# v0.6b Within-Branch Energy Audit Form Lock (Alignment-Breaking) Status: **VERDICT LANDED 2026-05-24**. The within-branch energy-quartile audit failed the registered p <= 0.01 threshold under the sparse-cell fallback's exact permutation test. Verdict: `within_branch_energy_fails_audit` at `chi^2 = 6.904`, `permutation_p = 0.0292` (`n_permutations = 10000`, `seed = 20260523`, `min_expected = 2.655 < 5`, `min_occupied_bin_count = 6 > 2`). The |L| sidecar likewise fell short of an independent loud-signal trigger (`chi^2 = 4.465`, `permutation_p = 0.0741`, report-only). The energy shadow does NOT carry stability information beyond labeling the v0.5a `(m_3 < 1, z_0 < 0.3)` branch on supp-B at the registered floor. Audience: v0.6b runner; v0.6c form-lock author (if v0.6b passes); paper-side reviewer of the alignment-breaking transition. Companions: - `kfacet_v06a_energy_quartile_audit_form.md` -- v0.6a parent audit with `energy_quartile_passes_audit_alignment_warning` verdict that licensed this alignment-breaking child. - `kfacet_v06_mechanism_preregistration.md` -- v0.6 parent registration (operational definitions of E and |L|, disallowed-feature list, inheritance discipline). - `kfacet_v05a_branch_map_form.md` -- v0.5a branch hash; the alignment-broken stratification axis. - `kfacet_v05b_branch_predictor_form.md` -- v0.5b held-out predictor pattern; inheritance source for the sparse-cell fallback discipline. Frame: v0.6a passed chi-squared (33.70 vs 11.34) but the alignment-tightness scalar fired hard (0.956 vs 0.8 threshold). Q4 holds 65 of 68 rows in the v0.5a `(m_3 < 1, z_0 < 0.3)` branch (95.6%), so the in-sample energy stratification is plausibly the same bin-locality phenomenon v0.5b exposed dressed in energy clothing. v0.6b breaks the alignment by **conditioning on branch**: within the v0.5a stable branch alone (113 rows / 63 S / 50 U), does Q_E carry stability information? ## Verdict (Landed, 2026-05-24) Receipt: `results/isotrophy/k-facet-v06b-within-branch-energy/manifest.json`. The within-branch audit cleared all sanity gates and triggered the sparse-cell fallback to an exact permutation test: ```text stratum rows: 113 (S = 63, U = 50) sanity_check: PASS (observed N, S, U match locked counts) primary E audit: contingency (Q_E, S/U within branch): Q1: N=0 (Q1 entirely outside the stratum -- tightest catalog orbits all sit in other branches) Q2: N=6 S=2, U=4 S_frac=0.333 expected_S=3.35 expected_U=2.65 Q3: N=42 S=18, U=24 S_frac=0.429 expected_S=23.42 expected_U=18.58 Q4: N=65 S=43, U=22 S_frac=0.662 expected_S=36.24 expected_U=28.76 chi^2: 6.904228 occupied_bin_count: 3 min_occupied_bin_count: 6 (>= 2, fallback test branch active) min_expected_cell: 2.655 (< 5 -> permutation test branch fires) test_branch_taken: permutation permutation_seed: 20260523 n_permutations: 10000 permutation_p_value: 0.0292 threshold: p <= 0.01 verdict: within_branch_energy_fails_audit ``` The within-branch direction is monotone in the same direction as the catalog-wide v0.6a finding (Q2: 33% S -> Q3: 43% S -> Q4: 66% S), but the permutation p-value of 0.0292 does not clear the registered p <= 0.01 floor. The audit is a clean fail at the pre-registered threshold. The |L| within-branch sidecar (report-only, no verdict claim): ```text sidecar |L| audit: contingency (Q_|L|, S/U within branch): Q1: N=66 S=41, U=25 S_frac=0.621 expected_S=36.80 expected_U=29.20 Q2: N=45 S=22, U=23 S_frac=0.489 expected_S=25.09 expected_U=19.91 Q3: N=2 S=0, U=2 S_frac=0.000 expected_S=1.12 expected_U=0.88 Q4: N=0 (Q4 entirely outside the stratum) chi^2: 4.464524 test_branch_taken: permutation permutation_p_value: 0.0741 threshold: p <= 0.01 (would be needed to trigger a fresh |L| form lock as v0.6c) status: report_only; |L| does NOT meet the loud-signal threshold for a fresh registration. ``` Structural reading: > Within the v0.5a stable branch, the within-branch energy quartile > distribution is dominated by Q4 (m_3 = 0.4, 52 of 65 Q4 rows) > and Q3 (mid-m_3 = {0.5..0.8}); Q1 is empty within the branch > (Q1 collects the tightest catalog orbits, all in other branches). > Energy is essentially a 1-to-1 label for m_3 sub-bin inside this > stratum, and the residual within-branch stratification of S/U is > below the registered chi-squared / permutation floor at p = 0.01. > The v0.6a in-sample positive (chi^2 = 33.70) was, as the alignment > warning anticipated, dominated by branch-shadow content. Energy > does not carry stability information beyond what the v0.5a branch > hash already provides on this catalog. Per-other-branch diagnostic audits (3 non-stratum branches, all report-only) and the within-branch Q_E x m_3 and Q_E x Q_|L| joint tables are emitted on the receipt. Receipts: ```text results/isotrophy/k-facet-v06b-within-branch-energy/manifest.json results/isotrophy/k-facet-v06b-within-branch-energy/per_row_table.csv results/isotrophy/k-facet-v06b-within-branch-energy/contingency_table_within_branch_E.csv results/isotrophy/k-facet-v06b-within-branch-energy/contingency_table_within_branch_L.csv results/isotrophy/k-facet-v06b-within-branch-energy/within_branch_Q_E_by_m3.csv results/isotrophy/k-facet-v06b-within-branch-energy/within_branch_Q_E_by_Q_L.csv scripts/v06b_within_branch_energy_audit.py ``` ## What v0.6b Is v0.6b is a **catalog-side audit**, NOT a predictor (continues v0.5a / v0.6a audit-not-predictor framing): ```text Stratum: v0.5a (m_3 < 1, z_0 < 0.3) branch only (113 rows). Test: chi-squared independence of Q_E vs S/U within the stratum. Default df: occupied_Q_E_bins - 1. Fallback: exact permutation test if any expected occupied cell < 5; within_branch_energy_inconclusive_sparse verdict if any occupied bin has < 2 rows. Critical: p <= 0.01 (chi-squared or permutation, depending on which fires). Pass: within_branch_energy_passes_audit -> energy carries within-branch stability information; license v0.6c as held-out predictor. Fail: within_branch_energy_fails_audit -> the v0.6a signal was bin-locality dressed in energy clothing; energy-shadow-as-mechanism sub-question closes. Sparse: within_branch_energy_inconclusive_sparse -> the within-branch Q_E partition is too thin for an honest test; v0.6c may register a power-conservative alternative (e.g., median-split E within branch). ``` A pass licenses v0.6c (separately-registered held-out predictor with appropriate alignment-broken partition). A fail closes the energy-shadow-as-mechanism question. A sparse-inconclusive verdict licenses neither and may motivate a re-registered v0.6b' with a coarser binning. ## Why Within-Branch on (m_3<1, z_0<0.3) The v0.5a branch contingency: ```text branch_label N S U S_fraction m_3 >= 1, z_0 >= 0.3 87 18 69 0.2069 m_3 >= 1, z_0 < 0.3 17 5 12 0.2941 m_3 < 1, z_0 >= 0.3 56 11 45 0.1964 m_3 < 1, z_0 < 0.3 113 63 50 0.5575 <-- v0.6b stratum ``` The stable branch (m_3 < 1, z_0 < 0.3) has: - the largest N (113 rows), - the only S majority on the catalog (S_fraction = 0.5575), - the highest alignment with v0.6a's Q4 (95.6%). Conditioning on this branch is the alignment-break with the highest statistical power to ask the substantive within-branch question. The other three branches have: - (m_3 >= 1, z_0 >= 0.3): N = 87, S_frac = 0.2069 (U-skewed, 18 S / 69 U, may have low S cell counts in some Q_E quartiles); - (m_3 >= 1, z_0 < 0.3): N = 17 (too small for a four-bin chi-squared); - (m_3 < 1, z_0 >= 0.3): N = 56, S_frac = 0.1964 (U-skewed). Per-other-branch audits are recorded as report-only diagnostics on the v0.6b receipt but are NOT gating. If the post-receipt residuals motivate a separately-registered within-branch audit on (m_3 >= 1, z_0 >= 0.3) under a coarser binning, that gets its own form lock. ## The Locked Form ```text stratum: rows r with branch_label(r) == "m_3 < 1, z_0 < 0.3" (113 rows from v0.5a per_row_table.csv). feature: Q_E(r) per v0.6a operational definition, using the GLOBAL supp-B quartile cutpoints from v0.6a's receipt: q25_E = -1.6446242186674882 q50_E = -1.2315294472828655 q75_E = -0.9947713598607784 No re-binning within branch. This is critical: v0.6b preserves v0.6a's feature definition; the within-branch Q_E distribution is whatever it is under the global cutpoints. stability: S/U per supp-B catalog (independent of Q_E). contingency: n_occupied_Q_E x 2 table over (Q_E, S/U) for the 113 stratum rows. primary test: chi-squared statistic chi^2 = sum over cells of (observed - expected)^2 / expected where expected_{kj} = (row_k_total * col_j_total) / 113. default df: occupied_Q_E_bins - 1. threshold: p <= 0.01 (one-sided in the standard chi-squared upper-tail sense). ``` ## Sparse-Cell Fallback Logic Pre-registered, deterministic. Evaluated BEFORE the verdict is read. ```text 1. Compute expected cell counts: E_{kj} = (N_k * C_j) / 113 where N_k = within-branch row count in Q_E bin k, C_j = within-branch row count in stability class j. 2. Compute min_occupied_bin_count: min over occupied Q_E bins of N_k. 3. Branch: a) If min_occupied_bin_count < 2: verdict = within_branch_energy_inconclusive_sparse do NOT report a p-value (the contingency is degenerate). b) Elif min over occupied cells of E_{kj} < 5: use exact permutation test: - permutation_seed = 20260523 (locked) - n_permutations = 10000 (locked) - statistic = chi-squared as computed above - p_value = (# permutations with stat >= observed) / n_permutations verdict at p_value <= 0.01. c) Else: use the asymptotic chi-squared(df) test: - df = occupied_Q_E_bins - 1 - critical at p = 0.01: df = 1 -> 6.63 df = 2 -> 9.21 df = 3 -> 11.34 verdict at chi^2 > critical(df). ``` Both the chi-squared and the permutation test compute the same test statistic (chi-squared). The permutation test resamples S/U labels across the 113 stratum rows uniformly without replacement; Q_E assignments are fixed. This is a labels-permutation test of independence under the null `Q_E independent of S/U within stratum`. The locked permutation seed (`20260523`) is the same seed registered for v0.5b's sidecar B random-half split. Re-using it preserves audit-traceability across the v0.5/v0.6 chain. ## Pre-Audit Sanity Surface Before v0.6b runs: ```text 1. v0.6a verdict must be on file: verdict in {energy_quartile_passes_audit, energy_quartile_passes_audit_alignment_warning}. If v0.6a's verdict is energy_quartile_fails_audit, v0.6b is NOT licensed under this form lock; the chapter closes on E. 2. v0.5a per_row_table.csv must be present and readable. 3. Stratum row count must equal 113. If different, block and inspect (could indicate a parser or branch-assignment drift). 4. Stratum S/U counts must equal (63, 50). Same gate as #3. 5. Q_E values for the 113 stratum rows must be reproducible from the v0.6a runner under the locked global cutpoints. Per-row Q_E residual must be 0 (exact match) against the v0.6a per_row_table. ``` Sanity surface receipts are emitted regardless of verdict outcome. ## Non-Circularity Audit ```text Within-branch conditioning uses the v0.5a branch_label only as a stratification variable. The stratum membership is determined by catalog parameters (m_3, z_0) alone; it does not inspect Q_E or S/U. Q_E(r) 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, - the v0.6a global quartile cutpoints (computed from the supp-B catalog E distribution BEFORE any S/U inspection). Q_E(r) is NOT a function of: - the row's stability label, - the row's branch_label (Q_E depends on E only, not on (m_3, z_0) bit indicators). The S/U label is supplied as the independent test column. Conditional independence under the null: H_0: P(S | Q_E, branch=(m_3<1, z_0<0.3)) = P(S | branch=(m_3<1, z_0<0.3)) H_1: P(S | Q_E, branch=(m_3<1, z_0<0.3)) != P(S | branch=...). The non-circularity provenance is the same as v0.6a: catalog-only features paired with the held-back stability label. ``` The within-branch test does NOT re-use S/U information beyond what v0.6a already used. The branch conditioning is a stratification choice that uses (m_3, z_0) only. ## Free Parameter Count ```text Number of free parameters fitted to the data: 0 ``` The stratum definition (v0.5a (m_3<1, z_0<0.3) branch), the feature (Q_E with v0.6a global cutpoints), the test statistic (chi-squared), the df formula (occupied_bins - 1), the fallback logic, the permutation seed, the permutation count, and the critical p-value are all locked pre-data. The global Q_E cutpoints were determined in v0.6a from the full supp-B E distribution, independently of the within-branch sub-table. ## Sidecar B: Within-Branch |L| Audit (Report-Only) Under the same form shape, the v0.6b runner ALSO emits a within-branch |L| audit: ```text feature: Q_|L|(r) per v0.6a operational definition, using the GLOBAL supp-B quartile cutpoints from v0.6a's receipt: q25_L = 0.11747388134200563 q50_L = 0.2645346632690697 q75_L = 0.4190703676328425 stratum: same 113-row v0.5a (m_3<1, z_0<0.3) branch. contingency: n_occupied_Q_|L| x 2 over (Q_|L|, S/U). test: same as primary (chi-squared with sparse-cell fallback to permutation or inconclusive_sparse). REPORT ONLY: NO VERDICT IS CLAIMED FROM THE SIDECAR. The sidecar's test statistic, p-value, and contingency are recorded on the receipt for diagnostic and motivation purposes only. re-registration: if v0.6b primary fails (within_branch_energy_fails_audit) but the within-branch |L| sidecar shows a strong test statistic at p <= 0.01, the within-branch |L| audit MUST be separately registered as a fresh v0.6b' (or v0.6c) form lock with its own paper-side document, signed off before any further compute on |L|. ``` ## Reserved Diagnostics (Not Gating) Beyond the primary and the sidecar, the v0.6b runner emits per-row and aggregated diagnostics for v0.6c form-lock design only: ```text per-other-branch audits (m_3 >= 1, z_0 >= 0.3 and others): - chi-squared statistic and contingency - sparse-cell status - REPORT-ONLY, no verdict claim. Within-branch Q_E x m_3 joint table: - tells whether within-branch Q_E is dominated by a single m_3 sub-bin. - if it is, v0.6c's partition design must account for it. Within-branch Q_E x Q_|L| joint table: - tells whether E and |L| are tightly coupled within the stratum. - input to a possible joint (E, |L|) within-branch registration. Per-row table emitted under v0.6b: (label, m_3, z_0, E, |L|, Q_E, Q_|L|, branch_label, in_v06b_stratum, stability) ``` ## Edge Cases (Pre-Registered) 1. **Stratum count drift (N != 113, S != 63, U != 50)**: block v0.6b runner; inspect for parser/branch-assignment drift before re-running. 2. **All 4 Q_E quartiles occupied within branch but one has count < 2**: triggers `within_branch_energy_inconclusive_sparse`. The pre-registered re-registration response is a v0.6b' with median-split (2-bin) Q_E or a constant-Q_E vs other binning that ensures all occupied bins have >= 2 rows. 3. **Two occupied Q_E quartiles within branch (Q_E distribution collapsed)**: chi-squared with df = 1, critical 6.63. Run normally. Receipt records the bin collapse. 4. **One occupied Q_E quartile within branch**: degenerate. Verdict `within_branch_energy_inconclusive_sparse`. Cannot distinguish E stratification within the branch under this binning. 5. **All sparse-cell expected counts < 5 but all occupied bins have >= 2 rows**: triggers the permutation test branch. The exact permutation p-value is the verdict-bearing statistic. 6. **Permutation test produces p <= 1 / n_permutations**: report p_value as "<= 1 / n_permutations" (a left-bound on the tail). This is rare given n_permutations = 10000; if it occurs, the audit passes at the registered threshold. 7. **The v0.6a verdict was energy_quartile_fails_audit**: v0.6b is NOT licensed under this form lock. The chapter closes on E. ## Pre-Mortem Expectation The within-branch Q_E distribution has NOT been computed before lock-in. The audit runs blind. Expected published statements: > v0.6b tests whether the v0.5a (m_3<1, z_0<0.3) branch's energy > quartile carries stability information after conditioning on branch. > The verdict is one of `within_branch_energy_passes_audit`, > `within_branch_energy_fails_audit`, or > `within_branch_energy_inconclusive_sparse` per the locked threshold > and sparse-cell fallback. Pre-mortem note: the v0.6a alignment-tightness reading of 0.956 makes the most likely outcome `within_branch_energy_fails_audit` — energy may not separate S/U once the branch is held fixed. If that lands, the chapter closes on a sharp methodological statement: *the energy shadow stratifies the catalog by labeling the v0.5a branch, not by carrying independent within-branch stability content.* A pass would be the more interesting outcome and would license v0.6c. ## Lock-In Statement This audit form is committed before: - the within-branch Q_E distribution has been computed, - the within-branch S/U cell counts have been computed, - the within-branch |L| sidecar test statistic has been computed, - the per-other-branch diagnostic tables have been computed. Any change to the stratum definition, feature, cutpoints, df formula, sparse-cell threshold, permutation seed, permutation count, or critical p-value after the v0.6b runner is executed is a re-registration, not a refinement. Implementation may proceed against this form lock using: ```text results/isotrophy/k-facet-v06a-energy-quartile-audit/manifest.json (per-row table with E, |L|, Q_E, Q_|L|, branch_label, stability) results/isotrophy/k-facet-v05a-branch-map/per_row_table.csv (branch_label provenance for the stratum definition) ``` as the sole inputs. No new dynamical compute is required. ## Receipt Schema (Planned) ```text results/isotrophy/k-facet-v06b-within-branch-energy/ manifest.json - mode = "v0.6b-within-branch-energy-audit" - form_lock = "docs/isotrophy/kfacet/kfacet_v06b_within_branch_energy_audit_form.md" - input_v06a_manifest, input_v05a_per_row_table - stratum = "v0.5a (m_3 < 1, z_0 < 0.3) branch" - stratum_row_count - stratum_S_count, stratum_U_count - sanity_checks: {N, S, U, Q_E reproducibility} - quartile_cutpoints_used_E (copied from v0.6a) - quartile_cutpoints_used_L (copied from v0.6a) - within_branch_contingency_E: n_occupied_Q_E x 2 - within_branch_min_bin_count - within_branch_min_expected_cell - test_branch_taken: "chi_squared" | "permutation" | "inconclusive_sparse" - chi_squared_E: test statistic (if applicable) - df_E: occupied_bins - 1 (if chi-squared) - critical_E: chi-squared critical (if applicable) - permutation_seed: 20260523 (if permutation) - n_permutations: 10000 (if permutation) - p_value_E: chi-squared upper-tail p OR permutation p - verdict: within_branch_energy_passes_audit | within_branch_energy_fails_audit | within_branch_energy_inconclusive_sparse - sidecar_L_contingency: same shape (report-only) - sidecar_L_test_branch_taken - sidecar_L_chi_squared_or_permutation_p - sidecar_L_status: "report_only" - diagnostic_per_other_branch: dict per non-stratum branch - diagnostic_within_branch_Q_E_x_m3 - diagnostic_within_branch_Q_E_x_Q_L contingency_table_within_branch_E.csv contingency_table_within_branch_L.csv (sidecar) per_row_table.csv (label, m_3, z_0, E, |L|, Q_E, Q_|L|, branch_label, in_v06b_stratum, stability) ``` ## Implementation Plan Suggested script: ```text scripts/v06b_within_branch_energy_audit.py ``` Suggested command: ```powershell python scripts\v06b_within_branch_energy_audit.py ` --v06a-manifest results\isotrophy\k-facet-v06a-energy-quartile-audit\manifest.json ` --v06a-per-row results\isotrophy\k-facet-v06a-energy-quartile-audit\per_row_table.csv ` --out results\isotrophy\k-facet-v06b-within-branch-energy ``` Expected runtime: seconds (chi-squared branch) or under a minute (permutation branch with n_permutations = 10000 over a 113-row stratum). ## Interpretation If v0.6b passes (`within_branch_energy_passes_audit`): > The v0.5a stable branch's energy quartile stratifies S/U beyond > what the branch label alone explains. Energy carries within-branch > stability information, and the v0.6a alignment-warning verdict > reflects necessary-but-not-sufficient bin-locality. This licenses > v0.6c: a separately-registered held-out predictor with a partition > design informed by the v0.6b per-branch diagnostics. If v0.6b fails (`within_branch_energy_fails_audit`): > The energy quartile carries no stability information beyond labeling > the v0.5a stable branch. The v0.6a in-sample chi-squared was bin- > locality dressed in energy clothing. The energy-shadow-as-mechanism > sub-question closes; the chapter records a clean > conditional-independence result. If v0.6b is sparse-inconclusive (`within_branch_energy_inconclusive_sparse`): > The within-branch Q_E partition is too thin under the locked > binning for an honest test. v0.6b' may register a coarser within- > branch binning (median-split or quantile=2) before re-running. ## Doc Trail - `kfacet_v06a_energy_quartile_audit_form.md` -- v0.6a parent with alignment-warning verdict. - `kfacet_v06_mechanism_preregistration.md` -- v0.6 parent registration. - `kfacet_v05a_branch_map_form.md` -- v0.5a stratification axis. - `kfacet_v05b_branch_predictor_form.md` -- inheritance source for sparse-cell discipline and McNemar conventions. - v0.6a receipt: `results/isotrophy/k-facet-v06a-energy-quartile-audit/manifest.json`. --- Alignment-breaking within-branch audit. Chi-squared by default, exact permutation at sparse cells, honest sparse-inconclusive verdict if the within-branch partition is degenerate. |L| sidecar report-only.