# PDE C1 Cell Set v0 - Kolmogorov-Flow Instance > First concrete instance of the pre-registered cell set required by > [`PDE_DETERMINING_MODES_POSTULATE1.md`](PDE_DETERMINING_MODES_POSTULATE1.md) > section "Pre-Registered Cell Set For The Next Pass." Status: drafted, > desk-auditable, unreviewed, no numerical verdict yet. Filed 2026-05-28. > This is the v0 instance, not the only possible one; alternative regimes > (free decay, Charney-DeVore, finite Galerkin NSE with random forcing) > remain admissible if a reviewer prefers them. This file pins the six items the C1 sidecar lists by name. It does **not** run a simulation, train a detector, or adjudicate the pre-registered negative. It defines a procedure that an external PDE reviewer can read end-to-end and an analyst can later execute. ## 1. Regime - **Equation.** 2D incompressible Navier-Stokes on the torus `T^2 = [0, 2*pi]^2` with viscosity `nu > 0` and steady Kolmogorov forcing `f(x, y) = sin(k_f y) e_x` for a fixed forcing wavenumber `k_f` (canonical choice: `k_f = 4`). - **Galerkin truncation.** Project onto the first `N` Fourier modes per axis; `N = 16`, so the nominal truncation has `N^2 = 256` mode slots and a truncated state vector `u^N(t) in R^{2N^2}` after the divergence-free and reality constraints are represented in real coordinates. The precise coordinate count is not load-bearing; what matters is that `K < N`. - **Grashof number.** Pin `G = ||f||_{L2} / nu^2 = 100` by choosing `nu` accordingly. `G = 100` is intended as a moderate chaotic Kolmogorov-flow desk-audit cell, not as a Sundog-original PDE regime claim. Literature determining-mode constants remain review inputs. - **Decision domain `X`.** The v0 comparator uses the certified absorbing-ball decision domain `B_abs` of the truncated system after burn-in. The numerical evaluation measure `mu` is the SRB-like distribution induced by long trajectories of the truncated system. If a reviewer requires the exact support of `mu` as `X`, the support-level non-injectivity certificate in section 4.1 becomes mandatory before the comparator is treated as closed. ## 2. Observation - **Map.** `Phi_K : X -> R^{2K^2}` is the projection onto the first `K` Fourier modes per axis, with `K = 4`. The signature carries `2K^2 = 32` real components: large enough to include the forced mode `k_f = 4` and a few neighbours, and strictly lower-dimensional than the finite-Galerkin state used by the comparator in section 4. - **Sigma-algebra.** `F_K = sigma(Phi_K)`, the sigma-algebra generated by `Phi_K`. A `Phi_K`-fiber is `C_sigma = { u in X : Phi_K(u) = sigma }`. ## 3. Objective **Name.** Low-band energy safety trigger. - **Low-band energy.** `E_K(u) = 1/2 ||Phi_K(u)||^2_{L2}`. - **Safety envelope.** Pin `E_max` as the 95th percentile of `E_K` along a `10^5`-step burn-in trajectory of the truncated system at the fixed `G`, `k_f`, `N`, and `K`. The percentile is fixed before any numerical fiber audit is interpreted. - **Look-ahead horizon.** `tau = 5` viscous turnover times computed at `G = 100`, fixed at file time. - **Action space.** `A_ctrl = { no_op, damp_low_band }`. - `no_op`: identity, no modification of the dynamics. - `damp_low_band`: add the term `-gamma Phi_K(u)` to the right-hand side of the Galerkin NSE during the look-ahead window, where `gamma = 1 / tau`. Damping is restricted to the low-band modes only. - **Provisional selector.** `pi*_J(u) = damp_low_band` iff the trajectory of the `no_op` evolution starting at `u` would carry `E_K` above `E_max` at any time `t in [0, tau]`; otherwise `no_op`. This is a deterministic label computed from the truncated dynamics. The objective is intentionally lower-dimensional than the state: only the low-band energy trajectory matters for the verdict. High-mode dynamics influence the verdict only through their projection onto `Phi_K` over the look-ahead window. **Open review item.** The selector above is not yet a fully derived `J`-optimal selector, because the cost of damping has not been specified. A later patch must either add an action cost / lexicographic objective that makes this selector optimal, or rename it as a tractable proxy rather than `pi*_J`. See [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md) section 3 for the proxy resolution adopted in the v0 patch (section 7 below). The cost branch remains admissible at review. ## 4. State-Reconstruction Comparator The comparator has two layers. The first is the actual v0 state-insufficiency witness; the second is the literature reference that says what a state-reconstructive observation would look like. ### 4.1 Finite-Galerkin Non-Injectivity Witness For the desk audit, the state-reconstruction question is not decided by the inequality `K < m_det`. An upper determining-mode bound proves that some larger observation can reconstruct state; it does **not** prove that a smaller observation cannot. The v0 comparator instead uses the Galerkin coordinates directly. Let `Q_K = I - Phi_K` be the complementary high-mode projection. Choose any nonzero divergence-free high-mode vector `h` with ```text Phi_K(h) = 0, Q_K h = h. ``` For any interior state `u` of the certified absorbing ball `B_abs` whose small high-mode perturbations remain in that ball, choose `epsilon > 0` such that ```text u_a = u + epsilon h in B_abs, u_b = u - epsilon h in B_abs. ``` Then: ```text Phi_K(u_a) = Phi_K(u_b) = Phi_K(u), u_a != u_b. ``` Thus `Phi_K` is not injective on the absorbing-ball decision domain. This is the only state-insufficiency claim v0 treats as desk-certified. It is an algebraic finite-Galerkin witness, not a new PDE theorem. **Attractor-support caveat.** If the reviewer requires the decision domain to be the exact support of the SRB-like measure rather than the absorbing ball or a thickened reachable set, the algebraic witness above is not enough. In that stricter reading, v0 must certify non-injectivity by a twin-state search on the sampled support: ```text ||Phi_K(u_i) - Phi_K(u_j)|| <= epsilon_K and ||Q_K(u_i) - Q_K(u_j)|| >= delta_H ``` with `epsilon_K`, `delta_H`, burn-in length, sampling interval, and acceptance count pinned before interpretation. Until that support-level certificate lands, the honest status is: ```text Phi_K is certified non-injective on B_abs; Phi_K is not yet certified non-injective on supp(mu_SRB). ``` The [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md) section 6 bridges this caveat with a one-line spec for the support-level twin-state certificate; instantiation is deferred to a dedicated artifact and is **not** part of the v0 patch (section 7 below). ### 4.2 Literature State-Reconstructive Reference The literature input (Foias-Olson-Robinson tradition; Olson-Titi-style Kolmogorov-flow bounds) supplies a state-reconstructive reference observation: a determining-mode / determining-observable count `m_det(G)` with a bound of the form `m_det <= C * G^p`, for fixed constants `C, p` taken from the literature rather than estimated here. For the v0 audit, this reference is used only as a **sufficiency ceiling**: ```text P_{m_det} = state-reconstructive reference observation. Phi_K = lower-dimensional test observation. ``` It is not used to prove `Phi_K` state-insufficient. The separation test is therefore: 1. `P_{m_det}` represents the literature state-reconstruction side. 2. `Phi_K` is certified non-injective on the finite-Galerkin absorbing-ball comparator by section 4.1. 3. The remaining C1 question is whether `Phi_K` can nevertheless be control-sufficient for the registered objective. This is the regime in which the Postulate-1 / state-reconstruction distinction can be operationally separated, per [`PDE_DETERMINING_MODES_POSTULATE1.md`](PDE_DETERMINING_MODES_POSTULATE1.md) Reading 2. ## 5. Fiber Classifier For a candidate `Phi_K`-fiber `C_sigma`, the desk-auditable procedure is: 1. **Construct two probe states** `u_a, u_b in C_sigma` with `Phi_K(u_a) = Phi_K(u_b) = sigma` and differing only in modes `K < |k| <= N`. This is trivial in Galerkin coordinates on `B_abs`: pick the high-mode tail freely while holding low modes equal. On exact `supp(mu_SRB)`, use the twin-state certificate in section 4.1 instead. 2. **Evolve both** under the `no_op` truncated Galerkin NSE for time `tau` from each probe state. 3. **Read low-band energy** `E_K(t)` along each trajectory. 4. **Determine fiber-local action label**: - `(crosses E_max, crosses E_max)` => `damp_low_band` for both => fiber-local common action exists. - `(stays below, stays below)` => `no_op` for both => fiber-local common action exists. - `(crosses, stays below)` or vice versa => **fiber-local incompatibility** => candidate Postulate-1 negative. The classifier is deterministic, finite-time, and bounded by the look-ahead `tau` plus the cost of one Galerkin integration. A reviewer can sanity-check the predicate without running it. ## 6. Negative Branch **`PDE-C1-NEG` is filed iff** there exists a positive-`mu`-measure set of fibers `{ C_sigma : sigma in S, mu(Phi_K^{-1}(S)) > 0 }` on which the fiber-local incompatibility outcome of section 5 is realised. Equivalently: the selector is not constant on `Phi_K`-fibers of positive measure, so it is not `F_K`-measurable, so Postulate-1 control-sufficiency fails at this regime / observation / objective. By the C1 reading ([`PDE_DETERMINING_MODES_POSTULATE1.md`](PDE_DETERMINING_MODES_POSTULATE1.md) section "Reading 2" plus section "Equivalence / Named-Negative Regimes"), this is the named-negative boundary for the C1 candidate at this cell. **`PDE-C1-NEG` is not filed iff** the fiber-local incompatibility outcome occurs only on a `mu`-null set of fibers. In that case the strictness witness is recorded as a candidate positive: `Phi_K` is control-sufficient at this cell despite being non-injective on the finite-Galerkin comparator in section 4.1. **Open measure item.** In the continuous setting, exact `Phi_K`-fibers are typically `mu`-null. A later patch must replace the finite-MDP phrase "positive-measure fibers" with either a regular-conditional/disintegration statement or a pre-registered numerical binning/tolerance scheme. Until then, section 6 is the intended branch logic, not yet an executable continuous-measure test. See [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md) sections 1–5 for the tolerance + binning scheme adopted in the v0 patch (section 7 below). The disintegration branch remains admissible at review. ## 7. Parameter Instantiation (Fiber Protocol Patch) > Filed 2026-05-28 as the v0 patch instantiating the parameter slots in > [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md). These are v0 > defaults; alternative values remain admissible at review, subject to the > same parameter-pinning discipline (no post-hoc retuning; changes after > section-5 aggregation file `PDE-C1-NEG-B`). ### 7.1 Integration parameters - **Integration step.** `dt = 0.01` time units. Combined with the `10^5`-step burn-in from section 3, this fixes the burn-in duration at `1000` time units — comfortably past steady state for Kolmogorov flow at `G = 100`. - **Lyapunov timescale (assumed).** `T_lyap ~ 5` time units at `G = 100`. Used only to defend the sub-sampling interval in section 7.4; not load-bearing for the predicate. ### 7.2 Tolerance object - **Signature tolerance.** `epsilon_K = 0.05 * sqrt(2 * E_max)`. Five percent of the typical signature scale `||Phi_K|| ~ sqrt(2 * E_max)`. Scale-free formula; numerically pinned once `E_max` is computed by the section 3 burn-in. ### 7.3 Binning instance - **Per-coordinate lattice spacing.** `h_K = epsilon_K / sqrt(d_K) = epsilon_K / sqrt(32) ~ 0.0088 * sqrt(2 * E_max)`. Chosen so the typical bin diameter is `~ epsilon_K`, per the protocol section 2 lattice convention. - **Lattice bounding box.** Empirical range of `Phi_K(u)` over the section 3 burn-in trajectory, computed once at burn-in completion. Lattice origin is pinned at the lower corner of that bounding box; alignment is not re-optimised after sampling. ### 7.4 Sample set - **Number of post-burn-in samples.** `N_sample = 50,000`. - **Sampling interval.** `0.5` time units `= 50` integration steps. Total post-burn-in integration: `25,000` time units `= 2.5 * 10^6` steps, approximately a few minutes on a laptop with a vanilla Galerkin integrator. - **Samples per Lyapunov time.** `~10` (over-sampled but cheap). Sub-sampling at every `T_lyap` for decorrelation gives an effective `N_eff ~ 5,000` for per-bin minority-fraction statistics; the full `50,000`-sample set is used for bin-occupation counts and coverage diagnostics. ### 7.5 Adjudication thresholds - **Per-bin minimum count.** `n_min = 30`. Ensures the per-bin binomial standard error on a minority fraction `p ~ 0.05` is `~ 0.04`, so the `delta_action = 0.10` threshold sits at least one standard error above the noise floor. - **Minority-fraction threshold.** `delta_action = 0.10`. A bin is flagged fiber-incompatible if more than `10%` of its evaluated samples disagree with the bin majority action. Conservative on the positive side: only flags fibers with substantial proxy-action disagreement. - **Coverage gate.** `S_pos = 0.50`. The verdict is interpreted only if at least half of the occupied bins have `n(b) >= n_min`. - **Vacuity gate.** `delta_proxy_min = 0.01`. The verdict is interpreted only if the global `damp_fraction` lies in `[0.01, 0.99]`. Outside this band the proxy selector is essentially constant on the entire sampled support, every fiber is trivially action-consistent, and the strictness predicate has no discriminative content — see [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md) section 5 vacuity-gate step. Outcomes outside the band file `DEFERRED_VACUITY`, not a strictness positive. - **Action tie-break.** Prefer `no_op` (do not damp unless damping is the strict majority). Safety-conservative; matches the cell-set v0 § 3 framing of `damp_low_band` as an intervention triggered only when needed. ### 7.6 Fall-back sample size (pre-registered) If the section 5 procedure defers due to insufficient coverage (`S_eval < S_pos`), the admitted fall-back is **one** increase of `N_sample` from `50,000` to `200,000`, with all other parameters held. This fall-back is pre-registered here, before any sampling. Any further increase, or any change to other parameters in response to a deferral, files `PDE-C1-NEG-B` per the protocol section 7 parameter-pinning discipline. ### 7.7 Spec self-consistency check - `dt`, `T_burnin`, `tau`, `N_sample`, sampling interval, `T_lyap` are all in the same time unit (the truncated Galerkin time). - `epsilon_K`, `h_K`, `E_max` are all in the same signature-space norm (`R^{2K^2}` with the standard `L^2` inner product induced from the modal basis). - The look-ahead window `tau = 5` viscous turnover times from section 3 is **not** redefined here; this patch inherits it unchanged. - `K = 4`, `N = 16`, `G = 100`, `k_f = 4` from sections 1–2 are inherited unchanged. - `delta_proxy_min = 0.01` is the v0 default for the protocol's vacuity gate; it is pre-registered here, not after lock execution. No locked figure from sections 1–6 is re-stated at a different value in this patch. ## 8. Execution Harness The runnable harness is: ```powershell python scripts\pde_c1_kolmogorov_cell.py ``` It implements the v0 cell as a pseudospectral vorticity-form Kolmogorov-flow integrator with explicit nonlinear advection and implicit viscous damping (`pseudo_spectral_vorticity_semi_implicit_euler` in the manifest). The registered lock uses the section-7 values above. Smoke and manual-override runs are non-verdict receipts; only `--preset lock` or the pre-registered `--preset fallback` may adjudicate the C1 branch. ### 8.1 Validation smoke ```powershell python scripts\pde_c1_kolmogorov_cell.py ` --preset smoke ` --out results\proof\c1-kolmogorov-v0-smoke ``` Current smoke receipt (2026-05-28): - Read-back path: `results/proof/c1-kolmogorov-v0-smoke/manifest.json`. - Result: `SMOKE_ONLY`; no C1 negative or positive may be filed. - Runtime: `4090` steps in `2.046` seconds (`~1999` steps/sec). ### 8.2 Capped rate probe Non-verdict probe used only to estimate the registered lock: ```powershell python scripts\pde_c1_kolmogorov_cell.py ` --preset lock ` --allow-unregistered-overrides ` --sample-count 100 ` --out results\proof\_c1-kolmogorov-rate-probe ``` Measured 2026-05-28: - `105450` steps in `48.113` seconds (`~2192` steps/sec). - Registered lock step count: `100000 + (50000 - 1) * 50 + 500 = 2600450` steps. - Extrapolated lock wall-clock: about `19.8` minutes. - Fallback step count: `100000 + (200000 - 1) * 50 + 500 = 10100450` steps. - Extrapolated fallback wall-clock: about `76.8` minutes. Per [`../../AGENTS.md`](../../AGENTS.md) "The ~10-minute rule", the lock and fallback are **not inline-agent commands**. They are operator / long-budget runner commands. ### 8.3 Registered lock command ```powershell python scripts\pde_c1_kolmogorov_cell.py ` --preset lock ` --out results\proof\c1-kolmogorov-v0-lock ``` Read-back: - `results/proof/c1-kolmogorov-v0-lock/manifest.json` - `results/proof/c1-kolmogorov-v0-lock/bin-summary.csv` - `results/proof/c1-kolmogorov-v0-lock/PDE_C1_KOLMOGOROV_RESULTS.md` Branches: - `PDE-C1-NEG-A`: at least one evaluated bin has minority fraction `> delta_action`. - `STRICTNESS_WITNESS_POSITIVE`: coverage gate passes and no evaluated bin crosses `delta_action`, under the proxy selector. - `DEFERRED_COVERAGE`: `S_eval < S_pos`; run the one pre-registered fallback. ### 8.4 Pre-registered fallback command Run only after a lock receipt returns `DEFERRED_COVERAGE`: ```powershell python scripts\pde_c1_kolmogorov_cell.py ` --preset fallback ` --out results\proof\c1-kolmogorov-v0-fallback ``` Any further sample-count increase, binning change, or threshold change after the lock/fallback aggregation files `PDE-C1-NEG-B` per [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md) section 7. ## Sample Fiber Pairs (Illustrative, Not Adjudicated) The four pairs below are *not* a verdict. They sketch the kind of construction the v0 procedure expects, so a reviewer can see the predicate has bite without running the procedure. | # | Common `Phi_4(u_a) = Phi_4(u_b)` | High-mode tail difference | Hypothesised `E_K(tau)` for `u_a` | Hypothesised `E_K(tau)` for `u_b` | Predicted outcome | |---|---|---|---|---|---| | 1 | low-band well below `E_max`, near-laminar | tiny | stays below | stays below | common `no_op` | | 2 | low-band near `E_max`, near-burst | tiny | crosses | crosses | common `damp_low_band` | | 3 | mid `E_K`, high-mode excitation only in `u_a` | medium, enstrophy-cascade-active in `u_a` | crosses (inverse-cascade pump-up) | stays below | **incompatible** => candidate `PDE-C1-NEG` | | 4 | mid `E_K`, high-mode excitation symmetric | medium | crosses | crosses | common `damp_low_band` | Pair 3 is the critical predicate-test instance. If the truncated Kolmogorov dynamics realises a positive-measure version of pair 3 under the eventual continuous-fiber protocol, the C1 candidate is negative at this cell. If pair-3-shaped fibers occupy only a `mu`-null set, the C1 candidate is a strictness-witness positive at this cell. ## What This v0 Closes And Does Not Close **Closes / improves (for the C1 promotion criteria):** - **Promotion criterion (d)** of [`SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md) is now closed at the predicate-and-parameter level: the pre-registered failure boundary is fixed at a specific cell, and the section 7 patch instantiates every parameter slot the [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md) requires (`epsilon_K`, `h_K`, `n_min`, `delta_action`, `S_pos`, `N_sample`, burn-in length, integration step, action tie-break order). Final close awaits the external PDE reviewer. - **Promotion criterion (b)** is closed at the procedure level: the section 7 patch makes the section 5 adjudication runnable end-to-end. Execution remains the next step. - The comparator no longer relies on the invalid inference `K < m_det => state-insufficient`. The desk-certified insufficiency claim is finite-Galerkin non-injectivity on `B_abs`. - The continuous-fiber gate and the provisional-selector gate are addressed by the protocol: tolerance + binning for the former, named proxy `\hat{pi}` with substitution conditions for the latter. **Does not close:** - Promotion criterion (a): Front-A vacuity rebuttal is the job of the C1 sidecar, not of this cell-set instance. The instance only enables the rebuttal to be tested. - Promotion criterion (c): external PDE reviewer is still unnamed. - Exact-support state insufficiency: `Phi_K` is not yet certified non-injective on `supp(mu_SRB)` unless the section 4.1 twin-state certificate is supplied (bridged in protocol section 6, instantiation deferred). - The pre-registered negative remains unadjudicated: it can be triggered (`PDE-C1-NEG-A` or `PDE-C1-NEG-B`), refuted, or left open by a future numerical run. ## Lock Execution Disposition (2026-05-28) The v0 lock cell was executed on 2026-05-28 via [`scripts/pde_c1_kolmogorov_cell.py`](../../scripts/pde_c1_kolmogorov_cell.py) with the section 7 patch values. Receipt and manifest are at `results/proof/c1-kolmogorov-v0-lock/`. The harness completed in 1464 seconds (~24 min) with `steps_per_second ≈ 1776` and produced a procedural `STRICTNESS_WITNESS_POSITIVE` under the verdict rule as written *before* the 2026-05-28 vacuity-gate amendment to the fiber protocol. ### Honest reading - `damp_fraction = 0`: zero of 50,000 sample windows crossed `E_max = 0.09765` in the look-ahead horizon. - `sample_energy_max = 0.09761 < e_max = 0.09765`: the post-burn-in trajectory never reached the burn-in 95th percentile. - `burnin_energy_max - burnin_energy_min = 0.0002` (~0.2% of the mean): the burn-in trajectory itself sits in a deeply relaxational regime near the linearly stable laminar Kolmogorov solution at `G = 100`, `k_f = 4`. - Bin distribution: 70% of samples (34,883 / 50,000) fall in a single fiber bin; the next-largest holds 13% (6,310). The attractor is concentrated near a fixed point. ### Disposition under the amended verdict rule The fiber-protocol vacuity gate added 2026-05-28 (`delta_proxy_min = 0.01`) re-reads this receipt as `DEFERRED_VACUITY`: the proxy selector is essentially constant on the sampled support, so the strictness predicate has no discriminative content. No verdict is filed against the v0 cell on this lock. The procedural `STRICTNESS_WITNESS_POSITIVE` in the harness output predates the amendment and is **superseded**; it is not promoted into the ledger as a positive. ### Why this is not `PDE-C1-NEG-B` `PDE-C1-NEG-B` covers post-hoc retuning of pinned parameters in response to seeing an aggregation. The amended verdict rule is a methodology amendment that closes a logical-completeness gap in the protocol (a constant proxy can never separate fibers); it does not relax a pinned parameter to flip the receipt's interpretation. The new `delta_proxy_min = 0.01` is pre-registered for **all future v0+ runs**, including any v0 re-run, and is not tuned against the specific damp_fraction observed here. ### Bridge to v1 The v0 cell is non-discriminative because 2D Kolmogorov flow at `G = 100`, `k_f = 4` is linearly stable. A cell-set v1 instance at `k_f = 2` (lower forcing-mode stability threshold, supercritical at the same Grashof number) is staged at [`PDE_C1_CELLSET_KOLMOGOROV_v1.md`](PDE_C1_CELLSET_KOLMOGOROV_v1.md); the harness exposes it as the `lock_v1` preset. v0 remains filed as a desk-auditable cell with an executed-and-deferred lock receipt; v1 is the discriminative-regime sibling, not a retune of v0. ## Status - **Drafted.** 2026-05-28. - **Comparator tightened.** The state-reconstruction comparator now separates the finite-Galerkin non-injectivity witness from the literature state-reconstructive sufficiency reference. - **Fiber protocol patched.** Section 7 instantiates every parameter slot required by [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md): `epsilon_K`, `h_K`, `n_min`, `delta_action`, `S_pos`, `N_sample`, burn-in length, integration step, action tie-break order. Pinning is pre-registered before any sampling. - **Desk-auditable.** A reviewer can read this file end-to-end and audit the predicate structure without running code. - **Unreviewed.** No external PDE reviewer has signed off on the regime / observation / objective / comparator / classifier / negative-branch quintuple or on the section 7 parameter values as a faithful instance of the Postulate-1 reading. - **Unrun.** No numerical execution of the section 5 classifier has been attempted. The "Hypothesised" column of the sample table is a predicate-shape demonstration, not a verdict. ## Cross-References - [`PDE_DETERMINING_MODES_POSTULATE1.md`](PDE_DETERMINING_MODES_POSTULATE1.md) - the C1 sidecar this file instantiates. The cell-set requirements list there is what this file pins. Cross-back: that file's "Pre-Registered Cell Set For The Next Pass" section now points here. - [`PDE_C1_FIBER_PROTOCOL.md`](PDE_C1_FIBER_PROTOCOL.md) - the fiber protocol whose parameter slots the section 7 patch instantiates. The protocol pins methodology (tolerance object, bin lattice, proxy selector, adjudication procedure, parameter-pinning discipline); section 7 pins numerical values. - [`../SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md) - the ledger. Promotion criteria for C1 are defined there. - [`../NAVIERSTOKES_LITPASS_MEMO.md`](../NAVIERSTOKES_LITPASS_MEMO.md) - the lit-pass record. Determining-mode / synchronization references supply the state-reconstructive sufficiency side of the section 4 comparator; the `Phi_K` state-insufficiency side is the finite-Galerkin non-injectivity witness in section 4.1. - [`PHASE2_MDP.md`](PHASE2_MDP.md) - the finite-MDP Phase 2 result this cell-set translates into a PDE/Galerkin instance. The fiber criterion in section 5 is the Phase 2 fiber-selector condition with the state space replaced by the admitted Galerkin domain and the action space by `A_ctrl`.