# K_facet v0.6 Methodology + Result Handoff Status: closed conditional-independence (catalog audit passes, within-branch audit fails), 2026-05-24. Audience: collaborators, paper-side writers, future coding agents. Canonical sources: `kfacet_v06_mechanism_preregistration.md`, `kfacet_v06a_energy_quartile_audit_form.md`, `kfacet_v06b_within_branch_energy_audit_form.md`, and [`../../CROSS_SUBSTRATE_NOTES.md`](../../CROSS_SUBSTRATE_NOTES.md) sections 6-7 (projection-language framing). Companion (closed v0.5 chapter): `kfacet_v05_writeup.md`. ## One-Line Read v0.6 closes as a **conditional-independence result** on the catalog-coordinate conserved-quantity (E, |L|) shadow of the Li-Liao supplementary-B piano-trio catalog. The univariate energy quartile audit passes the registered chi-squared gate decisively in-sample (`chi^2 = 33.70` vs critical 11.34, `p ~= 2.29e-7`, v0.6a) but trips the pre-registered alignment-tightness guard (`alignment = 0.956 > 0.8`). The alignment-breaking within-branch audit then fails the registered permutation-test floor (`chi^2 = 6.90`, `permutation p = 0.029`, v0.6b), confirming the v0.6a in-sample positive was dominated by branch-shadow content. Energy quartile is essentially a 1-to-1 label for m_3 sub-bin within the v0.5a stable branch; it does not carry stability information independent of the v0.5a branch hash on this catalog. Distinct from v0.4 (structural-negative) and v0.5 (projection-limit); a third chapter-close type. ## What v0.6 Tested v0.6 opened after v0.5's projection-limit close. The v0.5 chapter established that the 2-bit catalog branch shadow `(m_3 < 1, z_0 < 0.3)` stratifies supp-B in-sample (chi^2 = 34.99) but does not predict held-out stability across m_3 bins (accuracy delta -0.019). v0.6 promoted the projection from indicator-bit features to continuous orbit-level conserved quantities: > v0.6 promotes the projection from a 2-bit catalog hash to continuous > catalog-coordinate orbit invariants (E, |L|), and asks whether the > body's stability structure is visible in this richer projection. Three pre-registered chapters, all paper-side first: ```text v0.6a: univariate energy-quartile catalog audit Body: supp-B piano-trio orbit as primary conserved- quantity object. Projection: row -> E(row); supp-B-relative quartiles. Observable: 4 x 2 contingency Q_E vs S/U over 273 rows. Question: chi-squared independence of Q_E vs S/U at p=0.01, with pre-registered alignment-tightness guard. Result: energy_quartile_passes_audit_alignment_warning (chi^2 = 33.70 vs 11.34, p ~= 2.29e-7, alignment-tightness = 0.956 > 0.8). Pass with the guard tripped. v0.6b: within-branch energy-quartile audit (alignment-breaking) Stratum: v0.5a (m_3 < 1, z_0 < 0.3) branch (113 rows / 63 S / 50 U). v0.5a's most-stable branch. Feature: Q_E under v0.6a's GLOBAL supp-B cutpoints (no within-branch re-binning). Test: chi-squared with sparse-cell fallback to exact permutation test (seed = 20260523, n_permutations = 10000) or within_branch_energy_inconclusive_sparse verdict. Sidecar: within-branch Q_|L| audit (report-only). Question: does Q_E carry stability info beyond branch? Result: within_branch_energy_fails_audit (chi^2 = 6.90, permutation p = 0.029; |L| sidecar p = 0.074 below loud-signal floor). ``` The audit chain inherited and extended v0.5's discipline with one new register-shaping addition. ## Audit Chain Methodology The v0.6 chain inherited the v0.5 discipline (audit-then-predictor separation, asymmetric falsifier on the predictor, default leave-one-m_3-bin-out partition, conservative tie rule, constant-feature retirement) and added: **Alignment-tightness guard.** v0.6a pre-registered a scalar measuring how tightly the audit feature's bins align with the v0.5a branch hash. If `max over Q_E of (fraction in any single branch_label bucket) > 0.8`, the audit pass is **bin-locality-suspect** and a default held-out predictor partition is no longer inherited cleanly. The alignment guard fired hard (0.956), confirming the v0.6a in-sample positive was branch-shadow-aligned and licensing v0.6b as an alignment-breaking child audit instead of v0.6c as a held-out predictor. **Sparse-cell fallback discipline.** v0.6b pre-registered a three-way fallback tree before any compute: (a) chi-squared(occupied_bins - 1) if min expected cell >= 5; (b) exact permutation test (seed locked pre-data) if min expected cell < 5 but min occupied bin count >= 2; (c) `within_branch_energy_inconclusive_sparse` verdict if any occupied bin has < 2 rows. The permutation branch fired (min_expected = 2.655 < 5; min_occupied_bin_count = 6 >= 2), giving an exact distribution-free p-value of 0.029 — below the chi-squared(2) critical-implied threshold but above the registered p <= 0.01 floor. Three stages, all paper-side first: 1. **v0.6 parent registration** (`kfacet_v06_mechanism_preregistration.md`). Locked body / projection / observable, operational definitions of E and |L|, non-circularity provenance, disallowed-feature list (no Floquet, no stability, no v0.5a output, no K_fib decomposition), inheritance discipline from v0.5, candidate audit forms (A-E) and predictor forms (F-I). 2. **v0.6a child form lock + verdict** (`kfacet_v06a_energy_quartile_audit_form.md`). Selected candidate form A (univariate E quartiles) over B (|L| sidecar-only), C (joint, df dilution), D (K/|V|, reference scale unpinned), and E (|L|-scaled, reference scale unpinned). Locked supp-B-relative quartile cutpoints via `numpy.quantile(..., method='linear')`, 4 x 2 contingency, df = 3, critical = 11.34, alignment-tightness scalar with 0.8 threshold. Sanity gates (v0.3 cross-m_3 sentinel parity at < 1e-6, bound-orbit check E < 0 for all 273) passed. 3. **v0.6b child form lock + verdict** (`kfacet_v06b_within_branch_energy_audit_form.md`). Alignment- breaking by **conditioning on branch**: stratum = v0.5a (m_3 < 1, z_0 < 0.3), 113 rows. Reused v0.6a's GLOBAL supp-B cutpoints (no re-binning within branch). Pre-registered sparse-cell fallback. Within-branch |L| audit as report-only sidecar. ## Result And Conditional-Independence Verdict ```text v0.6a verdict: energy_quartile_passes_audit_alignment_warning chi^2_E = 33.70 vs critical 11.34 (chi-squared(3), p = 0.01). p_value ~= 2.29e-7 (~3x threshold). 273 rows, 97 S / 176 U; 4 quartile bins. Q-bin S fractions: Q1 30.4%, Q2 17.6%, Q3 30.9%, Q4 63.2%. alignment_tightness_scalar_E = 0.956 > 0.8. Q4 holds 65 of 68 rows in the v0.5a (m_3<1, z_0<0.3) branch (95.6%); the in-sample positive is bin-locality-suspect. |L| sidecar (report-only): chi^2 = 28.95, p = 2.29e-6, alignment 0.957. v0.6b verdict: within_branch_energy_fails_audit Stratum: 113 rows / 63 S / 50 U (v0.5a stable branch). Sanity passed. Within-branch Q_E contingency: Q1 N=0 (entirely outside stratum -- tightest catalog orbits all sit in other branches). Q2 N=6 (2 S / 4 U, 33.3% S) Q3 N=42 (18 S / 24 U, 42.9% S) Q4 N=65 (43 S / 22 U, 66.2% S) Within-branch direction MONOTONE in same direction as v0.6a. min_expected_cell = 2.655 (< 5) -> sparse-cell fallback fired. Permutation test (seed = 20260523, n_permutations = 10000): chi^2 = 6.904, permutation p = 0.0292. Verdict: fails registered p <= 0.01 floor. Within-branch Q_E x m_3 joint diagnostic: Q4 = m_3 in {0.4, 0.5} (52 of 65 Q4 rows are m_3 = 0.4) Q3 = m_3 in {0.5, 0.6, 0.7, 0.8} Q2 = m_3 in {0.8, 0.9} Within stratum, Q_E is essentially a 1-to-1 label for m_3 sub-bin. |L| sidecar (report-only): chi^2 = 4.465, permutation p = 0.0741. Does NOT meet loud-signal threshold (p <= 0.01) for a fresh v0.6c |L| form lock. ``` The publishable v0.6 statement: > v0.6 registered a univariate energy-quartile audit on the supp-B > piano-trio catalog and tested it under two pre-registered > protocols. The catalog audit (v0.6a, chi-squared independence with > alignment guard) passed the registered chi-squared gate decisively > (chi^2 = 33.70, p ~= 2.29e-7) but tripped the alignment-tightness > guard at 0.956 (> 0.8 threshold). The alignment-breaking > within-branch audit (v0.6b, on the v0.5a (m_3<1, z_0<0.3) stratum) > failed under the pre-registered sparse-cell fallback's exact > permutation test (chi^2 = 6.90, permutation p = 0.029 vs threshold > p <= 0.01). The within-branch direction is monotone in the same > direction as the catalog-wide finding, but Q_E within the stratum > is essentially a 1-to-1 label for m_3 sub-bin and does not > generalize as a within-branch stratifier at the registered floor. > **Energy quartile does not carry stability information beyond > labeling the v0.5a branch hash on supp-B.** The |L| sidecar at > permutation p = 0.074 likewise falls short of the loud-signal > threshold; no fresh |L| form lock is licensed. v0.6c (held-out > predictor) is not licensed; the chapter closes as a > conditional-independence result, distinct from v0.4 > (structural-negative) and v0.5 (projection-limit). Three structural sub-results preserved in receipts and worth carrying into v0.7: 1. **Audit-with-alignment-guard as a methodological register**. A chi-squared independence audit at p ~= 1e-7 is operationalized as bin-locality-suspect by a pre-registered alignment-tightness scalar against v0.5a's branch label. This is the third register- shaping discipline added to the isotrophy methodology after v0.5's audit-then-predictor separation and asymmetric McNemar+delta falsifier. The alignment guard fires BEFORE a held-out test runs, forcing alignment-breaking re-registration rather than letting bin-locality survive into a predictor stage. 2. **Sparse-cell fallback tree as a permanent discipline**. The v0.6b form lock introduced an explicit three-way fallback (chi- squared / exact permutation / sparse-inconclusive) with locked thresholds and a locked permutation seed. This generalizes to any future within-strata audit on small N and avoids the standard chi-squared assumption violations (expected cell >= 5) without silently inflating type-I error. 3. **Energy quartile = m_3 sub-bin within the v0.5a stable branch on supp-B**. Q4 (loosest catalog orbits) is dominated by m_3 = 0.4 (52 of 65 Q4 rows); Q3 is mid-m_3; Q2 is high-m_3 within the branch; Q1 is entirely outside the branch (the tightest catalog orbits all sit in other branches). The conserved-quantity shadow does not add information beyond the (m_3, z_0) branch hash on this catalog at the registered precision. Joint v0.4 + v0.5 + v0.6 envelope on supp-B (the publishable multi-chapter statement): > Three sequential pre-registered low-dimensional projections of the > supp-B body — the Z_2 symmetry shadow (tangent-isotypic and > orbit-gauge-rigidity, v0.4), the 2-bit catalog branch shadow > (v0.5), and the continuous orbit-level conserved-quantity shadow > E and |L| (v0.6) — none carries held-out or branch-conditional > stability information on this catalog at the registered floors. > Stability on supp-B is bin-local to the m_3 = 0.4 cluster; no > catalog-coordinate or symmetry-shadow projection tested so far > lifts that locality to a generalizable mechanism. Any v0.7 > mechanism must either leave catalog-coordinate space entirely, > target the m_3 = 0.4 cluster as a substrate-specific object, or > retire the program. ## Reproducibility Surface Primary scripts (catalog-only; total compute is seconds): ```bash # v0.6a (catalog-only, seconds): python scripts/v06a_energy_quartile_audit.py # v0.6b (catalog-only, seconds + sub-minute permutation): python scripts/v06b_within_branch_energy_audit.py ``` Key receipt directories: ```text results/isotrophy/k-facet-v06a-energy-quartile-audit/manifest.json results/isotrophy/k-facet-v06a-energy-quartile-audit/contingency_table_E.csv results/isotrophy/k-facet-v06a-energy-quartile-audit/contingency_table_L.csv results/isotrophy/k-facet-v06a-energy-quartile-audit/energy_quartile_by_branch.csv results/isotrophy/k-facet-v06a-energy-quartile-audit/energy_quartile_by_m3.csv results/isotrophy/k-facet-v06a-energy-quartile-audit/energy_quartile_by_z0_bucket.csv results/isotrophy/k-facet-v06a-energy-quartile-audit/l_quartile_by_branch.csv results/isotrophy/k-facet-v06a-energy-quartile-audit/per_row_table.csv results/isotrophy/k-facet-v06b-within-branch-energy/manifest.json 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 results/isotrophy/k-facet-v06b-within-branch-energy/per_row_table.csv ``` Load-bearing constants live in the form-lock documents: ```text kfacet_v06a_energy_quartile_audit_form.md chi-squared critical = 11.34 (df = 3, p = 0.01); alignment-tightness threshold = 0.8; operational definitions of E and |L|; quartile method = numpy linear interpolation. kfacet_v06b_within_branch_energy_audit_form.md sparse-cell fallback tree; permutation seed = 20260523; n_permutations = 10000; stratum = v0.5a (m_3<1, z_0<0.3) branch; sidecar = within-branch |L|, report-only. ``` ## Where v0.7 Opens The v0.6 conditional-independence close adds a third chapter-close register to the isotrophy methodology: ```text v0.3: domain-of-applicability (Gamma_i predicts 0 on D_3-strict domain, supp-B daughters are Z_2-or-smaller -> out of domain). v0.4: structural-negative (Z_2 shadow's two registered predictors both fail). v0.5: projection-limit (audit passes in-sample at p ~= 1e-7, predictor fails held-out by accuracy_delta = -0.019). v0.6: conditional-independence (catalog audit passes with alignment warning, within-branch audit fails permutation p = 0.029 vs p <= 0.01 floor). ``` Each chapter closes a distinct family of catalog-coordinate or symmetry-shadow projections. The cumulative envelope says no such projection tested so far lifts the m_3 = 0.4 stable cluster's bin-locality to a generalizable mechanism. **v0.7 opens with the gamma_1 direction-of-instability family** (codex direction, 2026-05-24). Re-registers the v0.3 gamma_1 sidecar (direction of largest-real-part Floquet eigenvector) as a primary mechanism. This required a careful non-circularity audit because v0.4b explicitly disallowed Floquet-derived features for stability circularity reasons. **Parent registration landed 2026-05-24** at `kfacet_v07_mechanism_preregistration.md`; **v0.7a child form lock landed 2026-05-24** at `kfacet_v07a_velocity_fraction_audit_form.md`. The parent locks body / projection / observable, sketches five candidate operational definitions (D1-D5), explicitly identifies three circularity risks, and inherits all v0.5/v0.6 discipline. The v0.7a form lock picked **D5 + B** (velocity-fraction quartile audit) over D1 + A because D5 minimizes the tie-break surface and avoids using eigenvalue ordering as the feature-selection mechanism; D1 + A is preserved as a named report-only sidecar. The locked non-circularity sentence: *"v0.7a 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. The tested scalar is a phase-space composition ratio of the selected direction, not the growth rate of that direction."* v0.7a runner pending implementation (`scripts/v07a_velocity_fraction_audit.py`). ## Doc Trail - `kfacet_v06_mechanism_preregistration.md` -- v0.6 parent registration. - `kfacet_v06a_energy_quartile_audit_form.md` -- v0.6a catalog audit form lock + verdict. - `kfacet_v06b_within_branch_energy_audit_form.md` -- v0.6b within-branch audit form lock + verdict. - `kfacet_v07_mechanism_preregistration.md` -- v0.7 parent registration (gamma_1 direction-of-instability; opens v0.7). - `kfacet_v05_writeup.md` -- v0.5 projection-limit chapter close (companion). - `kfacet_v04_writeup.md` -- v0.4 structural-negative chapter close (grand-companion). - `kfacet_v03h_writeup.md` -- v0.3 domain-of-applicability close (great-grand-companion). - `docs/CROSS_SUBSTRATE_NOTES.md` sections 6-7 -- projection- language framing (body / shadow / projection-limit / conditional- independence vocabulary). The chapter closes cleanly. Receipts are durable. v0.6a's catalog audit is preserved as a real finding (and as a methodological demonstration of the alignment-tightness guard); v0.6b's within-branch fail is preserved as the corresponding conditional-independence result. The chain has produced four sequential pre-registered chapters (v0.3, v0.4, v0.5, v0.6) of progressively richer projections on the Li-Liao supplementary-B catalog, each with reproducible receipts and explicit chapter-close discipline.