# K_facet v0.5 Methodology + Result Handoff Status: closed projection-limit (audit-passes-predictor-fails), 2026-05-23. Audience: collaborators, paper-side writers, future coding agents. Canonical sources: `kfacet_v05a_branch_map_form.md`, `kfacet_v05b_branch_predictor_form.md`, and [`../../CROSS_SUBSTRATE_NOTES.md`](../../CROSS_SUBSTRATE_NOTES.md) sections 6-7 (projection-language framing). Companion (closed v0.4 chapter): `kfacet_v04_writeup.md`. ## One-Line Read v0.5 closes as a **projection-limit on the catalog-coordinate branch shadow** of the Li-Liao supplementary-B piano-trio catalog. The 2-bit branch hash on `(m_3 < 1, z_0 < 0.3)` is decisively associated with stability in-sample (`chi^2 = 34.99` vs critical 11.34, `p ~= 1.23e-7`, v0.5a) but does **not** predict held-out stability across mass bins under a pre-registered fold-trained branch-majority rule (`accuracy_delta = -0.019` vs always-U, `McNemar p = 1.0`, v0.5b). The branch hash is a descriptive catalog partition, not a predictive mechanism. The asymmetric result is itself the methodological finding: a positive chi-squared audit does not license a held-out predictor. ## What v0.5 Tested v0.5 opened after v0.4 closed as a structural-negative on the Z_2 shadow as a stability projector. The v0.5 frame, locked at chapter open: > Stop asking what symmetry the row has. Start asking which branch the > row belongs to. This trades the Z_2 tangent/orbit projection for a catalog-coordinate indicator-bit projection (a "branch hash") and asks whether the supp-B body's stability structure is visible in that projection. Two pre-registered chapters, both paper-side first: ```text v0.5a: branch-shadow audit Body: supp-B piano-trio orbit as primary branch object Projection: catalog branch hash on indicator 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} with deterministic constant-bit retirement. Observable: contingency table[branch_label][stability] Question: chi-squared independence of branch_label vs S/U. Result: PASS at chi^2 = 34.99 vs critical 11.34 (chi-squared(3), p ~= 1.23e-7, 3.1x threshold). v0.5b: branch-shadow predictor (held-out) Body: same primary branch body. Projection: same active (b1, b2) branch hash. Predictor: fold-trained branch-majority on the 4-bucket hash. Partition: leave-one-m_3-bin-out over 12 bins with N >= 5. Baseline: always-U. Observable: pooled paired accuracy on 263 gating rows. Question: exact one-sided McNemar p <= 0.01 AND positive accuracy delta? Result: FAIL. accuracy_model = 0.6198 vs always-U 0.6388; accuracy_delta = -0.019; win=28/loss=33/p=1.0. ``` ## Audit Chain Methodology The v0.5 chain inherited v0.3 + v0.4 discipline (closure-relative gates, pre-registered constants, deterministic outcome categories, pre-mortem discipline) and extended it with one new register: **Audit-then-predictor separation.** v0.5a is registered as an AUDIT (chi-squared independence on the full catalog), explicitly NOT a predictor. A pass does not by itself establish a mechanism; it licenses a separately-registered held-out predictor. v0.5b is registered as the predictor with leave-one-m_3-bin-out folds, a pre-registered tie rule (predict U), a pre-registered absent-branch fallback (global majority, tie -> U), an asymmetric falsifier (McNemar p <= 0.01 AND positive accuracy delta), and a sidecar discipline that keeps the single-rule `S iff (m_3 < 1 and z_0 < 0.3)` and a deterministic random-half split as report-only. The asymmetric falsifier (both p AND positive delta required) prevents "differ-from-baseline" passes that aren't "better-than-baseline." The fold-trained majority avoids baking in the v0.5a observed direction. The leave-one-m_3-bin-out partition respects bifurcation structure; the deterministic random-half is sidecar only because random splits can leak mass-bin structure across train and test. Three stages, all paper-side first: 1. **v0.5a form lock (`kfacet_v05a_branch_map_form.md`)**. 4-bit candidate signature with deterministic constant-bit retirement. Pre-registered chi-squared(df) falsifier with `df = occupied_branch_count - 1`. 2. **v0.5a runner and verdict**. `scripts/v05a_branch_map_audit.py` ran the locked audit against the v0.4a manifest's 273 rows. Catalog degeneracy check retired b3 (constant FALSE on supp-B) and b4 (constant TRUE on supp-B); active signature `(b1, b2)` gave 4 occupied buckets {87, 17, 56, 113}. Verdict `branch_hash_passes_audit` at chi^2 = 34.99. 3. **v0.5b form lock + held-out predictor + verdict**. `kfacet_v05b_branch_predictor_form.md` locked the leave-one-m_3-bin-out fold-trained branch-majority rule before any held-out compute. Pre-mortem flagged m_3 = 0.4 as the load-bearing fold (35 S of 55 rows; the catalog's most-stable bin). `scripts/v05b_branch_predictor.py` reproduced the receipt under the locked rule. Verdict `branch_predictor_fails_heldout`; m_3 = 0.4 fold was indeed load-bearing. ## Result And Projection-Limit Verdict ```text v0.5a verdict: branch_hash_passes_audit chi^2 = 34.99 vs critical 11.34 (chi-squared(3), p = 0.01). p_value ~= 1.23e-7 (3.1x threshold). 273 rows, 97 S / 176 U; 4 occupied branch buckets. audit-dominant bucket: (m_3 < 1, z_0 < 0.3) at 113 rows / 55.75% S. other buckets: 87 / 20.69% S, 17 / 29.41% S, 56 / 19.64% S. catalog mean: 35.53% S. small-mass low-z corner concentrates stability. v0.5b verdict: branch_predictor_fails_heldout 263 gating rows over 12 leave-one-m_3-bin-out folds. accuracy_model = 0.6198 accuracy_always_U = 0.6388 accuracy_delta = -0.019 (predictor LOSES to always-U) McNemar win=28, loss=33, n_disc=61, p=1.0 load-bearing fold: m_3 = 0.4 (55 rows / 35 S / 63.6% S). on its held-out fold, the residual five m_3 < 1 bins train the (m_3 < 1, z_0 < 0.3) bucket as U-majority (28 S / 33 U); the predictor cannot capture the m_3 = 0.4 stable cluster on its own held-out fold. ``` The publishable v0.5 statement: > v0.5 registered a 2-bit catalog branch hash on the supp-B piano-trio > catalog and tested it under two pre-registered protocols. The audit > (v0.5a, chi-squared independence) passed decisively: the catalog > branch is associated with stability at p ~= 1.23e-7. The held-out > predictor (v0.5b, leave-one-m_3-bin-out fold-trained branch-majority > vs always-U baseline) failed: accuracy delta -0.019, McNemar p = 1.0. > **The branch hash describes the supp-B distribution but does not > generalize across mass bins.** The in-sample positive is bin-local > to the m_3 = 0.4 stable cluster; under leave-one-bin-out it cannot > reproduce its own observed direction. The catalog branch shadow is a > descriptive partition, not a predictive mechanism. Three structural sub-results preserved in receipts and worth carrying into v0.6: 1. **Bin-locality of the m_3 = 0.4 stable cluster on supp-B**. m_3 = 0.4 has 35 S of 55 rows (63.6% stable) — the catalog's most-stable bin by a wide margin. Held-out, the rest of the m_3 < 1 sub-catalog cannot reproduce this cluster's S-majority direction. Either the mechanism is mass-specific to m_3 = 0.4, or the relevant feature is along an axis not aligned with the m_3 < 1 indicator. 2. **Audit-vs-predictor asymmetry as a methodological finding**. A chi-squared independence audit at p ~= 1e-7 did not survive a leave-one-bin-out predictor with the same active feature set. This is a load-bearing register for subsequent isotrophy chapters: passing audits license predictors, they do not establish them. The pre-registered v0.5b form caught this cleanly because it locked the asymmetric falsifier (both p AND positive accuracy delta required) and the partition (leave-one-m_3-bin-out, not random-half) before any held-out compute. 3. **Joint v0.4 + v0.5 projection-limit envelope on supp-B**. Stability information on this catalog is NOT carried by the Z_2 shadow at either tangent-isotypic or orbit-gauge-rigidity granularity (v0.4), AND is NOT carried as a held-out generalizable signal by the catalog-coordinate 2-bit branch shadow (v0.5). Any predictor with held-out stability content must look beyond both projections. ## Reproducibility Surface Primary scripts: ```bash # v0.5a (catalog-only, seconds): python scripts/v05a_branch_map_audit.py # v0.5b (catalog-only, seconds): 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 ``` Key receipt directories: ```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 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 ``` Load-bearing constants live in the form-lock documents: ```text kfacet_v05a_branch_map_form.md chi-squared critical = 11.34 (df = 3, p = 0.01), 4-bit candidate with deterministic constant-bit retirement, active signature (b1, b2) on supp-B. kfacet_v05b_branch_predictor_form.md leave-one-m_3-bin-out partition, fold-trained branch-majority predictor, tie rule predict U, absent-branch fallback, asymmetric falsifier (McNemar p <= 0.01 AND positive accuracy delta), inconclusive verdict at n_disc < 10. ``` ## Where v0.6 Opens The v0.5 projection-limit close is asymmetric, not a pure structural-negative. The audit pass is a real catalog stratification finding; the held-out fail is a real generalization gap. Together they constrain the next chapter: > v0.6 must use a projection richer than a 2-bit catalog hash, or it > must be a different mechanism family entirely. Continuous-feature > promotion within the catalog-coordinate axis (v0.5c) is the obvious > path but inherits v0.5b's bin-locality risk. A different projection > — orbit-level conserved quantities, bifurcation tracks across m_3, > direction-of-instability sidecar, or action-angle / KAM-style > decomposition — is the natural alternative. Candidate v0.6 mechanism families, all paper-side first, in rough order of inheritance from existing receipts: 1. **Conserved-quantity (E, |L|) stratification.** Orbit-level invariants verified tangent to v_bridge directions in v0.3; their per-row joint distribution may stratify supp-B in a stability-relevant way without invoking new groups. Builds directly on v0.3 cross-m_3 receipts; per-row E and |L| are cheap when the v0.3 sentinel ran on the row. 2. **Bifurcation-track projection.** The per-m_3 S/U distribution shows visible discrete transitions (m_3 = 1.0 has 7 S / 31 U, m_3 = 1.1 has 1 S / 22 U, m_3 = 1.5 has 4 S / 4 U, m_3 = 1.7 has 0 S / 10 U). A bifurcation-track mechanism would project the catalog onto the transition structure rather than an equivariant subspace. Risk: the m_3 axis is precisely the axis that failed held-out in v0.5b. 3. **Direction-of-instability (gamma_1) sidecar.** The v0.3 gamma_1 sidecar tracked unstable directions per row; v0.4 retired the related gamma_3 family. A re-registration of gamma_1 as a primary mechanism would ask whether the direction of instability (not just its presence) carries predictive content. 4. **Action-angle / KAM-style decomposition.** The heaviest theoretical machinery: project orbits onto torus structure and ask whether nearly-integrable rows are stable. Theoretically the most aligned with the substantive question; technically the most expensive. 5. **Closing without v0.6.** If the next workstream targets a different substrate (mesa, geometry) rather than another K_facet refinement, the isotrophy program retires at end-of-v0.5 with publishable methodology + three negative findings + one descriptive positive + one explicit projection-limit finding. Codex direction will pick. No runner work proceeds until the v0.6 mechanism family is registered paper-side. **Update 2026-05-23:** v0.6 has opened with the **conserved-quantity (E, |L|) stratification** family. Parent registration locked at `kfacet_v06_mechanism_preregistration.md`. The chosen mechanism is strictly richer than the v0.5 2-bit branch hash (continuous orbit- level invariants vs. catalog-coordinate indicator bits) while remaining catalog-derivable (computed from initial conditions + the Hamiltonian; no orbit integration required). The parent registration locks body / projection / observable, operational definitions of E and |L|, non-circularity provenance, disallowed-feature inheritance, and the v0.5 discipline (audit-then-predictor, asymmetric falsifier, held-out partition, tie rule, constant-feature retirement). v0.6a (audit form lock with explicit binning) and v0.6b (held-out predictor form lock) are pending child registrations. ## Doc Trail - `kfacet_v05a_branch_map_form.md` -- v0.5a audit form lock + verdict. - `kfacet_v05b_branch_predictor_form.md` -- v0.5b held-out predictor form lock + verdict. - `kfacet_v06_mechanism_preregistration.md` -- v0.6 parent registration (conserved-quantity (E, |L|) stratification; opens v0.6). - `kfacet_v04_writeup.md` -- v0.4 chapter close (companion). - `kfacet_v03h_writeup.md` -- v0.3 chapter close (grand-companion). - `docs/CROSS_SUBSTRATE_NOTES.md` sections 6-7 -- projection- language framing (body / shadow / projection-limit vocabulary). The chapter closes cleanly. Receipts are durable. v0.5a's catalog stratification is preserved as a real finding; v0.5b's held-out failure is preserved as a real generalization-limit. The chain has now produced three sequential pre-registered chapters (v0.3, v0.4, v0.5) of projection-limit results on the Li-Liao supplementary-B catalog, each with reproducible receipts and explicit chapter-close discipline.