# v0.4b Mechanism Predictor Pre-Registration Status: **retired before full sweep**, 2026-05-23. Pre-registered on 2026-05-22; the `gamma_3` threshold-rule baseline was retired when the smoke/cross-m3 sanity rows showed its tangent-isotypic precondition is not met on supplementary-B. Audience: operator who executes the staged sweep; future paper-side writer; v0.4c comparison reader. Companions: - `kfacet_v04a_domain_map_preregistration.md` (verdict-landed domain audit that locked the v0.4 body). - `kfacet_v04a0_o434_anatomy.md` (methodology pre-mortem for gauge classifier). - `kfacet_v03h_writeup.md` (v0.3 closure + projection-language framing). - `docs/CROSS_SUBSTRATE_NOTES.md` §6-§7 (projection vocabulary). ## Retirement (Landed, 2026-05-23) Verdict: `form_precondition_failed`. The registered `gamma_3` baseline depended on an isotypic split of `K_fib = ker(M_i - I) / N_C` under the closed-form at-anchor `F_beta` operator. A seven-row sanity surface found systematic leakage out of `K_fib` instead: ```text row m_3 period stab k_dim K_fib_dim even+odd over F_beta_leak O_50 0.4 105.33 S 6 6 9 +3 0.7723 O_62 0.4 110.39 U 6 6 8 +2 0.4481 O_67 0.4 111.25 U 6 6 9 +3 0.3510 O_434 0.4 200.66 S 4 4 5 +1 0.1034 O_242 1.0 84.10 U 3 3 6 +3 0.5273 O_282 1.0 92.06 U 6 5 6 +1 0.3212 O_284 1.0 92.20 U 6 5 7 +2 0.5716 ``` The median `F_beta_leakage_inf` is approximately `0.45`, far above the `1e-3` projector floor. The naive projectors `(I +/- F_beta_K)/2` also overcount `K_fib` by `+1` to `+3` directions on every row, confirming that the split is not a valid isotypic decomposition. This retires the threshold-rule baseline **before** the staged 273-row sweep. The historical run commands below are retained as the original registration surface, but they are no longer authorized for the retired baseline. ## v0.4b Sub-Result: Orbit Symmetry vs Tangent Symmetry Gap The sanity surface exposes a new structural distinction: ```text v0.4a domain map: F_beta is an orbit-level symmetry of all 273 supplementary-B piano-trios under the two-pass gauge classifier. v0.4b sanity surface: F_beta is not, as implemented by the at-anchor operator used here, a tangent-level symmetry of K_fib on any of the seven tested piano-trio rows. ``` This gap is independent of the retired threshold rule. It says the `Z_2` ansatz is clean at orbit level, but its tangent realization on `ker(M_i - I) / N_C` is not the same object that was available in v0.3h's strict `D_3` choreography setting. A follow-on `gamma_3'` must first define a projection target that is actually invariant under the relevant `F_beta` action. Deferred `gamma_3'` design candidates: ```text gamma_3'_anticomm: Replace P_+/- on K_fib with a construction based on an operator that commutes with F_beta under F_beta M F_beta = M^-1, e.g. (M_i + M_i^-1)/2 restricted to the appropriate near-kernel structure. gamma_3'_full_kernel: Test whether raw ker(M_i - I), before the neutral quotient, is preserved better than K_fib. Requires its own precondition sanity check. gamma_3'_orbit_features: Step back from tangent isotypics and use orbit-level Z_2 invariants from the v0.4a domain map plus non-spectral row features. Loses the tangent representation-theory framing but keeps non-circularity. ``` No `gamma_3'` is registered here. The next paper-side step is to choose and pre-register one replacement object before any new catalog-wide compute. ## Historical One-Line Read (Superseded by Retirement) The following sections preserve the pre-registered form as originally filed. They are not launch authorization after the 2026-05-23 retirement above. With the v0.4a verdict locking all 273 supplementary-B piano-trios into the `Z2_clean` domain, v0.4b asks the **next non-circular question**: *Does the Z_2 neutral/tangent shadow of each orbit contain enough structure to predict that orbit's stability flag, stratified by m_3?* The primary predictor `gamma_3` uses only Z_2 tangent-structure invariants (kernel dimensions, isotypic dim split, neutral-sector conditioning) and explicitly **excludes** anything that inspects off-unit-circle Floquet multipliers. The falsifier is a single full-table chi-squared / likelihood-ratio test over m_3 bins with `N >= 5`. The non-circularity is load-bearing; a pre-registered no-variation retirement clause retires `gamma_3` as non-informative (rather than over-fitting) if the chosen features are nearly-constant across the catalog. ## Locked From v0.4a ```text Body: primary Z_2 piano-trio orbit Projection: orbit -> Z_2 neutral/tangent shadow Domain: all 273 supplementary-B rows (Z2_clean under the two-pass gauge classifier) ``` The v0.4a verdict simplifies v0.4b's scope: no domain triage, no smaller-symmetry side-channel. Every catalog row contributes to the S/U observable. ## Observable to Predict (Locked) ```text table[m_3][stability] = (S_count, U_count) per m_3 Observed (from v0.4a manifest): m_3 S U total 0.4 35 20 55 0.5 15 16 31 0.6 6 16 22 0.7 4 14 18 0.8 6 12 18 0.9 8 17 25 1.0 7 31 38 1.1 1 22 23 1.2 2 5 7 1.3 2 2 4 <- N < 5, descriptive only 1.4 0 4 4 <- N < 5, descriptive only 1.5 4 4 8 1.6 7 1 8 1.7 0 10 10 1.9 0 2 2 <- N < 5, descriptive only Gating bins (N >= 5): {0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.5, 1.6, 1.7} Descriptive-only: {1.3, 1.4, 1.9} ``` ## Primary Predictor: gamma_3 (Non-Circular Structural) `gamma_3` predicts the per-m_3 stable fraction `P_S(m_3)` from Z_2 tangent-structure invariants computed at each row's IC. The prediction is then compared row-by-row against the observed stability flag, aggregated per m_3, and tested against the binomial observation `(S_count, U_count)`. ### Allowed gamma_3 features (Z_2 neutral/tangent invariants) ```text 1. kernel_dim: dim ker(M_i - I) at the row's adaptive floor (after neutral quotient if used). 2. F_beta_even_dim: dim of F_beta = +I isotypic in ker(M_i - I). 3. F_beta_odd_dim: dim of F_beta = -I isotypic in ker(M_i - I). 4. F_beta_even_odd_ratio: F_beta_even_dim / max(F_beta_odd_dim, 1). 5. bridge_band_count: count of (M_i - I) singular values in (1e-7, 1e-3), i.e., near-1 bridge-band SVs. 6. F_beta_leakage_inf: ||(I - P_ker) F_beta P_ker||_inf, the F_beta leakage outside the recovered kernel. 7. neutral_sector_conditioning: condition number of the standard 8-dim Hamiltonian/translation/boost neutral block at y0. 8. period_T: stratification variable (NOT a fitted feature). 9. m_3: stratification variable. ``` ### Disallowed for prediction (would make gamma_3 circular) ```text - "has off-unit-circle Floquet pair" - spectral radius of M_i - unit-circle count / number of eigenvalues with |lambda| > 1 - max |lambda_i| over Floquet spectrum - Krein signature - any function of M_i eigenvalues (real or complex) beyond what is carried by the allowed kernel/isotypic-dim counts above - the stability label itself, or anything derivable from it without the orbit IC ``` If `gamma_3` is constructed from any feature in this disallowed list, the v0.4b registration is invalidated and must be re-issued. ### gamma_3 form (pre-registered, paper-side closure required) `gamma_3` is a function ```text gamma_3 : (allowed features) -> P_S in [0, 1] ``` The functional form is **registered paper-side before any compute** in a separate `kfacet_v04b_gamma3_form.md` sub-document. Acceptable forms: - Linear or low-order polynomial in the allowed features. - Logistic on a linear combination. - A pre-registered piecewise / threshold rule (e.g., "predict S iff `F_beta_even_dim >= k`"). Adaptive / fitted forms (fit the model to v0.4a's table and then test on the same table) are **disallowed**. The pre-registration must commit to a fixed functional form before reading the per-row table. ## Falsification Surface ### Primary falsifier (gating) A single full-table likelihood-ratio test over the **12 gating m_3 bins with `N >= 5`**: ```text For each gating m_3 bin: N(m_3) = total rows S_observed(m_3) = observed stable count P_S_predicted(m_3) = mean of gamma_3 over the rows in that m_3 bin H_0: S_observed(m_3) ~ Binomial(N(m_3), P_S_predicted(m_3)) Test statistic: -2 log Lambda (likelihood ratio) = sum over gating bins of -2 [ log L(S_obs | P_S_pred) - log L(S_obs | S_obs/N) ] Reference distribution: chi-squared with df = 12 (number of gating bins) minus the number of free parameters in gamma_3 (registered paper-side; e.g., for a 3-feature linear model, df = 9) Falsifier: p-value < 0.01 <--> gamma_3 prediction fails the table at the 99% level. ``` The test threshold (`p < 0.01`) is registered. Multiple-comparison correction is built into the single-table aggregation; no per-bin adjustment. ### Diagnostics (non-gating) For each m_3 bin (including the descriptive-only `{1.3, 1.4, 1.9}`): ```text P_S_observed(m_3) = S_count / N P_S_predicted(m_3) = mean(gamma_3 over rows in that m_3) binomial residual = (P_S_observed - P_S_predicted) / sqrt(P_S_pred (1-P_S_pred) / N) 2-sigma flag = |residual| > 2.0 ``` Flags are reported for inspection but **do not gate the verdict**. Per-bin 2-sigma flags on small bins are descriptive context, not falsification. ### Pre-Registered No-Variation Retirement Clause If any of the following holds when computed across all 273 rows: ```text For every allowed feature f in gamma_3's input set: coefficient_of_variation(f) = std(f) / |mean(f)| < 0.01 ``` then `gamma_3` is **retired as non-informative** and the v0.4b verdict becomes: ```text v0.4b verdict (retirement): The Z_2 neutral/tangent shadow features chosen for gamma_3 do not contain enough structural variation across the 273-row catalog to predict stability stratification. The projection is well-defined but the shadow does not preserve enough information to predict the observed S/U distribution at the granularity of m_3 bins. This is a clean projection-limit result, not a falsification of v0.4. The Z_2 mechanism may still apply; it just does not carry stability information at this body. ``` Retirement is preferable to over-fitting. The clause prevents post-hoc feature selection from rescuing a non-informative predictor. ### Compute-Reuse Sidecar: gamma_1 Decomposition (Non-Gating) After `gamma_3` runs and M_i is computed for all 273 rows, an **explicitly non-predictive** sidecar uses `gamma_1` features (the F_beta even/odd Floquet spectrum) to attribute instability to one F_beta block: ```text For each U row: Compute F_beta-even and F_beta-odd block restrictions of M_i. Count off-unit-circle eigenvalue pairs per block. Report which block carries the instability for this row. Aggregate: table[m_3][F_beta_even_carries_U vs F_beta_odd_carries_U] ``` The sidecar is **not a predictor**. It is a mechanism-decomposition diagnostic that reuses M_i computed for `gamma_3`. Its result feeds discussion in v0.4c paper-side write-up but does not enter the falsifier. ## Covariates (Recorded, Non-Gating) For each row, the receipt records: ```text - z_0 (continuous IC parameter; not used in gamma_3 fit) - T (period; not used in gamma_3 fit) - m_3 (stratification only) - gamma_3 features (allowed list above) - stability flag (observed) - gamma_3 prediction P_S for this row ``` Non-gating exploratory readout: residuals of `gamma_3` prediction vs `z_0` and vs `T` within each m_3 bin. Useful as a mechanism-discovery follow-up if the primary test passes; not part of the falsifier. ## Stratification Discipline ```text Primary stratification: m_3 only (15 discrete values, 12 gating bins) Covariates recorded: z_0 (continuous), T, stability flag Non-gating readout: residuals vs z_0 and T inside each m_3 bin ``` Conditioning on `(m_3, z_0)` joint bins is rejected as too sparse for the catalog (some `(m_3, z_0)` cells have 1-2 rows). The single-axis m_3 stratification provides the statistical power; finer-grained analyses are exploratory. ## Compute Discipline ```text gamma_3 needs: - M_i for all 273 rows (variational integration, 18-dim system over one period). - F_beta operator (catalog-style, IC-independent; reused from kfacet-sentinel construction). - Kernel basis at adaptive floor + F_beta isotypic projection. Per-row time estimate: - M_i variational integration alone at rtol=1e-12: ~10-30 sec/row - Kernel + isotypic post-processing: ~1 sec/row - Total: ~15-35 sec/row Total catalog estimate: 273 rows * ~25 sec average = ~115 min = ~2 hours staged. Compute is bounded by M_i computation; gamma_3 feature extraction is trivial relative to integration. gamma_1 sidecar reuses M_i with no additional integration. Above the inline ~10-minute rule. Stage as operator-driven sweep. ``` ## PowerShell Run Commands (Staged) **Retirement note (2026-05-23): do not run these commands for the retired threshold-rule baseline.** They are retained to document the original compute surface. A follow-on `gamma_3'` must issue a new staged command block after its projection target and precondition checks are registered. ```powershell # v0.4b primary sweep: M_i + Z_2 isotypic decomposition for all 273 supp-B rows $m3_values = @("0.4","0.5","0.6","0.7","0.8","0.9","1.0","1.1","1.2","1.3","1.4","1.5","1.6","1.7","1.9") foreach ($m3 in $m3_values) { python scripts/isotrophy_workbench.py kfacet-row-z2-sweep ` --source B ` --path docs/isotrophy/supplementary-B_piano-init-condit-3d.txt ` --m3 $m3 ` --limit 0 ` --rtol 1e-12 ` --atol 1e-12 ` --closure-floor 1e-7 ` --out "results/isotrophy/k-facet-v04b-mechanism/per_m3/m3eq$($m3)" } # v0.4b aggregator: emits per-row receipt + gamma_3 feature table + chi-squared # verdict + gamma_1 sidecar attribution. Pre-registered, no per-row knobs. python scripts/v04b_aggregator.py ` --gamma3-form docs/isotrophy/kfacet/kfacet_v04b_gamma3_form.md ``` **Note:** `kfacet-row-z2-sweep` is a new workbench subcommand to be written at v0.4b implementation time. Its job is to compute M_i for a row and emit the `gamma_3` allowed-feature row plus the M_i.npy for the sidecar. Spec is in **Aggregator + sweep-command schema** below. ## Aggregator + Sweep-Command Schema ```text kfacet-row-z2-sweep: Input: catalog row (source + path + m_3 + index/indices) Computes: - M_i via variational integration - kernel basis at --closure-floor - F_beta-even/odd isotypic dim split - bridge_band_count (SVs of (M_i-I) in (1e-7, 1e-3)) - F_beta_leakage_inf - neutral_sector_conditioning Saves: M_i.npy, F_beta_isotypic_basis.npy, gamma_3_features.json Receipt schema (per row): {row_index, label, m_3, z_0, period, stability, gamma_3_features: {kernel_dim, F_beta_even_dim, F_beta_odd_dim, F_beta_even_odd_ratio, bridge_band_count, F_beta_leakage_inf, neutral_sector_conditioning}, M_i_norm_inf, integration_seconds} v04b_aggregator: Input: per-row receipts under per_m3/m3eq*/{O*.json} + paper-side gamma_3 form spec Computes: - gamma_3 prediction P_S for each row - per-m_3 (P_S_pred, P_S_obs, residual) - likelihood-ratio test over gating bins - no-variation retirement check - gamma_1 sidecar: F_beta block Floquet attribution per U row Emits: manifest.json with verdict in {pass, fail, retired_no_variation} per_row_table.csv per_m3_table.csv (for diagnostics) gamma_1_sidecar_attribution.csv (non-gating) ``` ## Historical Sequencing After v0.4b (Superseded) This sequencing block is retained from the original registration. After the 2026-05-23 retirement, the active next step is `gamma_3'` design, not this sweep. ```text v0.4a (DONE): domain map; 273/273 Z2_clean. v0.4b (THIS): gamma_3 structural predictor + chi-squared falsifier. v0.4c: read verdict. - pass: v0.4 mechanism positive; paper-side write-up. - fail: gamma_3 form is wrong; propose gamma_3' for v0.4b'. - retired_no_variation: v0.4 projection-limit; paper-side record that the shadow does not carry stability info. ``` After v0.4c, regardless of outcome: - The gamma_1 sidecar attribution informs paper-side discussion of which F_beta block (even or odd) carries instability where present. - Recorded covariates (z_0, T) enable follow-up analyses if the primary verdict is `pass` or `retired_no_variation`. ## What This Pre-Registration Does NOT Do - **No on-the-fly fitting of gamma_3.** The functional form is locked in `kfacet_v04b_gamma3_form.md` paper-side before any per-row prediction is computed. - **No use of stability information for prediction.** Stability is observed, not predicted-from. - **No per-bin gating.** The chi-squared test aggregates over all gating bins; per-bin 2-sigma flags are diagnostic only. - **No revision of v0.4a.** The Z2_clean domain is locked. - **No revisit of v0.3.** The v0.3 epilogue's domain-of-applicability finding is unaffected. ## What Falsifies the v0.4b Registration Itself If at implementation time we discover any of: - A feature listed in "Allowed" is structurally equivalent to a forbidden feature (e.g., `bridge_band_count` is shown to be one-to-one with off-unit-circle pair count). - The gamma_3 form cannot be expressed without using a disallowed feature. - The chi-squared test's degrees of freedom calculation is wrong. then the pre-registration is re-issued. The verdict is not read against a pre-registration that itself is invalidated. ## How To Verify This Pre-Registration Is Still Current This section is historical after the 2026-05-23 retirement. It verifies the original registration trail, not launch authorization. 1. Re-parse supp-B: `npm run isotrophy:parse:b` returns 273 rows; per-m_3 counts match the observed table above. 2. Confirm v0.4a verdict: open `results/isotrophy/k-facet-v04a-domain-map/manifest.json` and check `verdict = outcome_A_all_Z2_clean`, `class_counts.Z2_clean = 273`. 3. Confirm `kfacet_v04b_gamma3_form.md` exists and registers the functional form before any per-row prediction is generated. If any of (1)-(3) drifts, this pre-registration should be re-issued before any v0.4b sweep launch.