# v0.5a Branch-Map Audit Registration (Branch-Shadow Audit) Status: **VERDICT LANDED 2026-05-23**. Registered and computed against the v0.4a manifest. Verdict: `branch_hash_passes_audit` with `chi^2 = 34.986` vs critical `11.34` (`df = 3`, p ~= `1.23e-7`). Audience: v0.5a runner; v0.5b reader of the verdict; paper-side reviewer of the projection-limit framing. Companion: `kfacet_v04_writeup.md` (chapter close that opened v0.5), `kfacet_v04b_gamma3prime_form.md` (prior falsified baseline for non-circularity provenance), `kfacet_v04a_domain_map_preregistration.md` (catalog source). Frame: v0.5 stops asking what symmetry the row has and starts asking which branch the row belongs to. ## Verdict (Landed, 2026-05-23) The locked branch hash passes the audit: ```text total rows: 273 observed S/U: 97 S / 176 U active bits: (m_3 < 1, z_0 < 0.3) occupied buckets: 4 df: 3 chi^2: 34.986253 critical: 11.34 (chi-squared(3), p = 0.01) p-value: 1.23e-7 verdict: branch_hash_passes_audit ``` 4x2 contingency table: ```text branch_label N S U S_fraction chi2_contrib m_3 >= 1, z_0 >= 0.3 87 18 69 0.2069 8.366 m_3 >= 1, z_0 < 0.3 17 5 12 0.2941 0.278 m_3 < 1, z_0 >= 0.3 56 11 45 0.1964 6.171 m_3 < 1, z_0 < 0.3 113 63 50 0.5575 20.171 ``` Structural reading: > The catalog-only branch shadow `(m_3 < 1, z_0 < 0.3)` carries stability > information on supp-B. This is not yet a predictor; it is the v0.5a audit > pass that licenses a separate v0.5b held-out prediction registration. Receipts: ```text results/isotrophy/k-facet-v05a-branch-map/manifest.json results/isotrophy/k-facet-v05a-branch-map/contingency_table.csv results/isotrophy/k-facet-v05a-branch-map/per_row_table.csv scripts/v05a_branch_map_audit.py ``` ## What v0.5a Is v0.5a is a **catalog-side audit**, NOT a predictor: ```text Test: chi-squared independence of branch_label vs S/U on supp-B df: occupied_branch_count - 1 Critical: chi-squared(df) at p = 0.01 Pass: chi^2 > critical -> branch hash carries stability information Fail: chi^2 <= critical -> branch shadow does not stratify S/U ``` A pass licenses v0.5b: a separately-registered predictor with held-out test. A fail is the third structural-negative in the isotrophy program — recorded, published, and the chain moves to the next mechanism family without contorting the rule. ## The Locked Form (4-bit candidate, with constant-bit retirement) ```text branch_label(row) = tuple of active nonconstant bits drawn from: b1 = (m_3 < 1) b2 = (z_0 < 0.3) b3 = (abs(v_z) < 1e-6) b4 = (m_3 * z_0^2 < 2) retirement rule (deterministic, pre-data): any bit that is constant (all True or all False) on the catalog under test is RECORDED on the receipt but RETIRED from the active signature and from df. ``` The 4-bit candidate is locked before the audit runs. The retirement rule is locked before the catalog degeneracy check (below). Active bit selection is therefore mechanical, not curated. ## Catalog Degeneracy Check (Pre-Audit, Landed) Per-bit truth counts over the v0.4a manifest's 273 supp-B rows: ```text bit true / 273 status b1 = (m_3 < 1) 169 / 273 ACTIVE b2 = (z_0 < 0.3) 130 / 273 ACTIVE b3 = (abs(v_z) < 1e-6) 0 / 273 RETIRED (constant FALSE) b4 = (m_3 * z_0^2 < 2) 273 / 273 RETIRED (constant TRUE) ``` Active signature on supp-B: `(b1, b2)`. Constant bits b3, b4 are preserved on the receipt for cross-substrate comparability (mesa, geometry, or freeze-A choreographies may light up the retired bits and require their own registration). ## Active Branch Buckets (Pre-Audit, Landed) ```text bucket N (m_3 >= 1, z_0 >= 0.3) 87 (m_3 >= 1, z_0 < 0.3) 17 (m_3 < 1, z_0 >= 0.3) 56 (m_3 < 1, z_0 < 0.3) 113 --- 273 ``` All 4 buckets occupied -> occupied_branch_count = 4 -> df = 3. ## Primary Test ```text H_0: branch_label and stability label are independent on supp-B. H_1: branch_label and stability label are not independent. Test statistic: chi^2 over 4-bucket x 2-class contingency table df: occupied_branch_count - 1 = 3 Critical value: chi-squared(3) at p = 0.01 = 11.34 verdicts (deterministic): chi^2 > 11.34: branch_hash_passes_audit chi^2 <= 11.34: branch_hash_fails_audit ``` ## Interpretation > *The branch hash is a coarse low-dimensional shadow of catalog parameters > (m_3, z_0) only. The audit asks whether this pre-registered shadow > stratifies the supp-B catalog's S/U distribution. A pass means the body's > stability structure is partly visible in these branch coordinates; a fail > joins the v0.4 Z_2-shadow result as another projection-limit negative.* ## Non-Circularity Audit ```text b1, b2 are functions of catalog parameters (m_3, z_0) only: - m_3 is the published Li-Liao mass ratio (catalog input). - z_0 is the published initial-condition coordinate (catalog input). - NEITHER is derived from the Floquet spectrum. - NEITHER inspects the stability label. b3, b4 (retired here, recorded for cross-substrate) likewise use catalog geometry only (v_z is initial velocity; m_3 * z_0^2 is a dimensionful geometric quantity). The S/U label is supplied as the independent test column. Assigning branch from catalog is information-disjoint from stability labeling: the audit uses no stability information to construct the branch hash. This is the same non-circularity provenance as gamma_3'_orbit_pass2. ``` ## Free Parameter Count ```text Number of free parameters fitted to the data: 0 ``` The 4-bit signature, the thresholds (`< 1`, `< 0.3`, `< 1e-6`, `< 2`), the retirement rule (constant -> retire), and the df formula (occupied_branch_count - 1) are all locked pre-data. The active-bit selection is mechanical and audit-traceable. The 4-bucket cross-classification follows from the active bits without further choice. ## Chi-Squared Degrees of Freedom ```text df = occupied_branch_count - 1 = 4 - 1 [on supp-B; future catalogs may differ] = 3 ``` Reference distribution `chi-squared(3)`. Critical value `11.34` at `p = 0.01`. Test statistic `chi^2 > 11.34` triggers `branch_hash_passes_audit`. Wording note: the pass condition is `branch_hash_passes_audit`, NOT "branch-shadow theorem confirmed". A pass licenses a v0.5b predictor registration; it does not on its own establish a predictive mechanism. ## Reserved Diagnostics (Not Gating) The following per-row continuous quantities are emitted on the receipt but are NOT used in v0.5a's hash or test: ```text T (period) -- catalog input, no normalization fixed yet abs(v_z) -- magnitude only sign(v_z) -- reserved for chirality diagnostic only E, |L| -- orbit-level conserved quantities (from v0.3 cross-m_3 receipts when available) m_3, z_0 raw values -- the underlying catalog coordinates ``` Per the v0.5 sign-off direction: *define T_kepler exactly and mu_eff exactly before using continuous ratios beyond diagnostics*. Quantities such as `T / T_kepler`, `log(T)`, `log(z_0)`, and `T * mu_eff^(1/2)` are DEMOTED to diagnostics until their reference scales are pinned in a separate registration. They do not enter the v0.5a hash. ## Edge Cases (Pre-Registered) 1. **Empty bucket on supp-B**: the pre-audit catalog check above shows all 4 buckets occupied; this case is inactive on supp-B but the df formula `occupied_branch_count - 1` would absorb empties on another catalog automatically. 2. **Cross-substrate bit re-activation**: if mesa, geometry, or freeze-A choreographies make b3 (`abs(v_z) < 1e-6`) or b4 (`m_3 * z_0^2 < 2`) non-constant, that catalog gets its OWN registration. supp-B is locked to the active `(b1, b2)` signature. 3. **Pass with mass-only structure**: if v0.5a passes (`chi^2 > 11.34`) but the residual structure is dominated by b1 (m_3 axis) and b2 (z_0 axis) is uninformative, v0.5b must re-register the predictor with explicit `b1`-only thresholds and a held-out test. ANY post-audit narrowing of the signature is a new registration, not a refinement. 4. **Marginal pass (`11.34 < chi^2 < ~16`)**: report as `branch_hash_passes_audit_marginal`. v0.5b registration must include leave-one-m_3-bin-out cross-validation as a gating step, not diagnostic. ## Pre-Mortem Expectation The per-bucket S/U counts have NOT been computed before lock-in. The audit runs blind to the actual chi-squared value. The expected published statement, regardless of outcome: > v0.5a tests whether the supp-B catalog's branch hash — formed by > retiring degenerate bits and using the active `(m_3 < 1, z_0 < 0.3)` > signature — stratifies stability. The verdict is reported as > `branch_hash_passes_audit` (chi^2 > 11.34) or > `branch_hash_fails_audit` (chi^2 <= 11.34) under the locked > chi-squared(3) threshold at p = 0.01. ## Why v0.5a Is An Audit, Not A Predictor A 4-bucket signature on 273 rows is too coarse to be honestly called a predictor on this catalog. It tests the **projection limit**: does this pre-registered low-dimensional branch shadow carry stability information? ```text v0.4a domain-shadow audit: well-defined (PASS) but not predictive v0.4b Z_2-shadow predictor: FALSIFIED (chi^2 = 1202 vs 26.22) v0.5a branch-shadow audit: catalog projection-limit test v0.5b branch-shadow predictor: registered ONLY if v0.5a passes ``` We get the Z_2 buddha in the posture, not the statistic: this is a projection-limit test. If the branch hash fails, we record that this low-dimensional branch shadow also does not carry stability and move to the next mechanism family without contorting the rule. ## Lock-In Statement This audit registration is committed before v0.5a's chi-squared statistic is computed against the v0.4a manifest's per-row catalog parameters. The 4-bit candidate was specified before the constant-bit degeneracy check; the active `(b1, b2)` signature follows from the recorded retirement rule. Any change to the bit candidates, thresholds, retirement rule, df formula, or critical value after the chi-squared statistic is computed is a **re-registration**, not a refinement. Implementation may proceed using the v0.4a manifest at `results/isotrophy/k-facet-v04a-domain-map/manifest.json` as the sole input. No new dynamical compute is required: the four bits are catalog-only. ## Receipts (planned) ```text docs/isotrophy/kfacet/kfacet_v05a_branch_map_form.md (this document) results/isotrophy/k-facet-v05a-branch-map/manifest.json (after audit) - bit_truth_counts: per-bit true/false counts and retirement status - bucket_counts: per-bucket N - contingency_table: 4 x 2 (bucket x S/U) - chi_squared: test statistic - df: occupied_branch_count - 1 - critical: chi-squared(df) at p = 0.01 - verdict: branch_hash_passes_audit | branch_hash_passes_audit_marginal | branch_hash_fails_audit - per_row_table: row_id, m_3, z_0, abs(v_z), m_3*z_0^2, branch_label, stability ``` ## Doc Trail - `kfacet_v04_writeup.md` -- v0.4 structural-negative chapter close (opens v0.5) - `kfacet_v04b_gamma3prime_form.md` -- prior baseline (falsified Z_2 shadow) - `kfacet_v04a_domain_map_preregistration.md` -- catalog source manifest - `docs/CROSS_SUBSTRATE_NOTES.md` sections 6-7 -- projection-language framing (body / shadow / projection-limit vocabulary) --- Branch-shadow audit. Projection-limit floor, not predictor. Three buckets of falsification at the threshold; a pass licenses v0.5b registration.