# v0.8a Purity-Quartile Audit Form Lock Status: **VERDICT LANDED 2026-05-24**. Verdict: **`purity_quartile_fails_audit`** at `chi^2 = 4.94`, p = 0.176 (df=3, critical 11.34). The unsigned purity transform `abs(vf - 0.5)` does NOT capture the v0.7a' signal at the registered floor. The diagnostic purity_signed contingency reproduces v0.7a' (chi^2 = 16.43; expected because purity_signed = vf - 0.5 is a linear transform of vf, preserving the quartile ordering). The asymmetry diagnostic (`|S_fraction(Q1_signed) - S_fraction(Q4_signed)|` = 0.064) is **below the 0.2 flag threshold**, so the asymmetric U-shape flag did NOT fire -- the S-fractions at the two pure ends are similar. The asymmetry that breaks the purity transform is in **row density**, not S-fraction: 184/250 rows have vf < 0.5 (max vf = 0.87, no rows near vf = 1), so the unsigned purity folding does not align with the v0.7a' U-shape structure. This document locks the v0.8a purity-quartile chi-squared audit on the 250-row analyzable supp-B subset under the parent registration's candidate slot **A** (purity-quartile audit). Candidate forms B (monotone purity-threshold), C (Spearman correlation), and D (median-split) are held back. ## Verdict (Landed, 2026-05-24) Receipt: `results/isotrophy/k-facet-v08a-purity-audit/manifest.json`. ```text verdict: purity_quartile_fails_audit domain: 250 analyzable rows (87 S / 163 U) purity_sd: 0.0961 (>> 0.01 retirement floor; purity has real variation, signal just isn't captured by quartile chi-squared) chi^2_purity: 4.943205 df: 3 critical (p=0.01): 11.34 p_value (chi-sq(3)): 0.1760 alignment_tightness: 0.5873 (well below 0.8 warning threshold) test_branch_taken: chi_squared ``` Primary purity quartile contingency (re-binned within the 250-row analyzable subset): ```text Q_purity N S U S_fraction chi2_contrib approx_vf_range 1 63 21 42 0.3333 0.06 vf in [0.44, 0.56] mid-vf 2 62 18 44 0.2903 0.91 vf in [0.39, 0.44] U [0.56, 0.61] 3 62 19 43 0.3065 0.47 vf in [0.28, 0.39] U [0.61, 0.72] 4 63 29 34 0.4603 3.50 vf in [0, 0.28] U [0.72, 0.87] ----- chi^2 = 4.94 ``` Q4 (highest purity; vf far from 0.5) shows 46% S -- elevated relative to the catalog mean 34.8%, but Q1, Q2, Q3 are essentially flat around 30%. The signal is concentrated at the high-purity end but does not distribute across the quartiles in a way that reaches chi-squared(3) > 11.34. Diagnostic purity_signed contingency (signed direction; report-only): ```text Q_signed N S U S_fraction chi2_contrib 1 63 31 32 0.4921 5.76 vf in [0, 0.30] pos pure 2 62 11 51 0.1774 7.95 vf in [0.30, 0.42] 3 62 18 44 0.2903 0.91 vf in [0.42, 0.50] 4 63 27 36 0.4286 1.80 vf in [0.50, 0.87] vel pure ----- chi^2 = 16.43 asymmetry: |S_fraction(Q1) - S_fraction(Q4)| = |0.4921 - 0.4286| = 0.064 < 0.2 (flag threshold) asymmetric_u_shape_flag: False ``` The diagnostic reproduces v0.7a's chi^2 = 16.43 exactly. This confirms the v0.7a' signal is real and lives in the **signed** vf direction; the unsigned purity transform loses access to it because the vf distribution is density-asymmetric (74% of rows have vf < 0.5). vf distribution diagnostic (250 analyzable rows): ```text vf_min: 0.0426 vf_q25: 0.2972 vf_q50: 0.4188 (median below 0.5) vf_q75: 0.5036 vf_max: 0.8653 vf < 0.5: 184 / 250 (73.6%) vf > 0.5: 66 / 250 (26.4%) ``` The vf distribution is heavily skewed toward low values (positional- leaning gamma_1). The "Q4 of vf" rows (vf > 0.5) are actually vf in [0.5, 0.87], not symmetric around vf = 0.5. The purity transform `abs(vf - 0.5)` treats vf = 0 and vf = 1 as equivalent, but the data doesn't have many vf-near-1 rows, so the symmetry assumption of "purity" is violated by the catalog's actual distribution. Per-m_3 analyzable counts (for v0.8b partition design, recorded but v0.8b is NOT licensed under the fails verdict): ```text m_3=0.4: 55 m_3=0.9: 21 m_3=1.4: 4 m_3=0.5: 30 m_3=1.0: 35 m_3=1.5: 4 m_3=0.6: 22 m_3=1.1: 23 m_3=1.6: 2 m_3=0.7: 18 m_3=1.2: 7 m_3=1.7: 5 m_3=0.8: 18 m_3=1.3: 4 m_3=1.9: 2 ``` High-m_3 bins (1.4-1.9) are very thin after attrition (2-5 rows each), confirming the v0.6/v0.7 pattern: attrition concentrates at long-period high-m_3, where the variational equation hits the precision wall. **Structural reading:** > v0.8a tested the hypothesis that v0.7a's U-shaped vf-quartile > signal is captured by an unsigned direction-purity scalar > `purity = abs(vf - 0.5)`. The hypothesis fails at the registered > floor: chi-squared(3) = 4.94 vs critical 11.34, p = 0.176. The > failure is NOT explained by feature constant-retirement (purity > has sd = 0.096) or alignment-tightness with the v0.5a branch hash > (alignment 0.587, well below 0.8). The failure is explained by > the **density asymmetry** of the vf distribution: 74% of rows lie > below vf = 0.5, max vf = 0.87, so the "vf near 1" end of the > "purity" interpretation is structurally under-populated. The > diagnostic purity_signed quartile contingency reproduces v0.7a's > chi^2 = 16.43 exactly (since vf - 0.5 is a linear transform of > vf), confirming the v0.7a' positive is a real signed-direction > signal that the unsigned purity transform discards. Per the locked outcome interpretation: > Direction-purity does not stratify stability on the analyzable > sub-catalog at the registered floor. The v0.7a' U-shape signature > was not purity-driven in the structural sense; the v0.8 chapter > closes on the purity shadow. The asymmetric_u_shape diagnostic flag did NOT fire (asymmetry 0.06 < 0.2), so the U-shape is not asymmetric in S-fraction terms. The asymmetry that broke the purity transform is in catalog row density, not stability outcome. v0.8b (held-out purity predictor) is NOT licensed under the fails verdict. **Receipts:** ```text docs/isotrophy/kfacet/kfacet_v08a_purity_quartile_audit_form.md (this document) results/isotrophy/k-facet-v08a-purity-audit/manifest.json results/isotrophy/k-facet-v08a-purity-audit/per_row_table.csv results/isotrophy/k-facet-v08a-purity-audit/contingency_table_purity.csv results/isotrophy/k-facet-v08a-purity-audit/contingency_table_purity_signed.csv results/isotrophy/k-facet-v08a-purity-audit/per_m3_analyzable_counts.csv scripts/v08a_purity_audit.py ``` Audience: v0.8a runner; v0.8b form-lock author (conditional on v0.8a passing); paper-side reviewer of the direction-purity transition. Companions: - `kfacet_v08_mechanism_preregistration.md` -- v0.8 parent registration. Locks body / projection / observable, operational definition of purity, non-circularity audit, disallowed-feature list, and inheritance discipline. - `kfacet_v07a_prime_restricted_scope_form.md` -- v0.7a' PASS verdict on the velocity-fraction quartile audit (source of the U-shape signature that licenses v0.8). - `kfacet_v07a_velocity_fraction_audit_form.md` -- v0.7a parent audit (source of the vf operational definition; non-circularity sentence carries forward). - `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). Frame: v0.7a' produced a non-monotone U-shape on velocity-fraction quartiles (Q1 49% S, Q2 18% S, Q3 29% S, Q4 43% S). The U-shape suggests **direction-purity** -- the distance of gamma_1's velocity-fraction from the maximally-mixed point at vf = 0.5 -- as the load-bearing axis. v0.8a tests whether the purity quartile stratifies S/U cleanly under the same statistical pipeline that v0.7a' used. ## Why Purity-Quartile (Form A) Over Alternatives Per the parent registration: ```text A. purity-quartile audit (4 x 2 chi-squared): SELECTED. - Most-direct analogue of v0.7a's B audit form (which v0.7a' also used). Inherits exactly the same statistical pipeline (4-bin chi-squared, df = 3, critical 11.34, alignment- tightness guard, sparse-cell fallback). - Preserves enough granularity to detect monotone-in-purity structure AND any residual asymmetry between Q1-pure and Q4-pure rows (recorded via the purity_signed diagnostic). - Cleanest extension of v0.7a' under inheritance discipline. B. monotone purity-threshold (2 x 2): held back. - Coarser test (df = 1, critical 6.63) with lower power if the effect is graded rather than discrete-threshold. - Threshold-choice flexibility (which quantile?) requires extra pre-registration work. C. Spearman rank correlation: held back. - Different test statistic; breaks the chi-squared lineage inherited from v0.6/v0.7. - Continuous; avoids binning but loses the chi^2 contribution breakdown that revealed the U-shape in v0.7a'. D. median-split + binomial fallback: held back. - Even coarser (2 x 2 with 125/125 split); same df-power tradeoff as B. - Power-conservative but loses the quartile-pattern visibility. ``` The locked choice prioritizes **inheritance fidelity** with v0.7a' plus **diagnostic granularity** (the four quartiles let us see the shape inside the analyzable subset, not just a monotone summary). ## What v0.8a Is v0.8a is a **catalog-side audit on the v0.7a analyzable subset**, NOT a predictor (continues v0.5a/v0.6a/v0.6b/v0.7a/v0.7a' audit-not-predictor framing): ```text Domain: 250 analyzable rows from v0.7a (rows where compute_monodromy_vectorized completed at the R1-amended sanity gates). Attrition (23 blocked + 11 R1-sanity-failed) is carried as a permanent domain restriction. Feature: purity(row) = abs(vf(row) - 0.5) where vf is the v0.7a velocity-fraction (locked selection rule + tie-break cascade + reference frame + non-circularity provenance). Binning: quartiles over the 250-row analyzable subset's purity distribution. Test: chi-squared independence of Q_purity vs S/U. df = 3 (or occupied_bins - 1 under sparse fallback). critical 11.34 at p = 0.01 (chi-squared(3)). Alignment guard: max over Q_purity 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). Constant-feat: if sd(purity) < 0.01, retire. Verdicts: Pass: chi^2 > 11.34 AND alignment <= 0.8 -> purity_quartile_passes_audit Warn pass: chi^2 > 11.34 AND alignment in (0.8, 0.95] -> purity_quartile_passes_audit_alignment_warning Severe pass: chi^2 > 11.34 AND alignment > 0.95 -> purity_quartile_passes_audit_severe_alignment Fail: chi^2 <= 11.34 -> purity_quartile_fails_audit Sparse: min_occupied_bin_count < 2 -> purity_quartile_inconclusive_sparse Retired: sd(purity) < 0.01 -> purity_quartile_retired_near_constant ``` A clean Pass licenses v0.8b: a separately-registered held-out purity predictor with the v0.5b discipline. A Warn/Severe Pass licenses only an alignment-breaking v0.8b. A Fail closes the purity-shadow sub-question; an asymmetric U-shape diagnostic (see below) may motivate a non-purity feature re-registration as v0.8a'. ## The Locked Form ### Operational definition (inherited from v0.8 parent) ```text For each row r in the 250-row analyzable subset: purity(r) = abs(vf(r) - 0.5) range: [0, 0.5] purity(r) = 0 iff vf(r) = 0.5 (maximally mixed) purity(r) = 0.5 iff vf(r) in {0, 1} (perfectly pure end) ``` ### Binning (locked) ```text cutpoints: q25, q50, q75 of the 250-row analyzable subset's purity distribution. method: numpy.quantile(purity_values, [0.25, 0.50, 0.75], method='linear'). assignment: right-closed lower intervals; ties to lower bin (matches v0.6a/v0.7a/v0.7a' convention). Q_purity(r) = 1 iff purity(r) <= q25 Q_purity(r) = 2 iff q25 < purity(r) <= q50 Q_purity(r) = 3 iff q50 < purity(r) <= q75 Q_purity(r) = 4 iff purity(r) > q75 ``` ### Expected direction (pre-registered) If v0.7a's U-shape is purity-driven, the expected ordering is **monotone increasing in Q_purity**: ```text expected: S_fraction(Q_purity=1) < S_fraction(Q_purity=2) < S_fraction(Q_purity=3) < S_fraction(Q_purity=4) ``` If the observed pattern is NOT monotone (e.g., Q4 has lower S than Q3), the U-shape is asymmetric and the purity transform may not be the cleanest mechanism. The purity_signed diagnostic (below) addresses this. ## Diagnostic: purity_signed Quartile (Asymmetry Check) The v0.8 parent registration recorded `purity_signed = vf - 0.5` as a diagnostic-only quantity. v0.8a's receipt MUST emit the purity_signed quartile contingency to check whether Q1-pure (vf near 0) and Q4-pure (vf near 1) orbits behave the same: ```text purity_signed quartiles (4-bin over [-0.5, 0.5]): Q1: vf in [0, q25_signed] near-positional pure end Q2: vf in (q25_signed, q50_signed] Q3: vf in (q50_signed, q75_signed] Q4: vf in (q75_signed, 1] near-velocity pure end asymmetry diagnostic: abs(S_fraction(Q1) - S_fraction(Q4)) If asymmetry > 0.2 (locked threshold): flag asymmetric_u_shape on receipt; v0.8b may need re-registration with a non-purity feature. ``` The diagnostic is **not gating**. The primary verdict is determined by the unsigned purity quartile audit. The diagnostic is recorded for v0.8b form-lock design. ## Per-m_3 Analyzable Row Counts (Required Diagnostic) The v0.8a receipt MUST report per-m_3-bin analyzable row counts on the 250-row subset, so v0.8b's leave-one-m_3-bin-out partition design is informed: ```text m_3 bin analyzable count 0.4 (count after attrition) 0.5 0.6 ... 1.9 ``` This is non-gating but mandatory on the receipt. Long-period high-m_3 bins (1.5, 1.6, 1.7) are expected to be thin after attrition; v0.8b's partition may need to drop bins with N < 5 to report-only status. ## Pre-Audit Sanity Surface Before v0.8a 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; flag any out- of-range row as a parser/computation bug. 5. Compute purity = abs(vf - 0.5) for all 250 rows. 6. Constant-feature retirement check: if sd(purity) < 0.01, verdict = purity_quartile_retired_near_constant, no audit run. ``` ## Non-Circularity Audit purity is a monotone deterministic function of vf; non-circularity provenance is inherited verbatim from v0.7a: ```text - purity depends on vf only, which depends on gamma_1, which is the eigenvector of the largest-real-part Floquet eigenvalue. - The eigenvalue ORDERING is used to disambiguate; the eigenvalue MAGNITUDE never enters purity. - The 250-row subset is filtered by integration_blocked, a runtime status of the integrator, NOT a function of stability. - The S/U label is the held-back test column, joined at the chi-squared step. - The v0.5a branch_label is joined ONLY for the alignment-tightness diagnostic, NOT used to construct purity. ``` The locked non-circularity sentence from v0.7a is re-asserted for v0.8a. ## Free Parameter Count ```text Number of free parameters fitted to the data: 0 ``` The feature (purity = abs(vf - 0.5)), the binning rule (quartiles), the quantile method (numpy linear interpolation), the bin assignment convention (right-closed lower), the df formula (occupied_bins - 1), the critical value (11.34), the alignment-tightness thresholds (0.8, 0.95), the constant-feature retirement threshold (sd < 0.01), the sparse-cell fallback parameters (min_expected = 5, min_occupied_bin_count = 2, permutation_seed = 20260523, n_permutations = 10000), and the asymmetry-diagnostic threshold (0.2) are all locked pre-data. ## 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 bins of N_k. 3. Branch: a) If min_occupied_bin_count < 2: verdict = purity_quartile_inconclusive_sparse no p-value emitted. b) Elif min over occupied cells of E_{kj} < 5: permutation test (seed = 20260523, n_permutations = 10000); p_value = empirical tail. c) Else: asymptotic chi-squared(df = occupied_bins - 1) at p = 0.01. ``` For quartile binning on 250 rows with 87 S / 163 U, expected counts per cell are approximately {22, 41} -- well above the asymptotic floor. The fallback is documented for completeness. ## Alignment-Tightness Guard (Inherited from v0.6a / v0.7a) ```text For each Q_purity in {1, 2, 3, 4}: - identify the v0.5a branch_label that dominates the bin. - compute dominant_fraction = (rows in bin in dominant branch) / (rows in bin). alignment_tightness_scalar = max over Q_purity of dominant_fraction. Thresholds: 0.8 (warning), 0.95 (severe). If chi^2 > 11.34: alignment <= 0.8 -> purity_quartile_passes_audit alignment in (0.8, 0.95] -> purity_quartile_passes_audit_alignment_warning alignment > 0.95 -> purity_quartile_passes_audit_severe_alignment ``` ## Edge Cases (Pre-Registered) 1. **Out-of-range vf**: any row with vf outside [0, 1] is a parser/computation bug; block v0.8a and inspect. 2. **Constant-feature retirement**: sd(purity) < 0.01 triggers `purity_quartile_retired_near_constant`. Pre-mortem estimate: v0.7a' vf_sd was 0.144 over 250 rows; purity sd should be similar order (>= 0.05); retirement is structurally unlikely. 3. **Empty quartile**: structurally impossible for quartile binning over 250 rows; not in scope. 4. **All-S or all-U quartile**: chi-squared still computes; expected cell counts handle the case. 5. **Asymmetric U-shape**: if `|S_fraction(purity_signed_Q1) - S_fraction(purity_signed_Q4)| > 0.2`, the receipt flags `asymmetric_u_shape = True`. The verdict is unchanged (still reflects unsigned purity quartile chi-squared); v0.8b form-lock author is informed to consider non-purity features. ## Pre-Mortem Expectation The per-row purity distribution has NOT been computed before lock-in. The audit runs blind. If the v0.7a' U-shape is purity-driven and roughly symmetric: the v0.8a verdict is **Pass** (chi^2 > 11.34) with alignment similar to v0.7a's 0.698 (below 0.8); the quartile S-fractions are expected to be monotone increasing in Q_purity, with Q4 (high purity) approximately matching v0.7a's combined Q1+Q4 S-fraction (about 46%) and Q1 (low purity, near vf=0.5) approximately matching v0.7a's Q2 S-fraction (about 18%). If the U-shape is asymmetric (e.g., Q4-pure has much higher S than Q1-pure), the purity transform conflates two mechanisms; the asymmetric_u_shape flag fires and v0.8b form-lock author is expected to register a non-purity feature. If purity does not stratify (Fail verdict), the v0.7a' positive was an artifact of the quartile boundary rather than a structural direction-purity effect; the chapter closes on the purity shadow. ## Lock-In Statement This audit form is committed before: - per-row purity has been computed, - the 250-row subset's purity distribution has been examined, - the quartile cutpoints have been calculated, - the alignment-tightness scalar has been computed, - the purity_signed diagnostic contingency has been computed. Any change to the feature, binning, df formula, critical value, alignment-tightness thresholds, constant-feature retirement threshold, sparse-cell fallback parameters, or asymmetry-diagnostic threshold after the v0.8a runner is executed is a re-registration, not a refinement. Implementation may proceed against this form lock using `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-v08a-purity-audit/ manifest.json - mode = "v0.8a-purity-quartile-audit" - form_lock = "docs/isotrophy/kfacet/kfacet_v08a_purity_quartile_audit_form.md" - input_v07a_per_row_table - domain_row_count = 250 - domain_S_count, domain_U_count - attrition_disclosure: {v07a_full_catalog_row_count, v07a_integration_blocked_count, v07a_sanity_failed_count} - cutpoints_purity: {q25, q50, q75} - cutpoints_purity_signed: {q25, q50, q75} - constant_feature_sd: sd(purity) - constant_feature_retired: bool - contingency_purity: 4 x 2 (Q_purity, S/U) - chi_squared: test statistic - test_branch_taken: "chi_squared" | "permutation" | "inconclusive_sparse" | "retired_near_constant" - df: 3 (or occupied_bins - 1) - critical: 11.34 - p_value: chi-squared tail OR permutation tail - alignment_tightness_scalar - alignment_records: per-quartile dominant branch + counts - verdict: purity_quartile_{passes_audit | passes_audit_alignment_warning | passes_audit_severe_alignment | fails_audit | inconclusive_sparse | retired_near_constant} - diagnostic_purity_signed_contingency: 4 x 2 (Q_purity_signed, S/U) - asymmetry_diagnostic: abs(S_fraction(Q1_signed) - S_fraction(Q4_signed)) - asymmetric_u_shape_flag: asymmetry > 0.2 - per_m3_analyzable_counts: dict {m3_key: N} per_row_table.csv contingency_table_purity.csv contingency_table_purity_signed.csv per_m3_analyzable_counts.csv ``` ## Implementation Plan Suggested script: `scripts/v08a_purity_audit.py`. Suggested command: ```powershell python scripts\v08a_purity_audit.py ` --v07a-per-row results\isotrophy\k-facet-v07a-velocity-fraction-audit\per_row_table.csv ` --out results\isotrophy\k-facet-v08a-purity-audit ``` Expected runtime: seconds. No new variational compute. ## Interpretation If v0.8a passes (chi^2 > 11.34, alignment <= 0.8): > The Floquet direction-purity scalar stratifies stability on the > 250-row analyzable supp-B subset under a pre-registered quartile > chi-squared audit, independent of the v0.5a branch hash. This > confirms the v0.7a' U-shape signature is purity-driven and > licenses v0.8b: a separately-registered held-out predictor with > the v0.5b discipline. If v0.8a passes with alignment warning (chi^2 > 11.34, alignment in (0.8, 0.95]): > Pass-but-suspect. The purity quartiles track the v0.5a branch > hash tightly; v0.8b's partition must be re-registered alignment- > breaking (within-branch test mirroring v0.6b). If v0.8a fails (chi^2 <= 11.34): > Direction-purity does not stratify stability on the analyzable > sub-catalog. The v0.7a' U-shape signature was not purity-driven > in the structural sense; the chapter closes on the purity > shadow. If the asymmetric_u_shape flag fired, the U-shape's > asymmetry suggests a non-purity feature mechanism; the v0.8a' > form-lock author may register a signed-direction or sub-quartile > feature with a fresh circularity audit. If purity retires near-constant: structurally unlikely given v0.7a' vf_sd; the chapter closes on the purity-feature shadow. ## Doc Trail - `kfacet_v08_mechanism_preregistration.md` -- parent registration. - `kfacet_v07a_prime_restricted_scope_form.md` -- v0.7a' PASS source of the U-shape signature. - `kfacet_v07a_velocity_fraction_audit_form.md` -- v0.7a parent audit (vf operational definition source). - `kfacet_v06a_energy_quartile_audit_form.md` -- alignment- tightness guard pattern. - `kfacet_v06b_within_branch_energy_audit_form.md` -- sparse-cell fallback tree pattern. --- Purity-quartile audit on the 250-row analyzable subset. df = 3, critical 11.34. Inherits v0.7a non-circularity sentence verbatim. Diagnostic purity_signed contingency for U-shape asymmetry check. Per-m_3 analyzable row counts emitted to inform v0.8b partition design.