# v0.5b Branch-Predictor Registration (Held-Out) Status: **VERDICT LANDED 2026-05-23**. The held-out predictor fails: `accuracy_model = 0.6198` vs always-`U` `0.6388`, with McNemar `win = 28`, `loss = 33`, `p = 1.0`. Verdict: `branch_predictor_fails_heldout`. Audience: v0.5b runner; paper-side reviewer of the v0.5 transition from audit to prediction. Companions: - `kfacet_v05a_branch_map_form.md` -- v0.5a audit verdict and active branch hash. - `kfacet_v04_writeup.md` -- v0.4 chapter close that opened the branch program. - `kfacet_v04a_domain_map_preregistration.md` -- supp-B domain map source. Frame: v0.5a showed that the branch shadow `(m_3 < 1, z_0 < 0.3)` carries stability information. v0.5b asks whether that shadow predicts held-out stability better than the catalog-skew baseline. ## Verdict (Landed, 2026-05-23) The primary leave-one-`m_3`-bin-out gate fails: ```text gating rows: 263 gating S/U: 95 S / 168 U model accuracy: 0.619772 always-U accuracy: 0.638783 accuracy delta: -0.019011 McNemar: win: 28 loss: 33 n_discordant: 61 p-value: 1.0 [win <= loss short-circuit] verdict: branch_predictor_fails_heldout ``` Structural reading: > v0.5a's branch hash stratifies stability in-sample, but the branch-majority > predictor does not generalize across held-out mass bins. The branch shadow > is a descriptive catalog partition, not a mass-bin-held-out predictive > mechanism under this first registered rule. The fold table localizes the failure. The `m_3 = 0.4` bin is the dominant stable fold (`35 S / 20 U`). When it is held out, the remaining low-mass, low-`z_0` rows train that branch as `U` (`28 S / 33 U`), so the model misses the large `m_3 = 0.4` stable block. In the other low-mass folds, the same branch trains as `S`, but the stable-row wins and unstable-row false positives nearly cancel (`28` wins vs `33` losses pooled). Sidecars: ```text single-rule sidecar: rule: predict S iff (m_3 < 1 and z_0 < 0.3) accuracy: 0.688213 vs always-U 0.638783 McNemar: win = 63, loss = 50, p = 0.129431 verdict: fails sidecar threshold random-half sidecar: seed: 20260523 accuracy: 0.688213 vs always-U 0.638783 McNemar: win = 63, loss = 50, p = 0.129431 verdict: fails sidecar threshold ``` Receipt paths: ```text results/isotrophy/k-facet-v05b-branch-predictor/manifest.json results/isotrophy/k-facet-v05b-branch-predictor/per_fold_table.csv results/isotrophy/k-facet-v05b-branch-predictor/per_row_predictions.csv results/isotrophy/k-facet-v05b-branch-predictor/branch_training_table.csv results/isotrophy/k-facet-v05b-branch-predictor/sidecar_single_rule.csv results/isotrophy/k-facet-v05b-branch-predictor/sidecar_random_half.csv scripts/v05b_branch_predictor.py ``` ## Four Decisions Locked By This Form 1. **Held-out partition**: leave-one-`m_3`-bin-out is the primary gate. The gating folds are the 12 `m_3` bins with `N >= 5`. This respects the bifurcation/cross-mass structure. A deterministic random half split is reserved as a non-gating robustness sidecar. 2. **Predictor form**: keep the full active 4-bucket `(b1, b2)` hash from v0.5a. Do NOT narrow to the single rule `S iff (m_3 < 1 and z_0 < 0.3)` as the primary form. The single rule is recorded as a sidecar only. 3. **Falsifier**: compare held-out branch-majority predictions against the always-`U` baseline using a one-sided exact McNemar/binomial test on discordant rows, with `p <= 0.01` and positive accuracy delta required for a pass. 4. **Diagnostic surface**: continuous quantities (`T`, `abs(v_z)`, `E`, `|L|`, and derived ratios) remain diagnostic-only. They are candidate v0.5c features, not v0.5b features. ## Body / Projection / Observable ```text Body: supplementary-B piano-trio orbit as primary branch object domain: 273 rows, already verified Z2_clean by v0.4a Projection: catalog branch shadow row -> active branch label from (m_3 < 1, z_0 < 0.3) Prediction target: stability label in {S, U} Primary observable: held-out row-level prediction accuracy vs always-U, pooled across leave-one-m_3-bin-out folds ``` The v0.5b predictor is intentionally catalog-side and orbit-input-only. It does not use Floquet spectra, monodromy data, Z2 tangent decompositions, or post-integration stability information to construct branch labels. ## Inputs Primary input: ```text results/isotrophy/k-facet-v05a-branch-map/per_row_table.csv ``` This table supplies, per row: ```text label, index, m3, z0, period, stability, vz, abs_vz, m3_z0_squared, branch_label, b1_m3_lt_1, b2_z0_lt_0p3, b3_abs_vz_lt_1e_minus_6, b4_m3_z0_sq_lt_2 ``` The v0.5b runner may also read: ```text results/isotrophy/k-facet-v05a-branch-map/manifest.json ``` only to verify that the active bits are exactly: ```text b1 = (m_3 < 1) b2 = (z_0 < 0.3) ``` No new orbit integration, tangent integration, or gauge minimization is authorized by this form. ## Gating Folds Primary leave-one-`m_3`-bin-out folds are the 12 bins with `N >= 5`: ```text m_3 N S U 0.4 55 35 20 0.5 31 15 16 0.6 22 6 16 0.7 18 4 14 0.8 18 6 12 0.9 25 8 17 1.0 38 7 31 1.1 23 1 22 1.2 7 2 5 1.5 8 4 4 1.6 8 7 1 1.7 10 0 10 ``` Total gating rows: `263`. The three small bins are report-only and excluded from the primary gate: ```text m_3 N S U 1.3 4 2 2 1.4 4 0 4 1.9 2 0 2 ``` They may be scored in a supplemental table using the predictor trained on all 263 gating rows, but they do not enter the primary p-value, accuracy, or verdict. ## Predictor Algorithm For each gating fold `f`: ```text test_f = rows with m_3 == f train_f = rows in the 12 gating bins with m_3 != f ``` For each active branch label `b` in the training set: ```text S_train(b) = number of training rows in branch b with stability S U_train(b) = number of training rows in branch b with stability U ``` Class prediction for a test row in branch `b`: ```text if S_train(b) > U_train(b): predict S else: predict U ``` Ties therefore predict `U`. This is the conservative direction against the always-`U` baseline. It avoids creating an `S` call when the training fold does not support one. Absent-branch fallback: ```text if branch b has no training rows: predict the global training majority class ties again predict U ``` The absent-branch fallback is not expected on supp-B, but it is specified to make the runner total. Probability diagnostics, not gating: ```text p_hat_S(b) = (S_train(b) + 0.5) / (S_train(b) + U_train(b) + 1.0) ``` This Jeffreys-smoothed branch probability is emitted for log-loss/Brier diagnostics only. The primary verdict uses the class prediction above. ## Primary Baseline The baseline predicts `U` for every held-out row: ```text baseline(row) = U ``` This is the catalog-skew baseline. On the full v0.5a catalog it achieves `176 / 273 = 64.47%`; on the 263-row gating subset the runner must compute and record the exact baseline accuracy before scoring v0.5b. ## Primary Test: Exact McNemar Against Always-U Pool predictions across the 12 leave-one-`m_3` folds. Define paired discordance counts: ```text win = # rows where v0.5b predictor is correct and always-U is wrong loss = # rows where v0.5b predictor is wrong and always-U is correct ``` Rows where both are correct or both are wrong do not enter McNemar's discordant-pair test. The one-sided exact McNemar p-value is: ```text n_discordant = win + loss p_value = P[ Binomial(n_discordant, 0.5) >= win ] ``` If `win <= loss`, set `p_value = 1.0`. Primary verdicts: ```text if n_discordant < 10: verdict = branch_predictor_inconclusive_low_discordance elif (accuracy_model > accuracy_always_U) and (win > loss) and (p_value <= 0.01): verdict = branch_predictor_passes_heldout else: verdict = branch_predictor_fails_heldout ``` The p-value threshold is locked at `0.01`. No multiple-comparison correction is needed for the primary gate because there is exactly one primary held-out test. ## Secondary Metrics (Report-Only) The runner records, but does not gate on: ```text accuracy_model accuracy_always_U accuracy_delta per_fold_accuracy_delta balanced_accuracy_model balanced_accuracy_always_U Brier score using p_hat_S log loss using p_hat_S per_branch training S fractions by fold ``` Balanced accuracy is useful because supp-B is U-skewed, but it is not the primary gate. The primary gate remains paired row-level improvement over always-`U`. ## Sidecar A: Fixed Single-Rule Predictor The following rule is recorded as a non-gating sidecar: ```text predict S iff (m_3 < 1 and z_0 < 0.3) predict U otherwise ``` This sidecar is NOT the v0.5b primary predictor because it bakes in the v0.5a observed direction of the effect. It is useful as a readability check: if it agrees with the fold-trained 4-bucket predictor on most rows, the branch-shadow effect has a simple public explanation; if it diverges, the full four-bucket form was necessary. ## Sidecar B: Deterministic Random-Half Robustness Split The random split is not primary. If implemented, it is a robustness sidecar only: ```text seed: 20260523 split: deterministic 50/50 row split over the 263 gating rows train/test: train branch-majority predictor on half A, test on half B; train on half B, test on half A; pool both directions primary-like test: same exact McNemar calculation, report-only ``` This sidecar probes whether the branch-majority rule survives a row-level split. It does not replace leave-one-`m_3`-bin-out because a random split can place near-duplicate mass-bin structure in both train and test. ## Continuous Diagnostics Are Held Back The following are emitted but not used: ```text T abs(v_z) sign(v_z) E |L| m_3 * z_0^2 K / |E| T-normalized quantities ``` Reason: v0.5a's pass was produced by the branch hash alone. Promoting continuous quantities in v0.5b would turn the first held-out predictor into a feature-search step. They are reserved for v0.5c if v0.5b fails or if the per-fold residuals suggest a specific next mechanism. Before any continuous ratio becomes gating, the reference scales must be registered explicitly (`T_kepler`, `mu_eff`, energy normalization, and any log/ratio transforms). ## Edge Cases 1. **Branch absent in training**: use global training majority, tie -> `U`; record `absent_branch_fallback_used = true`. 2. **Fold has only one stability class**: score normally. McNemar pooling absorbs this; the per-fold table records the one-class fold. 3. **Small `m_3` bins (`N < 5`)**: excluded from the primary gate, reported only. They must not be quietly added to the gate after seeing the result. 4. **Exact 50/50 branch training split**: branch predicts `U` by the tie rule. 5. **Metric disagreement**: if McNemar passes but accuracy delta is not positive, verdict is fail. If accuracy delta is positive but McNemar does not pass, verdict is fail. Both are required. ## Receipt Schema Planned output: ```text results/isotrophy/k-facet-v05b-branch-predictor/ manifest.json per_fold_table.csv per_row_predictions.csv branch_training_table.csv sidecar_single_rule.csv sidecar_random_half.csv (optional if implemented) ``` `manifest.json` fields: ```text mode version = "v0.5b-branch-predictor-heldout" form_lock = "docs/isotrophy/kfacet/kfacet_v05b_branch_predictor_form.md" input_per_row_table gating_m3_bins excluded_report_only_m3_bins active_bits primary_partition = "leave_one_m3_bin_out" predictor_form = "fold_trained_branch_majority_on_active_b1_b2" tie_rule = "U" baseline = "always_U" N_gate S_gate U_gate accuracy_model accuracy_always_U accuracy_delta win loss n_discordant mcnemar_exact_one_sided_p verdict absent_branch_fallback_used sidecar_single_rule_summary sidecar_random_half_summary diagnostics_summary ``` ## Implementation Plan Suggested script: ```text scripts/v05b_branch_predictor.py ``` Suggested command: ```powershell python scripts\v05b_branch_predictor.py ` --input results\isotrophy\k-facet-v05a-branch-map\per_row_table.csv ` --out results\isotrophy\k-facet-v05b-branch-predictor ``` Expected runtime: seconds. No staged compute is required. ## Interpretation If v0.5b passes: > The catalog branch shadow is not only associated with stability in-sample; > it predicts held-out mass-bin stability better than the always-unstable > catalog baseline under a pre-registered branch-majority rule. This would license v0.5c to ask *why* the `(m_3, z_0)` branch boundary carries stability, with continuous diagnostics promoted one at a time. If v0.5b fails: > v0.5a found real in-sample stratification, but the branch shadow does not > generalize across mass bins. The branch hash is a descriptive catalog > partition, not a predictive mechanism. This would be a clean projection-limit result, not a failed run. ## Lock-In Statement Any change to the held-out partition, active bits, branch-majority rule, tie rule, primary baseline, McNemar threshold, gating-bin set, or continuous feature exclusion after the v0.5b runner is executed is a re-registration, not a refinement. No v0.5b predictor claim is licensed until this form is signed off and the held-out runner writes its receipt.