# PDE C2 Cell Set v1 — Stationarity-Gated Sabra Burst Cell > **Cross-substrate failure-map entry:** NUMERICAL WALL (fixed-dt RK4 blows up > integrating the intermittent bursts the task is about) — see > [`../CROSS_SUBSTRATE_NOTES.md`](../CROSS_SUBSTRATE_NOTES.md) > "Cross-Substrate Generality Failure Map". > **Design proposal for sign-off**, filed 2026-05-29. Re-pose of > [`PDE_C2_CELLSET_SABRA_v0.md`](PDE_C2_CELLSET_SABRA_v0.md), whose > headline run returned `PDE-C2-DEFERRED-BASERATE` with a *block-dependent* > burst rate (train 0.138, val 0, test 0) → the trajectory was not > stationary / not representatively sampled across the labelled span. > v1 fixes stationarity rather than retuning the burst threshold (a v0 > retune would be `PDE-C2-NEG-B`). **Status: proposed, not built, not > run.** §8 lists the sign-off decisions; build/run only after sign-off. ## 1. Root-cause read of the v0 deferral The v0 base rates ran **0.138 → 0 → 0** across temporally-ordered blocks (calib → train → val → test). The monotonic decline to zero is the signature of **energy drift**, not random rare-cluster scatter (which would give non-monotonic noise). Most likely the constant *additive* forcing `f_1 = const` on shell 1 does not drive a statistically steady cascade on the labelled timescale: the large-scale energy drifts, so a calib-derived `E_burst` (set early, high) is rarely reached in later blocks. The integrator is verified correct (inviscid energy conservation), so this is a forcing/stationarity issue, not an integration bug. Two fixes are needed, and "longer blocks" alone is **not** sufficient if the cause is drift — the forcing must produce a stationary state first. ## 2. Fix A — forcing scheme for a statistically steady cascade **Default (recommended): fixed-amplitude forcing on shell 1.** Hold `|u_1|` at a fixed value each step (renormalise the forced shell's modulus, let its phase evolve), the standard shell-model device for a statistically steady energy input. This reliably removes large-scale energy drift. The inviscid energy-conservation self-test is unaffected (it tests the nonlinear term with forcing off). **Retained alternative (sign-off option):** keep v0's constant additive `f_1` but with a verified-long warmup (only if the §3 diagnostic shows it actually plateaus). Constant-power (fixed energy-input-rate) forcing is a third option. The forcing scheme is the **load-bearing sign-off decision** (§8.1): it is a model change, so it is named, not chosen unilaterally. ## 3. Fix B — stationarity diagnostic (cheap, run FIRST) Before any labelled cell, run a **diagnostic trajectory** under the chosen forcing and report, downsampled, the time series of: - total energy `Σ_n |u_n(t)|^2`; - `max_n |u_n(t)|^2` (the burst observable). From these, pre-registered estimates: - **equilibration time `T_eq`**: the time after which a sliding-window mean of total energy is flat to a pinned tolerance; - **burst recurrence time `T_burst`**: mean inter-exceedance interval of `max_n|u_n|^2` above a provisional high quantile. This is the cheap analogue of the base-rate-first discipline: diagnose the stationarity and burst timescales *before* committing a full cell. It also distinguishes drift (energy never flattens) from rare-cluster intermittency (flattens, but `T_burst` is long). ## 4. Cell parameters set by a pre-registered rule The labelled cell's warmup and block lengths are then pinned by a rule fixed here (concrete numbers go in a v1 patch *after* the diagnostic, before the verdict-bearing run): ```text warmup >= 3 * T_eq each block >= 50 * T_burst (so each of train/val/test contains O(50) burst events) calib block >= 50 * T_burst gaps >= 5 * T_burst ``` Everything else (Sabra `N=22`, `λ=2`, `k0=2^-4`, `ε=1/2`, `ν=1e-7`, `dt`, `q_burst=0.98`, channels, log-signature L2, the four matched-budget baselines, leakage-safe contiguous splits) is **inherited unchanged from v0**. ## 5. Fix C — per-block stationarity gate (new) The v0 base-rate gate (test base rate ∈ [0.05, 0.40]) is **necessary but not sufficient** — it missed the block-dependence. v1 adds, checked before any detector-vs-baseline read: ```text base_rate(train), base_rate(val), base_rate(test) each in [0.05, 0.40] AND pairwise |base_rate(i) - base_rate(j)| <= 0.10 ``` - Fail the band → `PDE-C2-DEFERRED-BASERATE` (as v0). - Pass the band but fail pairwise consistency → new **`PDE-C2-DEFERRED-NONSTATIONARY`** (the v0 failure mode, now named). - Both pass → proceed to the matched-budget comparison. No retune of `q_burst`/`E_burst`/blocks after reading a gate result (that is `PDE-C2-NEG-B`); a further re-pose is a v2 cell. The receipt reports **per-block diagnostics** (mean/median max-energy, burst count, base rate) so a deferral is interpretable (drift vs under-sampling). ## 6. Inherited from v0 Channel tiers (Tier 0 shell log-energies headline; Tier 1 +transfers; Tier 2 +hidden-self-similarity), log-signature level 2, the four matched-budget baselines (DMD / CSD / lacunarity / Rényi) with the matched-budget rule, the Pareto lead-time vs. false-positive evaluation, the two-sided negative (`PDE-C2-NEG-A` / `PDE-C2-NEG-B`), and the `PDE-C1-NEG` re-framing rule. The matched-budget 4-baseline comparison remains the next increment, now gated on **both** the base-rate band and the per-block stationarity gate. ## 7. Harness changes (for build — not yet built) In `scripts/pde_c2_sabra_cell.py`: - a `--forcing {additive, fixed-amplitude, constant-power}` option (default `fixed-amplitude` per §2); - a `--diagnostic` mode that runs §3 and writes the energy time series + `T_eq` / `T_burst` estimates, filing no verdict; - per-block base-rate + energy/burst diagnostics in the receipt; - the §5 per-block stationarity gate and the new `PDE-C2-DEFERRED-NONSTATIONARY` branch. The inviscid energy-conservation self-test stays the integrator gate and must still pass. ## 8. Open sign-off decisions 1. **Forcing scheme (load-bearing).** Default `fixed-amplitude` |u_1| forcing (recommended — guarantees a steady cascade), vs retaining v0 additive forcing with a long verified warmup, vs constant-power. 2. **Stationarity gate tolerance.** Pairwise base-rate consistency `≤ 0.10` and band `[0.05, 0.40]` — confirm. 3. **Diagnostic-then-cell staging.** Run the §3 diagnostic first (cheap), pin warmup/blocks from `T_eq`/`T_burst`, then the full cell — vs pin conservative large values up front and skip the diagnostic. 4. **Compute budget.** Long blocks (O(50) bursts each) + 4 baselines + hyperparameter search will be heavier than v0's ~32 min; confirm a trajectory length / target wall-clock. 5. **Build trigger.** On sign-off I add the forcing option + diagnostic mode + stationarity gate, re-run the energy self-test, run the §3 diagnostic, then the verdict-bearing cell. ## 10. Build + diagnostic-phase amendments (2026-05-29) Signed off and built (objective-validity/diagnostic layer; the verdict-bearing cell remains gated on the diagnostic + the §5 gate): - **Forcing = fixed-amplitude `|u_1| = 1.0`** (§8.1 resolved). The diagnostic smoke confirmed it removes the v0 drift: total energy plateaus with first/second-half drift **0.2%** (vs v0 drift-to-zero). Inviscid energy-conservation self-test still passes. - **Burst observable amended: `max_n|u_n|²` → dissipation rate `ε(t) = ν Σ_n k_n² |u_n|²`.** *Why:* the diagnostic immediately exposed that under fixed-amplitude forcing `max_n|u_n|²` is **degenerate** — it is pinned by the forced shell (`|u_1|` fixed), so median = q98 = 1.0, no burst signal. `ε` (the canonical shell-model intermittency observable; `k_n²` weighting concentrates it in the dissipation range, and it is the quantity the instanton/large-deviation literature we cited targets) is non-degenerate: smoke `ε` q98/median ≈ 5.8. This is a diagnostic-motivated refinement of the *objective* made **before any verdict-bearing run** (no verdict read), the legitimate purpose of the diagnostic phase — not a post-hoc retune. `E_burst` is now the held-out `q_burst = 0.98` quantile of `ε`. - **Harness built:** `--forcing {additive, fixed-amplitude}`, `--diagnostic` mode (energy series → `T_eq` / `T_burst` + suggested warmup/block lengths), `ε` burst observable + base-rate gate; `scripts/pde_c2_sabra_cell.py`. Self-test + smoke green. - **Headline diagnostic running** (fixed-amplitude, `ε`, 3M steps) to pin `T_eq` / `T_burst`; the verdict-bearing cell lengths (warmup ≥ 3·T_eq, blocks ≥ 50·T_burst) get pinned in a v1 patch from its output before the cell run, with the §5 per-block stationarity gate. ## 11. Diagnostic result (2026-05-29) and the burst-rarity finding Headline diagnostic (`results/proof/c2-sabra-v1-diagnostic/`, 3M steps / 300 time units, fixed-amplitude, ε observable): - **Stationarity solved.** Total energy first/second-half drift `0.0001` (0.01%), `T_eq ≈ 0`, plateaued. Fixed-amplitude forcing fixes v0's non-stationarity decisively. ε non-degenerate (q98/median ≈ 9.9). - **But ε-bursts at q98 are too rare for a detection cell.** Only ~2 burst events in 300 time units (`T_burst ≈ 294`). The rule blocks ≥ 50·T_burst → ~147M steps/block → ~700M total → ~46 h, infeasible; and ~2 burst events is too few positives regardless. **Root cause + fix (the C1 lesson, again).** The v0/v1 label pinned `E_burst` at the q=0.98 quantile of *instantaneous* ε — a rare extreme. That is NOT the C1 construction that worked: C1's portable objective set the threshold as a quantile of the *look-ahead-max* targeting a fixed base rate by construction. **Amend the C2 label the same way:** pin a **target query base rate** `r_burst` (proposed 0.15) and set `E_burst` = the `(1 − r_burst)`-quantile of the held-out calib block's look-ahead-max ε. This pins ~15% positive windows by construction (enough examples, spread across the timeline → far less clustered than 2 extremes), leakage-safe, with the §5 per-block stationarity gate confirming representativeness. This supersedes the fixed-`q_burst` pin in §3 (a diagnostic-phase refinement, pre-verdict). **Honest meta-note.** C2 has now surfaced three diagnostic-caught design issues — non-stationarity (→ fixed-amplitude forcing), observable degeneracy (→ ε), burst rarity (→ target-base-rate label). Each was caught cheaply before a verdict-bearing or infeasible run (the diagnostic-first discipline working), but the cumulative signal is that a clean shell-model burst-detection cell is genuinely thornier to pose than the C1 Kolmogorov cell. The target-base-rate amendment is the next concrete step; a pause to rethink the C2 framing is also reasonable. ## 12. v1 headline run (2026-05-29) — NUMERICALLY INVALID (integrator blow-up) The target-base-rate label + per-block stationarity gate were built and the v1 headline cell ran (`results/proof/c2-sabra-v1-headline/`, 6.3M steps). **The run is numerically invalid:** RuntimeWarning overflows in the nonlinear term appeared at ~step 3.5M (50–60%), and the gate output (`PDE-C2-DEFERRED-BASERATE`, base rates train/val/test 0.138/0/0) is a **blow-up artifact**, not a real base-rate read — `e_burst = 4.3e-9` is finite (computed from the clean calib block at 2–3M), but val/test = 0 because the post-blow-up blocks are `nan` (`nan > E_burst` is `False`). **Cause.** Fixed-dt RK4 (`dt = 1e-4`) cannot integrate through a large intermittent dissipation burst: when a small-scale spike drops the fastest nonlinear timescale below `dt` (CFL/stiffness; possibly compounded by the fixed-amplitude `|u_1|` rescale overcorrecting on a dip), the explicit step diverges. The v1 diagnostic (3M steps) was clean because the blow-up sits just past its window (~3.5M). The inviscid energy self-test does not stress bursts, so it passed throughout. **We are trying to detect the very events the integrator cannot survive.** **Cumulative C2 picture — four coupled obstructions:** (1) non-stationarity (→ fixed-amplitude forcing), (2) observable degeneracy (→ ε), (3) burst rarity (→ target-base-rate label), (4) **numerical blow-up through bursts (no quick fix)**. The first three each yielded to a clean diagnostic-motivated fix; the fourth needs an **adaptive CFL-limited timestep**, which breaks the uniform-`dt` assumption baked into the sampling/blocking/labeling machinery — a substantial integrator redesign, not an iteration. **Disposition: C2 paused at v1.** Not a verdict (no detector-vs-baseline comparison ran); the headline cell is numerically invalid and the burst objective remains unadjudicated. The honest banked finding is the four-obstruction catalogue: a clean, numerically-robust, representatively-sampled shell-model burst-detection cell is genuinely hard and requires real numerical-methods investment. **Resume path (if C2 is reopened):** a v2 harness with an adaptive/stiff integrator (CFL-limited dt or exponential-integrator with substepping) that records samples on a uniform *time* grid (decoupled from the variable step grid), re-run the diagnostic to confirm burst-robustness, then the cell. This is a design-for-sign-off, not a quick patch. C1 remains the strong, defensible NSE result; C2 is an honest, well-characterized open problem. ## 9. Cross-references - [`PDE_C2_CELLSET_SABRA_v0.md`](PDE_C2_CELLSET_SABRA_v0.md) — the deferred v0 cell (§12 disposition) this re-poses. - [`PDE_C2_SHELL_SIGNATURE_SCOPING.md`](PDE_C2_SHELL_SIGNATURE_SCOPING.md) — research object, channel taxonomy, baselines, two-sided negative. - [`PDE_C1_REGIME_GENERALITY_v1.md`](PDE_C1_REGIME_GENERALITY_v1.md) — the C1 portable-objective + gate discipline whose lineage this follows (deferral → diagnose → new pinned cell, never a same-cell retune). - [`../SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md) — ledger.