# K_facet v0.8 Methodology + Result Handoff Status: closed structural-negative (direction-purity does not capture the v0.7a' signal), 2026-05-24. Audience: collaborators, paper-side writers, future coding agents. Canonical sources: `kfacet_v08_mechanism_preregistration.md`, `kfacet_v08a_purity_quartile_audit_form.md`. Companion (closed v0.7 chapter): `kfacet_v07_writeup.md`. ## One-Line Read v0.8 closes as a **structural-negative on the unsigned direction- purity hypothesis**. The v0.7a' U-shape signature (chi^2 = 16.43, p = 9.3e-4 on the 250-row analyzable subset) was hypothesized to be captured by `purity = abs(vf - 0.5)`; v0.8a tested this with the same quartile chi-squared pipeline that produced v0.7a's positive and **failed at chi^2 = 4.94, p = 0.176**. The diagnostic `purity_signed` contingency reproduced v0.7a's chi^2 = 16.43 exactly (linear transform of vf; same ordering), confirming the v0.7a' signal is real but **does not live in the unsigned distance from 0.5**. The vf distribution is density-asymmetric (184/250 rows below vf = 0.5, max vf = 0.87, no rows near vf = 1), violating the purity transform's symmetry assumption. **The signal lives in the signed vf direction**, and the v0.8 chapter closes on the purity shadow as a structural-negative. v0.9 opens on signed direction-composition. ## What v0.8 Tested v0.8 opened after v0.7's qualified-positive close. v0.7a' produced a non-monotone U-shape on the velocity-fraction quartiles: Q1 49% S, Q2 18% S, Q3 29% S, Q4 43% S. The U-shape suggested **direction- purity** -- the distance of gamma_1's velocity-fraction from the maximally-mixed point at vf = 0.5 -- as the load-bearing axis. v0.8 tested this with a pre-registered chi-squared quartile audit on the purity scalar. Two pre-registered sub-chapters: ```text v0.8a: purity-quartile audit Domain: 250 analyzable supp-B rows (v0.7a integration_ blocked == False). Feature: purity(row) = abs(vf(row) - 0.5). Binning: quartiles within the 250-row subset. Test: 4 x 2 chi-squared (Q_purity vs S/U), df = 3, critical 11.34 at p = 0.01. Result: purity_quartile_fails_audit (chi^2 = 4.94, p = 0.176, alignment 0.587). v0.8b: held-out purity predictor (NOT licensed under v0.8a fails). ``` The audit form was pre-registered as candidate A of the v0.8 parent registration's four candidate forms (purity-quartile, monotone purity-threshold, Spearman correlation, median-split). A was selected for inheritance fidelity with v0.7a' (same quartile chi-squared pipeline). The diagnostic purity_signed contingency was required on the receipt as the asymmetric-U-shape diagnostic. ## Audit Chain Methodology The v0.8 chain inherited v0.7's discipline (audit-then-predictor separation, alignment-tightness guard at 0.8/0.95, sparse-cell fallback tree with seed 20260523, constant-feature retirement at sd < 0.01) and added one new register-shaping discipline: **Domain restriction as permanent.** The v0.7a integration-attrition verdict (23 blocked + 11 sanity-failed = 34/273 catalog rows unevaluable at variational precision 1e-12) is carried as a permanent domain restriction on v0.8. The v0.8 receipts explicitly record the v0.7a attrition counts; v0.8's claims apply only to the 250-row analyzable subset. This avoids the failure mode of treating attrition as a missing-data problem to be imputed past. Two stages, both paper-side first: 1. **v0.8 parent registration** (`kfacet_v08_mechanism_preregistration.md`). Locked body / projection / observable; operational definition of purity = `abs(vf - 0.5)`; v0.7a non-circularity sentence re-asserted verbatim; disallowed-feature list extended (vf itself, purity_signed as primary, v0.7a' Q_vf quartile assignments); five circularity-risk locks (eigenvalue-choice, well-definedness, feature-extraction, branch-shadow conflation, v0.7a'-conflation); sketched four candidate audit forms A-D and three candidate predictor forms E-G. 2. **v0.8a child form lock + verdict** (`kfacet_v08a_purity_quartile_audit_form.md`). Selected A (purity-quartile audit) over B/C/D for inheritance fidelity. Verdict landed: `purity_quartile_fails_audit`. The diagnostic purity_signed contingency reproduced v0.7a's chi^2 = 16.43 exactly, surfacing the density-asymmetry explanation. ## Result And Structural-Negative Verdict ```text v0.8a verdict: purity_quartile_fails_audit domain: 250 analyzable rows (87 S / 163 U). chi^2_purity: 4.943205 p_value: 0.1760 df: 3 critical: 11.34 alignment: 0.587 (well below 0.8 warning threshold) sd(purity): 0.0961 (above 0.01 retirement floor; feature has real variation, just isn't captured by this quartile chi-squared). test_branch: chi_squared (min_expected = 21.6, asymptotic conditions met). primary contingency (Q_purity, S/U): Q1 N=63 S=21 33.3% vf ~ [0.44, 0.56] mid Q2 N=62 S=18 29.0% vf ~ [0.39, 0.44] U [0.56, 0.61] Q3 N=62 S=19 30.6% vf ~ [0.28, 0.39] U [0.61, 0.72] Q4 N=63 S=29 46.0% vf ~ [0, 0.28] U [0.72, 0.87] Q4 (highest purity) elevated at 46%; Q1-Q3 flat ~30%. Pattern not strong enough to reach chi^2(3) > 11.34. diagnostic purity_signed contingency (signed direction): Q1 N=63 S=31 49.2% vf in [0, 0.30] positional pure Q2 N=62 S=11 17.7% vf in [0.30, 0.42] Q3 N=62 S=18 29.0% vf in [0.42, 0.50] Q4 N=63 S=27 42.9% vf in [0.50, 0.87] velocity pure chi^2 = 16.43 (reproduces v0.7a' exactly; linear transform of vf preserves quartile ordering). asymmetry: |S(Q1) - S(Q4)| = 0.064 (< 0.2 flag threshold) vf distribution diagnostic: vf < 0.5: 184 / 250 (73.6%) vf > 0.5: 66 / 250 (26.4%) vf max: 0.8653 (no rows near vf = 1) ``` The publishable v0.8 statement: > v0.8 tested the hypothesis that v0.7a's U-shaped velocity-fraction > signal is captured by an unsigned direction-purity scalar > `purity = abs(vf - 0.5)`. The hypothesis fails: chi-squared(3) = > 4.94 vs critical 11.34, p = 0.176, on the 250-row analyzable > subset. The diagnostic purity_signed contingency reproduces > v0.7a's chi-squared = 16.43 exactly, confirming the v0.7a' > positive is real but lives in the **signed** vf direction, not > in unsigned distance from 0.5. The failure is explained by the > vf distribution's density asymmetry: 74% of rows lie below > vf = 0.5 and max vf = 0.87, so the "vf near 1" pure end of the > "purity" interpretation is structurally under-populated by the > catalog. The unsigned purity transform treats vf = 0 and vf = 1 > as equivalent, but the catalog does not. The v0.8 chapter closes > on the purity shadow; v0.9 opens on signed direction-composition. Three structural sub-results preserved for v0.9: 1. **The v0.7a' signal is signed, not symmetric.** The chi^2 = 16.43 on the 250-row subset is real, branch-independent (alignment 0.698), and reproducible. But it is NOT captured by the unsigned purity transform. The signal lives in some signed-direction function of vf, not in |vf - 0.5|. 2. **The vf distribution is density-asymmetric.** 74% of analyzable rows have vf < 0.5; max vf = 0.87. Any v0.9 hypothesis that treats the two "pure ends" of vf as equivalent will continue to fail. The catalog's actual distribution is the constraint. 3. **The Q1-vs-Q4 S-fraction split is roughly symmetric.** Q1_signed (vf in [0, 0.30]) and Q4_signed (vf in [0.50, 0.87]) both concentrate stable orbits (49% and 43% S respectively). The "purity-of-direction correlates with stability" interpretation is correct in **outcome** but not in **density**. v0.9 needs to test a hypothesis that respects both. Joint v0.4 + v0.5 + v0.6 + v0.7 + v0.8 envelope: > Five sequential pre-registered low-dimensional projections of the > supp-B body have produced five distinct chapter-close types: > structural-negative on Z_2 symmetry (v0.4); projection-limit on > 2-bit catalog branch hash (v0.5); conditional-independence on > continuous catalog-coordinate (E, |L|) shadow (v0.6); integration- > attrition + qualified-positive on restricted domain for Floquet > velocity-fraction (v0.7); and structural-negative on Floquet > direction-purity (v0.8). The v0.7a' positive on the 250-row > analyzable subset is the only statistically-significant > non-branch-aligned signal in the chain, and v0.8a confirms it > is signed (not symmetric in vf). v0.9 opens on the signed > direction-composition hypothesis with explicit anti-circular > framing. ## Reproducibility Surface ```bash # v0.8a purity-quartile audit (catalog-only, seconds): python scripts/v08a_purity_audit.py ``` Key receipt directory: ```text 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 ``` Load-bearing constants live in the form-lock documents: ```text kfacet_v08_mechanism_preregistration.md body / projection / observable; operational definition of purity; five circularity-risk locks; disallowed-feature list (vf itself, purity_signed as primary, v0.7a' Q_vf assignments); inheritance discipline. kfacet_v08a_purity_quartile_audit_form.md Selected candidate A; quartile binning method (numpy linear); alignment-tightness thresholds 0.8/0.95; sparse-cell fallback; asymmetry-diagnostic threshold 0.2; verdict landed `purity_quartile_fails_audit`. ``` ## Where v0.9 Opens The v0.8 structural-negative confirms the v0.7a' signal is signed, not symmetric. v0.9 opens on the **signed Floquet direction- composition** mechanism (codex direction, 2026-05-24). Parent registration at `kfacet_v09_mechanism_preregistration.md`. The v0.9 parent's load-bearing methodological challenge is **anti-circular framing**: any chi-squared-of-independence test on vf-ordering (whether expressed as raw vf quartiles, signed quartiles, or binary vf-side threshold) will reproduce v0.7a's chi^2 = 16.43 by the linear-transform invariance. v0.9 cannot honestly use chi^2 of independence as the test statistic; it must test a **specific pattern hypothesis** that v0.7a' did not pre-register. The parent registration's four candidate v0.9a forms: ```text A. Ordered quartile audit (Q1/Q2/Q3/Q4 labels preserved): CIRCULAR RISK -- reproduces v0.7a' chi^2 exactly. B. Threshold audit at vf < 0.5 vs vf >= 0.5: CIRCULAR RISK -- binary collapse of the same Q ordering. C. Three-zone audit (positional-pure / mixed / velocity-heavy): Non-circular IF zone boundaries are pre-registered with physical motivation, NOT data-driven from v0.7a's quartile cutpoints. D. Ordinal trend + nonmonotonicity test: Non-circular. Tests "is vf vs S/U monotone?" -- the v0.7a' U-shape predicts FAIL-TO-FIT-MONOTONE. This is a hypothesis-specific test, not chi^2 of independence. ``` Forms A and B are excluded under the anti-circular discipline. Forms C and D are the live candidates; D is the cleanest hypothesis-specific test of the U-shape pattern. ## Doc Trail - `kfacet_v08_mechanism_preregistration.md` -- v0.8 parent registration. - `kfacet_v08a_purity_quartile_audit_form.md` -- v0.8a child form lock + structural-negative verdict. - `kfacet_v07_writeup.md` -- v0.7 qualified-positive chapter close (companion). - `kfacet_v07a_velocity_fraction_audit_form.md`, `kfacet_v07a_prime_restricted_scope_form.md` -- v0.7 form locks (predecessors). - `kfacet_v09_mechanism_preregistration.md` -- v0.9 parent registration (signed Floquet direction-composition; opens v0.9). The chapter closes cleanly. v0.7a's positive is preserved unchanged. v0.8a's structural-negative on unsigned purity is preserved as a methodological finding: the v0.7a' U-shape is real but NOT capturable by symmetric distance-from-mixed under the catalog's actual vf distribution. The chain has now produced six sequential pre-registered chapters with five distinct chapter-close types; v0.9 extends the chain with an explicitly anti-circular framing.