# PDE C1 — Robustness Wave (pre-registration)

> The `OPEN` tag of the C1 proposition
> ([`PDE_C1_PROPOSITION.md`](PDE_C1_PROPOSITION.md) §6): refinement-invariance.
> This wave tests whether the regime-2 separation survives modest changes in
> discretization and proxy construction — the difference between "survives
> perturbation" and "structural." Sweeps, in priority order: **N-refinement
> (first)**, then K-window, enstrophy objective, alternate projection. Each is
> verdict-bearing and pre-registered before it runs. Finite-Galerkin
> throughout — refinement *approaches* but never reaches the PDE.

## Sweep 1 — N-refinement (running)

**Cell `lock_v5_n48`.** Identical to the established G=200 witness cell
`lock_v5` (`k_f=2`, `G=200`, `K=3`, overshoot-burnin objective + last-25%
`E_max`, same `dt=0.01`, same sampling, same seed) **except** the Galerkin
resolution is refined:

| | `lock_v5` (baseline) | `lock_v5_n48` (refined) |
| --- | --- | --- |
| grid | 32 | 48 |
| dealias cutoff `|k|` | ~10 | ~16 |
| `n_modes` | 16 | 24 |
| high-mode DOF (`Q_K`) | 422 | **1070** |
| `ν` (so `G`) | 0.07071 | 0.07071 (unchanged) |
| `K`=3 signature (`Φ_K`) | 9 modes / d=18 | same 9 modes / d=18 |

**Only the resolved-scale count changes.** `ν`, `G`, `k_f`, the objective,
`dt`, the sampling schedule, and the K=3 observation map are all held fixed.
**Stability confirmed** before launch: grid 48 at `dt=0.01` is stable over
30k steps, `E_low` plateaus to ~0.73 (matching grid 32 — large scales
converged, refinement adds small-scale content), no blow-up (the C2-style
CFL check).

**Pre-registered pass/fail.** The separation is **refinement-invariant at
this step** iff all three clauses persist at N=24:

- **(i)** `twin-state` → `TWIN_STATE_CERTIFIED` (non-injectivity persists —
  expected *more* easily, the `Q_K` null space grew 2.5×).
- **(ii)** paired fiber-constancy → `PAIRED_FIBER_CONSTANCY_POSITIVE`
  (`D_witness ≤ delta_action = 0.10`).
- **(iii)** `mz-budget` → `COUPLING_SIGNATURE_SLAVED` (`R²(R|Φ_K) ≥ 0.70`,
  controls `R²(g) > 0.90`, `R²(perm) < 0.10`).

All three → **`REFINEMENT_INVARIANT`** (flips the proposition's `OPEN`
refinement tag to `DEMONSTRATED` for the N axis). Any clause failing → the
result is resolution-dependent at this step; **file honestly, do not rescue**
(no post-hoc dt / objective / K retune — that would be the C1-NEG-B
boundary).

**Runs.** `twin-state` (clauses i + paired-ii) and `mz-budget` (clause iii)
at `lock_v5_n48`, `results/proof/c1-n48-twin/` and `…/c1-n48-mz/`. ~55 min
each (grid 48 ≈ 2.5× the per-step cost of grid 32).

### Sweep 1 result (2026-05-29) — REFINEMENT_INVARIANT at N=24

Both clause-runs returned the regime-2 verdicts, with the mz-budget
calibration controls clean at the finer resolution. Despite `Q_K` growing
**422 → 1070 DOF** (2.5×), the numbers are **nearly identical** to N=16:

| clause | N=16 (`lock_v5`) | N=24 (`lock_v5_n48`) |
| --- | --- | --- |
| (i) twin-state | `CERTIFIED`, 693,795 witness pairs | `CERTIFIED`, **689,263** |
| (ii) paired `D_witness` / `D_candidate` | 0.0367 / 0.0319 | **0.0377 / 0.0319** |
| (iii) `R²(R\|Φ_K)` | 0.998 | **0.998** |
| (iii) controls `R²(g)` / `R²(perm)` | 0.999 / −0.001 | **0.9993 / −0.0012** |
| `T_LLL` check `rms(R)/rms(T_low)`; `corr(g,R)` | 1.0; −0.85 | 1.0; −0.85 |
| `damp_fraction` | 0.298 | 0.304 |

Results: `results/proof/c1-n48-twin/`, `…/c1-n48-mz/`.

**Verdict: `REFINEMENT_INVARIANT`** — all three clauses persist, and the
quantities barely move (`D_candidate` unchanged at 0.0319; slaving 0.998 →
0.998) even though the unresolved state space grew 2.5×. The separation does
**not** degrade with resolution: strong evidence it is *structural, not a
truncation artifact*. This flips the proposition's refinement tag from `OPEN`
to `DEMONSTRATED` on the **N axis** (one step; an N=32 trend is the registered
next increment). Sweeps over K, objective, and projection remain open.

## Sweep 2 — K-window (new axis: observation choice) (running)

**Cells `lock_v5_k2`, `lock_v5_k4`** = `lock_v5` with only the signature band
`K` changed (`K=2`, d=8; `K=4`, d=32). Tests whether regime-2 is robust to
the observation choice, and brackets it.

- **`K=2` (lower / coarser):** clauses (i) twin-state + paired-(ii) +
  (iii) mz-budget. *Pass* = all three persist at a coarser observation
  (window includes K=2). A clause failing here = the lower bracket `K_lo`
  (control needs more than 4 modes) — filed honestly.
- **`K=4` (upper probe):** twin-state. **Expected coverage-limited**: at
  K=4 (d=32) signature-near pairs become sparse (the curse-of-dimensionality
  finding from v2/v4 that motivated K=3), so twin-state may return
  `TWIN_STATE_DEFERRED_COVERAGE` — a *method boundary*, **not** a regime-2
  failure or an injectivity-return. The clean `m_det` upper bracket needs a
  coverage-free state-reconstruction measure (`R²(Q_K|Φ_K)` across K) — a
  registered follow-on, not this sweep.

So the K-window demonstrates regime-2 across `K∈{2,3}` (both clean); the
upper end is honestly flagged method-limited.

## Sweep 3 — Enstrophy objective (new axis: objective construction) (running)

**Cell `lock_v5_enstrophy`** = `lock_v5` except the safety trigger watches
**low-band enstrophy** `Z_low = Σ_low|ω̂/scale|²` (energy without the
`1/|k|²` weighting) instead of energy; same quantile/`E_max` construction,
**same signature `Φ_K`**. Run twin-state (the non-injectivity clause (i) is
objective-free and identical; the new content is the **paired fiber-constancy
(ii)** under the enstrophy-triggered action). *Pass* =
`PAIRED_FIBER_CONSTANCY_POSITIVE` under enstrophy → clause (ii) is robust to
the physical observable, not an artifact of the energy proxy.

### Sweeps 2 & 3 result (2026-05-29) — both PASSED

**K-window (energy, G=200).** Regime-2 holds across **K ∈ {2,3,4}**:

| K (d) | (i) twin | (ii) paired `D_witness`/`D_cand` | (iii) mz `R²` (controls) |
| --- | --- | --- | --- |
| 2 (8) | CERTIFIED, 849,229 | POSITIVE 0.0500 / 0.0375 | SLAVED 0.9989 (g 0.9994, perm −0.0004) |
| 3 (18) baseline | CERTIFIED, 693,795 | POSITIVE 0.0367 / 0.0319 | SLAVED 0.998 |
| 4 (32) | CERTIFIED, 1,140,366 | POSITIVE 0.0348 / 0.0324 | — |

**Pre-registration prediction falsified (honestly recorded):** K=4 was
predicted to defer on coverage. It did **not** — full coverage
(`candidate_sample_fraction = 1.0`), 1.14M witness pairs, clean POSITIVE.
The original K=4 coverage failure (v2/v4) was a property of the **bin**
adjudicator (hard cells); the twin-state's `ε_K`-ball neighbourhoods are
coverage-robust, so K=4 adjudicates cleanly. The regime-2 window is at least
`K ∈ {2,3,4}` (no upper bracket reached — twins still abundant at K=4).

**Enstrophy objective (`Z_low` trigger, K=3).** Clause (i) CERTIFIED
(identical 693,795 — objective-free); **clause (ii)
`PAIRED_FIBER_CONSTANCY_POSITIVE`, `D_witness = 0.0350` / `D_cand = 0.0315`**.
Control-sufficiency is robust to the physical observable, not an artifact of
the energy proxy. Results: `results/proof/c1-k2-twin|c1-k2-mz|c1-k4-twin|c1-enstrophy-twin/`.

**Net:** every `D_witness` clusters 0.035–0.05 (all < 0.10, all ≈ candidate
rate) across K and objective — the boundary-layer-not-`Q_K` disposition is
itself robust. Three axes demonstrated: **N** (16→24), **K** (2,3,4),
**objective** (energy + enstrophy).

## Sweep 4 — `m_det` upper bracket (state-recon, running)

The clean, **coverage-free** state-determination measurement that twin-state
cannot give (twin-state needs signature-near pairs; this is a regression).
New `state-recon` adjudicator: does the signature `Φ_K` determine the
unresolved high modes `Q_K`? Held-out R² (the validated HGB estimator, block
split) of:

- **`R²(E_high | Φ_K)`** — the high-band energy `E_high = Σ_high|·|²` (primary);
- **`FVE(top-16 high modes | Φ_K)`** — variance-weighted reconstruction of the
  most state-relevant (lowest-wavenumber) unresolved components;
- **`R²(E_high permuted)`** — negative control, must be `< 0.10` (else
  `STATE_RECON_ESTIMATOR_INVALID`).

Run across **K ∈ {3,4,5,6}** (`lock_v5` / `lock_v5_k4/k5/k6`, grid 32). As K
grows toward the determining / inertial-manifold threshold, `R²(E_high|Φ_K)`
rises toward 1; **the bracket `K*`** is the smallest K where it crosses ~0.90
(the attractor becomes a graph over `Φ_K` — state-determined). At K=3 it must
be `< 1` (twin-state non-injectivity is its complement). If the trend rises
but does not reach ~1 within the grid-32 range (`K ≲ 10` by dealiasing), that
*brackets `m_det` above the tested range* and confirms determination is
approached — the honest outcome if `m_det` exceeds the resolution. This flips
the proposition's `K*` tag from *registered* to *measured*.

### Sweep 4 result (2026-05-29) — the bracket finding (tempers clause (i))

Two catches, both recorded honestly (verify-before-file):

**(a) The first metric measured the wrong thing.** `R²(E_high|Φ_K)` across K
came out `0.997 / 0.918 / 0.878 / 0.878` (K=3/4/5/6) — **decreasing**, not
rising. Reason: `E_high` is the energy *above* K, which becomes the far-UV
tail as K grows (less predictable). So that metric tracks **cascade locality
/ local closure**, not state-determination, and does **not** bracket `m_det`.
`FVE_top16 ~0.99` at every K likewise just confirms the modes *just above* the
band are slaved. Neither is the determining bracket.

**(b) The corrected full-`Q_K` measure (energy-weighted, matches twin-state).**
Re-run at K=3 (`c1-recon-k3-fullQ`) over the 69 components covering 99.9% of
the variance:

| measure | K=3 |
| --- | --- |
| **`FVE(Q_K\|Φ_K)` variance-weighted** | **0.9994** (state residual **0.06%**) |
| median R² over energy modes (top-69) | 0.994 |
| **median R² over uniform component sample** (incl. small scales) | **0.728** |
| `R²(E_high)` / perm control | 0.997 / −0.001 |

**Reading — scale-selective, energy-marginal.** Energy-weighted, the K=3
attractor is `0.9994` a graph over `Φ_K` — an **approximate inertial
manifold**, so `K=3` is **near the determining threshold**. But per-DOF the
small scales are only `~0.73` determined — the **genuine** state-insufficiency
lives in the low-energy small-scale modes. Because the decision (and the
twin-state metric) are energy-weighted, the certified non-injectivity is real
but **marginal in the decision-relevant norm** (twin pairs sit at the ~2–3σ
tail of the 0.06% residual). This **tempers clause (i)**: the separation is
real and genuine per-DOF, but not a gross state/control gap. The honest
caveat — 2D NSE at moderate `G` is low-dimensional, so a non-marginal regime-2
separation likely needs a higher-dimensional regime — is now in the
proposition's "does/doesn't claim". No clean `K*` is reachable at grid 32; the
proposition tag is **MEASURED** (near-determining at K=3) rather than a
bracketed `K*`.

## Sweeps 5+ (registered, not yet run)

- **N=32** (grid 64) — a second N point for a convergence *trend*. ~1.5 h.
- **Alternate projection** — a different 9-mode low subset: "not a lucky basis."

## Cross-references

- [`PDE_C1_PROPOSITION.md`](PDE_C1_PROPOSITION.md) — the `OPEN` tag this closes.
- [`PDE_C1_TWIN_STATE_CERTIFICATE.md`](PDE_C1_TWIN_STATE_CERTIFICATE.md) / [`PDE_C1_PAIRED_FIBER_CONSTANCY.md`](PDE_C1_PAIRED_FIBER_CONSTANCY.md) / [`PDE_C1_MZ_ENERGY_BUDGET.md`](PDE_C1_MZ_ENERGY_BUDGET.md) — the clause adjudicators reused unchanged.