E quartile x z_0 axis joint table E quartile x v0.5a branch_label joint table |L| quartile x v0.5a branch_label joint table ``` The last three tables address the parent registration's Risk 1 (alignment with v0.5b's (m_3, z_0) buckets). If the E or |L| quartiles are tightly aligned with the v0.5a branch label, the audit pass does NOT license confidence in held-out generalization, and v0.6b's partition must be re-registered as something other than the default leave-one-m_3-bin-out. Specifically, the receipt MUST report: ```text max over E quartile bin of (fraction of bin in any single (m_3, z_0) branch_label bucket) ``` This single number is the alignment-tightness scalar. The parent registration requires it on every v0.6 receipt; v0.6a is the first form lock to enforce it. ## Edge Cases (Pre-Registered) 1. **Tied E values at cutpoints**: numerically impossible at double precision for distinct rows. Pre-registered convention: ties go to the lower bin (right-closed lower intervals). 2. **Quartile bin size variation**: with 273 rows divided into 4 quartiles, bin sizes will be approximately {68 or 69 each}. The small imbalance is absorbed by the chi-squared expected-count calculation; no Yates' correction is applied. 3. **Expected count < 5 in any cell**: with 97 S / 176 U over 273 rows and 4 bins, expected S count per bin is ~24 and expected U count per bin is ~44. All cells well above the chi-squared assumption threshold. No correction needed. 4. **Failed sanity check on v0.3 cross-m_3 sentinels**: block v0.6a; record the discrepancy and fix the operational-definition implementation before the audit runs. This is a hard gate on the non-circularity provenance (we cannot claim E was independently computable from IC if the implementation disagrees with v0.3). 5. **Bound-orbit check failure (E >= 0 for any row)**: same gate as #4. Periodic orbits are bound; an E >= 0 reading indicates a bug. 6. **Alignment-tightness scalar > 0.8**: pass v0.6a with a flagged verdict `energy_quartile_passes_audit_alignment_warning`. v0.6b's partition MUST be re-registered with an alternative that breaks the alignment (leave-one-E-quartile-out, or a sub-catalog at constant m_3). ## Pre-Mortem Expectation (Pre-Run) At lock-in, the per-quartile S/U counts had NOT been computed. The audit ran blind. The expected published statement, regardless of outcome: > v0.6a tests whether the supp-B catalog's energy quartile Q_E > stratifies stability. The verdict is reported as > `energy_quartile_passes_audit` (chi^2 > 11.34), > `energy_quartile_passes_audit_alignment_warning` (chi^2 > 11.34 AND > max-bin-alignment > 0.8), or `energy_quartile_fails_audit` (chi^2 > <= 11.34) under the locked chi-squared(3) threshold at p = 0.01. Risk-1 pre-mortem note from the parent registration: E and |L| may track (m_3, z_0) on supp-B. If the alignment-tightness scalar exceeds 0.8, the audit pass is bin-locality-suspect and v0.6b's default partition does not inherit cleanly. ## Lock-In Statement This audit form is committed before: - per-row E or |L| has been computed for the 273 supp-B rows, - the supp-B quartile cutpoints have been calculated, - the per-bin S/U distribution has been examined, - the |L| sidecar's chi-squared has been computed. Any change to the feature, binning, cutpoint method, bin-assignment convention, df, critical value, alignment-tightness threshold, or sidecar shape after the v0.6a runner is executed is a re-registration, not a refinement. Implementation proceeded against this form lock using: ```text results/isotrophy/k-facet-v04a-domain-map/manifest.json (row identity, m_3, stability/provenance metadata) results/isotrophy/k-facet-v05a-branch-map/per_row_table.csv (per-row branch_label for the alignment check) docs/isotrophy/supplementary-B_piano-init-condit-3d.txt (published z_0, v_x, v_y, v_z, T, stability IC fields) ``` as catalog-only inputs. No new dynamical compute is required; per-row E and |L| computation is catalog-only. ## Receipt Schema (Landed) ```text results/isotrophy/k-facet-v06a-energy-quartile-audit/ manifest.json - mode = "v0.6a-energy-quartile-audit" - form_lock = "docs/isotrophy/kfacet/kfacet_v06a_energy_quartile_audit_form.md" - input_manifest_v04a, input_per_row_table_v05a - input_catalog_supplementary_b - quartile_method = "numpy_quantile_linear" - cutpoints_E: {q25, q50, q75} - cutpoints_L: {q25, q50, q75} (sidecar) - bin_assignment_convention = "right_closed_lower_ties_to_lower" - sanity_check_v03_sentinels: - per_row_residual_E_max - per_row_residual_L_max - pass_threshold = 1e-6 - status = pass | fail - bound_orbit_check: - rows_with_E_ge_0 - status = pass | fail - contingency_E: 4 x 2 (Q_E, S/U) - chi_squared_E: test statistic - df_E: 3 - critical: 11.34 - verdict: energy_quartile_passes_audit | energy_quartile_passes_audit_alignment_warning | energy_quartile_fails_audit - alignment_tightness_scalar_E: max over Q_E of (fraction in any single branch_label bucket) - alignment_tightness_scalar_L: (sidecar) - contingency_L: 4 x 2 (Q_|L|, S/U) (sidecar) - chi_squared_L: test statistic (sidecar) - df_L: 3 (sidecar) - sidecar_status: report_only - diagnostic_tables_emitted: [E_vs_m3, E_vs_z0, E_vs_branch, L_vs_branch] contingency_table_E.csv contingency_table_L.csv (sidecar) per_row_table.csv (label, m_3, z_0, E, |L|, Q_E, Q_|L|, branch_label, stability) ``` ## Implementation / Command Suggested script: ```text scripts/v06a_energy_quartile_audit.py ``` Suggested command: ```powershell python scripts\v06a_energy_quartile_audit.py ` --manifest results\isotrophy\k-facet-v04a-domain-map\manifest.json ` --v05a-table results\isotrophy\k-facet-v05a-branch-map\per_row_table.csv ` --out results\isotrophy\k-facet-v06a-energy-quartile-audit ``` Observed runtime: seconds. No staged compute or operator interaction was required. ## Interpretation If v0.6a passes (chi^2 > 11.34, alignment-tightness <= 0.8): > The supp-B catalog's energy quartile stratifies stability under a > pre-registered univariate audit. The energy shadow carries > in-sample stability information that the v0.5a 2-bit branch hash did > not fully capture, AND the energy quartiles are not tightly aligned > with v0.5b's bin-locality failure mode. This licenses v0.6b: a > separately-registered held-out energy-quartile predictor with the > v0.5b discipline. If v0.6a passes with alignment warning (chi^2 > 11.34, alignment-tightness > 0.8): > Pass-but-suspect. The energy quartiles track (m_3, z_0) tightly on > supp-B, so the in-sample signal is plausibly the same bin-locality > phenomenon v0.5b exposed. v0.6b's partition must be re-registered to > break the alignment. If v0.6a fails (chi^2 <= 11.34): > The energy-shadow does not stratify stability on supp-B under the > univariate quartile audit. The chapter sub-question on E closes; the > |L| sidecar's chi-squared and the v0.6 mechanism family's remaining > projections (|L| univariate, joint, energy-ratio, angular-momentum- > scaled) are open for separate registration if any sidecar signal > warrants pursuit. ## Doc Trail - `kfacet_v06_mechanism_preregistration.md` -- parent registration. - `kfacet_v06b_within_branch_energy_audit_form.md` -- v0.6b alignment-breaking child (within-branch audit on the v0.5a (m_3<1, z_0<0.3) stratum, licensed by this form's `energy_quartile_passes_audit_alignment_warning` verdict). **Verdict landed 2026-05-24: `within_branch_energy_fails_audit`** (chi^2 = 6.90, permutation p = 0.029); confirms the v0.6a alignment warning was load-bearing. - `kfacet_v05_writeup.md` -- v0.5 chapter close (predecessor). - `kfacet_v05a_branch_map_form.md` -- v0.5a audit pattern (form template). - v0.3 cross-m_3 receipts / invariant implementation -- E, |L| sentinel sanity-check provenance. --- Audit floor, not predictor. Univariate energy quartile, df = 3, critical 11.34. |L| quartile sidecar report-only. Alignment-tightness scalar guards against v0.5b's bin-locality failure mode. Verdict landed 2026-05-23 as `energy_quartile_passes_audit_alignment_warning`, licensing v0.6b as an alignment-breaking within-branch audit. v0.6b landed 2026-05-24 as `within_branch_energy_fails_audit`, confirming the alignment warning was load-bearing: the v0.6a in-sample chi^2 was dominated by branch-shadow content.