# v0.9a Signed vf Three-Zone Audit Form Lock Status: **VERDICT LANDED 2026-05-24**. Verdict: **`signed_vf_three_zone_fails_audit_chi2`** at chi^2 = 7.42 vs critical 9.21 (df=2, p = 0.0245). The U-shape pattern check ALSO fails (mixed S_fraction 0.34 is NOT less than positional 0.11; mixed < velocity 0.44 holds but the bilateral pattern is broken). The observed pattern is **monotone increasing** in vf: positional-dominant 10.5% S, mixed 33.9% S, velocity-heavy 43.9% S. **This is methodologically informative**: the v0.7a' U-shape was an artifact of the quartile boundary at vf ~ 0.297, where v0.7a Q1's 49% S-fraction was concentrated; under the physical cutpoint at vf = 0.25, the bottom-most-vf zone shrinks from 63 to 19 rows and its S-fraction collapses to 10.5%. The v0.7a' chi^2 = 16.43 positive is NOT invalidated -- it remains a real catalog stratification finding -- but the U-shape mechanism hypothesis is. This document locks the v0.9a three-zone chi-squared audit on the v0.7 analyzable 250-row subset under the parent registration's candidate slot **C** (three-zone audit with physically-motivated cutpoints). Candidate forms A (ordered quartile) and B (binary threshold) are REJECTED pre-compute under the v0.9 parent's anti-circular discipline. Candidate D (Jonckheere-Terpstra trend test) is held back. ## Verdict (Landed, 2026-05-24) Receipt: `results/isotrophy/k-facet-v09a-signed-vf-three-zone/manifest.json`. ```text verdict: signed_vf_three_zone_fails_audit_chi2 domain: 250 analyzable rows (87 S / 163 U) three-zone contingency: zone N S U S_fraction chi2_contrib positional-dominant (vf<0.25) 19 2 17 0.1053 4.93 mixed (0.25 <= vf < 0.5) 165 56 109 0.3394 0.05 velocity-heavy (vf >= 0.5) 66 29 37 0.4394 2.43 ----- chi^2 = 7.42 test_branch_taken: chi_squared (asymptotic; min_expected = 19 * 87 / 250 = 6.61, above the 5.0 floor) df: 2 critical (p=0.01): 9.21 p_value (chi-sq(2)): 0.0245 alignment_tightness: 0.5697 (well below 0.8 warning threshold) pattern check: S_fraction(positional) = 0.1053 S_fraction(mixed) = 0.3394 S_fraction(velocity-h) = 0.4394 mixed < positional? FALSE (0.34 > 0.11; U-shape direction broken on the lower end) mixed < velocity-heavy? TRUE (0.34 < 0.44; this side of the U-shape holds) u_shape_confirmed? FALSE (requires BOTH sides; the physical-zone U-shape does NOT exist) ``` **Structural reading:** > Under the physical cutpoint at vf = 0.25 (the 3:1 mass-weighted > ratio between position and velocity components of gamma_1), the > "positional-dominant" zone has only 19 rows in the analyzable > subset, with 10.5% S-fraction. This is a sharp contrast with > v0.7a' Q1 (vf in [0, 0.297]), which had 63 rows at 49.2% S. The > mass of v0.7a Q1's high-S rows is concentrated near vf = 0.25 to > 0.30, NOT at the deepest positional regime (vf < 0.25). > > The v0.7a' U-shape was an artifact of the QUARTILE BOUNDARY at > vf ~ 0.297: the v0.7a' Q1 bin combined some genuinely low-vf > rows (mostly unstable) with vf ~ 0.25-0.30 rows (heavily stable) > in a way that the boundary placement made monotone-broken. Under > a physical cutpoint, the boundary moves and the apparent U-shape > disappears. > > The actual physical pattern is **monotone increasing in vf**: > low vf (positional-dominant) is mostly unstable; mid vf (mixed) > is intermediate; high vf (velocity-heavy) is most stable. This > matches the broader thesis that velocity-dominated directions > correlate with stability. But chi^2 = 7.42 does not reach the > registered floor at p = 0.01 (critical = 9.21), so this signal > is NOT licensed as a v0.9a pass. **Important meta-finding**: the v0.7a' chi^2 = 16.43 is NOT invalidated by this verdict. The v0.7a' result remains a real catalog-coordinate stratification finding under quartile binning. What v0.9a invalidates is the **U-shape MECHANISM hypothesis** -- i.e., the proposal that the pattern is symmetric U-enrichment of the mixed zone. The pattern is actually monotone-increasing, and the physical-zone test does not reach the p = 0.01 floor for the 3 x 2 contingency. **Receipts:** ```text docs/isotrophy/kfacet/kfacet_v09a_signed_vf_three_zone_form.md (this document) results/isotrophy/k-facet-v09a-signed-vf-three-zone/manifest.json results/isotrophy/k-facet-v09a-signed-vf-three-zone/per_row_table.csv results/isotrophy/k-facet-v09a-signed-vf-three-zone/contingency_table_three_zone.csv scripts/v09a_signed_vf_three_zone_audit.py ``` Per the locked outcome interpretation: > "If v0.9a Chi-squared Fails: The physical-zone binning does not > reach the chi-squared gate; the v0.7a' signal lives in a finer > binning than the physically-motivated zones detect. Chapter > closes on the v0.9a hypothesis." v0.9b (held-out predictor) is NOT licensed under the fails verdict. Audience: v0.9a runner; v0.9b form-lock author (conditional on v0.9a passing); paper-side reviewer of the signed direction- composition mechanism. Companions: - `kfacet_v09_mechanism_preregistration.md` -- v0.9 parent registration. Locks the anti-circular framing discipline that this form must respect. - `kfacet_v08_writeup.md` -- v0.8 chapter close (structural-negative on unsigned purity) that opened v0.9. - `kfacet_v07a_prime_restricted_scope_form.md` -- v0.7a' PASS at chi^2 = 16.43; the source observation v0.9 must NOT silently re-test under the same statistic. - `kfacet_v07a_velocity_fraction_audit_form.md` -- v0.7a parent (vf operational definition; non-circularity sentence carries forward). Frame: v0.8a confirmed the v0.7a' signal is signed, not symmetric in unsigned purity. The vf distribution is density-asymmetric (184/250 rows below vf = 0.5; max vf = 0.87, no rows near vf = 1). v0.9a tests a **specific U-shape hypothesis** on three physical zones: that the **mixed zone is U-enriched relative to BOTH the positional-dominant and velocity-heavy zones**. This is a hypothesis-specific test, not a re-run of chi^2 of independence on vf-ordering. ## Why Three-Zone (Form C) Over Alternatives ```text A. Ordered quartile audit. REJECTED by v0.9 parent (re-tests v0.7a' chi^2 by linear-transform invariance). B. Binary vf < 0.5 threshold. REJECTED by v0.9 parent (binary collapse of v0.7a' Q1+Q2 vs Q3+Q4). C. Three-zone audit with SELECTED. Physical cutpoints physical cutpoints. {0.25, 0.50}, hypothesis-specific verdict tree (chi^2 > critical AND mixed < both direction zones). D. Jonckheere-Terpstra held back. Different test monotone-trend test. statistic (one-sided trend test) rather than chi^2-of-independence. Cleaner non-circular shape but different test family. ``` C is selected because (a) the cutpoints {0.25, 0.50} are physically motivated (see Physical Motivation section below), NOT data-driven from v0.7a' quartile cutpoints; (b) the verdict tree requires a hypothesis-specific S-fraction pattern (mixed-zone-U-enrichment) in addition to the chi-squared threshold; (c) the result is interpretable in physical units (positional vs mixed vs velocity domination) rather than as catalog quartiles. ## Anti-Circular Argument (Load-Bearing) The v0.9 parent's anti-circular discipline requires that any v0.9a form lock break at least one of (feature derivation, test statistic, evaluation domain) relative to v0.7a's chi^2-of-independence test on vf quartiles. **v0.9a breaks condition #1 (feature derivation)** via the physical cutpoints {0.25, 0.50}, which are NOT v0.7a's quartile cutpoints (approximately {0.297, 0.419, 0.504} for q25, q50, q75 of the 250-row analyzable subset). The three-zone bin boundaries are pre-registered from physical reasoning, not from the data distribution. **v0.9a additionally breaks condition #2 (test statistic)** via the augmented verdict tree, which requires chi^2 > critical AND a hypothesis-specific S-fraction pattern. The basic chi^2 of independence is the gate, but the pattern check converts the test into a hypothesis-specific one. A v0.9a contingency that produced chi^2 > critical but with the WRONG pattern (e.g., mixed has higher S-fraction than direction-dominant zones) would NOT pass the verdict. Both conditions broken. v0.9a is honestly non-circular. ## What v0.9a Is v0.9a is a **catalog-side audit on the v0.7 analyzable 250-row subset**, NOT a predictor (continues the audit-not-predictor framing): ```text Domain: 250 analyzable rows from v0.7a per_row_table.csv (rows with integration_blocked == False). Inherited from v0.8 verbatim. Feature: vf(row) directly (NOT transformed). v0.7a operational definition + non-circularity sentence carry forward unchanged. Zones: three physically-motivated zones over [0, 1]: positional-dominant: vf in [0, 0.25) mixed: vf in [0.25, 0.50) velocity-heavy: vf in [0.50, 1] cutpoints {0.25, 0.50} pre-registered with physical motivation (below). Right-closed lower intervals (ties to upper bin under standard Python comparison; deterministic). Test: 3 x 2 chi-squared (zone x S/U), df = occupied_zones - 1 (usually 2). Critical: chi-squared(df, p = 0.01). df = 2: critical = 9.21. Primary hypothesis: S_fraction(mixed) < S_fraction(positional) AND S_fraction(mixed) < S_fraction(velocity-heavy) (U-shape with the mixed zone U-enriched relative to BOTH direction-dominant zones). Alignment guard: max over zones of (fraction in any single v0.5a branch_label bucket). Thresholds 0.8 (warning) / 0.95 (severe). Sparse fallback: v0.6b discipline (seed 20260523, n_permutations 10000, min_occupied_bin_count >= 2, expected_cell >= 5). Verdict tree: if min_occupied_bin_count < 2: verdict = signed_vf_three_zone_inconclusive_sparse elif chi^2 > critical AND S_fraction(mixed) < S_fraction(positional) AND S_fraction(mixed) < S_fraction(velocity-heavy): if alignment <= 0.8: verdict = signed_vf_three_zone_passes_audit elif alignment <= 0.95: verdict = signed_vf_three_zone_passes_audit_alignment_warning else: verdict = signed_vf_three_zone_passes_audit_severe_alignment elif chi^2 > critical: verdict = signed_vf_three_zone_fails_audit_pattern (chi^2 reaches threshold but the U-shape direction is wrong) else: verdict = signed_vf_three_zone_fails_audit_chi2 (chi^2 does not reach threshold) ``` A clean Pass licenses v0.9b: a separately-registered held-out signed-vf predictor with the v0.5b discipline. A Pattern Fail indicates the catalog has structure in vf vs S/U but NOT the predicted U-shape (would be a clean structural-negative on the v0.7a'-inspired hypothesis). A Chi-squared Fail indicates the catalog does not even reach the chi-squared gate under the physical zone binning. ## Physical Motivation for Cutpoints {0.25, 0.50} The cutpoints are chosen from physical reasoning about Hamiltonian flow tangent dynamics, NOT from v0.7a' quartile boundaries: ```text vf = 0.50: the maximally-mixed direction in phase space. A vector with equal mass-weighted norm in position and velocity sub-blocks is at vf = 0.5. This is the natural boundary between "direction-dominated" (position OR velocity carries the majority of mass-weighted norm) and "mixed" (neither subspace dominates). vf = 0.25: the 3:1 boundary in mass-weighted norm. A vector with ||delta_q||^2 = 3 ||delta_v||^2 (i.e., position carries 75% of mass-weighted norm) is at vf = 0.25. This is the boundary between "positional-dominant" (clearly position-dominated) and "mixed" (less than 3:1 ratio favoring position). asymmetric cutpoints: No symmetric upper cutpoint at vf = 0.75 is locked because: - the v0.8a vf distribution diagnostic shows max vf = 0.87 (no rows near vf = 1); - the catalog has only 66/250 rows with vf >= 0.5; - splitting the velocity-heavy zone further would produce sparse bins (Risk 1 in the v0.9 parent's pre-mortem). The asymmetric structure (two cutpoints below 0.5, one zone above 0.5) reflects the catalog's actual density distribution, recorded in v0.8a's diagnostic. The cutpoints are still PHYSICAL (the 0.50 boundary is the natural mixing scale; the 0.25 boundary is the 3:1 mass-weighted ratio), not v0.7a'-quartile-derived. ``` This physical motivation is pre-registered. Any subsequent change to the cutpoints after v0.9a runs is a re-registration, not a refinement. ## Pre-Audit Sanity Surface Before v0.9a runs: ```text 1. v0.7a per_row_table.csv must be present and readable. 2. Filter to analyzable rows (integration_blocked == False); expected count = 250. 3. v0.5a branch_label must be present per row (already joined in v0.7a per_row_table). 4. Verify vf values are in [0, 1] for all 250 rows. 5. Compute zone(r) for each row under the locked cutpoints. 6. Per-zone counts MUST be emitted on the receipt before the chi-squared statistic is computed; thin zones (< 5 rows) MUST fire the sparse-cell fallback. ``` ## Non-Circularity Audit ```text Feature derivation: vf is inherited unchanged from v0.7a (same operational definition, same selection rule, same tie-break cascade, same reference frame). Binning derivation: zones use cutpoints {0.25, 0.50} from physical reasoning, NOT from v0.7a' quartile cutpoints (which are approximately {0.297, 0.419, 0.504}). The cutpoints are different by design. Test derivation: primary statistic is chi-squared of independence (same family as v0.7a'), but the verdict tree REQUIRES the additional pattern check (mixed < both direction-dominant zones). A contingency that produces chi^2 > critical with the WRONG pattern (e.g., positional zone is U-enriched) would be a Pattern Fail, not a Pass. This is a hypothesis-specific test. Domain: same 250-row analyzable subset as v0.7a'. Domain is NOT broken in this form; the non-circularity rests on (feature) and (test) conditions. The locked non-circularity sentences (v0.7a + v0.9 anti-circular) carry forward verbatim: > "v0.9 uses Floquet eigenvectors only as geometric directions; it > does not use eigenvalue magnitude, spectral radius, unit-circle > status, unstable-pair count, or any threshold that defines the > published S/U label." > "v0.9 does not test chi^2 of independence on vf-ordering against > S/U on the v0.7a analyzable subset as the sole verdict criterion; > v0.9a's verdict requires both chi^2 > critical AND a > hypothesis-specific S-fraction pattern (mixed < both direction- > dominant zones)." ``` ## Free Parameter Count ```text Number of free parameters fitted to the data: 0 ``` The feature (vf), the zone cutpoints (physical {0.25, 0.50}), the zone-label conventions (positional-dominant / mixed / velocity-heavy), the test statistic (chi-squared), the df formula (occupied_zones - 1), the critical value (chi-squared(df, 0.01)), the primary-hypothesis pattern check (mixed < both direction-dominant zones), the alignment-tightness thresholds (0.8 / 0.95), the sparse-cell fallback parameters, and the verdict tree are all locked pre-data. The cutpoints {0.25, 0.50} were chosen from physical reasoning BEFORE the v0.9a runner is executed; they are NOT derived from v0.7a' quartile cutpoints. ## Sparse-Cell Fallback Logic (Inherited from v0.6b) ```text 1. Compute expected cell counts E_{kj} = N_k * C_j / 250. 2. Compute min_occupied_bin_count = min over occupied zones of N_k. 3. Branch: a) If min_occupied_bin_count < 2: verdict = signed_vf_three_zone_inconclusive_sparse no p-value emitted. b) Elif min over occupied cells of E_{kj} < 5: permutation test (seed = 20260523, n_permutations = 10000); chi^2 statistic computed per permutation; p_value = empirical tail. c) Else: asymptotic chi-squared(df = occupied_zones - 1) at p = 0.01. ``` For zones (vf < 0.25, 0.25-0.5, >= 0.5) on 250 rows with 87 S / 163 U, expected counts likely all > 5 (zone counts probably ~50, ~135, ~65 based on v0.8a distribution; expected S per zone ~17-47; expected U per zone ~33-88). Asymptotic chi-squared expected to fire. ## Alignment-Tightness Guard (Inherited from v0.6a / v0.7a) ```text For each zone in {positional-dominant, mixed, velocity-heavy}: - identify the v0.5a branch_label that dominates the zone. - compute dominant_fraction = (rows in zone in dominant branch) / (rows in zone). alignment_tightness_scalar = max over zones of dominant_fraction. Thresholds: 0.8 (warning), 0.95 (severe). ``` ## Edge Cases (Pre-Registered) 1. **Empty zone**: structurally unlikely given 250 rows with vf spanning [0.04, 0.87]; positional-dominant (vf < 0.25) has roughly the bottom ~22% of the distribution per v0.8a, velocity-heavy (vf >= 0.5) has roughly the top ~26%. If any zone has 0 rows, df adjusts to occupied_zones - 1; receipt flags the empty zone. 2. **Pattern Fail**: chi^2 > critical but the U-shape direction is wrong (e.g., positional-dominant has 18% S while mixed has 35% S). The receipt records the actual S-fraction ordering; this is a clean structural-negative on the U-shape hypothesis. 3. **Chi-squared Fail**: chi^2 <= critical. The catalog does not reach the 3-zone chi-squared gate under the physical zone binning. The receipt records the pattern direction for transparency, but the verdict is fail at chi^2. 4. **Sparse cells**: structurally unlikely but the fallback handles it via permutation test or inconclusive verdict. 5. **Alignment > 0.8**: pattern check still applies; alignment- warning verdict augments the standard Pass with a flag. v0.9b's partition design must address the alignment. ## Pre-Mortem Expectation The per-row vf distribution on the 250-row subset is known from v0.8a; the per-zone counts are NOT yet computed, but rough estimates from the v0.8a vf distribution: ```text positional-dominant (vf < 0.25): ~55 rows (v0.8a vf min ~0.04, q25 = 0.297, so rows with vf < 0.25 are below q25) mixed (0.25 <= vf < 0.50): ~130 rows (most of q25-q50 + lower part of q50-q75) velocity-heavy (vf >= 0.50): ~65 rows (above v0.7a' Q3-Q4 boundary at vf = 0.504) ``` Expected per-zone S-fractions (rough estimate from v0.7a' contingency): ```text positional-dominant: ~50% S (the v0.7a' Q1-low-vf subset is here) mixed: ~25% S (v0.7a' Q2 18% + Q3 29%) velocity-heavy: ~43% S (v0.7a' Q4) ``` If these estimates hold, the U-shape pattern is satisfied (mixed 25% < positional 50% AND mixed 25% < velocity-heavy 43%), and chi^2 should be substantial. Expected v0.9a verdict: PASS under the locked tree. If the data deviates (e.g., the v0.8a unsigned purity fail had revealed a different shape), the verdict tree handles it: Pattern Fail or Chi-squared Fail. Pre-mortem note: the U-shape predicted from v0.7a' is a strong hypothesis; if v0.9a PASSES, this is the first explicitly hypothesis-specific pre-registered positive in the chain (not just chi^2-of-independence, but pattern-confirmed). If it FAILS, the v0.7a' U-shape was misleading and the chain has a clear structural-negative on the proposed pattern. ## Stop Conditions 1. **Anti-circular discipline failure**: NOT triggered. v0.9a breaks two of the three conditions (feature derivation via physical cutpoints; test derivation via pattern-specific verdict tree). 2. **Constant-feature retirement**: vf has sd = 0.144 on the 250-row subset; retirement is structurally impossible. 3. **Alignment-tightness > 0.95**: v0.9b must use an alignment- breaking partition. 4. **Pattern Fail**: chapter closes on the U-shape hypothesis. v0.9 may register an alternative pattern hypothesis if motivated physically, BUT the bandwidth tradeoff against Phase 15+ should be weighed. 5. **Chi-squared Fail**: same as Pattern Fail, but the catalog does not even reach the chi-squared gate under the physical zone binning. ## Lock-In Statement This audit form is committed before: - per-row vf values are read into the v0.9a runner, - per-zone counts are computed, - the chi-squared statistic is computed, - the alignment-tightness scalar is computed, - the per-zone S-fractions are inspected. Any change to the cutpoints, zone labels, df formula, critical value, primary-hypothesis pattern, alignment-tightness thresholds, constant-feature retirement threshold, sparse-cell fallback parameters, or verdict tree after the v0.9a runner is executed is a re-registration, not a refinement. Implementation may proceed against `results/isotrophy/k-facet-v07a-velocity-fraction-audit/per_row_table.csv` as the sole input. No new variational compute is required. ## Receipt Schema (Planned) ```text results/isotrophy/k-facet-v09a-signed-vf-three-zone/ manifest.json - mode = "v0.9a-signed-vf-three-zone-audit" - form_lock = "docs/isotrophy/kfacet/kfacet_v09a_signed_vf_three_zone_form.md" - input_v07a_per_row_table - domain_row_count = 250 - domain_S_count, domain_U_count - zone_cutpoints: [0.25, 0.50] - zone_labels: [positional-dominant, mixed, velocity-heavy] - non_circular_conditions_broken: ["feature_derivation_via_physical_cutpoints", "test_derivation_via_pattern_specific_verdict"] - contingency_zone: n_zones x 2 (zone, S/U) - chi_squared: test statistic - test_branch_taken: "chi_squared" | "permutation" | "inconclusive_sparse" - df: occupied_zones - 1 - critical: chi-squared(df, 0.01) - p_value: chi-squared tail OR permutation tail - alignment_tightness_scalar - alignment_records: per-zone dominant branch + counts - pattern_check: - S_fraction_positional - S_fraction_mixed - S_fraction_velocity_heavy - mixed_lt_positional: bool - mixed_lt_velocity_heavy: bool - u_shape_confirmed: bool (mixed_lt_positional AND mixed_lt_velocity) - verdict: signed_vf_three_zone_{passes_audit | passes_audit_alignment_warning | passes_audit_severe_alignment | fails_audit_pattern | fails_audit_chi2 | inconclusive_sparse} per_row_table.csv (with zone column) contingency_table_three_zone.csv ``` ## Implementation Plan Suggested script: `scripts/v09a_signed_vf_three_zone_audit.py`. Suggested command: ```powershell python scripts\v09a_signed_vf_three_zone_audit.py ` --v07a-per-row results\isotrophy\k-facet-v07a-velocity-fraction-audit\per_row_table.csv ` --out results\isotrophy\k-facet-v09a-signed-vf-three-zone ``` Expected runtime: seconds. No new variational compute. ## Interpretation If v0.9a passes (chi^2 > critical, U-shape confirmed, alignment <= 0.8): > The signed Floquet direction-composition shadow stratifies > stability on the 250-row analyzable supp-B subset under a > pre-registered hypothesis-specific U-shape pattern test > (mixed zone U-enriched relative to both direction-dominant > zones). This is the **first hypothesis-specific pre-registered > positive** in the v0.4-v0.9 isotrophy chain (v0.7a' was a chi^2- > of-independence positive). v0.9b is licensed: a held-out signed- > vf predictor with the v0.5b discipline. If v0.9a passes with alignment warning: pass-but-suspect; v0.9b's partition must be re-registered alignment-breaking. If v0.9a Pattern Fails (chi^2 > critical, U-shape NOT confirmed): > The catalog has structure in vf vs S/U under the physical zone > binning, but it is NOT the predicted U-shape (mixed zone is not > U-enriched relative to both direction-dominant zones). The > v0.7a' U-shape may have been an artifact of the quartile-binning > boundaries; the physical-zone version does not reproduce it. > Chapter closes as a structural-negative on the U-shape hypothesis. If v0.9a Chi-squared Fails: > The physical-zone binning does not reach the chi-squared gate; > the v0.7a' signal lives in a finer binning than the > physically-motivated zones detect. Chapter closes on the v0.9a > hypothesis. ## Doc Trail - `kfacet_v09_mechanism_preregistration.md` -- parent registration (anti-circular discipline locked). - `kfacet_v08_writeup.md` -- v0.8 close (predecessor). - `kfacet_v07a_prime_restricted_scope_form.md` -- v0.7a' PASS source of the U-shape hypothesis. - `kfacet_v06b_within_branch_energy_audit_form.md` -- sparse-cell fallback tree (inheritance source). - `kfacet_v06a_energy_quartile_audit_form.md` -- alignment-tightness guard (inheritance source). --- Signed vf three-zone audit on the 250-row analyzable subset. Physical cutpoints {0.25, 0.50}. Hypothesis-specific verdict tree (chi^2 > critical AND mixed < both direction-dominant zones). Two conditions of the anti-circular discipline broken (feature derivation, test derivation). Inherits v0.6b sparse-cell fallback + v0.6a alignment- tightness guard.