# v0.4b gamma_3' Functional Form Lock (Orbit-Pass2 Baseline) Status: **VERDICT LANDED 2026-05-23**. Locked paper-side 2026-05-23 after the tangent-isotypic gamma_3 baseline retired and the cocycle-rescue probe surfaced `(R_i, phi_i) = (I, 0)` uniformly. The locked orbit-pass2 baseline was tested against the v0.4a manifest directly (no new compute). **Falsified by chi^2 = 1202.32 vs critical value 26.22** (46x threshold). Rule's accuracy 63.74% is slightly below the always-U baseline 64.47%, confirming gauge-rigidity is statistically independent of stability on supp-B. See the **Verdict (Landed)** section below. Audience: v0.4b implementation; v0.4c reader of the verdict. Companion: `kfacet_v04b_gamma3_form.md` (retired tangent baseline), `kfacet_v04b_mechanism_preregistration.md` (parent registration), and the v0.4a manifest at `results/isotrophy/k-facet-v04a-domain-map/manifest.json`. ## Verdict (Landed, 2026-05-23) The locked baseline was tested against the v0.4a manifest directly. No new compute. Per-row pass2_required boolean was read from `results/isotrophy/k-facet-v04a-domain-map/manifest.json#per_row_table[*].provenance`. ```text Total rows: 273 Predicted S (pass2_required = True): 24 ( 8.79% of catalog) Predicted U: 249 (91.21%) Observed S: 97 (35.53%) Observed U: 176 (64.47%) Accuracy: 174 / 273 = 63.74% Always-U baseline: 176 / 273 = 64.47% (rule is SLIGHTLY WORSE than always-U) chi^2 test statistic: 1202.32 critical value: 26.22 (chi-squared(12), p = 0.01) verdict: falsifies_baseline_orbit_pass2_rule (chi^2 / critical = 45.86) ``` Per-m_3 contributions to chi-squared: ```text m_3 N S_obs U_obs pass2_count P_S_pred chi2_contrib 0.4 55 35 20 1 0.0182 1177.41 0.5 31 15 16 0 0.0000 7.26 0.6 22 6 16 0 0.0000 1.64 0.7 18 4 14 0 0.0000 0.89 0.8 18 6 12 0 0.0000 2.00 0.9 25 8 17 0 0.0000 2.56 1.0 38 7 31 0 0.0000 1.29 1.1 23 1 22 0 0.0000 0.04 1.2 7 2 5 0 0.0000 0.57 1.5 8 4 4 8 1.0000 2.00 1.6 8 7 1 7 0.8750 0.00 1.7 10 0 10 4 0.4000 6.67 --------- chi^2 = 1202.32 ``` Diagnostic bins (N < 5, non-gating): m_3 in {1.3, 1.4, 1.9}. The m_3 = 0.4 bin alone contributes `1177` of the total chi-squared. 35 observed stable orbits at m_3 = 0.4 vs only 1 predicted stable (O_434, the worked-example Pass 2 rescue). The rule under-predicts S catastrophically at low m_3 and over-predicts S at m_3 in {1.5, 1.6, 1.7}. **Structural reading:** > Orbit-level gauge-rigidity, as measured by Pass 2 reclassification in the > v0.4a two-pass classifier, is statistically independent of stability on the > supp-B catalog. The two pieces of information do not predict one another; > they are orthogonal projections of the body. This falsifies the registered baseline. The wording per Codex sign-off: **falsifies this baseline orbit-pass2 rule**, not gamma_3'_orbit_features globally. A subsequent gamma_3'_T_threshold, gamma_3'_z0_threshold, or gamma_3'_combined registration could be tested separately if the diagnostic residuals suggest a different orbit feature carries stability information. ### Receipts ```text docs/isotrophy/kfacet/kfacet_v04b_gamma3prime_form.md results/isotrophy/k-facet-v04b-gamma3prime-orbit-pass2/manifest.json - per_m3_table: per-m_3 contingency + chi-squared contributions - per_row_predictions: 273 rows with pass2_required + predict + correct - verdict: falsifies_baseline_orbit_pass2_rule ``` ## The Locked Form `gamma_3'_orbit_pass2` is a **zero-parameter threshold rule** on a single orbit-level Z_2 feature recorded by the v0.4a two-pass classifier: ```text gamma_3'_orbit_pass2(row) = +1.0 (predict S) iff pass2_required(row) == True 0.0 (predict U) otherwise ``` Where: ```text pass2_required(row) == True iff the row's Pass 1 classification (default gauge tolerances: identity_rotation_tolerance = 1e-6, phase_grid = 73) placed it in any band other than Z2_clean, AND the row's final classification (after the Pass 2 tight rerun at identity_rotation_tolerance = 1e-9, phase_grid = 361) is Z2_clean. Equivalently: the v0.4a manifest's per_row_table[row]["provenance"] field reads "pass2". ``` 24 of 273 supplementary-B rows have `pass2_required == True`; 249 do not. ## Interpretation > *Orbits whose F_beta closure required tight gauge minimization to land in > `Z2_clean` are predicted stable; orbits whose closure was already clean at > default tolerances are predicted unstable.* This tests the **orbit-level gauge-rigidity hypothesis**: that an orbit's resistance to coarse-gauge classification is a structural property correlating with stability. The hypothesis is registered, not asserted. ## Non-Circularity Audit ```text pass2_required is determined entirely by F_beta closure residual at two gauge tolerances, BEFORE any stability label is consulted. The pass2_required boolean was computed by the v0.4a two-pass sweep which ran with no knowledge of stability and emitted a Z_2-clean/not decision per row. The stability label S/U is from the supp-B catalog, an independent input parameter. Predicting stability from pass2_required uses no stability information in the predictor. m_3 enters only as a stratification variable (chi-squared bins), NOT as a fitted feature. The rule fires on pass2_required alone. ``` The rule could be wrong; it cannot be circular. ## Free Parameter Count ```text Number of free parameters fitted to the data: 0 ``` The threshold `pass2_required == True` is a binary feature defined pre-data by the v0.4a two-pass tolerance schedule. The functional form maps that boolean directly to a S/U prediction. ## Chi-Squared Degrees of Freedom ```text df = (number of gating m_3 bins with N >= 5) - (number of free parameters) = 12 - 0 = 12 ``` Reference distribution `chi-squared(12)`. Critical value `26.22` at `p = 0.01`. Test statistic `chi^2 > 26.22` **falsifies this baseline orbit-pass2 rule**. Wording note (per Codex sign-off): the failure condition is recorded as "falsifies this baseline orbit-pass2 rule", **not** "falsifies gamma_3'_orbit_features globally". A subsequent gamma_3'_T_threshold or gamma_3'_z0_threshold could be separately registered if the diagnostic residuals suggest a different orbit feature. ## Bin-Level Predicted Stable Fraction For each m_3 bin: ```text P_S_predicted(m_3) = (count of rows in bin where pass2_required is True) / N(m_3) ``` This is a deterministic function of v0.4a manifest data; no additional fitting. ## Edge Cases (Pre-Registered) 1. **m_3 bins with 0 pass2_required rows**: predicted_S = 0 for all rows in that bin. If observed_S > 0, the bin contributes positively to chi-squared. 2. **m_3 bins with all pass2_required rows**: predicted_S = 1 for all rows in that bin (m_3=1.5, m_3=1.6, m_3=1.7 all have pass2 fractions visible in the v0.4a Pass 2 reclassification table). If observed_S < N(m_3), the bin contributes positively to chi-squared. 3. **Small bins (N < 5)**: m_3 in {1.3, 1.4, 1.9}. Reported as diagnostic only; not gating. ## Pre-Mortem Expectation Before the verdict computation, the registered baseline has a known pre-mortem signal: ```text 24 pass2_required rows, 11 S + 13 U: the rule labels all 24 as S; accuracy on these is 11/24 = 45.8%. 249 non-pass2_required rows, 86 S + 163 U: the rule labels all 249 as U; accuracy on these is 163/249 = 65.5%. Overall accuracy: (11 + 163) / 273 = 63.7%. Always-U baseline accuracy: 176 / 273 = 64.5%. The rule is slightly worse than the trivial always-U predictor. So the baseline is expected to fail chi-squared by a substantial margin; the question is how much. ``` The registered baseline is therefore expected to falsify cleanly. That falsification IS the v0.4b sub-result: "orbit-level gauge-rigidity as measured by Pass 2 reclassification does not predict stability." ## Lock-In Statement This functional form is committed before the verdict is computed. Any change to the form after the verdict is a re-registration, not a refinement. Implementation may proceed against this form lock using the v0.4a manifest at `results/isotrophy/k-facet-v04a-domain-map/manifest.json` as the sole input; no new compute required.