# v0.4a Domain Map Pre-Registration Status: **VERDICT LANDED 2026-05-22**. The pre-registered two-pass sweep executed 22:21:48 -> 02:46:24 (~4 h 25 min) against the registration below. Final verdict: **outcome_A_all_Z2_clean** (273/273 rows in `Z2_clean` after Pass 2; `marginal_Z2 / smaller_symmetry / undefined` all empty). Pass 2 rescued 24 rows from provisional `smaller_symmetry` to `Z2_clean`, confirming the O_434 gauge-artifact pre-mortem and validating the two-pass design. The v0.4 body is now locked as **"supplementary-B piano-trio orbit as primary Z_2 object; domain: all 273 rows"** (no longer "Z_2-or-smaller"). See the **Verdict (Landed)** section below. Audience: operator who executes the staged sweep; v0.4b paper-side writer; future coding agents. Companions: - `kfacet_v04a0_o434_anatomy.md` (methodological lesson that motivates the two-pass design; the run receipt lives at `results/isotrophy/k-facet-v04a0-o434-anatomy/anatomy_receipt.json`). - `kfacet_v03h_writeup.md` (v0.3 closure + projection-language framing). - `kfacet_v03_gamma_crossm3_preregistration.md` (II sentinel + symmetry probe that produced the original O_434 misclassification). - `docs/CROSS_SUBSTRATE_NOTES.md` §6-§7 (projection vocabulary). ## One-Line Read v0.4a builds the **domain map** of the supplementary-B piano-trio body under the `Z_2 = (12)`-swap projection. Each of the 273 catalog rows gets classified into one of four bands (`Z2_clean / marginal_Z2 / smaller_symmetry / undefined`) by a **two-pass gauge classifier with no per-row knobs**: a coarse Pass 1 over the full catalog, then a tight Pass 2 on every row that does not land cleanly in `Z2_clean` at Pass 1. The deliverable is a single table `table[m_3][stability][class]` that serves as v0.4b's audit ground truth. v0.4a is **not a mechanism test**; it audits the projection's well-definedness so that v0.4b's mechanism prediction (m_3 / stability stratification, registered separately) can be compared against an honest classifier rather than against a self-referential audit. ## Verdict (Landed, 2026-05-22) The pre-registered two-pass sweep ran against this registration. Headline: ```text v0.4a verdict: Well-definedness PASS. 273/273 supplementary-B piano-trios classify as Z2_clean after the pre-registered two-pass gauge protocol. marginal_Z2 = 0, smaller_symmetry = 0, undefined = 0. Pass 2 rescued 24 rows, confirming the O_434 gauge-artifact pre-mortem and validating the two-pass classifier. ``` Provisional Pass 1 classification (default tolerances, all 273 rows): ```text Z2_clean: 249 marginal_Z2: 0 smaller_symmetry: 24 undefined: 0 ``` Pass 2 (tight tolerances, 24 reruns): all 24 reclassifications were `smaller_symmetry -> Z2_clean`. No row remained in any non-clean band. Final classification (per-row provenance recorded): ```text Z2_clean: 273 <-- 100% of catalog marginal_Z2: 0 smaller_symmetry: 0 undefined: 0 pass2_invoked: 24 pass2_reclassification: 24 <-- all 24 rescued cleanly ``` ### Pass 2 reclassification details 24 rows rescued, with original/tight `F_beta` residuals: ```text original (Pass 1) tight (Pass 2) rescue ratio m_3 = 0.4: O_434(0.4) T=200.66 S 2.525e-01 1.060e-07 4.20e-07 m_3 = 1.4: O_191(1.4) T=189.96 U 1.111e-01 3.718e-07 3.35e-06 O_202(1.4) T=195.43 U 9.474e-02 2.316e-07 2.44e-06 O_203(1.4) T=195.58 U 1.177e-01 3.400e-07 2.89e-06 O_210(1.4) T=201.26 U 1.125e-01 2.246e-07 2.00e-06 m_3 = 1.5: 8 rows 9.10e-02 .. 1.66e-01 1.07e-07 .. 4.59e-07 m_3 = 1.6: 7 rows 9.04e-02 .. 1.50e-01 2.39e-07 .. 7.14e-07 m_3 = 1.7: 4 rows 1.50e-01 .. 1.99e-01 1.06e-07 .. 3.30e-07 Period range of rescues: T in [172.82, 201.82] (all long-period) Stability split: 12 S + 12 U (no correlation with stability) Rescue ratios: 4.2e-7 to 6.8e-6 (6-7 orders of magnitude) ``` ### Methodology rule worth carrying forward The 24-row rescue is not noise -- it is the validation of the two-pass design and an empirically-identified domain of fragility: ```text Long-period rows, especially T >= ~170 and m_3 in {1.4, 1.5, 1.6, 1.7}, stress the coarse SO(3) gauge minimizer. The default identity_rotation_tolerance = 1e-6 and phase_grid = 73 will misclassify ~8-9% of a piano-trio catalog spanning T up to ~200. Default gauge is a screening pass, not a final classifier. For any future audit chain that needs gauge-minimized residuals on long-period orbits at varied m_3, treat tight tolerances (identity_rotation_tolerance = 1e-9, phase_grid = 361) as the working classifier, with default tolerances reserved as the cheap screening pre-filter. ``` ### v0.4 body locked The verdict simplifies the v0.4 design: ```text v0.4 body (locked, 2026-05-22): supplementary-B piano-trio orbit as primary Z_2 object. domain: all 273 rows. smaller_symmetry / undefined bands: empty, verified. outlier lane: stays in the receipt vocabulary as a four-band classifier slot, but no rows currently populate it. ``` ### Subsequent action Per the pre-registered Outcome A action: **stage v0.4b mechanism test against the clean table**. The v0.4b registration is a separate deliverable; see "Sequencing After v0.4a" below for the locked observable target. ### Receipts ```text results/isotrophy/k-facet-v04a-domain-map/manifest.json results/isotrophy/k-facet-v04a-domain-map/pass1/aggregator_manifest.json results/isotrophy/k-facet-v04a-domain-map/pass1/flagged_for_pass2.csv results/isotrophy/k-facet-v04a-domain-map/pass1/m3eq{0.4..1.9}/ {manifest.json, residuals.csv} results/isotrophy/k-facet-v04a-domain-map/pass2/m3eq{0.4, 1.4, 1.5, 1.6, 1.7}/ O{idx}/{manifest.json, residuals.csv} ``` ## What's Being Tested In projection language (`CROSS_SUBSTRATE_NOTES.md` §6.3): ```text Body: supplementary-B piano-trio orbit as a primary Z_2 object. (not a daughter of strict S_3; not a v0.3 shadow.) Projection: orbit -> Z_2 symmetry-class shadow. The shadow is the orbit's gauge-minimized F_beta closure residual after SO(3) phase + rotation optimization, plus the band assignment in {Z2_clean, marginal_Z2, smaller_symmetry, undefined}. Well-definedness gate: the projection's image is a deterministic function of (orbit IC, period). The two-pass classifier below pre-registers that determinism by removing per-row tolerance choices. Output observable: table[m_3][stability][class] over the 273 rows. Rigidity check: the classification at Pass 2 (tight tolerances) is stable across the sentinel-to-full transition. Sentinel rows whose classification changes between the II 7-sentinel probe and the v0.4a full sweep get flagged. ``` The v0.4a deliverable is **the table**, not a mechanism. A separate v0.4b pre-registration will register an independent predictor (m_3/stability stratification per Codex anchor `gamma`) and compare it against the table. ## Pre-Registered Scope ```text Total rows: 273 m_3 values (15): 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 Per-m_3 row counts: 0.4: 55 0.5: 31 0.6: 22 0.7: 18 0.8: 18 0.9: 25 1.0: 38 1.1: 23 1.2: 7 1.3: 4 1.4: 4 1.5: 8 1.6: 8 1.7: 10 1.9: 2 Stability: 97 S / 176 U ``` ## The Two-Pass Gauge Classifier (Pre-Registered) The methodological lesson from the O_434 anatomy probe was that the default sigma_3-scan tolerances (`identity_rotation_tolerance = 1e-6`, `phase_grid = 73`) misclassified one row by SIX orders of magnitude. The two-pass design encodes that lesson as a deterministic procedure. ### Pass 1 (coarse, full catalog) ```text For each of the 273 rows: sigma3-scan with: --identity-rotation-tolerance 1e-6 --phase-grid 73 --n-samples 1009 --rtol 1e-12 --atol 1e-12 --sigma-tolerance 1e-5 --sigma-closure-multiple 3 ``` Pass 1 emits a per-m_3 manifest + `residuals.csv` covering every row in that m_3 slice. The Pass 1 classification is the four-band assignment read from each row's `F_beta_residual_inf`. ### Pass 2 (tight, conditional) ```text For every row whose Pass 1 classification is NOT Z2_clean (i.e., F_beta_residual_inf > 1e-4 OR undefined / integration failure): sigma3-scan with: --identity-rotation-tolerance 1e-9 --phase-grid 361 (all other settings identical to Pass 1) ``` The final classification for each row is: - If Pass 1 already classifies it as `Z2_clean`, that is the final classification. Pass 2 is not invoked. - Otherwise, Pass 2's `F_beta_residual_inf` produces the final band assignment. ### Four-band classifier ```text Z2_clean: F_beta_residual_inf <= 1e-4 marginal_Z2: 1e-4 < F_beta_residual_inf <= 1e-2 smaller_symmetry: 1e-2 < F_beta_residual_inf <= 1 undefined: F_beta_residual_inf > 1, integration failure, or sigma_3-scan precondition violated ``` **Important wording:** *No `smaller_symmetry` rows are currently confirmed; the band exists to prevent silent coercion if the full sweep finds one.* The O_434 anatomy probe was the only row tested at tight tolerances and it landed at `1.06e-7` (well inside `Z2_clean`); the band is therefore empty-allowed under the pre-registration. If Pass 2 populates it, v0.4b must revise to track those rows explicitly as a separate theoretical concern. No per-row knobs. No tolerance dialing. The only two thresholds are the pre-registered Pass 1 / Pass 2 tolerances and the four band edges above. ## Compute Discipline ```text Pass 1 estimate: ~17 sec / row * 273 = ~77 min staged Pass 2 estimate: ~85 sec / row * N_flagged N_flagged worst case: bounded above by 273 - rows that Pass 1 already places in Z2_clean. From the 7-row sentinel evidence we expect most rows pass at default tolerances; a typical N_flagged budget is 5-30, giving Pass 2 ~7-42 min. Aggregator + classification: seconds. Total expected: ~85-120 min staged, all operator-driven. ``` Above the inline ~10-minute rule. Stage as operator-driven commands. ## PowerShell Run Commands (Staged) ### Pass 1: 15 commands, one per m_3 ```powershell $m3_values = @("0.4","0.5","0.6","0.7","0.8","0.9","1.0","1.1","1.2","1.3","1.4","1.5","1.6","1.7","1.9") foreach ($m3 in $m3_values) { python scripts/isotrophy_workbench.py sigma3-scan ` --source B ` --path docs/isotrophy/supplementary-B_piano-init-condit-3d.txt ` --m3 $m3 ` --limit 0 ` --n-samples 1009 ` --phase-grid 73 ` --rtol 1e-12 ` --atol 1e-12 ` --sigma-tolerance 1e-5 ` --sigma-closure-multiple 3 ` --identity-rotation-tolerance 1e-6 ` --out "results/isotrophy/k-facet-v04a-domain-map/pass1/m3eq$($m3)" } ``` ### Aggregator (paper-side): identify Pass 2 candidates After Pass 1 finishes, run a small classifier script (write at v0.4a implementation time; spec is in **Aggregator schema** below) to: 1. Load every Pass 1 `residuals.csv` across all 15 m_3 values. 2. Apply the four-band classifier on `F_beta_residual_inf`. 3. Emit a list of `(m_3, index)` rows that landed outside `Z2_clean`. 4. Write a draft `table[m_3][stability][class]` reflecting Pass 1 provisional bands. The aggregator output is a single JSON file at `results/isotrophy/k-facet-v04a-domain-map/pass1/aggregator_manifest.json`. ### Pass 2: conditional re-runs For each `(m_3, index)` in the aggregator's flagged set: ```powershell python scripts/isotrophy_workbench.py sigma3-scan ` --source B ` --path docs/isotrophy/supplementary-B_piano-init-condit-3d.txt ` --m3 $m3 ` --indices $idx ` --n-samples 1009 ` --phase-grid 361 ` --rtol 1e-12 ` --atol 1e-12 ` --sigma-tolerance 1e-5 ` --sigma-closure-multiple 3 ` --identity-rotation-tolerance 1e-9 ` --out "results/isotrophy/k-facet-v04a-domain-map/pass2/m3eq$($m3)/O$($idx)" ``` ### Final aggregator: emit the table After Pass 2 finishes, re-run the aggregator with the Pass 2 receipts overriding the Pass 1 bands for flagged rows. The aggregator emits: ```text results/isotrophy/k-facet-v04a-domain-map/manifest.json - mode: kfacet_v04a_domain_map - version: v0.4a-domain-map - thresholds: {Z2_clean, marginal_Z2, smaller_symmetry, undefined} - pass1_tolerances + pass2_tolerances - per_row_table: 273 entries with final_class + provenance (pass1 / pass2) - table_m3_by_stability_by_class: nested counts - summary: total_rows class_counts: {Z2_clean: ?, marginal_Z2: ?, smaller_symmetry: ?, undefined: ?} pass2_invoked_count pass2_reclassification_count: rows where Pass 2 changed the band ``` ## Pre-Registered Decision Gates The v0.4a verdict is the final `manifest.json` table. Three structurally distinct outcomes: ### Outcome A: `Z2_clean` is the only populated band All 273 rows land in `Z2_clean` (possibly through Pass 2 rescue). Verdict: the projection is well-defined and uniform across the catalog; the v0.3 II symmetry probe's near-universal Z_2 finding scales to 273. v0.4b is then the mechanism test on a clean Z_2-only catalog. ### Outcome B: marginal_Z2 band is populated; smaller_symmetry / undefined are empty Some rows land in `marginal_Z2` after Pass 2, but none populate the smaller or undefined bands. Verdict: the projection is well-defined but not uniform; v0.4b's predictor must explain the marginal/clean split. ### Outcome C: smaller_symmetry and/or undefined are populated after Pass 2 Verdict: the projection has structural exceptions. v0.4b must revise to treat those rows as a separate theoretical category (the "outlier lane as receipt category" stays as a placeholder until v0.4b decides whether they need a second mechanism or whether they are catalog edge cases). A failure to converge in Pass 2 (`undefined` rows) is also a methodology finding: the catalog admission tolerance for supplementary-B may be looser than the sigma_3-scan can resolve. This would push back into a catalog-reconstruction conversation rather than a v0.4b mechanism conversation. ## Falsification Surface ```text Pre-registered claim: with two-pass gauge classification, the supplementary-B piano-trio body decomposes cleanly into the four-band shadow defined above, and the decomposition is a deterministic function of the orbit (no per-row tolerance choices). Falsifier: - Any row whose Pass 2 classification is inconsistent across re-runs at the registered tolerances (i.e., a determinism failure at fixed tolerances), OR - Any persistent `smaller_symmetry` or `undefined` row after Pass 2, populating bands the pre-registration declared empty-allowed. ``` Either falsifier is a real result. The first invalidates the "deterministic projection" framing and pushes back into a methodology review of the gauge minimizer's stability. The second populates a band that v0.4b must then treat as a structural concern, not as a discard category. ## What v0.4a Does NOT Do - **No mechanism prediction.** v0.4a's role is to build the table; the predictor (m_3 / stability stratification) is registered separately in v0.4b. - **No variational / `kfacet-sentinel` runs.** The full v0.3h audit chain is not invoked for v0.4a. The sigma_3-scan alone produces the classification. - **No tightening of the catalog admission criterion.** The four-band classifier reads off `F_beta_residual_inf` at the registered tolerances; catalog admission tightening (e.g., requiring all 273 rows to satisfy `sigma_3` closure tighter than the published catalog did) is out of scope. - **No retraction of any v0.3 verdict.** v0.3 closed cleanly as a domain-of-applicability finding; v0.4a is a fresh chapter on the supp-B body, not a revisit. ## Where This Builds On The O_434 Anatomy Probe The two-pass design is the encoded form of the O_434 anatomy probe's finding (`results/isotrophy/k-facet-v04a0-o434-anatomy/anatomy_receipt.json`): ```text O_434 anatomy verdict: gauge_artifact Original F_beta residual: 2.525e-01 (Pass 1 tolerances) Tight-rerun F_beta residual: 1.060e-07 (Pass 2 tolerances) Reduction factor: 4.20e-07 (six orders of magnitude) ``` The methodological lesson: a coarse SO(3) gauge minimizer can miss the correct rotation axis by enough that a structurally Z2_clean row reads as smaller_symmetry. The two-pass classifier rechecks every non-clean row at tight tolerances exactly to prevent that silent coercion. The O_434 receipt is the worked example that motivates the design. ## Sequencing After v0.4a ```text v0.4a (THIS): build the table[m_3][stability][class] from the two-pass classifier. Pre-registered. v0.4b: register an independent mechanism predictor (m_3 / stability stratification per Codex anchor gamma). Compare predicted table vs v0.4a's observed table. v0.4c: read the verdict. If predictor matches, v0.4 closes as a positive mechanism on the supp-B body. If not, v0.4 closes as a structural-negative on the chosen predictor; propose v0.4b' / v0.5 alternatives. ``` v0.4a's output is required input for v0.4b. v0.4b should not be staged until v0.4a's manifest has landed. ## How To Verify This Pre-Registration Is Still Current Re-parse supp-B before any sweep launch: ```powershell npm run isotrophy:parse:b ``` Confirm `total_rows = 273` and the per-m_3 counts match the scope table above. If the local mirror has been updated and the counts drift, this pre-registration should be re-issued. Also confirm the O_434 anatomy receipt is in place: ```text results/isotrophy/k-facet-v04a0-o434-anatomy/anatomy_receipt.json results/isotrophy/k-facet-v04a0-o434-anatomy/q5_tighter_gauge/residuals.csv ``` The methodological motivation for the two-pass design lives there.