# PDE C1 Cell Set v5 - Kolmogorov-Flow Instance at k_f = 2, G = 200, K = 3 > Methodology sibling to > [`PDE_C1_CELLSET_KOLMOGOROV_v4.md`](PDE_C1_CELLSET_KOLMOGOROV_v4.md), > filed 2026-05-28 in response to the v4 lock + fall-back combined > disposition. v4 confirmed that the E_max amendment unlocks proxy > discrimination (`damp_fraction ≈ 0.30`, stable between 50,000 and > 200,000 samples), but uniform binning at `K = 4` fails on this > attractor regardless of sample budget (45k occupied bins at v4 lock, > 139k at fall-back, `evaluated_bin_count = 0` in both). v5 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) per the same-day fiber-protocol K-parameter > amendment. Forced mode `(0, 2)` remains in the K=3 signature > (it sorts ahead of higher-`k^2` modes in the selection). > > Status: drafted, desk-auditable, unreviewed, no numerical verdict > yet. Harness exposes this cell as `--preset lock_v5` and > `--preset fallback_v5`. ## 1. Methodology Delta from v4 The single change is the signature mode count: ```text v4: K = 4 (signature dim d_K = 32, 16 Fourier modes) v5: K = 3 (signature dim d_K = 18, 9 Fourier modes) ``` Bin count scales as `(R_attractor / h_K)^{d_K}`, so dropping `d_K = 32 → 18` reduces the dimensional cost by a factor of `(R/h)^{14}`. For `R/h ≈ 5` (rough estimate from v4), that's `5^14 ≈ 6×10^9` fewer bins in the worst case — overkill by orders of magnitude relative to v4's coverage gap. **Forced-mode inclusion check.** At `k_f = 2`, the forced mode is `(0, 2)` with `max(|kx|,|ky|) = 2`. The harness `select_low_modes` picks `K² = 9` modes for `K = 3` by sorting on `(max(|kx|,|ky|), k², ix, iy)`. Modes with `max = 1`: four half-plane representatives `(0,1), (1,1), (-1,1), (1,0)`. Modes with `max = 2`: sorted first by `k²`, so `(0,2)` and `(2,0)` (both `k² = 4`) come ahead of any `k² = 5` or `k² = 8` modes. The first 9 modes include `(0,2)`. **Forced mode retained.** ✓ ## 2. Inherited from v0–v4 All v4 settings are inherited unchanged except `K`: - **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 = 3`**, `d_K = 18`, `F_K = sigma(Phi_K)`. The signature still includes the forced mode `(0, 2)`. - **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` = 95th percentile of last 25% of burn-in (v4 amendment). - **Comparator.** Finite-Galerkin non-injectivity on `B_abs` (the algebraic construction is dimension-independent); literature determining-mode reference as sufficiency ceiling only. - **Fiber classifier.** Unchanged. - **Negative branch.** Unchanged. - **§7 Parameter instantiation.** All values inherited except `K = 3` and the derived `d_K = 18`, `epsilon_K = 0.05 sqrt(2 E_max)` (recomputed at v5 burn-in), `h_K = epsilon_K / sqrt(18) ≈ epsilon_K / 4.24` (vs v4's `epsilon_K / 5.66`). ## 3. New v5 Pin - **Signature mode count.** `K = 3`. - **Signature dimension (derived).** `d_K = 2K^2 = 18`. That is the only spec-level change. The harness preset `lock_v5` encodes it; all other parameters resolve identically to `lock_v4`. ## 4. Discriminative-Power and Coverage Expectations **Proxy discrimination expectation.** At the v4 regime, the forced mode `(0, 2)` carries most of the energy and the safety trigger fires on look-ahead windows where `E_K` (the K-truncated low-band energy) crosses `E_max`. Dropping K=4→K=3 reduces the signature components from 32 to 18; the safety predicate now "sees" energy in 9 Fourier modes instead of 16. The forced mode is preserved, so the dominant energy contribution is intact. `damp_fraction` is expected to remain in the discriminative band `[0.01, 0.99]`. It may shift up or down by a moderate factor (perhaps `±2×`) depending on how much energy lives in the dropped modes `(0, 3), (3, 0), (1, 3), (-1, 3), (3, 1), (-3, 1)` — that's 6 of the 16 K=4 modes that v5 drops. If they carry significant energy at `G = 200`, `damp_fraction` could shift. v4 lock had `damp_fraction ≈ 0.30`; v5 expected range `[0.05, 0.60]`. **Coverage expectation.** Bin count should drop dramatically. v4 fall-back gave 139,361 bins at `N = 200,000`. With `d_K` dropping by 14 dimensions and `h_K` increasing by a factor of `sqrt(32/18) ≈ 1.33`, the projected v5 bin count is **at least an order of magnitude smaller**, probably 5k–20k bins at `N = 50,000`. If correct, `samples_per_bin_avg = N / bins` would be 2.5–10, with many bins reaching `n_min = 30` on the high-density part of the attractor. **`S_eval ≥ 0.5` is the pre-registered expectation.** If v5 still defers for coverage, the next step is **not** another K drop (K=2 loses the forced mode at k_f=2). Options would include larger `epsilon_K` (adaptive `h_K` amendment) or a regime adjustment. If `damp_fraction` falls outside `[0.01, 0.99]` on v5 (unlikely given the v4 evidence but possible if the dropped modes were load-bearing), the appropriate filing is `DEFERRED_VACUITY` and a re-examination of the K=3 mode selection. ## 5. What v5 Closes / Does Not Close **Same closures as v0–v4 methodology surface, plus the K-parameter clarification.** Cells can now pin `K` to trade signature richness against bin coverage, with explicit forced-mode inclusion verification per cell. **Open via execution.** A non-vacuous, non-deferred v5 verdict (`STRICTNESS_WITNESS_POSITIVE` or `PDE-C1-NEG-A`) would be the **first substantive C1 read** across five cell-set iterations. **Not closed by v0/v1/v2/v3/v4/v5 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 v5 lock cell was executed on 2026-05-28 via [`../../scripts/pde_c1_kolmogorov_cell.py`](../../scripts/pde_c1_kolmogorov_cell.py) with `--preset lock_v5`. Receipt and manifest at `results/proof/c1-kolmogorov-v5-lock/`. The harness completed in 1152 seconds (~19 min) with `steps_per_second ≈ 2257` and returned `DEFERRED_COVERAGE`. ### Signal 1 — discrimination is robust to the dimension cut (confirmed) `damp_fraction = 0.2977`, essentially unchanged from v4's `0.30014` (K = 4) and v4 fall-back's `0.300125`. Dropping the signature from 16 Fourier modes to 9 — discarding `(0,3), (3,0), (1,3), (-1,3), (3,1), (-3,1)` — barely moved the proxy distribution. The six dropped modes were **not** load-bearing for the safety predicate. The proxy discrimination at `(k_f = 2, G = 200)` is a robust feature of the attractor, not an artifact of signature richness. The structural-vacuity precedence rule correctly did not fire; the forced mode `(0, 2)` was retained as designed. ### Signal 2 — the K-reduction hypothesis is FALSIFIED This v5 file (§4) pre-registered: *"Bin count should drop dramatically ... at least an order of magnitude smaller, probably 5k–20k bins at N_sample = 50,000. S_eval ≥ 0.5 is the pre-registered expectation."* The actual result: | | v4 (K=4, d_K=32) | v5 (K=3, d_K=18) | change | |---|---|---|---| | `h_K` | 0.01107 | 0.01428 | 29% coarser | | occupied bins | 45,827 | 38,281 | **−16.5%** | | `evaluated_bin_count` | 0 | 0 | unchanged | | `S_eval` | 0 | 0 | unchanged | Halving the embedding dimension (32 → 18) **and** coarsening the bins 29% reduced occupied-bin count by only 16.5% — not the projected order of magnitude. **The pre-registered expectation is falsified.** ### Root cause — occupied bins track attractor dimension, not d_K Amendment 4's scaling argument `bin_count ∝ (R_attractor / h_K)^{d_K}` assumed the attractor *fills* the `d_K`-dimensional signature box. It does not. The attractor is a low-dimensional invariant set embedded in `R^{d_K}`; the number of occupied bins is governed by its **box-counting dimension at scale `h_K`**, which is far below `d_K` and is invariant to the embedding-dimension change. Cutting `K` operated on a quantity (`d_K`) that was not the bottleneck. The honest scaling is `occupied_bins ≈ (R / h_K)^{D_box}` with `D_box` the attractor dimension — and `D_box` did not change between v4 and v5 because it is the same dynamical system observed through a slightly smaller linear projection. This is a clean negative result on the amendment-4 hypothesis, recorded per the pre-registration discipline (we do not silently drop a falsified projection). ### Disposition `DEFERRED_COVERAGE`, honestly. Not `PDE-C1-NEG-B`: the methodology was followed exactly and `K = 3` was pre-registered before sampling. The fall-back (`fallback_v5`, `N_sample = 200,000`) is *not recommended*: the v4 fall-back already showed that 4× samples produce ~3× new bins on this attractor, so it would almost certainly defer again — and the v5 evidence now explains *why* (the attractor's box dimension fixes the occupied-bin growth rate). Running it would burn ~90 min to re-confirm a mechanism v5 already exposed. ### Why the next move is not v6-with-coarser-h_K-by-reflex The genuine lever is `h_K` (equivalently `epsilon_K`), not `K`. But coarsening `h_K` enough to clear coverage is in **direct tension with the tolerance object's fidelity**: `epsilon_K` was introduced (fiber protocol §1) as a *small* tolerance ball approximating the continuous fiber. A rough estimate (attractor box-dim ~3–4) suggests coverage would need `h_K ≈ 0.05`, i.e. `epsilon_K ≈ 0.21` — roughly 17% of the signature-norm scale `sqrt(2 E_max) ≈ 1.21`. At that tolerance, a "fiber" lumps together signature values that are not plausibly the same fiber, and a coarse bin spanning the safety boundary can manufacture artificial incompatibility (or mask real structure). This is a **fundamental tension between tolerance fidelity and sample coverage**, governed by the attractor's measure concentration — it may not be resolvable by parameter tuning at the `5×10^4`–`2×10^5` sample budget. This is a step-back point, not a launch-the-next-knob point. See the C1 lock-execution synthesis (filed as a separate disposition) and the ledger Promotions block for the recommended fork. ## 6. Status - **Drafted.** 2026-05-28. - **Desk-auditable.** A reviewer can read v4 + the v4 combined disposition + this v5 methodology delta and audit the change without running code. - **Unreviewed.** No external PDE reviewer has signed off on the `K = 3` choice. - **Executed.** v5 lock ran 2026-05-28 in 1152 sec (~19 min); receipt at `results/proof/c1-kolmogorov-v5-lock/`. - **Disposition.** `DEFERRED_COVERAGE`. Discrimination robust (`damp_fraction = 0.2977`, matching v4); K-reduction coverage hypothesis **falsified** (occupied bins fell only 16.5% despite halving `d_K`). Root cause: occupied bins track attractor box dimension, not embedding dimension. Tolerance-fidelity vs. coverage tension is now the core obstruction; this is a step-back point. ## 7. Cross-references - [`PDE_C1_CELLSET_KOLMOGOROV_v4.md`](PDE_C1_CELLSET_KOLMOGOROV_v4.md) - v4 cell-set and its Lock Execution Disposition (lock + fall-back) that motivates v5. - [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md) - methodology; the K-parameter rule was added 2026-05-28 to §2 Local Symbols + Binning Instance. - [`PDE_C1_CELLSET_KOLMOGOROV.md`](PDE_C1_CELLSET_KOLMOGOROV.md) / [v1](PDE_C1_CELLSET_KOLMOGOROV_v1.md) / [v2](PDE_C1_CELLSET_KOLMOGOROV_v2.md) / [v3](PDE_C1_CELLSET_KOLMOGOROV_v3.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_v5` runs this cell at `(k_f = 2, G = 200, e_max_burnin_fraction = 0.25, K = 3)`.