# K_facet v0.9 Methodology + Result Handoff Status: closed structural-negative (U-shape mechanism invalidated; monotone-increasing meta-finding preserved), 2026-05-24. Audience: collaborators, paper-side writers, future coding agents, isotrophy-program lab initiates. Canonical sources: `kfacet_v09_mechanism_preregistration.md`, `kfacet_v09a_signed_vf_three_zone_form.md`. Companion (closed v0.8 chapter): `kfacet_v08_writeup.md`. Companion (program pause): `kfacet_isotrophy_program_pause.md`. ## One-Line Read v0.9 closes as a **structural-negative on the U-shape mechanism hypothesis**, with a **substantive meta-finding** preserved: the v0.7a' chi^2 = 16.43 positive was an artifact of the v0.7a' quartile boundary at vf ~ 0.297. Under the physical cutpoint vf = 0.25, the positional-dominant zone shrinks from 63 to 19 rows and its S-fraction collapses from 49% to 10.5%. The actual physical pattern across the three signed-direction zones is **monotone-increasing** (positional 11% S, mixed 34% S, velocity-heavy 44% S) — chi^2 = 7.42, p = 0.0245, below the registered floor at p = 0.01 (critical 9.21). The U-shape MECHANISM is invalidated; the v0.7a' chi^2 = 16.43 catalog-coordinate positive remains a valid quartile-binning stratification finding. v0.9 is the **last chapter** in the isotrophy program before the formal program pause; v0.9b is not licensed. ## What v0.9 Tested v0.9 opened after v0.8's structural-negative on unsigned direction- purity. The v0.8a diagnostic purity_signed contingency reproduced v0.7a's chi^2 = 16.43 exactly (linear transform of vf), confirming the v0.7a' signal lives in the signed direction. v0.9 promoted signed direction-composition with an **anti-circular framing discipline** locked at the v0.9 parent: ```text Anti-circular discipline (locked permanently): any v0.9 child form lock that tests vf-ordering vs S/U on the v0.7a' analyzable subset via chi^2 of independence reproduces v0.7a' by linear-transform invariance. To be honest, v0.9a must break at least one of: 1. feature derivation, 2. test statistic, 3. evaluation domain. ``` The v0.9a child form lock selected candidate C (three-zone audit with physically-motivated cutpoints {0.25, 0.50}) and broke **two** of the three anti-circular conditions: ```text Condition #1 (feature derivation) BROKEN: physical cutpoints {0.25, 0.50}, NOT v0.7a' quartile cutpoints (which were approximately {0.297, 0.419, 0.504}). Condition #2 (test statistic) BROKEN: the verdict tree requires chi^2 > critical AND a hypothesis-specific pattern (mixed S-fraction < BOTH positional-dominant AND velocity-heavy S-fractions; i.e., the U-shape direction). ``` The hypothesis: under physical-zone binning, the v0.7a' U-shape generalizes to a true U-shape where the mixed zone is U-enriched relative to both direction-dominant zones. ## Audit Chain Methodology The v0.9 chain inherited all prior register-shaping disciplines (audit-then-predictor separation, alignment-tightness guard, sparse-cell fallback tree, conservative tie rule, constant-feature retirement, attrition-as-domain-restriction) and added one new permanent discipline: **Anti-circular framing for sequential audits on the same data.** After v0.7a' produced a chi^2 = 16.43 positive on vf-quartile vs S/U, any subsequent test on the same 250-row subset using chi^2-of-independence on vf-ordering would silently reproduce v0.7a' by linear-transform invariance of chi^2 under monotone re-binning of an ordered feature. v0.9 explicitly identifies this mechanism-laundering risk and rejects two candidate v0.9a forms (A: ordered quartile audit; B: binary vf<0.5 threshold) pre-compute under the locked anti-circular discipline. The framework is permanent: any future v0.10+ child form lock that tests vf vs S/U must respect the same discipline. Stages, all paper-side first: 1. **v0.9 parent registration** (`kfacet_v09_mechanism_preregistration.md`). Locked body / projection / observable; vf operational definition inherited from v0.7a; anti-circular framing discipline locked. 2. **v0.9a child form lock + verdict** (`kfacet_v09a_signed_vf_three_zone_form.md`). Selected C (three-zone audit with physical cutpoints) over A/B/D. Locked verdict tree includes both chi^2 gate and hypothesis-specific pattern check. ## Result And Structural-Negative Verdict ```text v0.9a verdict: signed_vf_three_zone_fails_audit_chi2 three-zone contingency (locked physical cutpoints {0.25, 0.50}): zone N S U S_frac 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 df: 2 (3 occupied zones - 1) 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) mixed < velocity-heavy? TRUE (0.34 < 0.44) u_shape_confirmed? FALSE ``` The publishable v0.9 statement: > v0.9a tested whether the v0.7a' chi^2 = 16.43 positive is captured > by a hypothesis-specific U-shape on three physically-motivated > direction zones. The hypothesis fails on both gates: the > chi-squared(2) statistic of 7.42 does not reach the registered > critical value 9.21 (p = 0.0245 vs threshold p = 0.01), AND the > U-shape pattern direction is broken (mixed zone is NOT > U-enriched relative to positional-dominant; mixed is in fact > U-enriched relative to ONLY velocity-heavy). The substantive > meta-finding: the v0.7a' U-shape was an artifact of the > quartile boundary at vf ~ 0.297. The actual physical pattern > under non-data-driven zones is **monotone-increasing in vf** > (positional 11% S < mixed 34% S < velocity-heavy 44% S), which > matches the broader thesis that velocity-dominated Floquet > directions correlate with stability — but does not clear the > registered chi-squared floor under three-zone physical binning. Three structural sub-results preserved: 1. **The v0.7a' chi^2 = 16.43 positive is a quartile-binning stratification finding, NOT a structural mechanism**. Under the v0.7a' Q1 bin (vf in [0, 0.297]), 63 rows / 49% S. Under the physical zone (vf < 0.25), only 19 rows / 11% S. The high-S rows that drove v0.7a' Q1's S-fraction live near the quartile boundary (vf in [0.25, 0.30]), NOT in the deepest positional regime. 2. **The actual physical pattern is monotone-increasing in vf**. Across the three signed-direction zones, S_fraction goes 0.105 -> 0.339 -> 0.439. This is consistent with the broader thesis that velocity-dominated gamma_1 directions correlate with stability. A J-T trend test (candidate D) would test this directly and would likely pass at p < 0.01 given the magnitude of the increase, but it is NOT registered under v0.9. Lab initiates may pursue this (see `kfacet_isotrophy_program_pause.md`). 3. **Anti-circular discipline pattern locked as permanent**. Any future sequential audit on the same data after a positive audit has produced an in-sample chi-squared must break at least one of (feature derivation, test statistic, evaluation domain) to be non-circular. This pattern is now part of the isotrophy methodology surface. Joint v0.4 + v0.5 + v0.6 + v0.7 + v0.8 + v0.9 envelope (the publishable multi-chapter statement at the program pause): > Seven sequential pre-registered low-dimensional projections of > the supp-B body have produced seven distinct chapter-close types: > domain-of-applicability (v0.3), structural-negative on Z_2 > symmetry (v0.4), projection-limit on 2-bit catalog branch hash > (v0.5), conditional-independence on continuous (E, |L|) shadow > (v0.6), integration-attrition + qualified-positive on restricted > domain (v0.7), structural-negative on unsigned direction-purity > (v0.8), and structural-negative on U-shape mechanism with > monotone-increasing meta-finding (v0.9). The v0.7a' > chi^2 = 16.43 quartile-binning positive remains the chain's > load-bearing positive substantive finding. v0.8 and v0.9 > contextualized but did not invalidate it: the signal is signed > (v0.8 confirmed), not symmetric (v0.8 disconfirmed), and is > quartile-boundary-driven rather than physical-zone-driven > (v0.9 disconfirmed; monotone-increasing pattern preserved as > meta-finding). The methodology surface (closure-relative > discipline + two-pass classifier + projection language + > audit-then-predictor + asymmetric falsifier + alignment-tightness > guard + sparse-cell fallback + per-row integration-failure > fallback + append-per-row resume + anti-circular framing for > sequential audits) is the load-bearing publishable contribution. ## Reproducibility Surface ```bash # v0.9a three-zone audit (catalog-only, seconds): python scripts/v09a_signed_vf_three_zone_audit.py ``` Key receipt directory: ```text 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 ``` Load-bearing constants live in the form-lock documents: ```text kfacet_v09_mechanism_preregistration.md body / projection / observable; anti-circular framing discipline (load-bearing); disallowed-feature list; inheritance discipline; candidate forms A-D with A and B rejected pre-compute. kfacet_v09a_signed_vf_three_zone_form.md Selected candidate C; physical cutpoints {0.25, 0.50}; verdict tree requires chi^2 > critical AND U-shape pattern; verdict landed `signed_vf_three_zone_fails_audit_chi2` at chi^2 = 7.42. ``` ## Where the Isotrophy Program Pauses The v0.9 chapter close marks the end of the active isotrophy program. The program PAUSES (does not retire) at end-of-v0.9. See `kfacet_isotrophy_program_pause.md` for: - The seven-chapter envelope summary. - The publishable load-bearing findings (v0.7a' positive, monotone-increasing meta-finding, full methodology surface). - Concrete reopening avenues for lab initiates ("waste compute on" candidates). - Bandwidth redirect to three-body Phase 15+ (per the principal's pre-v0.7 direction call). The program may be unpaused if a lab initiate produces clean evidence on one of the reopening avenues (e.g., a J-T trend test that passes the locked anti-circular discipline; a relaxed-precision re-run of v0.7a that recovers the attrited 23 rows; or a targeted m_3 = 0.4 sub-catalog mechanism investigation). ## Doc Trail - `kfacet_v09_mechanism_preregistration.md` -- v0.9 parent registration (anti-circular discipline lock). - `kfacet_v09a_signed_vf_three_zone_form.md` -- v0.9a child form lock + structural-negative verdict. - `kfacet_isotrophy_program_pause.md` -- program pause document (concrete reopening avenues for lab initiates). - `kfacet_v08_writeup.md` -- v0.8 structural-negative chapter close (companion). - `kfacet_v07_writeup.md` -- v0.7 qualified-positive chapter close (load-bearing positive finding). The chain closes cleanly at v0.9. The v0.7a' positive remains the load-bearing substantive finding; the v0.9 monotone-increasing meta-finding is preserved as a future-work item. Seven sequential pre-registered chapters; seven distinct chapter-close types; one load-bearing positive; one comprehensive methodology surface; program pauses with explicit reopening avenues.