# PDE C1 Cell Set v2 - Kolmogorov-Flow Instance at k_f = 2, G = 300 > Sibling cell-set instance to > [`PDE_C1_CELLSET_KOLMOGOROV.md`](PDE_C1_CELLSET_KOLMOGOROV.md) (v0) > and [`PDE_C1_CELLSET_KOLMOGOROV_v1.md`](PDE_C1_CELLSET_KOLMOGOROV_v1.md) > (v1), filed 2026-05-28 in response to the v1 lock disposition. v1 > escaped v0's laminar-vacuity but found a new structural vacuity: > the post-transient attractor at `k_f = 2`, `G = 100` is non-trivial > in signature space (1397 occupied bins) but **`E_K` is exactly > conserved** on it (sample energy constant to 13 decimal places), > so the safety-trigger objective is inaccessible. v2 triples the > Grashof number to `G = 300` while keeping `k_f = 2` to inject > enough energy that the attractor symmetry breaks and `E_K` > fluctuates non-trivially. This is a new cell-set instance, not a > retune of v0 or v1. > > Status: drafted, desk-auditable, unreviewed, no numerical verdict > yet. The harness exposes this cell as `--preset lock_v2` (and > `--preset fallback_v2` for the coverage-gate fall-back). ## 1. Regime Delta from v1 The single regime change is the Grashof number: ```text v1: G = 100 (supercritical at k_f = 2 but with E_K-conserving attractor) v2: G = 300 (3x energy injection, expected to break the conservation) ``` The viscosity scales accordingly: `nu = sqrt(forcing_amplitude / G) = sqrt(1/300) ≈ 0.0577` (v1 had `nu = 0.1`). All other parameters are inherited from v1 unchanged. `G = 300` is chosen as the smallest tripling that comfortably stays within the `N = 16` Galerkin truncation's resolution envelope. The viscous-mode wavenumber `k_d ~ 1/sqrt(nu) ~ 4.16` sits inside the dealiased cutoff (`N/3 ≈ 5.3`), so spectral pile-up is not a concern at this regime. Higher `G` (e.g. `G = 1000`) would require a larger truncation and is deferred to a possible v3 if v2 is also vacuous. ## 2. Inherited from v0 and v1 All of the following are inherited unchanged from v0 / v1: - **§1 Regime (excluding `G`).** 2D NSE on `T^2 = [0, 2*pi]^2`, `N = 16` modes per axis (`32 × 32` grid), `k_f = 2` (from v1), dimensionless forcing amplitude `1`, decision domain `B_abs` of the truncated system after burn-in, evaluation measure the SRB-like distribution. - **§2 Observation.** `K = 4`, signature dimension `d_K = 32`, σ-algebra `F_K`. The signature includes the forced mode `(0, 2)`. - **§3 Objective.** Low-band energy safety trigger; `E_max` = 95th percentile of `E_K` along the v2 burn-in (recomputed for v2 because the dynamics differ); look-ahead horizon `τ = 5` time units; action space `{no_op, damp_low_band}`; deterministic provisional selector / proxy `\hat{pi}` per the fiber protocol section 3. - **§4 Comparator.** Finite-Galerkin non-injectivity witness on `B_abs` (regime-independent); literature determining-mode reference as sufficiency ceiling only. The attractor-support twin-state certificate remains a deferred bridge. - **§5 Fiber classifier.** Unchanged. - **§6 Negative branch.** Unchanged; `PDE-C1-NEG-A` and `PDE-C1-NEG-B` receipts apply. - **§7 Parameter instantiation.** All section 7 values are inherited: `dt = 0.01`, `T_burnin = 10^5` steps, `epsilon_K = 0.05 sqrt(2 E_max)` (numerically re-pinned at v2 burn-in completion since `E_max` is regime-dependent), `h_K = epsilon_K / sqrt(d_K)`, `n_min = 30`, `delta_action = 0.10`, `S_pos = 0.50`, `delta_proxy_min = 0.01`, `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 v2 Pin - **Grashof number.** `G = 300`. - **Viscosity (derived).** `nu = sqrt(1/300) ≈ 0.0577`. That is the only spec-level change. The harness preset `lock_v2` encodes it; everything else in `build_config` resolves identically to `lock_v1`. ## 4. Discriminative-Power Expectation The v1 lock disposition identified energy-exact conservation as the v1 failure mode. The v2 hypothesis: at higher Grashof, the forced-dissipative balance breaks energy conservation on the attractor, producing non-trivial `E_K` fluctuations that exercise the safety trigger. Expected v2 behaviour: - `burnin_energy_max - burnin_energy_min` should be O(`burnin_energy_mean`) rather than O(0.0001) (as v1 showed post-burn-in). Burn-in trajectory reaches statistical steady state with non-trivial spread. - `sample_energy_max - sample_energy_min` should also be O(mean): the attractor itself has energy fluctuations, not just transients. - `damp_fraction` should land in `[0.01, 0.99]`. The 95th-percentile threshold gives, by construction, a `damp_fraction ~ 0.05` at steady state if the look-ahead window is comparable to the energy-fluctuation autocorrelation time. - If `damp_fraction` again falls outside `[0.01, 0.99]`, v2 also fires `DEFERRED_VACUITY`. The next sibling v3 would raise `G` further (e.g. `G = 1000` with `N >= 32` to maintain resolution) or change the objective. - If `0.01 ≤ damp_fraction ≤ 0.99`, the predicate has discriminative content and the verdict is `STRICTNESS_WITNESS_POSITIVE`, `PDE-C1-NEG-A`, or `DEFERRED_COVERAGE` (and on coverage deferral, the fall-back at `N_sample = 200,000` is admissible because the vacuity isn't structural). This expectation is **pre-registered**, not a result. ## 5. What v2 Closes / Does Not Close **Same closures as v0 / v1 methodology surface:** the fiber protocol's binning predicate, the proxy selector substitution rule, the parameter-pinning discipline, the structural-vacuity precedence rule (added 2026-05-28). **Open via execution.** A non-vacuous v2 verdict would be the first substantive C1 read. If v2 also defers, the next step is either a v3 at higher `G` with a larger truncation, or a re-objective at the sidecar level (e.g. a different `J` not based on low-band energy). **Not closed by v0 / v1 / v2:** - 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) The v2 lock cell was executed on 2026-05-28 via [`../../scripts/pde_c1_kolmogorov_cell.py`](../../scripts/pde_c1_kolmogorov_cell.py) with `--preset lock_v2`. Receipt and manifest at `results/proof/c1-kolmogorov-v2-lock/`. The harness completed in 1336 seconds (~22 min) with `steps_per_second ≈ 1946` and reported a procedural `DEFERRED_COVERAGE` verdict. ### Energy conservation broken — partial success The v1 disposition's diagnosis was correct: tripling Grashof broke the energy-conserved attractor. - `burnin_energy_min / max / mean = 0.79 / 4.69 / 1.13`. Burn-in spans a 5.9× range — substantive chaotic excursions, not just transients. - `sample_energy_min / max / mean = 0.78 / 1.52 / 0.89`. Post-burn-in itself has a ~95%-of-mean energy spread, qualitatively different from v1's 13-decimal-locked constant. - `damp_low_band_count = 430`, `no_op_count = 49,570`. The safety trigger actually fires — 430 of 50,000 sample windows cross `E_max = 1.40`. `damp_fraction = 0.0086` is structurally non-zero. - The structural-vacuity precedence rule correctly does **not** fire. ### Two new failure modes But v2 surfaces two issues v1 didn't reach: 1. **Coverage collapse.** `occupied_bin_count = 45,103`, `evaluated_bin_count = 0`, `S_eval = 0`. With 50,000 samples spread across 45,103 bins, the average bin has ~1.1 samples; no bin reaches `n_min = 30`. The pre-registered binning resolution at `h_K ≈ 0.0148` is too fine for the attractor extent in 32-dim signature space — the curse of dimensionality, not a methodological bug. 2. **Near-vacuity.** `damp_fraction = 0.0086 < delta_proxy_min = 0.01`. Even if coverage cleared, the statistical vacuity gate would fire (`DEFERRED_VACUITY` for proxy near-constancy on the sampled support). The safety predicate is barely discriminative. ### Disposition The procedural verdict `DEFERRED_COVERAGE` is honest at face value. The pre-registered fall-back (`N_sample = 200,000`) would give ~4.4 samples/bin on average — still well below `n_min = 30` for most bins — so it would almost certainly defer again. The deeper diagnosis is that v2's regime is *too* discriminative-by-binning: the attractor spreads across more of signature space than the fixed `h_K` can resolve at 50k–200k samples. This is not a `PDE-C1-NEG-B` retune candidate. The cell-set v0/v1/v2 binning prescription `h_K = ε_K / √d_K` adapts to `E_max` but not to attractor *extent*; the prescription is what it is. A coverage failure on a high-dimensional attractor is an honest receipt under the pinned methodology. ### Bridge to v3 v0 was sub-critical (laminar). v1 was super-critical but energy-conserved. v2 broke conservation but landed in a curse-of- dimensionality coverage regime. **v3 backs off the Grashof number to `G = 200`** — between v1's invariant set and v2's high-dimensional chaos — looking for the sweet spot where the attractor is discriminative *and* bin-resolvable at the pre-registered budget. Staged at [`PDE_C1_CELLSET_KOLMOGOROV_v3.md`](PDE_C1_CELLSET_KOLMOGOROV_v3.md); harness exposes `--preset lock_v3`. If v3 also fails for coverage or near-vacuity, the natural next move is **not** another G knob — it is to consider a methodology amendment that lets `h_K` adapt to attractor extent (or a smaller K that reduces the signature space dimensionally). Both are substantive changes warranting their own sign-off. ## 6. Status - **Drafted.** 2026-05-28. - **Desk-auditable.** A reviewer can read v0 + v1 + this v2 delta and audit the regime change without running code. - **Unreviewed.** No external PDE reviewer has signed off on the `G = 300` choice as a faithful symmetry-breaking instance. - **Executed.** v2 lock ran 2026-05-28 in 1336 sec (~22 min); receipt at `results/proof/c1-kolmogorov-v2-lock/`. - **Disposition.** Procedural `DEFERRED_COVERAGE` recorded honestly; diagnosis is two-fold (coverage collapse + near-vacuity on high-dimensional chaos). v3 at `G = 200` is the next cell-set instance to test the discriminative-and-resolvable hypothesis. ## 7. Cross-references - [`PDE_C1_CELLSET_KOLMOGOROV.md`](PDE_C1_CELLSET_KOLMOGOROV.md) - v0 cell-set and "Lock Execution Disposition (2026-05-28)" section recording the laminar-vacuity outcome. - [`PDE_C1_CELLSET_KOLMOGOROV_v1.md`](PDE_C1_CELLSET_KOLMOGOROV_v1.md) - v1 cell-set and "Lock Execution Disposition (2026-05-28)" section recording the energy-conservation-vacuity outcome that motivates v2. - [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md) - methodology; the structural-vacuity-precedence rule (section 5 step 8, added 2026-05-28) is what filed v1 as `DEFERRED_VACUITY` and would catch a same-mode failure on v2. - [`PDE_DETERMINING_MODES_POSTULATE1.md`](PDE_DETERMINING_MODES_POSTULATE1.md) - the C1 sidecar; v0, v1, v2 are concrete instances of the pre-registered cell set described there. - [`../SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md) - the ledger. - [`../../scripts/pde_c1_kolmogorov_cell.py`](../../scripts/pde_c1_kolmogorov_cell.py) - harness. The `--preset lock_v2` switch runs this v2 cell with the inherited v0/v1 parameter values plus `G = 300`.