# PDE C1 Cell Set v3 - Kolmogorov-Flow Instance at k_f = 2, G = 200 > Sibling cell-set instance to v0, v1, and v2, filed 2026-05-28 in > response to the v2 lock disposition. v2 escaped v1's > energy-conservation vacuity by tripling Grashof to `G = 300`, but > landed in a curse-of-dimensionality coverage regime: 45,103 > occupied bins from 50,000 samples (~1.1 samples/bin) with > `damp_fraction = 0.0086` just below `delta_proxy_min = 0.01`. > v3 backs off to `G = 200` to find a sweet spot where the attractor > is discriminative *and* bin-resolvable at the pre-registered budget. > Everything else inherited from v0/v1/v2. This is a new cell-set > instance, not a retune of v0/v1/v2. > > Status: drafted, desk-auditable, unreviewed, no numerical verdict > yet. The harness exposes this cell as `--preset lock_v3` (and > `--preset fallback_v3` for the coverage-gate fall-back). ## 1. Regime Delta from v2 The single regime change is the Grashof number: ```text v2: G = 300 (chaotic with broken conservation, but coverage collapses) v3: G = 200 (intended middle ground: chaotic + bin-resolvable) ``` Derived parameters: - `nu = sqrt(forcing_amplitude / G) = sqrt(1/200) ≈ 0.0707` (between v1's `0.1` and v2's `0.0577`). - Viscous-dissipation scale `k_d ~ 1/sqrt(nu) ~ 3.76`, comfortably inside the `N/3 ≈ 5.3` dealiased cutoff at `N = 16`. `G = 200` is a 2× step from v1 and a 1.5× step *below* v2. The expectation is that the linear-stability threshold for `k_f = 2` is already exceeded at `G = 100` (v1), so `G = 200` remains supercritical; but the chaotic attractor at `G = 200` should be less spread out in signature space than v2's, since attractor dimension typically grows with Reynolds number. ## 2. Inherited from v0, v1, and v2 All of the following are inherited unchanged from v0 / v1 / v2: - **§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 v3 burn-in (recomputed for v3 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. - **§7 Parameter instantiation.** All section 7 values inherited: `dt = 0.01`, `T_burnin = 10^5` steps, `epsilon_K = 0.05 sqrt(2 E_max)` (numerically re-pinned at v3 burn-in completion), `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 v3 Pin - **Grashof number.** `G = 200`. - **Viscosity (derived).** `nu = sqrt(1/200) ≈ 0.0707`. That is the only spec-level change. The harness preset `lock_v3` encodes it; everything else in `build_config` resolves identically to `lock_v2`. ## 4. Discriminative-Power Expectation The v3 hypothesis is that `G = 200` sits in a Goldilocks zone: - **Above v1's structural-vacuity floor.** Energy conservation should be broken; `damp_fraction` should be substantively non-zero. Order of magnitude expectation: 0.01 to 0.2. - **Below v2's coverage cliff.** The attractor extent in signature space should be moderate enough that 50,000 samples occupy O(1000) bins, with O(50) samples/bin on average — comfortably above `n_min = 30` for most bins. Expected v3 behaviour: - `burnin_energy_mean` somewhere between v1's `0.64` and v2's `1.13` (rough scaling argument; not load-bearing). - `sample_energy_max - sample_energy_min` should be O(`sample_energy_mean`), matching v2's qualitative behaviour but at smaller amplitude. - `damp_fraction` should land in `[0.01, 0.99]`. If `damp_fraction` falls below `0.01` again, the **near-vacuity** gate fires `DEFERRED_VACUITY` (not structural, but statistical). - `occupied_bin_count` should be in the low thousands (between v1's 1397 and v2's 45,103), giving `S_eval ≥ 0.5` at `N_sample = 50,000`. - If coverage fails, the pre-registered fall-back at `N_sample = 200,000` is admissible (this is statistical insufficiency, not structural — unlike v1's case). If v3 still defers (for either gate), the natural next steps are either: - a methodology amendment letting `h_K` adapt to attractor extent (e.g. `h_K = ε_K * (R_eff / R_ref)` for some adaptive scale), or - a smaller `K` to reduce signature-space dimensionality (e.g. v4 at `K = 3` or `K = 2` keeping `k_f = 2`, `G = 200`). Both would be substantive methodology changes warranting amendment-level discussion. This expectation is **pre-registered**, not a result. ## 5. What v3 Closes / Does Not Close **Same closures as v0 / v1 / v2 methodology surface.** **Open via execution.** A non-vacuous, non-deferred v3 verdict would be the first substantive C1 read. If v3 also defers, the diagnosis sharpens (which gate fires) and the next-move options above narrow. **Not closed by v0 / v1 / v2 / v3 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) The v3 lock cell was executed on 2026-05-28 via [`../../scripts/pde_c1_kolmogorov_cell.py`](../../scripts/pde_c1_kolmogorov_cell.py) with `--preset lock_v3`. Receipt and manifest at `results/proof/c1-kolmogorov-v3-lock/`. The harness completed in 1198 seconds (~20 min) with `steps_per_second ≈ 2171` and reported `DEFERRED_VACUITY` (`proxy_selector_structurally_constant`) under the 2026-05-28 structural-vacuity precedence rule. ### Outcome - `burnin_energy_min / max / mean = 0.7338 / 3.1263 / 0.8583`. Burn-in has substantive chaotic excursions (max 4.3× the min) — `G = 200` *is* supercritical at `k_f = 2`, as the v3 expectation predicted. - `e_max` (95th percentile of full burn-in) = `1.0674`. - `sample_energy_min / max / mean = 0.7649 / 0.7843 / 0.7747`. The post-burn-in attractor sits in a narrow band ~2.5% of the mean — much tighter than v2's 95% spread, and far below `e_max`. - `damp_low_band_count = 0`, `damp_fraction = 0` exactly. No look-ahead window crosses `e_max` because the post-burn-in steady-state attractor's max (`0.7843`) is below the transient-contaminated threshold (`1.0674`). - 44,065 occupied bins — comparable to v2 (45,103) — so the attractor is still high-dimensional in signature space, but the coverage gate is moot because structural-vacuity precedence fires first. ### Cross-cell pattern recognition Three of four executed cells now have `damp_fraction = 0` *exactly* (v0, v1, v3), and v2's was only `0.0086`. The cross-cell signal is sharp: > Across regimes, the burn-in 95th-percentile `E_max` sits **above** > what the post-burn-in attractor typically reaches in lookahead > windows. For v3, burn-in transient peaks at 3.1263 inflate the percentile to 1.0674, while the steady-state attractor only reaches 0.7843. This is a methodology problem (transient contamination of the threshold), not a regime problem (the regime is chaotic and discriminative-ready). The v3 file's pre-registered next moves (adaptive `h_K` or smaller `K`) addressed *coverage* failure, but v3 didn't fail for coverage. The actual failure mode is `E_max` over-estimation across regimes. ### Methodology amendment The fiber protocol §3 was amended same day with an **E_max windowing** rule: cells may pin `e_max_burnin_fraction ∈ (0, 1]` and define `E_max` from the last fraction of burn-in samples, excluding transients. Default remains `1.0` (full burn-in) for backwards compatibility with v0–v3 receipts. ### Disposition The v3 lock receipt stands as a `DEFERRED_VACUITY` under the original methodology (`e_max_burnin_fraction = 1.0`); the structural-vacuity precedence rule correctly identified the proxy as zero. The amendment does **not** retroactively re-read v3 — it introduces a new parameter that v4+ can pin. ### Bridge to v4 v4 at [`PDE_C1_CELLSET_KOLMOGOROV_v4.md`](PDE_C1_CELLSET_KOLMOGOROV_v4.md) holds the v3 regime constant (`k_f = 2`, `G = 200`) and pins `e_max_burnin_fraction = 0.25` (last quarter of burn-in only). Expected behaviour: `E_max` drops from 1.07 to ~0.78 (close to the post-burn-in 95th percentile), making the safety trigger fire on ~5% of windows — well inside `[delta_proxy_min, 1 - delta_proxy_min]`. This is the clean methodology test: does fixing E_max alone unlock the discriminative regime, holding everything else constant? ## 6. Status - **Drafted.** 2026-05-28. - **Desk-auditable.** A reviewer can read v0 + v1 + v2 + this v3 delta and audit the regime change without running code. - **Unreviewed.** No external PDE reviewer has signed off on the `G = 200` choice as a faithful discriminative-and-resolvable instance. - **Executed.** v3 lock ran 2026-05-28 in 1198 sec (~20 min); receipt at `results/proof/c1-kolmogorov-v3-lock/`. - **Disposition.** `DEFERRED_VACUITY` under structural-vacuity precedence (proxy selector zero on the sampled support). Cross-cell pattern across v0/v1/v3 reveals burn-in transient contamination of `E_max`. Methodology amendment landed same day (E_max windowing as cell-set parameter); v4 tests the amendment at the v3 regime. ## 7. Cross-references - [`PDE_C1_CELLSET_KOLMOGOROV.md`](PDE_C1_CELLSET_KOLMOGOROV.md) - v0. - [`PDE_C1_CELLSET_KOLMOGOROV_v1.md`](PDE_C1_CELLSET_KOLMOGOROV_v1.md) - v1. - [`PDE_C1_CELLSET_KOLMOGOROV_v2.md`](PDE_C1_CELLSET_KOLMOGOROV_v2.md) - v2, including its Lock Execution Disposition that motivates this v3 sibling. - [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md) - methodology. - [`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_v3` runs this cell at `(k_f = 2, G = 200)`.