# PDE C1 Cell Set v4 - Kolmogorov-Flow Instance at k_f = 2, G = 200, E_max from steady-state burn-in

> Methodology sibling to
> [`PDE_C1_CELLSET_KOLMOGOROV_v3.md`](PDE_C1_CELLSET_KOLMOGOROV_v3.md),
> filed 2026-05-28 in response to the v3 lock disposition and the
> same-day fiber-protocol E_max windowing amendment. v4 holds the v3
> regime constant (`k_f = 2`, `G = 200`) and pins
> `e_max_burnin_fraction = 0.25` (last quarter of burn-in only for
> the 95th-percentile threshold). This isolates the methodology
> question from the regime question: does fixing E_max alone unlock
> the discriminative regime at v3's settings?
>
> This is the first cell-set in the C1 chain whose delta is a
> *methodology* parameter rather than a *regime* parameter. Same
> instance-of-pre-registration discipline applies.
>
> Status: drafted, desk-auditable, unreviewed, no numerical verdict
> yet. The harness exposes this cell as `--preset lock_v4` (and
> `--preset fallback_v4` for the coverage-gate fall-back).

## 1. Methodology Delta from v3

The single change is the `E_max` windowing:

```text
v0..v3: e_max_burnin_fraction = 1.0   (95th pct of full burn-in,
                                        includes transient peaks)
v4:     e_max_burnin_fraction = 0.25  (95th pct of last 25% of burn-in,
                                        steady-state portion only)
```

For v3's burn-in length of `10^5` steps, the last 25% is the final
`25,000` steps (250 time units at `dt = 0.01`). For 2D Kolmogorov at
`G = 200`, the viscous-mode timescale is `1 / (nu k_f^2) ≈ 3.5` time
units; 250 time units is ~70 viscous turnovers, well past the
transient-decay window. The last 25% should be a faithful sample of
the steady-state attractor.

No other parameter changes.

## 2. Inherited from v3 (and transitively v0/v1/v2)

All v3 settings are inherited unchanged:

- **Regime.** 2D NSE on `T^2 = [0, 2*pi]^2`, `N = 16` modes per axis,
  `k_f = 2`, `G = 200`, `nu = sqrt(1/200) ≈ 0.0707`, forcing amplitude
  `1`, decision domain `B_abs`, evaluation measure SRB-like.
- **Observation.** `K = 4`, `d_K = 32`, `F_K = sigma(Phi_K)`.
- **Objective.** Low-band energy safety trigger; look-ahead horizon
  `tau = 5` time units; action space `{no_op, damp_low_band}`;
  deterministic proxy `\hat{pi}`. `E_max` is **the only objective
  parameter whose computation changes**: it is now the 95th
  percentile of the *last 25%* of burn-in.
- **Comparator.** Finite-Galerkin non-injectivity on `B_abs` (§4.1);
  literature determining-mode reference as sufficiency ceiling only.
- **Fiber classifier.** Unchanged.
- **Negative branch.** Unchanged; `PDE-C1-NEG-A` and `PDE-C1-NEG-B`
  apply.
- **§7 Parameter instantiation.** All values inherited except the new
  E_max windowing parameter: `dt = 0.01`, `T_burnin = 10^5` steps,
  `epsilon_K = 0.05 sqrt(2 E_max)` (recomputed under the windowed
  E_max), `h_K = epsilon_K / sqrt(d_K)`, `n_min = 30`, `delta_action
  = 0.10`, `S_pos = 0.50`, `delta_proxy_min = 0.01`,
  **`e_max_burnin_fraction = 0.25`**, `N_sample = 50,000`, sampling
  interval `0.5` time units, fall-back to `N_sample = 200,000` on
  coverage deferral, action tie-break favouring `no_op`.

## 3. New v4 Pin

- **E_max windowing.** `e_max_burnin_fraction = 0.25`.

That is the only spec-level change. The harness preset `lock_v4`
encodes it; all other parameters resolve identically to `lock_v3`.

## 4. Discriminative-Power Expectation

The v3 lock execution gave concrete numbers we can project from:

- Burn-in 95th percentile = `1.0674` (current `E_max`, contaminated
  by transient peak at `3.1263`).
- Post-burn-in (steady-state) 95th percentile is not directly
  reported, but `sample_energy_max = 0.7843`. The steady-state 95th
  percentile is somewhere in `[0.78, 0.79]`.
- Projected v4 `E_max` ≈ `0.78–0.79`. Then `epsilon_K = 0.05 sqrt(2
  × 0.78) ≈ 0.0625` (vs v3's `0.0731`) and `h_K ≈ 0.011` (vs v3's
  `0.0129`).
- With `E_max ≈ 0.78`, samples crossing the threshold in lookahead
  should approximate 5% by construction (95th percentile of the same
  distribution we're testing against). Expected `damp_fraction ≈ 0.05`,
  comfortably inside `[delta_proxy_min, 1 - delta_proxy_min] =
  [0.01, 0.99]`.
- Coverage: `h_K` shrinks slightly (~15% smaller), so bin count may
  rise slightly above v3's 44,065. Still likely too many bins for
  `n_min = 30` at `N_sample = 50,000`. **Coverage failure is a real
  possibility**, in which case the pre-registered fall-back to
  `N_sample = 200,000` becomes admissible (statistical coverage gap,
  not structural vacuity).

This pre-registers two predictions:

1. The structural-vacuity precedence rule does **not** fire on v4
   (because the steady-state-windowed E_max is close to the
   steady-state attractor's typical excursion, so the proxy is not
   structurally constant).
2. v4 may land `STRICTNESS_WITNESS_POSITIVE`, `PDE-C1-NEG-A`, or
   `DEFERRED_COVERAGE`. Coverage failure would be statistical
   (insufficient samples per bin), not the high-dimensional curse of
   v2.

If v4 *does* fire structural vacuity, the E_max-only amendment was
insufficient; further methodology rethinking (smaller K, different
objective) is the next move.

## 5. What v4 Closes / Does Not Close

**Same closures as v0..v3 methodology surface, plus the E_max windowing
clarification:** cells can now pre-register a burn-in sub-window for
threshold computation. This addresses the transient-contamination
pattern observed across v0/v1/v3.

**Open via execution.** A non-vacuous v4 verdict (any of POSITIVE /
NEG-A / DEFERRED_COVERAGE) would be the first substantive C1 read.

**Not closed by v0/v1/v2/v3/v4 alone:**

- Criterion (a) Front-A vacuity rebuttal (sidecar-level).
- Criterion (c) named external PDE reviewer.
- Attractor-support twin-state certificate.

## Lock Execution Disposition (2026-05-28 — lock + fall-back)

The v4 cell was executed in two runs on 2026-05-28 via
[`../../scripts/pde_c1_kolmogorov_cell.py`](../../scripts/pde_c1_kolmogorov_cell.py):
`--preset lock_v4` and the pre-registered fall-back
`--preset fallback_v4`. Receipts at
`results/proof/c1-kolmogorov-v4-lock/` and
`results/proof/c1-kolmogorov-v4-fallback/`. Both runs returned
`DEFERRED_COVERAGE`. **The disposition is a single combined read of
the two runs**, because they tell one story.

### E_max amendment confirmed — methodology win

The 2026-05-28 fiber-protocol E_max windowing amendment
(`e_max_burnin_fraction = 0.25` for v4) successfully unlocked
proxy discrimination at the v3 regime:

| Parameter | v3 (full burn-in) | v4 (last 25%) |
|---|---|---|
| `e_max` | 1.0674 | **0.7838** |
| `damp_low_band_count` | 0 | 15,007 (lock) / 60,025 (fall-back) |
| `damp_fraction` | 0 exactly | **0.30014** (lock) / **0.300125** (fall-back) |

The cross-cell pattern of `damp_fraction = 0 exactly` (v0, v1, v3) is
broken at v4. Both proxy actions fire substantively. The
structural-vacuity precedence rule correctly does **not** fire.
Critically, `damp_fraction` is **stable to 4 decimal places** between
the 50,000-sample lock and the 200,000-sample fall-back. This is
strong evidence that the proxy distribution is well-estimated and
`0.300` is the true rate on this attractor, not a sampling artifact.
**The cell IS substantively discriminative.**

### Coverage gate fires for the v2 reason

But coverage fails in both runs, and the fall-back behaviour is
diagnostic:

| Run | `N_sample` | occupied bins | avg samples/bin | `evaluated_bin_count` |
|---|---|---|---|---|
| v4 lock | 50,000 | 45,827 | 1.09 | 0 |
| v4 fall-back | 200,000 | **139,361** | 1.43 | **0** |

**4× samples produced 3.04× bins.** Most additional samples land in
*new* bins, not fill existing ones. Even at `N = 200,000` no bin
reached `n_min = 30`. Empirically, the bin distribution at
`(k_f = 2, G = 200, K = 4)` is heavy-tailed with no concentrated
mode — uniform binning at `h_K ≈ 0.011` in 32-dim signature space is
genuinely too fine for the attractor extent.

### Disposition

Both v4 receipts file `DEFERRED_COVERAGE` honestly. The pre-registered
fall-back was the protocol-correct response to the lock's
`DEFERRED_COVERAGE`; the fall-back's identical verdict closes the
fall-back option. This is *not* a `PDE-C1-NEG-B` retune — the
methodology was followed exactly. The diagnosis is empirical: at
`K = 4`, uniform binning fundamentally fails on this regime's
attractor regardless of sample budget.

### Methodology amendment 4 (added 2026-05-28)

The fiber protocol §2 was amended same day to make **`K`** (signature
mode count) a *cell-set parameter*, with explicit documentation of
the coverage vs. discrimination tradeoff: `bin_count ∝ (R_attractor
/ h_K)^{d_K}` where `d_K = 2K^2`. Lowering `K` reduces `d_K`
exponentially, addressing the curse of dimensionality.

### Bridge to v5

v5 at
[`PDE_C1_CELLSET_KOLMOGOROV_v5.md`](PDE_C1_CELLSET_KOLMOGOROV_v5.md)
holds the v4 regime constant (`k_f = 2`, `G = 200`,
`e_max_burnin_fraction = 0.25`) and pins `K = 3` (signature
dim 18 instead of 32). Forced mode `(0, 2)` remains in the signature
(it sorts ahead of higher-`k^2` modes in the K=3 selection).
Harness exposes `--preset lock_v5`.

### Why this is not `PDE-C1-NEG-B`

The K amendment is a methodology change introducing a new cell-set
parameter; it does **not** relax a pinned parameter to flip the v4
verdict. v4 stands as a `DEFERRED_COVERAGE` receipt under K = 4. v5
is a new cell-set instance with `K = 3` pre-registered before
sampling, not a v4 retune.

## 6. Status

- **Drafted.** 2026-05-28.
- **Desk-auditable.** A reviewer can read v3 + the v3 disposition +
  this v4 methodology delta and audit the change without running code.
- **Unreviewed.** No external PDE reviewer has signed off on the
  `e_max_burnin_fraction = 0.25` choice.
- **Executed.** v4 lock ran 2026-05-28 in 1140 sec (~19 min); v4
  fall-back ran 2026-05-28 in 5264 sec (~88 min). Receipts at
  `results/proof/c1-kolmogorov-v4-lock/` and
  `results/proof/c1-kolmogorov-v4-fallback/`.
- **Disposition.** Both runs `DEFERRED_COVERAGE`. Methodology
  win: E_max amendment unlocked proxy discrimination
  (`damp_fraction ≈ 0.30`, stable between 50k and 200k samples).
  Methodology limit: K = 4 binning empirically infeasible on this
  attractor regardless of N_sample. v5 at K = 3 is the next cell-set
  instance.

## 7. Cross-references

- [`PDE_C1_CELLSET_KOLMOGOROV_v3.md`](PDE_C1_CELLSET_KOLMOGOROV_v3.md)
  - v3 cell-set and its Lock Execution Disposition that motivates v4.
- [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md) -
  methodology; the E_max windowing rule was added 2026-05-28 to §3
  Proxy Selector.
- [`PDE_C1_CELLSET_KOLMOGOROV.md`](PDE_C1_CELLSET_KOLMOGOROV.md) /
  [`PDE_C1_CELLSET_KOLMOGOROV_v1.md`](PDE_C1_CELLSET_KOLMOGOROV_v1.md)
  / [`PDE_C1_CELLSET_KOLMOGOROV_v2.md`](PDE_C1_CELLSET_KOLMOGOROV_v2.md)
  - prior cell-set siblings.
- [`PDE_DETERMINING_MODES_POSTULATE1.md`](PDE_DETERMINING_MODES_POSTULATE1.md)
  - the C1 sidecar.
- [`../SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md) - the
  ledger.
- [`../../scripts/pde_c1_kolmogorov_cell.py`](../../scripts/pde_c1_kolmogorov_cell.py)
  - harness. `--preset lock_v4` runs this cell at
  `(k_f = 2, G = 200, e_max_burnin_fraction = 0.25)`.