# PDE C1 Regime Generality v1 — Portable Objective > **Pre-registration and result**, filed 2026-05-29. Successor to > [`PDE_C1_REGIME_GENERALITY_v0.md`](PDE_C1_REGIME_GENERALITY_v0.md), > whose G=300 probe returned `PDE-C1-RG-DEFERRED_VACUITY`: the > kNN/twin-state machinery stayed usable, but the *registered overshoot > proxy went near-vacuous* (`damp_fraction ≈ 0.004`) under > heavy-tailed/intermittent energy statistics. The regime-generality > question was therefore **not adjudicated** — the failure was objective > construction, not fiber-locality coverage or support-level > non-injectivity. This artifact replaces the fixed-percentile objective > with a **regime-portable** one and re-poses the Grashof-axis generality > test cleanly. > > **Status: executed.** §12 records `PDE-C1-RG-POS`: the portable > objective passes its portability gate at G=200 and G=300, the control > half is POSITIVE at both regimes, and the G=300 twin-state companion > is CERTIFIED. C1 remains unpromoted because proxy faithfulness and > external PDE review remain open. ## 1. The fix in one line Replace "exceed the **burn-in** 95th-percentile `E_K` in lookahead" with "be in the top `(1−q)` fraction of **own-regime** τ-excursions," calibrated on a **held-out post-burn-in** window. This pins `damp_fraction = 1−q` by construction at every regime, removing the v0 vacuity failure mode. ## 2. Why the v0 objective was not portable `E_max` was the 95th percentile of the (last-25%) burn-in `E_K`. Two coupled defects under intermittency: - **Calibration/sampling mismatch.** A single burst in the burn-in calibration window shifts the 95th percentile far out; the post-burn-in sampling window is comparatively quiescent → almost no look-ahead window reaches `E_max`. - **Heavy-tailed shape-dependence.** At G=200 the energy distribution is tight, so the 95th percentile sits near the bulk (damp_fraction 0.30); at G=300 it is intermittent, so the 95th percentile sits on a rare-burst tail (damp_fraction 0.004). The *same* percentile rule means different things at different regimes. Confirmed regime property, not artifact: v2 (G=300, full-burnin E_max) also gave sub-1% damp. The objective, not the machinery, fails to transfer. ## 3. Portable objective (pinned) For a state `u` with no-op τ-lookahead trajectory, define the **lookahead-max excursion** ```text M(u) = max_{t in [0, tau]} E_K(u(t)), tau = 5.0 time units. ``` Calibrate the threshold on a held-out post-burn-in calibration window `C` (disjoint from the adjudication sample `A`, see §4): ```text E_max = quantile_q( { M(u) : u in C } ), q = 0.70. ``` The proxy selector, applied to the adjudication sample `A`: ```text pi_hat(u) = damp_low_band iff M(u) > E_max = no_op otherwise, for u in A. ``` By construction `damp_fraction` on `C` is exactly `1 − q = 0.30`; on the disjoint stationary `A` it is `≈ 0.30` (the **portability gate**, §6, verifies this at both regimes). `q = 0.70` is chosen so the new objective's damp scale matches v5's emergent `0.30`, making the G=200 re-run a clean comparison to the existing witness. Everything else is inherited from v5/v6 unchanged: `k_f = 2`, `K = 3`, `d_K = 18`, `dt = 0.01`, `tau = 500` steps, `epsilon_K = 0.05·sqrt(2 E_max)` (now with the portable `E_max`), kNN sweep `k ∈ {10,15,20,25,30,40,50}` with `a_mm` thresholds (≤0.005 POSITIVE, ≥0.015 NEG-A), twin-state `k_twin=50`, `delta_H = max(1e-6, 0.05·median‖Q_K‖)`, gates `0.01` / `100`. ## 4. Held-out split (pinned) The post-burn-in trajectory is partitioned into two disjoint blocks with a decorrelation gap: ```text calibration window C : 50,000 samples at interval 50 steps decorrelation gap : 5,000 steps (~a few Lyapunov times) adjudication sample A: 50,000 samples at interval 50 steps ``` `E_max` is computed from `C` only; labelling, kNN convergence, and twin-state run on `A` only. `A` is held at 50,000 to match v5's adjudication N for apples-to-apples comparison. Total post-burn-in integration roughly doubles vs v5 (~40 min/run expected; see §7 cost). Rationale for held-out rather than in-sample: although `E_max` is a single global scalar (so in-sample calibration would not obviously leak *fiber* structure into the local kNN test), held-out removes all doubt at low cost and matches the agreed design. *Cost-reduced alternative (not pinned; a sign-off option):* `C = A = 30,000` with the same gap, ~1.2× v5 cost. Decide at sign-off. ## 5. Harness objective-mode A new objective mode, selected by a flag, leaves the v0–v6 fixed- percentile path untouched: ```text --objective {overshoot-burnin (default), portable-quantile} ``` `portable-quantile` config additions (pinned in the preset, pre- registered): `objective_quantile q = 0.70`, `calibration_sample_count = 50000`, `calibration_gap_steps = 5000`. New presets `lock_v7_g200` (`k_f=2, G=200, K=3`) and `lock_v7_g300` (`k_f=2, G=300, K=3`), both with `e_max_burnin_fraction` irrelevant under this objective (E_max no longer from burn-in) and `objective = portable-quantile`. Implemented by extending `run_cell` to integrate the extra calibration block + gap before the adjudication block; compute `M(u)` for calibration samples; set `E_max = quantile(M_C, q)`; then label/adjudicate the adjudication block exactly as today. All adjudicators (`knn`, `knn-sweep`, `twin-state`) consume the adjudication block unchanged. Smoke parity and overshoot-burnin regression were checked before the lock runs. ## 6. Portability gate (new, pre-registered) **Before** interpreting any control-sufficiency verdict, confirm the objective is actually portable: ```text damp_fraction(A) in [0.20, 0.40] at BOTH G=200 and G=300. ``` - If both pass → the objective is portable; proceed to read the kNN verdicts as a genuine regime comparison. - If either fails → **`PDE-C1-RG-PORTABILITY-FAIL`**: even the held-out-quantile objective does not transfer (e.g. severe non-stationarity between `C` and `A`). This is itself a finding; do **not** retune `q` or the split to rescue it (that would be `PDE-C1-RG-NEG-B`). A new construction (burst-onset, §9) would be a separate v2 pre-registration. `[0.20, 0.40]` is `0.30 ± 0.10`; by construction `A` should sit very near `0.30`, so this gate mainly catches stationarity/leakage failures. ## 7. Program and run order (pinned) ```text 1. lock_v7_g200 --adjudicator knn-sweep (positive control + de-confound) 2. lock_v7_g300 --adjudicator knn-sweep (the generality test) [portability gate checked on both before interpreting either] 3. IF both return STRICTNESS_WITNESS_POSITIVE: lock_v7_g300 --adjudicator twin-state (support companion) (lock_v7_g200 twin-state optional, for a complete re-witness) ``` The **G=200 re-run is mandatory and runs first**: the portable objective is a *new* objective, so the v5 witness must be re-established under it before G=300 generality means anything. If G=200 flips (does **not** return POSITIVE under the portable objective), that is a critical finding — it would mean the v5 POSITIVE was partly an artifact of the old fixed-percentile threshold — and the generality program pauses to reconcile before G=300 is interpreted. Expected wall-clock ~40 min per kNN-sweep run (≈2× v5 from the doubled post-burn-in integration), ~35 min per twin-state run. Up to ~4 runs. None inline under the ~10-minute rule. ## 8. Branches and interpretation Per-regime kNN verdict feeds the existing [`PDE_C1_REGIME_GENERALITY_v0.md`](PDE_C1_REGIME_GENERALITY_v0.md) §6 branch family, gated by §6 portability here: | G=200 kNN | G=300 kNN | program outcome | |---|---|---| | POSITIVE | POSITIVE | `PDE-C1-RG-POS` (control-suff replicates across G under a portable objective); proceed to twin-state | | POSITIVE | NEG-A | `PDE-C1-RG-NEG-A` (regime-2 is a localized window on the Grashof axis — the informative failure) | | POSITIVE | deferral | `PDE-C1-RG-INCONCLUSIVE_CONTROL` | | not POSITIVE | — | **program pauses**: portable objective overturns the v5 witness; reconcile before any generality claim | `PDE-C1-RG-POS` here means: the v5 Reading-2 regime-2 witness has a higher-Grashof replication **under an objective that is portable by construction** — strictly stronger than a same-objective coincidence. It still does **not** mean generality across all `G`, across `k_f`, an infinite-dimensional NSE theorem, `J`-optimality, or promotion without external review. ## 9. Pre-registration discipline - All values in §3–§6 are fixed before any verdict-bearing run. - Post-hoc change to `q`, the split, the gap, `epsilon_K`, the kNN/ twin-state thresholds, or the regime after reading a receipt → `PDE-C1-RG-NEG-B`. - The portability gate is a **precondition**, not a tunable: a `PORTABILITY-FAIL` is filed, not rescued. - Burst-onset (a `dE_K/dt` or relative-jump onset criterion) is the **documented v2 alternative**, not part of this artifact; it carries more free parameters and would need its own pre-registration. ## 10. Build Decisions Closed The sign-off choices landed as: 1. **Split sizes:** 50k calibration / 50k adjudication, preserving apples-to-apples comparison with v5. 2. **Quantile:** `q = 0.70`, targeting `damp_fraction ≈ 0.30` to match the v5 action scale. 3. **Build:** `--objective portable-quantile` plus `lock_v7_g200` / `lock_v7_g300` presets, with regression and smoke checks before the verdict-bearing runs. ## 12. Result (2026-05-29) — PDE-C1-RG-POS The full program executed; all three verdict-bearing runs landed. The portable objective fixed the v6 vacuity and the complete regime-2 witness **replicates at G=300**. | run | portability gate (adj damp ∈ [0.20,0.40]) | verdict | |---|---|---| | `lock_v7_g200` kNN (positive control) | 0.3003 ✓ (calib 0.300) | `STRICTNESS_WITNESS_POSITIVE`, `a_mm = −0.00079`, slope 0.736 | | `lock_v7_g300` kNN (generality test) | 0.2688 ✓ (calib 0.300) | `STRICTNESS_WITNESS_POSITIVE`, `a_mm = +0.00058`, slope 0.564 | | `lock_v7_g300` twin-state | — | `TWIN_STATE_CERTIFIED` (100% witness coverage, 942,834 unique pairs, `δ_H = 0.0111` from real median ‖Q_K‖ = 0.222) | Receipts: `results/proof/c1-rg-v1-g200-knn-sweep/`, `results/proof/c1-rg-v1-g300-knn-sweep/`, `results/proof/c1-rg-v1-g300-twin-state/`. ### What it establishes - **Portability gate passed at both regimes.** Held-out `damp_fraction` = 0.300 (G=200) and 0.269 (G=300), both in [0.20,0.40]. The portable objective is genuinely regime-portable: v6's `damp_fraction = 0.004` vacuity at G=300 is gone (0.269 now). The small G=300 calib-vs-adj gap (0.300 → 0.269) is a mild, honest signature of intermittency non-stationarity, well inside the gate. - **Positive control (G=200) re-establishes the v5 witness under a different, sounder objective.** `a_mm = −0.00079`, slope 0.736 vs v5's −0.00078 / 0.737 — near-identical despite a different `E_max` construction (held-out look-ahead-max quantile 0.7344 vs burn-in 95th-percentile). The v5 control-sufficiency result is **not** an artifact of the old threshold rule. - **Generality test (G=300) POSITIVE.** `mean_minority` extrapolates to ~zero (`a_mm = +0.00058 ≤ 0.005`), through-origin, 10–22× below the random-label floor → clean decision surface, control-sufficient. - **Twin-state CERTIFIED at G=300**, non-degenerately, at the same `ε_K = 0.0664` as the control read, so the two halves **compose** into a complete regime-2 witness at G=300. **Net: regime-2 (state-insufficient AND control-sufficient) is now a two-regime result on the Grashof axis — (k_f=2, G=200) and (k_f=2, G=300) — under a portable objective, no longer cell-local.** The control half is also dimension-robust (v4/v5 across d=32/d=18) and now objective-robust (fixed-percentile and portable-quantile both POSITIVE at G=200). ### What it does NOT establish (scope held) - **Two points on ONE axis.** Grashof only; `k_f = 2` fixed throughout. The objective family now has two members but both are energy-overshoot-type. This is not full substrate-generality. - **Finite-Galerkin, sampled-support, numerical** — not a theorem about the infinite-dimensional NSE attractor. - **Proxy faithfulness strengthened, not closed.** Two independent objective constructions agreeing helps, but `π̂` is still a proxy, not a derived `J`-optimal selector with an explicit action cost. - **External PDE review (criterion c) open.** **C1 remains UNPROMOTED** — but this is the strongest the candidate has been: a two-regime, dimension-robust, objective-robust, end-to-end regime-2 witness. ### Next firm-up axes (none required, none urgent) `k_f`-axis generality (vary forcing geometry at fixed G); a genuinely different objective family (enstrophy / burst-onset, the documented v2 alternative); proxy-faithfulness via a derived `J`-optimal selector; external review. Any of these would further harden C1; none is needed to bank the present two-regime result. ## 11. Cross-References - [`PDE_C1_REGIME_GENERALITY_v0.md`](PDE_C1_REGIME_GENERALITY_v0.md) — the G=300 fixed-percentile probe that deferred for objective vacuity; this artifact is its portable-objective successor. - [`PDE_C1_KNN_CONVERGENCE_CHECK.md`](PDE_C1_KNN_CONVERGENCE_CHECK.md) / [`PDE_C1_TWIN_STATE_CERTIFICATE.md`](PDE_C1_TWIN_STATE_CERTIFICATE.md) — the adjudicators reused unchanged on the adjudication block. - [`PDE_C1_CELLSET_KOLMOGOROV_v5.md`](PDE_C1_CELLSET_KOLMOGOROV_v5.md) — the completed `G=200, K=3` witness the G=200 re-run must re-establish under the portable objective. - [`../SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md) — the ledger; §12's `PDE-C1-RG-POS` is reflected there as a two-regime, still-unpromoted C1 witness.