# K_facet v0.3h Methodology + Result Handoff Status: current, 2026-05-22. Audience: collaborators, paper-side writers, future coding agents. Canonical sources: `docs/isotrophy/kfacet/kfacet-runner-spec.md`, `docs/isotrophy/sundog_v_isotrophy.md`, and `docs/SUNDOG_V_THREEBODY.md`. ## What v0.3h Tested v0.3h tested the catalog-level structural-zero hypothesis for `K_facet`. After the v0.2 containment count collapsed to the equivariance-null, the test was rebuilt around the rank-matrix `Gamma_i`: look inside `ker(M_i - I)` for an `F_beta`-even standard `D3` sector. If that sector is present, `Gamma_i` can carry a branch count. If it is absent, then `c_i = d_i = 0` by structure, not by tolerance or post-hoc pruning. The question for the 21 strict G.2 single-curve choreographies at `m_3 = 1` was therefore sharp: does any row contain a valid standard `D3` isotypic block, or is the v0.3h prediction structurally zero on the catalog? ## Audit Chain Methodology The audit chain was deliberately receipt-first and closure-relative, matching the discipline that resolved G.2. It proceeded in three stages. First, the sentinel/Gamma runner built `M_i`, typed `D3` operators, reduced `omega`, and the closed-form `partial_eps M_i`, then gated D3 relations, leakage, and finite-difference consistency. Second, the adaptive-floor reprocessor reused the saved `.npy` artifacts, with no integration, and chose the smallest pre-registered floor satisfying all three guards: D3 leakage `<= 1e-3`, gap ratio `<= 1e-3`, and first-rejected singular value `>= 1e-3`. Third, the bridge audit examined any row not resolved by that floor rule, testing bridge vectors for neutral overlap, Jordan behavior, and defective D3 signatures. No row-specific knobs were introduced. Constants and outcome categories were registered in code/spec before interpretation, so each row either became a structural-zero receipt, a named refinement case, or an excluded defective case. ## Result And Quarantine Final result: ```text 20 rows no_bridge_present T(2) + S(5 or 6) + E(0) 1 row defective_E_block_confirmed O_617 ``` The load-bearing v0.3h catalog statement is: **20 of 21 strict rows are confirmed structural zeros** under the adaptive-floor discipline. For those rows the standard `D3` sector is empty, so `c_i = d_i = 0`. `O_617` is quarantined, not counted for or against the prediction. Its bridge singular value is `7.8359e-4`; neutral overlap is `1.81e-4`; the Jordan-chain test amplifies rather than drops (`90.04`). At the bridge-admitted kernel the representation reads `T(2) + S(6) + E(1)`, which is not a valid real standard `D3` block, and `||sigma3^3 - I||_inf = 3.96e-2` shows the order-3 relation fails at about 4%. Correction from the companion deep dive: `O_617` is a clean opposite-strict catalog row. Its admission residual is `1.01e-8`; the earlier `1.62e-1` value is the canonical residual and is not the admitting orientation. The quarantine therefore reflects a bridge direction outside the valid `D3` representation, not an admission weakness and not a failure of the `Gamma_i` audit chain. ## Freeze + Supplementary-B Comparison The follow-on freeze is now explicit. The resolved v0.3h `Gamma_i` mechanism predicts zero piano-trio daughters from the strict G.2 seeds: ```text 20 resolved structural-zero rows: sum d_i = 0 O_617: quarantined, excluded from evidence K_facet_v0.3h(resolved Gamma mechanism) = 0 ``` Supplementary-B parses as a nonempty piano-trio catalog: ```text supplementary-B rows: 273 m3 = 1 rows: 38 m3 != 1 rows: 235 stability: 97 S, 176 U ``` This closes v0.3 as a **structural-null mechanism** against supplementary-B: the audit chain measured a real prediction, and that prediction does not explain the published piano-trio rows. This does not mean "no piano-trios"; it means the v0.3h `Gamma_i` standard-sector mechanism contributes zero resolved daughters. ## Reproducibility Surface Primary scripts: ```bash npm run isotrophy:kfacet:sentinel:gamma npm run isotrophy:kfacet:reprocess-floor npm run isotrophy:kfacet:bridge-audit ``` Key receipt directories: ```text results/isotrophy/k-facet-v03-sentinel-calibration-O62-gamma/ results/isotrophy/k-facet-v03-sentinel-sweep-calibrated/ results/isotrophy/k-facet-v03-sentinel-sweep-adaptive-floor-21/ results/isotrophy/k-facet-v03-bridge-audit-21/ results/isotrophy/k-facet-v03-freeze-supplementary-b-comparison/ ``` Load-bearing constants include the adaptive floor ladder, projector/leakage floor `1e-3`, gap ratio `1e-3`, `k_gamma = 3`, and `k_int = 10`. The final claim should be cited as **20/21 structural zeros plus one quarantined O_617 defective-D3 bridge**, not as a closed 21/21 theorem-facing result. ## v0.3 Epilogue (closed, 2026-05-22) The `alpha` freeze + supplementary-B comparison ran on 2026-05-22 and the II cross-m_3 sentinel sweep + symmetry probe followed on the same day. The combined verdict closes v0.3 as a **domain-of-applicability finding**, not a within-domain falsification: ```text v0.3 Gamma_i mechanism: - Defined: rank gate on F_beta-even standard D_3 isotypic of ker(M_i - I). - Domain of applicability: D_3-symmetric orbits. - Verified domain: the 21 strict G.2 single-curve choreographies at m_3 = 1 (catalog A). - Empirical result on domain: 20 of 21 structural zero; 1 quarantined row (O_617) with the `bridge_approx_sign_isotypic` diagnosis. - Predicted daughter count from G.2 strict 21: 0. Supplementary-B catalog: - 273 piano-trio orbits at varied m_3. - Symmetry class verified by sigma_3-scan + cross-m_3 sentinel: all seven tested rows are NOT sigma_3-symmetric (residual ~0.7, 7--9 orders above catalog-strict); six of seven are F_beta = (12)-swap symmetric (residual ~1e-8, comparable to strict G.2); O_434(0.4) also breaks F_beta. Symmetry class is Z_2-or-smaller, not D_3. - Lies outside the v0.3 prediction's domain of applicability. Comparison verdict: Predicted 0 daughters from G.2 strict (verified). Observed 273 piano-trios (verified). The mismatch is a **domain-of-applicability** finding: v0.3 cannot predict piano-trios by construction because they do not carry the assumed symmetry. The audit chain is intact; the prediction is intact; the catalog is in a different symmetry class. ``` The publishable v0.3 result is therefore: > v0.3's Gamma_i rank gate predicts zero daughter piano-trios from the > 21 strict G.2 D_3-symmetric choreographies at m_3 = 1, with 20 of 21 > structural-zero on the domain and the 21st (`O_617`) quarantined with > a precise causal diagnosis (`bridge_approx_sign_isotypic`). > Supplementary-B's piano-trio sentinels are empirically Z_2-or-smaller, > not D_3; the predicted-vs-observed mismatch reflects > domain-of-applicability, not within-domain falsification. Sub-investigation candidates carried forward, not blocking the epilogue: - `O_617` (m_3 = 1 strict G.2): defective-D3 boundary, see `kfacet_v03h_o617_deep_dive.md`. - `O_434(0.4)` (supp-B): breaks both `sigma_3` and `F_beta` closure; possibly a sub-Z_2 symmetry class. Cheap row-anatomy probe candidate. - Typed transport lemma (III): paper-side rigor; deferred since v0.3c, no longer blocking the empirical close. The next chapter is **v0.4**: design a candidate Z_2-equivariant mechanism (or an a-symmetric one) for piano-trio prediction. This is fresh paper-side work; do not scaffold runner code until the representation theory closes on Z_2. Companion docs: - `kfacet_v03_freeze_b_comparison.md` -- alpha freeze + domain addendum. - `kfacet_v03_gamma_crossm3_preregistration.md` -- II pre-registration + (Q1.D, Q2.D) verdict. - `kfacet_v03h_o617_deep_dive.md` -- quarantined-row companion. Follow-on: v0.4a treats supplementary-B piano-trios as primary `Z_2` objects and pre-registers a two-pass gauge domain map over all 273 rows; see `docs/isotrophy/kfacet/kfacet_v04a_domain_map_preregistration.md`. ## Open Polish Item (Run-Friendly) A ~10-hour tooling-polish initiation is registered in [`../../TODO.md`](../../TODO.md) under **Onboarding / Polish (Run-Friendly) -> V0.3h K_facet Tooling Polish**. It promotes the three one-shot diagnostic scripts (`scripts/o617_deep_dive.py`, `scripts/o617_why_dive.py`, `scripts/catalog_near_t_separator.py`) to discoverable workbench subcommands with `--row` parameters and npm entries, fixes the projector double-counting artifact in the catalog separator via simultaneous (sigma_3, F_beta) eigendecomposition, and bakes the "always signed inner products" lesson from the (II) round into a shared helper. The polish does **not** alter the load-bearing v0.3h evidence. The TODO entry includes the full paper trail (this writeup, the deep dive, the spec, the public mirrors, and the workbench script) as its table of contents.