# v0.10a Jonckheere–Terpstra Trend Test Form Lock Status: **VERDICT LANDED 2026-05-29: `jt_trend_monotone_registered`** — exact fixed-margin enumeration one-sided **p = 7.304e-3 < 0.01**. Locked 2026-05-29; the adjustment pass made the primary p-value a deterministic **exact fixed-margin enumeration** (the 1,340-table multivariate-hypergeometric null over S-allocations across zone sizes 19/165/66 at fixed total S=87), demoted the 10,000-shuffle permutation to a **secondary sanity check**, and strengthened the frozen-input gate to assert **exact** zone/S/U counts and recompute each row's zone from `velocity_fraction` under {0.25, 0.50}. No v0.10a runner has been written and no v0.10a command has been run at lock time. This document locks the test, statistic, exact-enumeration primary p-value, secondary permutation sanity check, significance floor, one-sided direction, non-circularity argument, and verdict tree **before** the J statistic is computed. **Naming.** This is **v0.10a**, the program's canonical post-pause reopening number (the v0.9 program-pause doc registers reopening avenues 4/5 as "a v0.10 chapter"). It executes **candidate D** of the v0.9 parent registration (Jonckheere–Terpstra), which v0.9a held back. The name `v0.9b` is left **reserved** for the held-out predictor (avenue 2), which v0.9a's fail did not license; that is the planned follow-on, not this chapter. ## Verdict (Landed 2026-05-29) Receipt: `results/isotrophy/k-facet-v10a-jt-trend/manifest.json`. ```text verdict: jt_trend_monotone_registered frozen-input check: PASSED (zones N=19/165/66, S=2/56/29, U=17/109/37 = 250/87/163; zones recomputed from velocity_fraction under {0.25,0.50} match) J_observed: 8760.5 exact fixed-margin enumeration (PRIMARY, binding): feasible tables: 1340 probability mass: 1.0000000000 (enumeration self-check clean) one-sided p: 7.304e-3 < alpha = 0.01 -> PASS permutation sanity (secondary, seed 20260523, n=10000): one-sided p: 7.499e-3 (agrees with exact) normal-approx (context, moments from the exact null): z = 2.471, one-sided p = 6.731e-3 ``` **Reading.** The right statistic finds the signal the blunt one missed: v0.9a's χ²-of-independence gave p=0.0245 (missed the 0.01 floor); the ordered J-T trend test gives **p=0.0073** (clears it). The monotone-increasing velocity-fraction → stability relationship (S-fraction 0.105 → 0.339 → 0.439) is now a **registered ordered trend**. **Bound (per the Integrity Caveat).** This is an **in-sample** registration of a post-hoc-discovered ordered direction — NOT held-out prediction, NOT an independent confirmation, NOT a theorem-facing claim, NOT a change to the v0.3h K_facet structural-null. The held-out predictor (the reserved `v0.9b`) remains the only path to a predictive claim. ## Frame The v0.4–v0.9 isotrophy chain produced one load-bearing positive: **v0.7a′ velocity-fraction → stability** (χ²=16.43, p=9.3e-4, branch-independent, on the 250-row analyzable supp-B subset). v0.9a then tested a **U-shape** hypothesis on three physical zones and **failed at χ²=7.42** (vs critical 9.21, p=0.0245) — but discovered the real pattern is **monotone-increasing** in velocity-fraction: ```text zone N S_fraction positional-dominant (vf<0.25) 19 0.1053 mixed (0.25 ≤ vf < 0.50) 165 0.3394 velocity-heavy (vf ≥ 0.50) 66 0.4394 ``` χ²-of-independence is a **blunt tool against an ordered claim**: it spends df on arbitrary departures from independence and ignores the ordering of the zones. A one-sided **Jonckheere–Terpstra trend test** is the statistic that matches the hypothesis "S-fraction increases monotonically with velocity-fraction." This is the phenomenon-aligned-test discipline (cf. the Navier-Stokes lesson: a fixed-threshold diagnostic went vacuous until replaced by a phenomenon-aligned objective). ## Integrity Caveat (Load-Bearing — Disclosed Pre-Lock) The **monotone direction was discovered in v0.9a**, not pre-registered before the data. Candidate D was originally framed in the v0.9 parent under the v0.7a′ **U-shape** hypothesis ("pass if J-T *fails* to find monotonicity"); the v0.9a discovery flipped the expectation. Therefore: > v0.10a is an **in-sample, post-hoc** trend test of a data-derived ordered > hypothesis on the same 250 rows. A PASS registers an **in-sample ordered trend** > under the correct statistic; it is **NOT** held-out prediction and **NOT** an > independent confirmation. The held-out predictor (avenue 2, the reserved `v0.9b`) > remains the only path to a predictive claim. This is allowed under the v0.9 parent's anti-circular rule because it breaks the **test statistic** (J-T ≠ χ²-of-independence). The one-sided direction (increasing) is post-hoc and is disclosed here; it is locked **before** the v0.10a runner reads the data. ## What v0.10a Is A trend test on the **frozen v0.9a receipt** — no new variational compute, no re-derivation of velocity-fraction. ```text Frozen input: results/isotrophy/k-facet-v09a-signed-vf-three-zone/per_row_table.csv 250 analyzable rows, columns: zone, stability (S/U), velocity_fraction. Inherited verbatim from v0.9a (which inherited vf + the 250-row analyzable subset from v0.7a). No re-binning: the three ordered zones are consumed as written. Ordered groups: positional-dominant (0) < mixed (1) < velocity-heavy (2). Ordering is the locked vf-zone ordering, NOT data-driven. Response: stability coded S = 1, U = 0 (S = "more stable"). Hypotheses: H0: no increasing monotone trend in S across the ordered zones. H1 (one-sided): S-fraction increases across positional → mixed → velocity-heavy. Statistic: Jonckheere–Terpstra J = sum over ordered zone pairs (u < v) of U_{uv}, where U_{uv} = #{(i in u, j in v): y_j > y_i} + 0.5 * #{(i in u, j in v): y_j = y_i}. With binary y this is U_{uv} = (#U in u)(#S in v) + 0.5 * [(#S in u)(#S in v) + (#U in u)(#U in v)]. Larger J = stronger increasing trend. p-value (primary): EXACT ENUMERATION of the finite fixed-margin permutation null. Enumerate every feasible allocation of S counts (s0, s1, s2) across the frozen zone sizes (19, 165, 66), preserving total S = 87 and U = 163. Weight each table by its multivariate- hypergeometric probability under random assignment of fixed S/U labels to fixed zone sizes: P(s0,s1,s2) = C(19,s0) C(165,s1) C(66,s2) / C(250,87) with ui = ni - si. Recompute J for each table and report the one-sided upper tail: p_exact = sum_{J_table >= J_obs} P(table) This is deterministic, seedless, assumption-free under the fixed-margin null, and cheap here because only 1,340 feasible tables exist. p-value (secondary): SEEDED MONTE CARLO PERMUTATION SANITY CHECK. Shuffle the zone labels across the 250 rows (preserving the per-zone counts and S/U multiset), recompute J, and take the one-sided upper tail p_mc = (1 + #{J_perm >= J_obs}) / (1 + n_permutations). Locked for reproducibility only: seed = 20260523, n_permutations = 10000 (the program's v0.6b-locked permutation parameters). This is NOT binding unless the exact-enumeration implementation fails its probability-mass sanity check. p-value (context): Normal approximation z = (J - E[J]) / sqrt(Var[J]) with the tie-corrected J-T null moments, reported for context only. Given the binary response, the exact-enumeration p is the binding verdict; the normal-approx p is disclosed as approximate. Significance: alpha = 0.01, ONE-SIDED (increasing). The program's locked floor. Frozen-input cross-check: Before computing J, the runner recomputes the per-zone counts and S/U counts and ASSERTS exact count matches against the v0.9a manifest: positional-dominant: N=19, S=2, U=17 mixed: N=165, S=56, U=109 velocity-heavy: N=66, S=29, U=37 total: N=250, S=87, U=163 The runner recomputes S-fractions from those exact counts. It also recomputes each row's zone from `velocity_fraction` using the frozen cutpoints {0.25, 0.50} and ASSERTS it matches the stored `zone` column. A mismatch aborts the run (the input is not the frozen v0.9a receipt). ``` ## Verdict Tree ```text if exact per-zone counts / S-U counts != v0.9a manifest: ABORT (frozen-input cross-check failed; not the v0.9a receipt) elif recomputed zones from velocity_fraction != stored zone column: ABORT (frozen-input cross-check failed; stale or inconsistent zone labels) elif exact-enumeration p < 0.01 (one-sided increasing): verdict = jt_trend_monotone_registered -> the velocity-fraction -> stability trend is a registered monotone (increasing) ordered trend under the aligned statistic (IN-SAMPLE; not held-out). else: verdict = jt_trend_not_significant -> the ordered trend does not clear the p=0.01 floor even under the phenomenon-aligned test; the v0.9a meta-finding does not promote. ``` ## Non-Circularity Argument The v0.9 parent requires any reopening to break at least one of (feature derivation, test statistic, evaluation domain) relative to v0.7a′'s χ²-of-independence on vf quartiles. ```text Feature derivation: inherited (vf zones from v0.9a). NOT broken. Evaluation domain: inherited (same 250 analyzable rows). NOT broken. Test statistic: BROKEN. Jonckheere-Terpstra is a one-sided ordered-trend test, NOT chi^2 of independence. It uses the zone ORDERING (which chi^2 discards) and tests a directional alternative. ``` Exactly one condition broken (statistic), which the locked rule permits. The **direction** of the one-sided alternative is post-hoc from v0.9a and is disclosed in the Integrity Caveat; the verdict is explicitly an in-sample trend registration, never an independent or held-out positive. The locked Floquet non-circularity sentence carries forward verbatim: > v0.10a 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. ## Free Parameter Count ```text Number of free parameters fitted to the data: 0 ``` The statistic (J-T), the zone ordering (positional < mixed < velocity-heavy), the binary coding (S=1/U=0), the exact fixed-margin enumeration, the secondary Monte Carlo permutation parameters (seed 20260523, n=10000), the significance floor (0.01), the one-sidedness (increasing), and the verdict tree are all locked before the runner reads the data. The one-sided direction is post-hoc-from-v0.9a but is disclosed and fixed before this runner executes. ## Claim Boundary ```text A PASS (jt_trend_monotone_registered): - registers an IN-SAMPLE monotone (increasing) ordered trend of velocity-fraction -> stability on the 250-row supp-B subset, under the phenomenon-aligned statistic that v0.9a's chi^2 lacked the power to detect. - does NOT claim held-out prediction (the reserved v0.9b held-out predictor is the next rung). - does NOT make isotrophy theorem-facing or restore a "sundog theorem predicts the catalog" claim. - does NOT change the v0.3h K_facet structural-null (this is the supp-B Z_2-projection thread; the m_3=1 D_3 K_facet gate is untouched). - does NOT touch three-body, Phase 15+, or any other substrate. A FAIL (jt_trend_not_significant): - the v0.9a monotone meta-finding does not promote even under the correct test; the chapter closes the trend thread honestly. ``` ## Receipt Schema (Planned) ```text results/isotrophy/k-facet-v10a-jt-trend/ manifest.json - mode = "v0.10a-jt-trend" - form_lock = "docs/isotrophy/kfacet/kfacet_v10a_jt_trend_form.md" - input_v09a_per_row_table - frozen_input_crosscheck: { per_zone_N, per_zone_S, per_zone_U, per_zone_S_fraction, zones_recomputed_from_velocity_fraction: bool, matches_v09a_manifest: bool } - zone_order: [positional-dominant, mixed, velocity-heavy] - response_coding: { S: 1, U: 0 } - J_observed - exact_enumeration: { feasible_table_count, probability_mass_total, J_null_mean, J_null_p95, p_value_one_sided } - permutation_sanity: { seed: 20260523, n: 10000, J_null_mean, J_null_p95, p_value_one_sided } - normal_approx: { E_J, Var_J_tie_corrected, z, p_value_one_sided } (context) - alpha: 0.01 - verdict: jt_trend_monotone_registered | jt_trend_not_significant jt_trend_receipt.json (same content, flat) ``` ## Lock-In Statement This form is committed before the J statistic, the exact null, or the normal-approximation moments are computed. Any change to the statistic, zone ordering, response coding, exact-enumeration rule, secondary permutation parameters, significance floor, one-sidedness, frozen-input cross-check, or verdict tree after the v0.10a runner is executed is a re-registration, not a refinement. Implementation may proceed against `results/isotrophy/k-facet-v09a-signed-vf-three-zone/per_row_table.csv` as the sole input. No new variational compute is required. Expected runtime: seconds. ## Doc Trail - `kfacet_v09_mechanism_preregistration.md` — parent registration; candidate D (J-T) source; anti-circular discipline. - `kfacet_v09a_signed_vf_three_zone_form.md` — v0.9a (χ² three-zone, failed; monotone meta-finding); the frozen input for v0.10a; reserved `v0.9b` for the held-out predictor. - `kfacet_v07a_prime_restricted_scope_form.md` — v0.7a′ PASS (χ²=16.43); the load-bearing positive whose shape this trend test characterizes. - `kfacet_isotrophy_program_pause.md` — reopening avenues 1 (this) and 2 (held-out predictor); v0.10 reopening-chapter convention. --- v0.10a: one-sided Jonckheere–Terpstra trend test on the frozen v0.9a 250-row receipt. Breaks the test statistic (ordered-trend, not χ²-of-independence); direction post-hoc-from-v0.9a and disclosed. Deterministic exact fixed-margin enumeration primary; seeded 10,000-shuffle permutation sanity check and normal-approx secondary; α=0.01 one-sided. PASS = registered in-sample monotone trend (not held-out); held-out predictor (`v0.9b`) is the next rung. Zero free parameters fitted to the data.