# K_facet Isotrophy Program Pause (2026-05-24) Status: **PAUSED 2026-05-24** at end-of-v0.9, then **ADVANCED off-pause 2026-05-29 -> 2026-06-03** with the v0.10 frontier pair (v0.10a + v0.10b), the v0.11 conditional-rank close, and the v0.12 -> v0.20 external-transfer / reliability mechanism arc, which reached a **replicated and mechanistically explained Tier-2 external PASS at v0.18, explained frame-fragility at v0.19, and bridged the spectral gap directly to AUC at v0.20**: the velocity-fraction signal transfers to an independent catalog once read as a continuous tail-resolved score rather than the coarse v0.11 zone, the per-cell heterogeneity repeats on fresh rows, and a fresh wider-cell reliability chapter confirmed that label-blind frame reliability predicts per-cell AUC (`AUC_cond = 0.620208`, `p = 1e-5`, `rho = 0.5975`, `p_reliability = 0.00523`, reversal guard `0` stable-decisive-negative cells). v0.19 then confirmed the first-principles H1 link from spectral near-degeneracy to frame-fragility (`rho(re_gap, frame_spread) = -0.836`, `p = 1e-5`, falsifier 15/2880), while preserving the procedural `spectral_gap_mechanism_partial` verdict because the direct per-cell median-gap H2 bridge to AUC missed. v0.20 then closed that aggregation gap as a confirmatory re-analysis on the same 18 cells: a pre-registered q10 low-tail gap summary restored the direct gap -> AUC bridge (`tail_gap_bridge_supported_confirmatory`, `rho = 0.882`, `p = 1e-5`) -- the median washed the mechanism out of view, it did not break the bridge. The chain ran from v0.3 (December 2025 / January 2026, pre-anniversary) through v0.9 (May 2026) as seven sequential pre-registered chapters with seven distinct chapter-close types and one load-bearing in-sample positive (v0.7a' chi^2 = 16.43 on signed velocity-fraction quartiles, 250 analyzable rows), then the frontier-advance matured that positive into a **publishable conditional-positive sequence**: - **v0.10a (the positive):** the in-sample monotone velocity-fraction -> stability trend, registered by the phenomenon-aligned J-T one-sided trend test (exact fixed-margin enumeration, p = 7.3e-3 < 0.01) that the blunt chi^2-of-independence had missed. - **v0.10b (the boundary):** the trend does NOT survive as a mass-MARGINAL held-out predictor (pooled leave-one-m_3-bin-out AUC = 0.41 < 0.5), yet it DOES rank within each held-out mass bin (per-fold AUC mean ~0.65; the observed pooled AUC still beats the within-m_3-shuffled null of 0.265 at p = 1e-4). The signal lives at the (m_3, zone) JOINT level. - **v0.11 (the conditional close):** the explicit within-m_3 rank test passes on the locked 229-row primary domain (`AUC_cond = 0.6783`, exact stratified `p = 2.046e-7`; permutation sanity consistent). This registers the conditional rank signal while preserving v0.10b's global held-out null. - **v0.12 (the external-transfer boundary):** the conditional rule does NOT yet transfer to an independent catalog. The nearest target, same-paper supplementary-A, is `external_transfer_blocked_by_attrition`: an unbiased uniform probe (300 rows, 6 shards) reads attrition 0.3433, Wilson95 [0.2919, 0.3987] -- the whole CI above the locked 0.20 block gate. NOT a falsification: the frozen v0.7 D5 measurement is numerically intractable on ~1/3 of supp-A. The ~2.5 h probe spared the projected 6.7-day full run. - **v0.13 (the target search):** a signal-blind search returned an independent-target landscape negative -- only Tier-2 Li/Liao 2021 (135,445 non-hierarchical orbits, same lineage) was schema-viable. v0.13a bounded cross-ansatz leakage at 0.0 but found the raw dominant-direction vf FEATURE frame-relative; v0.13b priced the coarse v0.11 ZONE as frame-stable enough to test (supp-B 4.35%, liao2021 pooled ~1.3%). - **v0.14 (coverage-undecidable):** the frozen coarse-zone rule on a 1280-row sampled liao2021 draw landed `sample_transfer_undecidable_coverage` -- only 7/16 mass cells could host a within-cell stable-vs-unstable comparison (liao2021 is overwhelmingly unstable and zone-2-saturated). - **v0.15 (directional-weak):** an outcome-balanced 80/80 case-control draw over the 7 stable-support cells defeated the coverage wall but landed `stable_support_transfer_directional_weak` (AUC_cond 0.5125, p 3e-4): direction transfers and is significant, but the coarse zone is saturated (98.4% zone-2) so the effect sits below the 0.55 floor -- the binning, not the projection, is the bottleneck. - **v0.16 (the external PASS):** a tail-resolved continuous 4-frame ensemble-median vf score on a fresh doubly-held-out 80/80 draw landed `tail_resolved_transfer_passes_clean` (AUC_cond 0.647, p 1e-5, attrition 0, frame gate clean). On the SAME rows the coarse zone reads 0.510 -- direct proof the binning was the bottleneck. Pooled and mass-cell-heterogeneous (5 of 7 cells in-direction, 2 reversed; leave-one-out 0.593 still clears the floor), Tier-2 and bounded, not theorem-facing. - **v0.17 (the heterogeneity replication):** a fresh triple-holdout 80/80 draw over the same seven cells landed `heterogeneous_transfer_replicates_clean` (AUC_cond 0.646875, p 1e-5, attrition 0, frame gate clean). The v0.16 per-cell AUC rank order reproduced exactly (Spearman rho=1.0, exact p_rho=1/5040); all seven directions relative to 0.5 repeated. Post-hoc anatomy reads the two sub-0.5 cells as the frame-fragile cell (`qA2_qB1`) and a near-null cell (`qA3_qB0`), not a competing physics. Reliability-map interpretation only; the registered claim is replicated heterogeneity. - **v0.18 (the mechanism capstone):** a fresh quadruple-holdout 8 x 8 reliability chapter landed `reliability_drives_per_cell_auc_supported` (18 cells, 2880 rows, 0 attrition). The pooled transfer replicated again (`AUC_cond = 0.620208`, `p = 1e-5`), label-blind frame reliability predicted per-cell AUC (`rho = 0.5975`, `p = 0.00523`), and the reversal guard found 0 stable-decisive-negative cells. The post-hoc v0.17 reliability-map interpretation is now a registered finding. - **v0.19 (the first-principles reliability mechanism):** exact v0.18 re-measure plus identity-frame spectrum dump landed `spectral_gap_mechanism_partial`. H1 is strong and label-blind: small Re-part Floquet gap predicts high frame spread (`rho = -0.836`, `p = 1e-5`), with a clean large-gap/high-spread falsifier (15/2880). H2's median `gap_reliability` -> AUC bridge misses (`rho = 0.063`), so the locked verdict stays partial. Interpretation: reliability is spectral near-degeneracy at the orbit level; median gap is the wrong cell aggregation. - **v0.20 (the tail-aggregated direct bridge):** a confirmatory re-analysis (no new measurement) on the exact v0.18/v0.19 18 cells + AUC. Swapping v0.19's median gap for a pre-registered q10 low-tail summary `log10(q10(re_gap))` landed `tail_gap_bridge_supported_confirmatory`: the direct gap -> AUC bridge returns at full strength (`rho = 0.882`, `p = 1e-5`; leave-one-cell-out 0.863-0.922; coherence guard tail -> frame_reliability `rho = 0.707`). Confirms the v0.19 diagnosis exactly -- the median was the wrong aggregation, not a missing mechanism. Bounded: same cells, not a Tier-2 evidence upgrade. The mature, falsifiable claim is **"yes, but only after conditioning"**: velocity-fraction stratifies three-body stability, but as a function of the mass ratio, not marginally over it. That is a stronger and more honest statement than either a bare in-sample positive or a bare held-out null — the held-out chapter localized the signal rather than killing it. The program remains **paused at the lab-bandwidth level**, not retired. The within-m_3 conditional rung that v0.10b motivated is closed (v0.11); the external-catalog rung is now **CONFIRMED, replicated, and mechanism-explained at Tier-2** -- v0.12's supp-A D5-attrition wall and v0.13's landscape-negative left Li/Liao 2021 as the only viable target, and the v0.14 (undecidable) -> v0.15 (directional-weak) -> v0.16 (clean PASS) -> v0.17 (heterogeneity replication) -> v0.18 (reliability mechanism) -> v0.19 (spectral-gap fragility mechanism) -> v0.20 (tail-gap direct bridge) arc carried it to a held-out external confirmation of the continuous velocity-fraction projection plus a registered explanation of its per-cell heterogeneity, its orbit-level frame fragility, and the direct spectral-gap -> AUC bridge under the failure-matched tail aggregation. Remaining axes: a larger supported-region or multi-feature chapter, or new-mechanism work -- not another rephrasing of the same table. Audience: paper-side writers, future coding agents, isotrophy- program lab initiates, anyone who wants to know what's left. ## What the Isotrophy Program Did Sundog v. Isotrophy tested whether a "sundog theorem" makes non-tautological predictions about Li & Liao 2025's three-body catalog (specifically the supplementary-B piano-trio catalog of 273 m_3-parametrized periodic orbits, half stable and half unstable). The program ran as seven sequential pre-registered chapters testing progressively richer low-dimensional projections of the catalog body: ```text v0.3: Gamma_i rank gate (D_3-equivariant; m_3 = 1 strict G.2). Outcome: domain-of-applicability. 20/21 m_3 = 1 strict-G.2 rows resolve as structural zeros; O_617 is a clean opposite-strict row with a bridge direction outside the valid D3 representation. Daughter catalog (supp-B) is Z_2-or-smaller and OUT of D_3-domain. v0.4: Z_2 shadow on supp-B (tangent-isotypic + orbit-gauge- rigidity). Outcome: structural-negative. Both registered Z_2-shadow predictors fail. gamma_3 retired pre-sweep (F_beta does not preserve K_fib on supp-B). gamma_3'_orbit_pass2 falsified at chi^2 = 1202 (vs critical 26.22). v0.5: 2-bit catalog branch shadow on (m_3 < 1, z_0 < 0.3). Outcome: projection-limit (audit-passes / predictor-fails). v0.5a audit passes chi^2 = 34.99, p = 1.23e-7. v0.5b held-out predictor fails: accuracy delta -0.019, McNemar p = 1.0. Branch hash describes but does not predict. v0.6: Continuous catalog-coordinate (E, |L|) shadow. Outcome: conditional-independence. v0.6a catalog audit passes chi^2 = 33.70 but alignment-tightness 0.956 trips the warning guard. v0.6b within-branch audit (alignment- breaking) fails permutation p = 0.029. Energy quartile is a 1-to-1 label for m_3 sub-bin within the stable branch. v0.7: Floquet velocity-fraction direction shadow. Outcome: integration-attrition + qualified-positive on restricted domain. v0.7a catalog-wide audit attrited at 8.4% (variational integrator step-size wall at high m_3). v0.7a' restricted-domain confirmation on 250 analyzable rows PASSES at chi^2 = 16.43, p = 9.3e-4, alignment 0.698. FIRST non-branch-aligned positive in the chain. v0.8: Unsigned direction-purity = abs(vf - 0.5). Outcome: structural-negative. v0.8a fails at chi^2 = 4.94. Diagnostic confirms the v0.7a' signal is signed (linear- transform invariance of vf preserves the quartile chi^2). Density-asymmetric vf distribution (74% below 0.5) violates the purity transform's symmetry assumption. v0.9: Signed three-zone direction-composition under physical cutpoints {0.25, 0.50}. Outcome: structural-negative on U-shape hypothesis + monotone-increasing meta-finding. v0.9a fails at chi^2 = 7.42 (vs critical 9.21); U-shape pattern check also fails. Under physical zones, the actual pattern is monotone-increasing in vf (positional 11% S, mixed 34% S, velocity-heavy 44% S). The v0.7a' U-shape was a quartile- boundary artifact at vf ~ 0.297. v0.10a: Jonckheere-Terpstra one-sided trend test on the v0.9a three-zone contingency (phenomenon-aligned statistic; the Navier-Stokes "stop using a blunt diagnostic against the wrong shape" lesson applied in-program). Outcome: registered-positive (in-sample). Exact fixed- margin enumeration (1,340-table multivariate-hypergeometric null) one-sided p = 7.304e-3 < 0.01. The monotone trend the chi^2-of-independence (p = 0.0245) missed is registered under the aligned ordered statistic. Direction post-hoc from v0.9a -> in-sample only, never held-out. v0.10b: leave-one-m_3-bin-out held-out predictor (zone-only weighted-PAVA monotone risk score) on the 234 primary rows. Outcome: conditional-positive / mass-marginal null. Pooled held-out AUC = 0.4125 <= 0.5 FAILS the registered gate, but per-fold (mass-conditioned) AUC is mostly > 0.5 (mean ~0.65; within-m_3 permutation p = 1e-4 beats the shuffled null 0.265). The vf -> stability signal is real but lives at the (m_3, zone) JOINT level, not the marginal zone level; a zone-only score cannot use the mass base rate that dominates marginal ranking. Pairs with v0.10a as the publishable "yes, but only after conditioning" result. v0.11: within-m_3 conditional vf rank test on the locked 229-row both-class primary domain. Outcome: conditional-rank positive. AUC_cond = 0.6783, exact stratified p = 2.046e-7, exact probability mass = 1.0, and the within-stratum permutation sanity sidecar is consistent. This registers the conditional signal explicitly, while leaving v0.10b's mass-marginal held-out null intact. v0.12: external transfer of the frozen v0.11 rule to same-paper supplementary-A (10,059 rows; 8,320 candidates after overlap quarantine + strict-m3=1 exclusion; reflection-overlap 0). Outcome: external_transfer_blocked_by_attrition. An unbiased uniform probe (300 rows, 6 shards, seed 20260523) reads attrition_fraction = 0.3433, Wilson95 [0.2919, 0.3987] -- the entire CI above the locked 0.20 block gate, systemic across m3. The frozen v0.7 D5 measurement (DOP853, 1e-12) is numerically intractable on ~1/3 of supp-A. NOT a falsification of the vf hypothesis. The ~2.5 h probe spared the projected 6.7-day run. v0.13: external target-search chapter (pre-feature source selection). Outcome: independent-target landscape NEGATIVE -- every Tier-3 catalog failed a hard gate (equal-mass / restricted / too small); only Tier-2 Li/Liao 2021 (135,445 non-hierarchical orbits, same lineage) was schema-viable. Four-layer anti-target-shopping firewall held. v0.13a: liao2021 adapter + cross-ansatz leakage preflight. Leakage bounded at 0.0 (expansion-only adapter), BUT the raw dominant-direction vf FEATURE is frame-relative (select_gamma_1 argmax flips under rotation). The coarse v0.11 ZONE, not raw vf, is the transferable object. v0.13b: frame-zone stability audit. coarse_zone_rule_frame_stable_enough_to_test (supp-B zone-change 4.35%, liao2021 pooled ~1.3%; fragility localized to two mass bands). Cleared the coarse zone for transfer. v0.14: sampled coarse-zone transfer (1280 rows, 16 mass cells). sample_transfer_undecidable_coverage -- only 7/16 cells hosted both stability classes (liao2021 is overwhelmingly unstable + zone-2-saturated); coverage gate (>=10 cells / >=800 rows) not met. No transfer read licensed. v0.15: stable-support case-control transfer (7 cells x 80 S + 80 U, v0.14-held-out). Coverage wall defeated (7/7 primary, 0 attrition) -> stable_support_transfer_directional_weak (AUC_cond 0.5125, p 3e-4). Direction transfers + significant, but the coarse zone is saturated so the effect is below the 0.55 floor. Diagnosis: the binning, not the projection. v0.16: tail-resolved transfer (4-frame ensemble-median continuous vf, fresh double-holdout 80/80). tail_resolved_transfer_passes_clean: AUC_cond 0.647, p 1e-5, attrition 0, frame-spread gate clean. SAME-rows coarse zone reads 0.510 -> the coarse binning was the bottleneck; the continuous projection transfers. First clean external (Tier-2, held-out) confirmation. Pooled + mass-cell-heterogeneous (5/7 in-direction, 2 reversed; leave-one-out 0.593); Tier-2, bounded, not theorem-facing. v0.17: tail-resolved heterogeneity replication (fresh triple-holdout 80/80). heterogeneous_transfer_replicates_clean: AUC_cond 0.646875, p 1e-5, attrition 0, frame gate clean; exact Spearman rho=1.0, p_rho=1/5040 against the pinned v0.16 per-cell vector. Replicated external transfer plus reproducible seven-cell heterogeneity. Anatomy suggests a reliability map (qA2_qB1 frame-fragile, qA3_qB0 near-null), but that "why" is descriptive, not a registered mechanism. v0.18: reliability-conditioned per-cell AUC chapter (fresh quadruple-holdout 8 x 8 sorted-mass grid, 18 cells, 2880 rows). Verdict: reliability_drives_per_cell_auc_supported. Pooled AUC_cond 0.620208, p 1e-5; label-blind frame reliability predicts per-cell AUC (rho 0.5975, p 0.00523); reversal guard passed with 0 stable-decisive- negative cells. Mechanism capstone for the v0.16/v0.17 heterogeneity. ``` ## The Load-Bearing Findings After seven chapters plus the v0.10-v0.20 frontier-advance, three findings carry the program's substantive output (the second is now a publishable conditional profile with a replicated and mechanism-explained Tier-2 external transfer): 1. **The v0.7a' chi^2 = 16.43 positive**. On the 250-row analyzable supp-B subset (post v0.7a integration-attrition), the velocity- fraction quartile contingency is statistically significant (p = 9.3e-4) AND branch-independent (alignment 0.70 against the v0.5a branch hash). This is the FIRST non-branch-aligned positive in the chain and remains uncontradicted by v0.8 and v0.9 (both of which contextualized it without invalidating it). 2. **The v0.10a + v0.10b + v0.11 conditional profile -- "yes, but only after conditioning."** The v0.9a monotone-increasing meta-finding (S_fraction 11% / 34% / 44% across positional / mixed / velocity- heavy zones under physical cutpoints {0.25, 0.50}) was matured into a registered result and then bounded: - **v0.10a (the positive).** The Jonckheere-Terpstra one-sided trend test — the ordered-alternative statistic the chi^2-of- independence was too blunt to see — registers the trend at exact fixed-margin enumeration p = 7.304e-3 < 0.01 (permutation sanity 7.50e-3; normal-approx 6.73e-3). The signal is real and ordered, in-sample. (Direction post-hoc from v0.9a, disclosed; in-sample only, never a held-out claim.) - **v0.10b (the boundary).** As a mass-MARGINAL held-out predictor the signal FAILS: a zone-only weighted-PAVA monotone risk score under leave-one-m_3-bin-out lands at pooled AUC = 0.4125 < 0.5, missing the registered AUC>0.5 gate. But CONDITIONED on mass it holds — per-fold held-out AUC is mostly > 0.5 (m_3 0.8->0.84, 0.9->0.90, 1.0->0.81, 1.1->0.82; mean ~0.65), and the observed pooled AUC still beats the within-m_3-shuffled null of 0.265 at permutation p = 1e-4. The failure is cross-bin CALIBRATION, not a within-bin signal failure: the vf -> stability relationship lives at the (m_3, zone) JOINT level, and the mass base rate — which a zone-only score cannot use — dominates any marginal ranking. - **v0.11 (the conditional close).** The explicit within-m_3 conditional rank test PASSES on the frozen 229-row primary domain: `AUC_cond = 0.6783`, exact stratified `p = 2.046e-7`, exact-null mass = 1.0, permutation sanity consistent. This is still in-sample and conditional, not an external-catalog predictor and not a v0.10b rescue. Together these are a **publishable conditional profile** with a mature, falsifiable shape: **velocity-fraction stratifies three-body stability, but only after conditioning on the mass ratio — it is a conditional predictor, not a marginal one.** This is a stronger and more honest statement than either a bare in-sample positive or a bare held-out null; the held-out chapter LOCALIZED the signal rather than killing it, and v0.11 registered that localization under the exact conditional rank gate. **The external-transfer arc (v0.12 -> v0.20): from supp-B-internal to a replicated, mechanism-explained clean Tier-2 PASS.** v0.12 took the frozen rule to same-paper supplementary-A and hit `external_transfer_blocked_by_attrition` (uniform probe attrition 0.3433, the whole CI above the 0.20 gate; a measurement-feasibility limit, not a falsification). v0.13's signal-blind search found only Tier-2 Li/Liao 2021 viable; v0.13a bounded leakage at 0.0 but exposed raw vf as frame-relative; v0.13b certified the coarse v0.11 zone as frame-stable enough to test. The transfer then ran in three rungs on liao2021: v0.14 `undecidable_coverage` (only 7/16 cells hostable), v0.15 `directional_weak` (AUC_cond 0.5125, p 3e-4 -- direction transfers but the coarse zone is 98.4% saturated, below the 0.55 floor), and v0.16 `tail_resolved_transfer_passes_clean` (**AUC_cond 0.647, p 1e-5**) once the feature was read as a continuous 4-frame ensemble-median vf rather than the coarse zone. On the SAME held-out rows the coarse zone reads 0.510 -- direct proof the binning, not the projection, was the bottleneck. v0.17 then replicated the transfer on fresh triple-held-out rows (**AUC_cond 0.646875, p 1e-5**) and reproduced the seven-cell heterogeneity exactly by rank (**rho=1.0, exact p_rho=1/5040**). **The conditional positive now has a replicated held-out external (Tier-2) confirmation of the underlying velocity-fraction projection** -- pooled and mass-cell-heterogeneous. v0.18 then promoted the v0.17 reliability-map anatomy into a registered mechanism: fresh 8 x 8 quadruple-holdout sample, **AUC_cond 0.620208, p 1e-5**, frame-reliability **rho=0.5975, p=0.00523**, and 0 stable-decisive-negative cells. v0.19 then explained frame-fragility itself: exact v0.18 re-measure plus spectrum dump gave **rho(re_gap, frame_spread) = -0.836, p = 1e-5**, with a clean large-gap/high-spread falsifier, while the median-gap -> AUC H2 bridge missed (**rho=0.063**). v0.20 then closed that bridge as a confirmatory re-analysis on the same cells: the pre-registered q10 low-tail gap reads **rho=0.882, p=1e-5** (`tail_gap_bridge_supported_confirmatory`), so the median was the wrong aggregation, not a missing mechanism. The coarse v0.11 zone FORM does not transfer (v0.15); its continuous tail-resolved relative does, replicates, and is now mechanistically explained by frame reliability plus spectral near-degeneracy read at the failure-matched tail (v0.16/v0.17/v0.18/v0.19/v0.20). Bounded to Tier-2 / stable-support / not theorem-facing. 3. **The methodology surface**. The chain produced and locked a comprehensive set of pre-registration discipline tools that are reusable across mechanism-search programs: - Closure-relative gate discipline (v0.3 origin). - Two-pass classifier with empirically-calibrated tolerances (v0.4a). - Projection-language vocabulary (body / shadow / projection- limit, from CROSS_SUBSTRATE_NOTES.md). - Audit-then-predictor separation (v0.5). - Asymmetric McNemar + positive-delta falsifier for held-out predictors (v0.5b). - Alignment-tightness guard against prior-chapter feature shadows (v0.6a, with thresholds 0.8 warning / 0.95 severe). - Sparse-cell fallback tree (v0.6b: chi-squared / exact permutation / inconclusive_sparse, with locked permutation seed 20260523 and n_permutations = 10000). - Non-circularity sentence template for Floquet-derived features (v0.7a: "use eigenvectors as geometric directions only"). - Per-row integration-failure fallback with attrition gate at 5% of catalog (v0.7a R2.A). - Append-per-row resume mode for crash resilience (v0.7a R2.C). - Anti-circular framing discipline for sequential audits on the same data (v0.9 parent). - Phenomenon-aligned statistic substitution: replace a blunt omnibus test with the ordered-alternative statistic that matches the discovered shape (v0.10a J-T — the Navier-Stokes lesson applied in-program), using exact fixed-margin enumeration as a deterministic primary p-value with permutation/normal-approx as sanity sidecars. - Pooled-vs-per-fold held-out diagnostic: report per-fold AUC alongside the pooled gate so a cross-bin calibration artifact is exposed as a conditional positive rather than hidden inside a failing marginal statistic — and so the goalposts stay locked (pooled AUC primary) while the per-fold structure is read as a diagnostic, not a post-hoc rescue (v0.10b). ## Concrete Reopening Avenues for Lab Initiates Things that could be done if a lab initiate has bandwidth to "waste compute on." Each is a real research direction that the principal deemed not worth the bandwidth tradeoff against three-body Phase 15+, but which would yield publishable substance if followed up. ### 1. Jonckheere-Terpstra trend test on v0.9a three-zone data (LOW EFFORT, HIGH SUBSTANCE) **Status**: **EXECUTED 2026-05-29 as v0.10a** — `kfacet_v10a_jt_trend_form.md`, `scripts/isotrophy_jt_trend.py`, `results/isotrophy/k-facet-v10a-jt-trend/`. Verdict **`jt_trend_monotone_registered`**: exact fixed-margin enumeration (1,340-table multivariate-hypergeometric null) one-sided **p = 7.304e-3 < 0.01** (permutation sanity 7.50e-3; normal-approx 6.73e-3). The monotone velocity-fraction → stability trend that v0.9a's χ² (p=0.0245) missed is registered under the aligned ordered statistic — **in-sample only** (the direction was post-hoc from v0.9a). The held-out predictor (avenue 2, reserved `v0.9b`) remains the next rung for a predictive claim. **Hypothesis**: the three-zone contingency `S_fraction = 0.105 / 0.339 / 0.439` shows a monotone-increasing trend in vf that would pass at p < 0.01 under a J-T trend test (which has more power than chi-squared for ordered alternatives). **Implementation**: ~30 minutes paper-side + seconds runner. Read `results/isotrophy/k-facet-v09a-signed-vf-three-zone/per_row_table.csv`, apply J-T trend test on zone-ordered S/U using scipy.stats. Pre-register the one-sided test against H_0 = no monotone trend. **Non-circularity**: this breaks condition #2 (test statistic) of the v0.9 parent's anti-circular discipline; J-T is NOT chi- squared of independence. Allowed under the locked framework. **Why deferred**: another chapter's worth of paper-side work after seven chapters had already been written; bandwidth shifted to Phase 15+. ### 2. v0.7a' held-out predictor on the analyzable subset (LOW-MEDIUM EFFORT) **Status**: **EXECUTED 2026-05-30 as v0.10b** — `kfacet_v10b_monotone_vf_heldout_predictor_form.md`, `scripts/isotrophy_vf_heldout_predictor.py`, `results/isotrophy/k-facet-v10b-monotone-vf-heldout/`. Verdict **`monotone_vf_predictor_fails_heldout`** — a clean **projection-limit** result, the audit-vs-predictor pattern for the THIRD time (cf. v0.5). The pre-registered primary gate (pooled leave-one-`m_3`-bin-out held-out AUC of a zone-only weighted-PAVA risk score) lands at **AUC = 0.4125 ≤ 0.5 → FAIL** (gate needs AUC>0.5 AND permutation p≤0.01). The mandated per-fold diagnostic shows the failure is a **cross-bin calibration artifact, not a within-bin signal failure**: per-fold held-out AUCs are mostly >0.5 (m_3 = 0.8→0.84, 0.9→0.90, 1.0→0.81, 1.1→0.82; mean ~0.65), and the within-`m_3` permutation p is tiny (observed 0.41 beats the shuffled-label null of 0.265 → genuine zone structure). What fails is the **pooled** cross-bin ranking — the zone-only score is not comparable across `m_3` bins because the mass-bin base rate (which the predictor cannot use) dominates global ranking. The v0.10a in-sample / within-bin monotone trend is real but does **not** yield a globally-calibrated held-out risk score. Per the locked verdict tree the goalposts stay where locked (pooled AUC primary); the encouraging per-fold AUCs are diagnostic, not a rescue. A within-`m_3` or `m_3`-conditional predictor would be a **fresh** chapter; that fresh chapter is now v0.11, recorded above as the conditional-rank close. The exact-`m_3`=0.4 risk noted below materialized as predicted (its fold AUC 0.671 is strong but does not transfer pooled). Hard-label sidecar (non-gating) confirms the base-rate trap: accuracy delta vs always-U = −0.167, McNemar one-sided p = 0.99999. _Original registration text (pre-execution) follows._ **Status (original)**: parent registration mentions v0.7b as a candidate; NEVER licensed under the v0.7a attrition verdict; could be licensed under v0.7a' restricted-scope PASS retrospectively if re-registered. **Hypothesis**: the v0.7a' chi^2 = 16.43 in-sample positive is held-out-predictive at p <= 0.01 under leave-one-m_3-bin-out on the 250 analyzable rows. **Implementation**: re-register `kfacet_v07b_*` paper-side as a held-out predictor on the restricted scope (v0.7a' domain). Use v0.5b's discipline verbatim (asymmetric McNemar, positive-delta gate, leave-one-m_3-bin-out partition). Per-fold accuracy + pooled McNemar at p <= 0.01. **Risk**: m_3 = 0.4 is the load-bearing fold; held-out it may flip the bucket-majority direction (same failure mode v0.5b exhibited). Lab initiate should run v0.5b-style diagnostic per fold. **Non-circularity**: domain is the v0.7a' subset (already used in v0.7a' for in-sample chi-squared); held-out partition breaks the in-sample-vs-held-out condition cleanly. ### 3. v0.7a relaxed-precision re-run (MEDIUM-HIGH EFFORT) **Status**: never registered. The v0.7a R1 amendment locked symplecticity gate at 1e-4 (relaxed from 1e-6) on the variational precision 1e-12 substrate; the integration attrition came from DOP853 step-adapter failures, not sanity-gate failures. **Hypothesis**: at variational precision 1e-10 or 1e-9 (matching v0.4a Pass 2), the DOP853 step adapter clears the 23 attrited rows, allowing a catalog-wide v0.7a verdict. **Implementation**: register a v0.7c paper-side at relaxed precision. Re-run the full audit (~3-5 hours expected). The restricted-domain v0.7a' verdict would be either confirmed (the attrited rows match the analyzable subset's pattern) or contested (the attrited rows have systematically different vf values). **Risk**: precision change affects every per-row vf value; the catalog-wide chi-squared at the new precision is its own audit and cannot be compared to v0.7a' directly without explicit precision disclosure on every receipt. **Non-circularity**: catalog-wide claim under the new precision is a fresh audit, not a re-test of v0.7a'. ### 4. m_3 = 0.4 sub-catalog targeted mechanism investigation (MEDIUM EFFORT, HIGH SUBSTANCE) **Status**: never registered. The m_3 = 0.4 cluster has been the load-bearing positive throughout the chain (v0.7a Q1 49% S; v0.5a (m_3<1, z_0<0.3) branch 56% S). **Hypothesis**: the m_3 = 0.4 sub-catalog (55 rows / 35 S / 20 U) has a substrate-specific mechanism that the catalog-wide audits average out. A within-m_3-bin audit on this single mass ratio would reveal what makes the bin special. **Implementation**: register a v0.10 chapter on the m_3 = 0.4 sub-catalog. Use any of the prior mechanism families (vf, |L|, joint, gamma_1 direction) but restrict the domain to 55 rows. Apply v0.6b-style within-stratum discipline. **Risk**: low N (55 rows) limits chi-squared power; sparse-cell fallback will likely fire frequently. The substantive return is high if the mechanism is clean; the statistical return is power-limited. **Non-circularity**: the m_3 = 0.4 sub-catalog is a domain restriction; conditioning on m_3 = 0.4 uses catalog parameter only, not the stability label. ### 5. Joint (vf, Q_E) audit on the 250 analyzable rows (MEDIUM EFFORT) **Status**: never registered. v0.6a (E quartile, chi^2 = 33.70) and v0.7a' (vf quartile, chi^2 = 16.43) both passed in-sample; the relationship between the two has not been characterized. **Hypothesis**: vf and Q_E carry overlapping but partially independent stability information. A joint 4x4x2 contingency (Q_vf x Q_E x S/U) would reveal whether one is a proxy for the other or whether they explain different variance in S/U. **Implementation**: register a v0.10 chapter on joint (vf, Q_E) stratification on the 250 analyzable rows. Compute the 4x4x2 contingency; run a stratified chi-squared (Cochran-Mantel- Haenszel) to test conditional independence of Q_vf and S/U given Q_E. **Risk**: joint contingency is high-df; many cells will be sparse; sparse-cell fallback fires. **Non-circularity**: both Q_vf and Q_E are inherited from prior chapters; the joint test is a NEW hypothesis (conditional independence) that v0.6a and v0.7a' did not pre-register. ### 6. Cross-substrate transfer test (HIGH EFFORT) **Status**: cross-substrate notes exist at `docs/CROSS_SUBSTRATE_NOTES.md` sections 6-7. **Hypothesis**: the v0.7a' velocity-fraction shadow is substrate- specific to supp-B; applying the same mechanism to a different three-body sub-catalog (e.g., free-fall, choreography, or mesa analog) would test whether the signal is supp-B-specific or generalizes. **Implementation**: identify a different three-body sub-catalog with comparable structure (stability labels + periodic-orbit ICs). Apply the v0.7a operational definition + the v0.9a three-zone audit. Compare verdicts. **Risk**: identifying a comparable catalog is non-trivial; re-implementation of variational integration substrate is real work. High effort, but high publishable substance if it works. **Non-circularity**: cross-substrate transfer is by design a fresh test on new data. ### 7. Action-angle / KAM-style decomposition (HIGH EFFORT, THEORETICAL) **Status**: mentioned as a candidate v0.7 mechanism family before v0.7 picked gamma_1 direction-of-instability; never registered. **Hypothesis**: the supp-B orbits have action-angle representations that stratify stability via integrability proximity. KAM theory says nearly-integrable orbits are stable; computing action-angle variables per row would test this directly. **Implementation**: heavy theoretical machinery. Per-row action-angle decomposition requires extended variational integration over multiple periods (not just one) plus diagonalization in canonical coordinates. Compute cost: weeks. **Non-circularity**: action-angle variables are new geometric quantities, not derived from prior-chapter features. ### 8. Full-catalog audit at variational precision 1e-10 with v0.7a R2.A discipline **Status**: candidate alternative to avenue 3. If attrition at 1e-10 is below 5%, the catalog-wide v0.7a verdict could be recovered without re-registering precision. **Implementation**: re-run `scripts/v07a_velocity_fraction_audit.py` with rtol = atol = 1e-10 (modify the locked constants paper-side first). Compare attrition rates and pattern stability against the 1e-12 result. **Risk**: precision change is a re-registration; needs paper-side discipline. ## Reusable Methodology Surface (For Programs Beyond Isotrophy) The discipline tools listed under "load-bearing findings" item 3 are not isotrophy-specific. They apply to any mechanism-search program where: - Multiple candidate mechanisms compete on the same data, - Sequential audits risk silent re-testing under different labels, - Catalog-coordinate vs orbit-dynamics axes need to be distinguished for non-circularity, - Integration-precision walls may limit catalog-wide claims, - Held-out generalization is the publishable predictor gate. Future mesa, geometry, or three-body Phase 15+ work that follows the same pattern is invited to inherit these disciplines as appropriate. ## Bandwidth Redirect Per the principal's pre-v0.7 direction (which the v0.7a' surprise positive temporarily overrode), bandwidth redirects to: - **three-body Phase 15+** (survival pocket investigation, the current high-ROI path). - **Coarse-graining proof trunk** (P0/P1 priority per `docs/TODO.md`). - **Public surface updates** including `threebody.html` (flagged as seriously stale during v0.7 close). The isotrophy section's pause does NOT release the program's discipline; any future re-opening must respect the locked methodology surface AND the anti-circular framing discipline for sequential audits on the v0.7a' analyzable subset. ## Doc Trail - `kfacet_external_transfer_capstone.md` -- consolidated v0.11-v0.20 external-transfer capstone and claim boundary. - `kfacet_v03h_writeup.md` -- v0.3 chapter close. - `kfacet_v04_writeup.md` -- v0.4 chapter close. - `kfacet_v05_writeup.md` -- v0.5 chapter close. - `kfacet_v06_writeup.md` -- v0.6 chapter close. - `kfacet_v07_writeup.md` -- v0.7 chapter close (load-bearing qualified-positive). - `kfacet_v08_writeup.md` -- v0.8 chapter close. - `kfacet_v09_writeup.md` -- v0.9 chapter close (this chapter's companion). - `kfacet_v10a_jt_trend_form.md` -- v0.10a ordered-trend positive. - `kfacet_v10b_monotone_vf_heldout_predictor_form.md` -- v0.10b mass-marginal held-out null / conditional diagnostic. - `kfacet_v11_m3_conditional_vf_rank_form.md` -- v0.11 within-m3 conditional rank pass. - `kfacet_v12_external_frozen_transfer_form.md` -- v0.12 external transfer blocked by frozen-measurement attrition on supp-A. - `kfacet_v13_external_target_search_form.md` -- v0.13 signal-blind external target search. - `kfacet_v13a_liao2021_adapter_leakage_preflight_form.md` -- v0.13a liao2021 adapter/leakage preflight. - `kfacet_v13b_frame_zone_stability_audit_form.md` -- v0.13b frame-zone stability audit. - `kfacet_v14_liao2021_sampled_transfer_form.md` -- v0.14 sampled transfer coverage wall. - `kfacet_v15_liao2021_stable_support_transfer_form.md` -- v0.15 stable-support directional-weak transfer. - `kfacet_v16_liao2021_tail_resolved_transfer_form.md` -- v0.16 clean Tier-2 tail-resolved transfer pass. - `kfacet_v17_liao2021_heterogeneity_scope.md` -- v0.17 replicated heterogeneity. - `kfacet_v18_liao2021_reliability_auc_form.md` -- v0.18 wider-cell reliability chapter; frame-reliability map positive. - `kfacet_v19_spectral_gap_reliability_form.md` -- v0.19 first-principles reliability mechanism; H1 spectral-gap -> frame-fragility positive, H2 median gap -> AUC bridge not confirmed. - `kfacet_v20_tail_gap_auc_bridge_form.md` -- v0.20 confirmatory tail-gap bridge; q10 low-tail gap -> AUC `rho = 0.882` (median 0.063 on the same cells). - `docs/CROSS_SUBSTRATE_NOTES.md` -- cross-substrate framing (avenue 6). - `docs/isotrophy/sundog_v_isotrophy.md` -- full chapter-by-chapter dated log. --- Isotrophy program paused at end-of-v0.9, then reopened through the v0.10-v0.20 conditional/external-transfer frontier. The live result is now sharper: velocity-fraction is a real conditioned stability rank signal, externally confirmed, replicated, and mechanism-explained at Tier-2 when read as a continuous tail-resolved score. v0.18 confirmed the reproduced per-cell heterogeneity as a frame-reliability map under a registered label-blind test. v0.19 then explained frame-fragility itself as spectral near-degeneracy (`rho(re_gap, frame_spread) = -0.836`, `p = 1e-5`), while preserving the procedural `partial` label because the direct per-cell median-gap H2 bridge missed. v0.20 then closed that aggregation gap (confirmatory, same cells): a pre-registered q10 low-tail gap restored the direct gap -> AUC bridge (`rho = 0.882`, `p = 1e-5`, `tail_gap_bridge_supported_confirmatory`). The mechanism is strong at the orbit/reliability level and bridges directly to AUC at the failure-matched tail aggregation; the median simply washed it out of view. Reopening is invited; mechanism-laundering is not.