# K_facet v0.4 Methodology + Result Handoff Status: closed structural-negative, 2026-05-23. Audience: collaborators, paper-side writers, future coding agents. Canonical sources: `kfacet_v04a_domain_map_preregistration.md`, `kfacet_v04b_gamma3_form.md` (retired baseline), `kfacet_v04b_gamma3prime_form.md` (falsified baseline), `kfacet_v04b_mechanism_preregistration.md`, and [`../../CROSS_SUBSTRATE_NOTES.md`](../../CROSS_SUBSTRATE_NOTES.md) sections 6-7 (projection-language framing for the v0.3 -> v0.4 transition). Companion (closed v0.3 chapter): `kfacet_v03h_writeup.md`. ## One-Line Read v0.4 closes as a **structural-negative on the Z_2 shadow as a stability projector** for the Li-Liao supplementary-B piano-trio catalog. The Z_2 orbit-level domain map is well-defined (v0.4a, 273/273 in Z2_clean after the two-pass classifier), but every registered Z_2-shadow stability predictor either has its precondition fail (gamma_3 tangent-isotypic, retired) or is empirically falsified against the per-m_3 stability distribution (gamma_3'_orbit_pass2, chi^2 = 1202 vs critical 26.22). Stability information on this catalog is not carried by the Z_2 projection at either the tangent-isotypic or orbit-gauge-rigidity granularity. ## What v0.4 Tested v0.4 opened after v0.3's domain-of-applicability finding: v0.3's `Gamma_i` rank gate is `D_3`-equivariant and structurally inapplicable to supplementary-B piano-trios, which carry only the `Z_2 = (12)`-swap symmetry. v0.4 re-framed the supp-B catalog as a primary `Z_2` body (not a daughter of strict G.2) and tested whether the `Z_2` shadow contains stability-predictive structure. Two pre-registered chapters: ```text v0.4a: domain-of-applicability audit Body: supp-B piano-trio orbit as primary Z_2 object Projection: orbit -> Z_2 symmetry-class shadow Observable: table[m_3][stability][class] over 273 rows Result: well-definedness PASS (all 273 land Z2_clean after the two-pass gauge classifier) v0.4b: mechanism-test audit (independent predictor) Body: same primary Z_2 body Predictor: non-circular gamma_3 family on Z_2 shadow features Observable: per-m_3 stability distribution Result: gamma_3 (tangent-isotypic) retired pre-sweep; gamma_3'_orbit_pass2 (orbit-gauge-rigidity) FALSIFIED by chi^2 = 1202 vs critical 26.22 ``` ## Audit Chain Methodology The v0.4 chain inherited and extended the v0.3h discipline: closure-relative gates, pre-registered constants, deterministic outcome categories. Three stages: 1. **v0.4a two-pass gauge classifier**. Pass 1 over all 273 rows at default tolerances (`identity_rotation_tolerance = 1e-6`, `phase_grid = 73`). Pass 2 tight rerun on every row not landing in `Z2_clean` at Pass 1 (`1e-9`, `phase_grid = 361`). Four-band classifier: `Z2_clean / marginal_Z2 / smaller_symmetry / undefined`. No per-row knobs. The two-pass design encodes the O_434 anatomy probe's methodological lesson, which found that default tolerances misclassified one sentinel row by six orders of magnitude. 2. **v0.4b paper-side form lock**. Two registered baseline predictors, both with zero free parameters and pre-registered chi-squared(12) falsifiers: - `gamma_3` (tangent-isotypic): predict S iff `F_beta_even_dim >= F_beta_odd_dim` on `K_fib` (neutral-quotiented kernel). - `gamma_3'_orbit_pass2` (orbit gauge-rigidity): predict S iff the row required Pass 2 rescue in the v0.4a classifier. 3. **Sanity probes between registration and sweep**. A 7-row sample on existing v0.3 cross-m_3 receipts surfaced two findings that foreclosed the tangent path before any 2-hour sweep was committed: - **K_fib leakage**: F_beta does not preserve K_fib at tangent level on supp-B (leakage 0.10 -- 0.77 across all 7 rows). The gamma_3 baseline's precondition fails uniformly. Retired pre-sweep with verdict `form_precondition_failed`. - **Anti-commutation cocycle**: typed `(R_i, phi_i) = (I, 0)` for all 7 rows; the at-anchor `F_beta` already IS the typed reversing operator. No cocycle correction is available. Cocycle-rescue hypothesis (gamma_3_anticomm) ruled out. The discipline successfully prevented two hours of compute being spent against precondition-broken or rescue-foreclosed predictors. The probe chain itself is a methodology contribution: cheap pre-sweep checks of predictor preconditions are a load-bearing discipline that v0.4 implemented for the first time. ## Result And Structural-Negative Verdict ```text v0.4a verdict: outcome_A_all_Z2_clean 273 / 273 rows in Z2_clean after the two-pass classifier. marginal_Z2 = 0, smaller_symmetry = 0, undefined = 0. Pass 2 rescued 24 rows (8.8% of catalog) at the long-period high-m_3 regime; the two-pass design is validated. v0.4b verdict (gamma_3 tangent baseline): form_precondition_failed F_beta does not preserve K_fib at tangent precision on supp-B. 7/7 sanity rows showed K_fib leakage 0.10 -- 0.77 (median 0.45). Projector decomposition is not an isotypic split; rule retired pre-sweep. v0.4b' verdict (gamma_3'_orbit_pass2 baseline): falsified chi^2 = 1202.32 vs critical 26.22 (chi-squared(12), p = 0.01). Rule accuracy 63.74% is slightly below always-U baseline 64.47%. Gauge-rigidity is statistically independent of stability on supp-B. ``` The publishable v0.4 statement: > v0.4 establishes the Li-Liao supplementary-B piano-trio catalog as a > primary `Z_2`-symmetric body with a well-defined orbit-level Z_2 > domain map (v0.4a). Two non-circular Z_2-shadow stability predictors > were registered before compute: gamma_3 on the tangent-isotypic > projection, and gamma_3'_orbit_pass2 on the orbit-gauge-rigidity > projection. The tangent predictor's precondition failed (F_beta does > not preserve the neutral-quotiented kernel on supp-B). The orbit > predictor was empirically falsified against the per-m_3 stability > distribution at chi^2 = 1202 vs critical 26.22. **Stability > information on this catalog is not carried by the Z_2 shadow** at > either tested granularity. Any predictor with stability content must > look beyond the Z_2 projection. Three structural sub-results preserved in receipts and worth carrying into v0.5: 1. **Orbit-tangent gap on supp-B**: F_beta is exactly an orbit-level symmetry of supp-B piano-trios but does not lift to a clean tangent-level symmetry of K_fib at this precision. The cocycle that would conceivably mediate the lift is identity for all tested rows; no transport-correction rescue is available. 2. **Long-period high-m_3 gauge stress**: 24 of 273 catalog rows require tight gauge minimization (`identity_rotation_tolerance = 1e-9`, `phase_grid = 361`) to land cleanly in `Z2_clean`. The pattern clusters at `m_3 in {1.4, 1.5, 1.6, 1.7}` and period `T >= ~170`. Methodology note carried into the v0.4a runner spec. 3. **Gauge-rigidity vs stability orthogonality**: empirically established via gamma_3'_orbit_pass2 chi^2 = 1202. These two orbit-level features are statistically independent. ## Reproducibility Surface Primary scripts (workbench subcommands + one-shot aggregators): ```bash npm run isotrophy:parse:b # supp-B parse, 273 rows # v0.4a (operator-staged, ~85-120 min compute): # pass1: 15 commands, one per m_3, default tolerances # aggregator: identify Pass 2 candidates # pass2: per-flagged-row tight rerun # aggregator final: emit verdict + table[m_3][stability][class] # see kfacet_v04a_domain_map_preregistration.md for exact PowerShell. # v0.4b (no compute, inline from v0.4a manifest): python scripts/v04b_aggregator.py # if implementing the # row-z2-sweep path # (Path retired pre-sweep; gamma_3'_orbit_pass2 was instead computed # directly from the v0.4a manifest, no separate sweep.) ``` Key receipt directories: ```text results/isotrophy/k-facet-v04a-domain-map/manifest.json results/isotrophy/k-facet-v04a-domain-map/pass{1,2}/... results/isotrophy/k-facet-v04b-gamma3prime-orbit-pass2/manifest.json results/isotrophy/k-facet-v04a0-o434-anatomy/anatomy_receipt.json (the row anatomy that motivated the two-pass design) ``` Load-bearing constants live in the form-lock documents: `kfacet_v04a_domain_map_preregistration.md` for the gauge thresholds (`1e-6` / `1e-9` / `phase_grid 73` / `361`), classifier bands (`Z2_clean: 1e-4`, `marginal_Z2: 1e-2`, `smaller_symmetry: 1`); and `kfacet_v04b_gamma3prime_form.md` for the chi-squared falsifier (`df = 12`, `critical = 26.22` at `p = 0.01`). ## Where v0.5 Opens The v0.4 structural-negative is a falsifier-clean close: both registered predictors fired and produced their pre-registered verdicts. The catalog-level claim is "Z_2 shadow does not carry stability"; the follow-on question is what does. Three candidate v0.5 directions, all paper-side first: 1. **Richer-symmetry projection.** The piano-trio catalog's `Z_2` shadow is too coarse. What ADDITIONAL structural invariants could plausibly enrich it? Candidate: orbit-level conserved quantities `(E, |L|)` already verified tangent to v_bridge directions; their per-row joint distribution may stratify supp-B in a stability-relevant way without invoking new groups. 2. **Bifurcation-track projection.** Across the m_3 axis, supplementary-B has a continuous parameter family with discrete stability transitions visible in the per-m_3 S/U distribution (e.g., m_3 = 1.1 has 1S/22U, m_3 = 1.6 has 7S/1U). A v0.5 mechanism could track stability transitions as the body crosses `m_3` bifurcation thresholds, projecting the catalog onto a transition structure rather than an equivariant subspace. 3. **Closing without v0.5.** The v0.3 + v0.4 epilogues together establish a clean methodology surface (closure-relative discipline + two-pass classifier + pre-registration cocycle audit + structural sub-results) and two structural-negative results. If the next round targets a different substrate (mesa, geometry) rather than another K_facet refinement, the isotrophy program retires at end-of-v0.4 with publishable methodology + two negative findings. Codex direction will decide which path opens. No runner work proceeds until the v0.5 mechanism family is registered paper-side. **Update 2026-05-23:** v0.5 has opened with a **branch-shadow audit** that LANDED with verdict `branch_hash_passes_audit`. The catalog-only branch hash on `(m_3 < 1, z_0 < 0.3)` carries stability information on supp-B: `chi^2 = 34.986` vs critical `11.34` (`chi-squared(3)`, p ~= `1.23e-7`). The audit-dominant bucket is `(m_3 < 1, z_0 < 0.3)` with 113 rows at 55.75% stable (catalog mean 35.5%); the other three buckets sit at 20-29%. See `kfacet_v05a_branch_map_form.md`. **v0.5b LANDED 2026-05-23 (predictor fails held-out).** The leave-one-m_3-bin-out fold-trained branch-majority predictor was tested against the always-U baseline on the 263-row gating subset (12 bins with N >= 5). Verdict: **`branch_predictor_fails_heldout`**. Model accuracy `0.6198` vs always-U `0.6388`; accuracy delta `-0.019` (predictor LOSES to always-U). McNemar discordance `win=28, loss=33`, `n_disc=61`, `p=1.0`. The load-bearing fold is m_3 = 0.4 (as pre-mortem flagged): when held out, the low-mass/low-z_0 branch trains as U on the residual five `m_3 < 1` bins (`28 S / 33 U`), so the predictor cannot capture the m_3 = 0.4 stable cluster on its own held-out fold. In other low-mass folds, the branch predicts S but wins and false positives cancel. **Joint v0.5 reading.** The catalog branch shadow is associated with stability in-sample (`chi^2 = 34.99`, p ~= `1.23e-7`) but does not predict held-out stability across mass bins (`accuracy_delta = -0.019`). This is a clean **projection-limit result**: the branch hash describes the supp-B distribution but does not lift to a mechanism. Combined with v0.4: stability on this catalog is not in the Z_2 shadow at either tangent or orbit-gauge-rigidity granularity AND is not out-of-sample predictable from the catalog-coordinate branch shadow. The branch hash is a descriptive catalog partition, not a predictive mechanism. v0.5c remains open if a continuous feature (with explicit `T_kepler` / `mu_eff` references) is to be promoted; otherwise the v0.5 chapter closes as an audit-passes-predictor-fails projection limit. ## Doc Trail - `kfacet_v04a_domain_map_preregistration.md` -- two-pass classifier registration + verdict. - `kfacet_v04a0_o434_anatomy.md` -- methodological pre-mortem (placed the two-pass design under the right precondition). - `kfacet_v04b_gamma3_form.md` -- tangent-isotypic baseline, retired. - `kfacet_v04b_gamma3prime_form.md` -- orbit-pass2 baseline, falsified. - `kfacet_v04b_mechanism_preregistration.md` -- parent registration. - `kfacet_v05a_branch_map_form.md` -- v0.5a branch-shadow audit (opens v0.5). - `kfacet_v05b_branch_predictor_form.md` -- v0.5b held-out predictor. - `kfacet_v05_writeup.md` -- v0.5 chapter close (projection-limit; companion). - `kfacet_v03h_writeup.md` -- v0.3 chapter close (grand-companion). - `kfacet_v03_freeze_b_comparison.md`, `kfacet_v03_gamma_crossm3_preregistration.md` -- the v0.3 alpha and II registrations that opened v0.4. The chapter is closed cleanly. Receipts are durable. Pre-mortem discipline saved ~4 hours of compute against broken preconditions.