# PDE C1 — Non-Marginal Regime Probe

> 2026-05-29. The `m_det` probe (robustness wave §4) found the G=200/K=3
> separation **energy-marginal** (the attractor is an approximate inertial
> manifold). This note pursues a *non-marginal* regime-2 separation. Two
> paths were considered; the high-G path hit a numerical wall, so the chosen
> path is a **norm reframing** at G=200.

## 1. High-G path — attempted, hit the C2 numerical wall

Rationale: a higher Grashof gives a higher-dimensional attractor, so a fixed
K=3 signature sits far below the determining count → genuine (all-norm)
state-insufficiency, if control still holds.

**Result: numerical wall (the same one C2 hit).** A `lock_hidim_g1000`
probe (G=1000, grid 64, K=3, `ν=0.0316`) was stability-checked first:

| dt | 40k-step transient |
| --- | --- |
| 0.010 | **blow-up** (overflow in advection within a few thousand steps) |
| 0.004 | blow-up |
| 0.002 | blow-up |
| 0.001 | stable (`E_low → 2.2`) |

So fixed-dt high-G is feasible only at `dt = 0.001` — **10× the steps,
~7 h/run** at grid 64 (before a grid-96 resolution check) — **and fragile**:
dt=0.001 survived a short transient, but C2's blow-up occurred at ~3.5M steps
on an *intermittent* event, so a full ~27M-step high-G run has a real chance
of dying partway. **Decision: not pursued by brute force** (it repeats the
C2 fixed-dt trap). The robust high-G path is the **adaptive/stiff integrator
on a uniform-time grid** — the C2 resume design — which would unblock *both*
this lane and C2; deferred to that build.

## 2. Chosen path — norm reframing at G=200 (no numerical wall)

The marginality is **norm-dependent**. 2D NSE concentrates *energy* at large
scales (always near-determined), so the **energy norm** makes the separation
look marginal — but the small-scale degrees of freedom are genuinely
under-determined, and they carry the *enstrophy*. Measure state-determination
`FVE(Q_K|Φ_K)` in three norms on a uniform sample of high-mode components
(`state-recon`, K=3, G=200):

- **energy-weighted** (component variance) — large scales;
- **enstrophy-weighted** (`|k|²`) — emphasizes small scales;
- **equal-weight** (median per-component R²) — every DOF counted equally.

Already in hand: energy-weighted `FVE = 0.9994` (marginal); equal-weight
median `R² = 0.73` (non-marginal). The **enstrophy-weighted FVE** is the new
crisp number.

**Pre-registered reading.** The separation is **non-marginal in the
enstrophy / per-DOF norm** iff the enstrophy-weighted state residual
`1 − FVE_enstrophy` is materially larger than the energy residual (0.06%) —
i.e. the small scales are genuinely under-determined. Combined with the
**already-established enstrophy control-sufficiency** (paired
`PAIRED_FIBER_CONSTANCY_POSITIVE` under the `Z_low` trigger, robustness wave
§"sweep 3"), that is a legitimate non-marginal regime-2 separation in a
matched (enstrophy) norm. **Honest caveat:** it is norm-dependent — marginal
in energy, non-marginal in enstrophy/per-DOF; a skeptic can note the norm
choice, so both must be reported side by side.

## 3. Result (2026-05-29) — REFRAME FAILED; G=200 is marginal in all physical norms

Three-norm state-determination at K=3, G=200 (`c1-recon-k3-norms`, uniform
sample over the high-mode spectrum, perm control −0.001):

| norm | `FVE(Q_K\|Φ_K)` | state residual |
| --- | --- | --- |
| energy-weighted (large scales) | 0.9972 | 0.28% |
| **enstrophy-weighted** (`\|k\|²`) | **0.9934** | **0.66%** |
| equal-weight per-component (median R²) | 0.71 | — |

**The hypothesis is falsified.** The enstrophy-weighted FVE (0.9934) is only
marginally below the energy-weighted (0.9972) — both ~0.99, both **marginal**.
The enstrophy norm does *not* rescue non-marginality. The only non-marginal
number is the **equal-weight per-DOF** median (0.71), but that weights every
Fourier mode equally, and most modes are far-UV **dissipation-range** modes
carrying negligible energy *and* enstrophy. So the under-determined part of
the state is **viscous dissipation-range noise** — physically irrelevant, read
by no physical objective.

**Honest conclusion.** `Φ_K` determines all the *physically-relevant*
(energy- and enstrophy-carrying) state content at G=200; it fails to
reconstruct only dynamically-negligible dissipation-range noise. The
control-vs-reconstruction separation is therefore **marginal in every
physical norm** — strictly non-vacuous (certified positive-measure
non-injectivity) but physically marginal, i.e. a *weak* example of the
regime-2 target. **A genuine non-marginal separation requires a
higher-dimensional regime** (high G / 3D) where physically-relevant content is
under-determined — which needs the adaptive/stiff integrator (§1). There is
no shortcut at G=200. Proposition clause (i) updated accordingly; the
robustness wave stands (N/K/objective robust), but the separation it is robust
*about* is physically marginal at this regime.

## 4. Cross-references

- [`PDE_C1_ROBUSTNESS_WAVE.md`](PDE_C1_ROBUSTNESS_WAVE.md) §4 — the m_det probe that found the energy-marginality.
- [`PDE_C1_PROPOSITION.md`](PDE_C1_PROPOSITION.md) — clause (i) to be reframed.
- [`PDE_C2_CELLSET_SABRA_v1.md`](PDE_C2_CELLSET_SABRA_v1.md) — the same numerical wall; the adaptive integrator is the shared resume path.