# PDE C1 Fiber Protocol — Tolerance and Binning > Methodology artifact for the continuous-fiber / tolerance gate of > [`PDE_C1_CELLSET_KOLMOGOROV.md`](PDE_C1_CELLSET_KOLMOGOROV.md) > section 6 ("Open measure item") and the provisional-selector gate of > section 3 ("Open review item"). Status: drafted, unreviewed, > 2026-05-28. This file pins parameter *roles* and the adjudication > *procedure*; concrete parameter *values* belong in the cell-set v0 (or > a v0 patch). Alternative protocols remain admissible if a reviewer > prefers them; this is the v0 default. ## Entry And Gate This protocol enters from [`SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md) Promotions: > The v0 comparator has been tightened: ... Promotion criterion (d) is > staged but still needs the continuous-fiber / tolerance protocol; > criterion (b) is partially closed (procedure defined, not executed). and from [`PDE_C1_CELLSET_KOLMOGOROV.md`](PDE_C1_CELLSET_KOLMOGOROV.md) section 6: > 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. The protocol below is the **binning + tolerance** branch of the disjunction. The disintegration branch remains an admissible alternative; if a reviewer prefers it, the cell-set's Reading-2 fiber criterion can be ported to a regular-conditional statement without discarding this artifact (the binning instance then serves as the numerical approximation of the disintegration). ## Claim Boundary This protocol does **not** claim: - a measure-theoretic disintegration theorem for the Kolmogorov-flow attractor measure `mu_SRB`; - that binning resolves all continuous-measure issues — only that it produces an executable adjudication predicate under a pre-registered parameter set; - that the proxy selector `\hat{pi}` is `J`-optimal — it is named here as a proxy with explicit substitution conditions, not as a derived optimum; - that the support-level twin-state certificate (cell-set v0 § 4.1 attractor-support caveat) is closed — it is named here as a separate open item with a one-line spec, instantiation deferred. It claims only a runnable predicate structure: 1. a tolerance object on signature space defines a continuous fiber relaxation; 2. a binning lattice defines an executable numerical approximation of the tolerance object; 3. a per-bin majority-action selector with a minority-fraction sub-predicate adjudicates the Reading-2 fiber criterion on bins with positive sample count; 4. a pre-registered coverage gate decides when the verdict is interpretable. ## Local Symbols | Symbol | Definition | | --- | --- | | `Phi_K` | Signature map from `X` to `R^{d_K}` (cell-set v0 § 2). The signature mode count `K` is a **cell-set parameter** (added 2026-05-28): cells may pin different `K` values to trade signature richness against bin-coverage cost in `R^{d_K}` (signature dim `d_K = 2K^2` for 2D Fourier signatures). The forced-mode inclusion constraint must be checked when changing `K`. | | `epsilon_K` | Tolerance radius on signature space `R^{d_K}`. | | `B_K` | Pre-registered bin partition of `R^{d_K}` (lattice spacing, alignment, bounding box). | | `b` | A bin in `B_K`. | | `n(b)` | Number of sampled states from `mu_SRB` falling in bin `b`. | | `n_min` | Pre-registered minimum sample count per bin required to evaluate that bin. | | `\hat{pi}(u)` | Proxy action label per state, computed from the cell-set v0 § 3 deterministic `E_K` overshoot check. | | `\hat{pi}_bin(b)` | Majority proxy action in bin `b` over states with `n(b) >= n_min`. | | `delta_action` | Pre-registered minority-fraction threshold; a bin with minority fraction above this is flagged as fiber-incompatible. | | `S_pos` | Pre-registered minimum fraction of evaluated bins required before the verdict is interpreted (coverage gate). | | `delta_proxy_min` | Pre-registered lower bound on the proxy-action discrimination floor; if the global `damp_fraction` is outside `[delta_proxy_min, 1 - delta_proxy_min]`, the proxy is essentially constant and the predicate is non-discriminative (vacuity gate). | | `e_max_burnin_fraction` | Cell-set parameter in `(0, 1]`; `E_max` is the 95th percentile of the last `e_max_burnin_fraction × burnin_steps` samples of the burn-in trace. Default `1.0` (full burn-in). Smaller fractions are pinned by cells that suspect transient contamination of the percentile. | | `N_sample` | Pre-registered number of post-burn-in samples drawn from `mu_SRB`. | All symbols above are parameter **slots**. Their values are pinned in the cell-set v0 (or a patch), not here. ## 1. Tolerance Object (Theoretical) The continuous-fiber predicate of the Reading-2 fiber criterion is relaxed to a tolerance ball on signature space: ```text C_sigma^{epsilon_K} = { u in X : ||Phi_K(u) - sigma||_{R^{d_K}} <= epsilon_K }. ``` The Reading-2 condition becomes: the proxy selector is constant on `epsilon_K`-tolerance balls, up to a pre-registered minority fraction `delta_action`, on a pre-registered coverage fraction `S_pos` of the support. This is the object the C1 sidecar's fiber criterion now refers to in the continuous setting. It is named here to give the criterion an executable continuous-measure surface; section 2 below provides the executable approximation. ## 2. Binning Instance (Runnable) The runnable predicate replaces `epsilon_K`-tolerance balls with bins from a pre-registered lattice partition `B_K` of `R^{d_K}`: - **Lattice.** `B_K` is a uniform cubic lattice on `R^{d_K}` with per-coordinate spacing `h_K` chosen so that the typical bin diameter is approximately `epsilon_K`. The bounding box is the empirical range of `Phi_K(u)` over a pre-registered burn-in trajectory. - **Alignment.** Lattice origin is pinned at the bounding-box corner computed during burn-in; alignment is *not* re-optimised after sampling. - **Bin** `b in B_K` is the open lattice cell. The bin lattice approximates the tolerance ball; for a state `u`, the bin containing `Phi_K(u)` is an `O(epsilon_K)`-tolerance neighbourhood of `Phi_K(u)` in signature space. Approximation faithfulness is itself review-checkable: a reviewer who prefers exact tolerance balls can replace `B_K` with a ball-cover at the cost of slower adjudication. **Coverage vs. discrimination tradeoff (added 2026-05-28).** Bin count scales as `(R_attractor / h_K)^{d_K}` where `R_attractor` is the attractor extent in signature space and `d_K = 2K^2` for 2D Fourier signatures. Uniform binning at fixed `h_K` becomes infeasible — bin count exceeds `N_sample` — when the attractor is high-dimensional in `R^{d_K}`. The C1 v2 / v4 lock and v4 fall-back executions surfaced this empirically: at `(k_f = 2, G ≥ 200, K = 4)`, the attractor occupied ~45k–139k bins depending on sample count, with `N_sample / bin_count ≈ 1.1–1.4` regardless of how `N_sample` was scaled. The pre-registered binning was empirically too fine. Cells facing this regime may pin a smaller `K` (per the Local Symbols amendment above) to reduce `d_K`, at the cost of a less rich signature. **Correction (2026-05-28, after C1 v5).** The `(R / h_K)^{d_K}` scaling above is **wrong as a remedy guide**, and the "pin smaller `K`" prescription it motivated was **falsified** by the C1 v5 lock. The attractor does not fill the `d_K`-dimensional signature box; it is a low-dimensional invariant set, so occupied-bin count tracks the attractor's **box-counting dimension `D_box` at scale `h_K`** — `occupied_bins ≈ (R / h_K)^{D_box}` with `D_box << d_K` — which is invariant to the embedding dimension. v5 halved `d_K` (32 → 18) and coarsened `h_K` 29%, yet occupied bins fell only 16.5% (45,827 → 38,281) and coverage still failed. The genuine lever is `h_K` (equivalently `epsilon_K`), but coarsening it enough to clear coverage conflicts with the tolerance-fidelity requirement of §1 — a tension that may not be resolvable by binning-parameter tuning. See [`PDE_C1_LOCK_EXECUTION_SYNTHESIS.md`](PDE_C1_LOCK_EXECUTION_SYNTHESIS.md) §4–5 for the obstruction and the kNN/kernel adjudication fork that targets it. `K` remains a legitimate cell-set parameter (it affects signature richness and discrimination), but it is **not** the coverage lever it was introduced as. ## 3. Proxy Selector The cell-set v0 § 3 deterministic `E_K` overshoot check is renamed here as a proxy label, with explicit substitution conditions: ```text \hat{pi}(u) = damp_low_band iff the no_op evolution from u carries E_K above E_max at some t in [0, tau]; = no_op otherwise. ``` **Proxy status.** `\hat{pi}` is *not* derived from a `J`-optimal calculation; it is a deterministic label computed from the truncated dynamics under the registered objective. The protocol adjudicates the Reading-2 fiber predicate under `\hat{pi}`. If an external reviewer treats `\hat{pi}` as faithful to the intended objective, the protocol result reads as the Postulate-1 predicate. If the reviewer requires a derived `J`-optimal selector, this protocol's result reads as a proxy adjudication and is replaced by re-running steps 4–6 with the reviewer's selector substituted for `\hat{pi}`. This resolves cell-set v0 § 3 "Open review item" by **naming the proxy explicitly** and pinning the substitution procedure, rather than by adding a cost-of-damping term. Both branches remain admissible at review. **E_max windowing (added 2026-05-28).** The cell-set's safety envelope `E_max` is a *cell-set parameter*, not a protocol parameter. When burn-in includes transient excursions that bias the percentile estimate above the steady-state attractor's typical excursions (observed in C1 cell-set v0/v1/v3 lock executions), the cell may pin an `e_max_burnin_fraction ∈ (0, 1]` and define `E_max` as the 95th percentile of the *last* `e_max_burnin_fraction × burnin_steps` samples of the burn-in trace. The default `e_max_burnin_fraction = 1.0` reproduces the original "full burn-in percentile" rule. Cells that suspect transient contamination may pin a smaller fraction (e.g. `0.25` for the last quarter). This is a cell-set choice pre-registered before sampling, **not** a post-hoc retune; changing the fraction after a receipt files is `PDE-C1-NEG-B`. ## 4. Per-Bin Majority Selector And Minority Fraction For a bin `b` with `n(b) >= n_min`: ```text \hat{pi}_bin(b) = argmax_{a in A_ctrl} |{u in b : \hat{pi}(u) = a}|. m(b) = 1 - |{u in b : \hat{pi}(u) = \hat{pi}_bin(b)}| / n(b). ``` `\hat{pi}_bin(b)` is the bin-local majority action; ties broken deterministically by action ordering pinned in the cell-set v0. `m(b) in [0, 1/2]` is the bin-local minority fraction. **Fiber-incompatibility flag.** A bin is flagged fiber-incompatible iff `m(b) > delta_action`. ## 5. Adjudication Procedure Pre-registered, in this order: 1. **Pin parameters.** `epsilon_K`, `h_K`, `n_min`, `delta_action`, `S_pos`, `N_sample`, burn-in length, integration step, action tie-break order. Fixed in the cell-set v0 (or v0 patch) before any sampling. 2. **Burn-in.** Integrate the truncated Galerkin NSE from a fixed initial condition; discard the burn-in window. 3. **Sample.** Draw `N_sample` post-burn-in states approximating independent samples from `mu_SRB` (e.g. sub-sampling at intervals of one Lyapunov time, pinned in v0). 4. **Bin.** Assign each sample to its bin in `B_K`. 5. **Label.** Compute `\hat{pi}(u)` for each sample using the deterministic overshoot check. 6. **Per-bin aggregation.** For each bin with `n(b) >= n_min`, compute `\hat{pi}_bin(b)` and `m(b)`. 7. **Coverage gate.** Let `S_eval = |{ b : n(b) >= n_min }| / |{ b : n(b) >= 1 }|`. If `S_eval < S_pos`, the result is **deferred** for insufficient coverage; record and stop without verdict. 8. **Vacuity gate.** Compute the global proxy-action distribution: `damp_fraction = (sum over samples of [\hat{pi}(u) == damp_low_band]) / N_sample`. If `damp_fraction < delta_proxy_min` or `damp_fraction > 1 - delta_proxy_min`, 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. Record as **`DEFERRED_VACUITY`** and stop without verdict. **Structural-vacuity precedence (added 2026-05-28).** If `damp_fraction` is *exactly* `0` or exactly `1` (no statistical noise can resolve the proxy to either action), the vacuity is structural rather than statistical: increasing `N_sample` cannot shake it. In this case the vacuity gate takes precedence over the step-7 coverage gate — file `DEFERRED_VACUITY` even if `S_eval < S_pos` would otherwise fire `DEFERRED_COVERAGE`. The standard `DEFERRED_COVERAGE` fall-back to `N_sample = 200,000` is **not admissible** for structural-vacuity cases, because the proxy is structurally inaccessible to the safety predicate on this attractor regardless of sample count. Re-pinning to a different regime (a new cell-set instance) is the only honest path. 9. **Verdict.** - If any evaluated bin has `m(b) > delta_action`, **file `PDE-C1-NEG-A`** at this cell. - Otherwise, the cell is a **strictness-witness positive**: under the proxy `\hat{pi}` and the registered binning, the fiber criterion holds on the evaluated coverage *and* the proxy selector exhibits non-trivial action discrimination on the sampled support. Both deferral branches in steps 7 and 8 are structurally distinct from the filed-negative and filed-positive branches: they are the protocol's named ways of refusing to interpret an under-sampled or non-discriminative adjudication attempt, and they do **not** count as a positive. ## 5b. kNN / Disintegration Adjudication (adopted 2026-05-28) The binning adjudicator (§2, §4, §5) was exhausted across C1 cell-sets v0–v5: faithful (`h_K ≈ epsilon_K`) bins are under-populated on high-dimensional attractors, and occupied-bin count tracks the attractor's box dimension rather than `d_K`, so neither sample-budget nor `K` reduction clears coverage (see [`PDE_C1_LOCK_EXECUTION_SYNTHESIS.md`](PDE_C1_LOCK_EXECUTION_SYNTHESIS.md) and the §1 design). This section adopts the **disintegration branch** named in §1 as an admissible alternative, realized nonparametrically by k-nearest-neighbour estimation. Design proposal and sign-off: [`PDE_C1_KNN_ADJUDICATION_DESIGN.md`](PDE_C1_KNN_ADJUDICATION_DESIGN.md) (Fork A, signed off 2026-05-28). It is an **alternative adjudicator, not a replacement**: the binning procedure remains valid for cells where it achieves coverage; a cell pins `--adjudicator {bin,knn}`. The kNN adjudicator estimates the disintegration of `mu` over `Phi_K`: the `k` nearest neighbours of a sample approximate a draw from `mu` conditioned on `Phi_K ≈ Phi_K(u_i)` within radius `r_k(u_i)`, and the local minority fraction estimates the conditional proxy non-constancy `1 - max_a mu_sigma(a)`. **Per-sample statistic.** For each sample `u_i` (L2 metric, the tolerance-ball metric of §1): - `N_k(u_i)` = the `k` nearest neighbours including `u_i` (so `|N_k| = k`); - `r_k(u_i)` = distance to the `k`-th neighbour (the per-sample fidelity radius); - `a_maj(u_i)` = majority proxy action in `N_k` (ties → `no_op`); - `m_i` = local minority fraction `= 1 - |{u in N_k : \hat{pi}(u) = a_maj}| / k`. **Gates and verdict** (pre-registered, parallels §5): 1. **Vacuity gate** — global `damp_fraction` in `[delta_proxy_min, 1 - delta_proxy_min]`; exact `0`/`1` → `DEFERRED_VACUITY` (structural precedence unchanged). 2. **Fidelity-coverage gate** — let `F = {i : r_k(u_i) <= epsilon_K}`. If `|F| / N < S_pos`, record **`DEFERRED_FIDELITY_COVERAGE`** (a receipt name distinct from the binning `DEFERRED_COVERAGE`, so the two adjudicators' deferrals are never conflated) and stop — the attractor is too sparse at scale `epsilon_K` for a faithful local fiber test at this `k` and `N`. No automatic fall-back; admissible responses (larger `N`, a pre-registered larger `epsilon_K` with the same fidelity caveat, or reconsidering the continuous-fiber object) are recorded, not defaulted. 3. **Verdict on `F`** — let `incompat_fraction = |{i in F : m_i > delta_action}| / |F|`. If `incompat_fraction > delta_incompat`, **file `PDE-C1-NEG-A`** (fiber incompatibility). Otherwise **strictness-witness positive**: the proxy is locally constant on fibers across the fidelity-passing coverage and the vacuity gate passed. **New pre-registered constant.** `delta_incompat` (default `0.01`) — the positive-mass threshold for filing `PDE-C1-NEG-A`, so a lone incompatible sample does not flip the verdict but a robust positive-mass region does. Post-hoc change → `PDE-C1-NEG-B`, same discipline as every other pinned parameter (§7). **Honest-deferral property.** Unlike the binning `S_eval = 0` (which reports only that coverage failed), the kNN deferral ships the full `r_k` distribution (a radius histogram), quantifying *how far from fidelity* the attractor sits — a measurable map even when no verdict is filed. **Implementation.** Exact neighbour search (`sklearn.neighbors.BallTree`, robust for the `d_K = 18` and `d_K = 32` regimes where KD-trees degrade); shares the deterministic integration via the harness `--adjudicator knn` flag, so only the post-sampling backend swaps. ## 6. Support-Level Certificate (Bridge) The cell-set v0 § 4.1 attractor-support caveat introduces a twin-state search certificate to upgrade the non-injectivity claim from `B_abs` to `supp(mu_SRB)`. That certificate is a separate kind of artifact — a sampling-based numerical certificate, not a methodology document — and is named here as a follow-up item, not instantiated. **One-line spec for the deferred certificate.** Sample `N_twin >= N_sample` states from `mu_SRB`; for each pair `(u_i, u_j)` with `||Phi_K(u_i) - Phi_K(u_j)|| <= epsilon_K`, compute the high-mode separation `||Q_K(u_i) - Q_K(u_j)||`; if a pre-registered fraction of such pairs has high-mode separation at least `delta_H`, non-injectivity on `supp(mu_SRB)` is certified. Parameters `N_twin`, `delta_H`, sampling interval, acceptance count pinned at file time of the certificate, not here. Instantiation started 2026-05-28 in [`PDE_C1_TWIN_STATE_CERTIFICATE.md`](PDE_C1_TWIN_STATE_CERTIFICATE.md): the harness now exposes `--adjudicator twin-state`, with `k_twin = 50`, `delta_H = max(1e-6, 0.05 * median ||Q_K||)`, a 1% witness-sample threshold, and a 100 unique-pair floor. The smoke receipt is plumbing only; the support-level bridge remains open until a verdict-bearing lock receipt returns. This protocol's verdict in section 5 remains valid on `B_abs` without the support-level certificate. If a reviewer requires `supp(mu_SRB)`-level state insufficiency before the C1 strictness witness is treated as load-bearing, the certificate is the precondition. ## 7. Parameter Pinning Discipline The protocol commits to the following non-negotiable order of operations, per the spec self-consistency rule ([`../SCIENTIFIC_CRITERIA.md`](../SCIENTIFIC_CRITERIA.md)): - `epsilon_K`, `h_K`, `n_min`, `delta_action`, `S_pos`, `delta_proxy_min`, `N_sample`, burn-in length, integration step, action tie-break order, lattice bounding box, lattice alignment are pinned **before** step 3 (sampling). - Re-tuning any parameter after step 6 (aggregation) and re-running steps 7–8 counts as filing `PDE-C1-NEG-B` (cell-set drift / post-hoc tuning). Note: this introduces the negative-receipt label `PDE-C1-NEG-B` on the C1 side, parallel to the `PDE-C2-NEG-B` receipt on the C2 side ([`PDE_C2_SHELL_SIGNATURE_SCOPING.md`](PDE_C2_SHELL_SIGNATURE_SCOPING.md) § Pre-Registered Negative). The original `PDE-C1-NEG` becomes `PDE-C1-NEG-A` retroactively for parallelism; existing references to `PDE-C1-NEG` should be read as `PDE-C1-NEG-A`. - Refining the binning lattice after step 6 to make a marginal verdict flip is `PDE-C1-NEG-B`. - Adding bins post hoc that move the verdict is `PDE-C1-NEG-B`. The deferred-insufficient-coverage branch (step 7 of section 5) is **not** `PDE-C1-NEG-B`. Increasing `N_sample` to escape a deferral and re-running, with all other parameters held, is admissible iff the increase is pre-registered as a fall-back path before sampling. The deferred-vacuity branch (step 8 of section 5) is also **not** `PDE-C1-NEG-B`. It indicates the cell is in a non-discriminative regime — the proxy selector is constant on `mu_SRB`-support so the strictness predicate has no operational content. **No fall-back is admissible to escape a vacuity deferral on the same cell.** Relaxing `delta_proxy_min` post hoc to re-interpret a constant-proxy run as a strictness-witness positive files `PDE-C1-NEG-B` (the predicate was moved, not the data). Re-pinning to a discriminative regime (higher `G`, different `k_f`, alternative forcing) requires a *new cell-set instance* (e.g. v1), filed as a separate artifact rather than as a retune of the original cell. ## 8. What This Protocol Closes And Does Not Close **Closes:** - **Criterion (b)(ii) — continuous-fiber measure protocol.** The Reading-2 fiber criterion now has a runnable predicate on bins, with the tolerance object as its continuous-measure relaxation. - **Criterion (b)(i) — provisional selector.** `\hat{pi}` is named explicitly as a proxy with a pinned substitution procedure under reviewer redefinition. The cost-of-damping branch remains admissible but is not required for adjudication. - **Verdict-rule completeness gap (added 2026-05-28).** Section 5's vacuity gate distinguishes a constant-proxy run from a fiber-coherent run. Without it, a cell whose objective is trivially satisfied by a single action would file a vacuous strictness-witness positive. **Bridges (named, not closed here):** - **Criterion (b)(iii) — support-level certificate.** Named in section 6 with a one-line spec; instantiation deferred to a dedicated certificate artifact. **Does not close:** - Criterion (a) — Front-A vacuity rebuttal remains an external-review question on the C1 sidecar, not on this protocol. - Criterion (c) — named external PDE reviewer remains open. - Criterion (d) — pre-registered failure boundary moves from "staged" to **"closed pending external review"**. The protocol pins the parameter roles, and the Kolmogorov cell-set v0 section 7 now pins their concrete values. ## 9. Cross-File Consequences The cell-set v0 patch requested by this protocol landed in [`PDE_C1_CELLSET_KOLMOGOROV.md`](PDE_C1_CELLSET_KOLMOGOROV.md) section 7. If this protocol survives review: - update [`../SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md) Promotions to keep criterion (d) closed at the artifact level and to adjudicate criterion (b) once the registered lock receipt lands; - introduce `PDE-C1-NEG-A` / `PDE-C1-NEG-B` as the parallel labels to `PDE-C2-NEG-A` / `PDE-C2-NEG-B`. Reading `PDE-C1-NEG` as `PDE-C1-NEG-A` is the retroactive convention. If this protocol fails review: - file the named negative against this artifact (not against the C1 sidecar); - explore the disintegration branch of the cell-set v0 § 6 disjunction as an alternative to binning + tolerance; - consider whether a smaller observation than `Phi_K` (e.g. low-band energy `E_K` directly, treating it as a scalar fiber predicate) sidesteps the continuous-measure issue by reducing fiber dimension. ## 10. External Review Path Promotion of this protocol requires a reviewer to confirm: 1. the tolerance object in section 1 is a faithful continuous-measure relaxation of the Reading-2 fiber criterion; 2. the binning instance in section 2 is a faithful numerical approximation of the tolerance object on `R^{d_K}`; 3. the proxy selector `\hat{pi}` is either an acceptable proxy for the intended objective or substitutable per section 3; 4. the parameter pinning discipline in section 7 closes the post-hoc-tuning attack surface. The reviewer for this protocol can be the same PDE analyst named for the C1 sidecar at promotion, or a separate methodologist (statistical methodologist familiar with measure-theoretic discretisation, or a data-assimilation researcher familiar with observer-design parameter pre-registration). Reviewer named at promotion. ## Status - **Drafted.** 2026-05-28. - **Desk-auditable.** A reviewer can read this file end-to-end and audit the predicate structure without running code. - **Unreviewed.** No external reviewer has signed off on the tolerance object, binning instance, proxy selector, or parameter pinning discipline. - **Unrun.** No numerical execution of the section 5 procedure has been attempted as a verdict-bearing lock. A non-verdict smoke receipt exists at `results/proof/c1-kolmogorov-v0-smoke/manifest.json`. - **v0 patch landed.** The cell-set v0 section 7 instantiates `epsilon_K`, `h_K`, `n_min`, `delta_action`, `S_pos`, `N_sample`, burn-in length, integration step, action tie-break order, and section 8 stages the registered lock / fallback commands. ## Cross-references - [`PDE_C1_CELLSET_KOLMOGOROV.md`](PDE_C1_CELLSET_KOLMOGOROV.md) — the cell-set v0 this protocol services. Sections 3 and 6 of the cell-set v0 are the gates this protocol addresses; section 4.1's attractor-support caveat is bridged here (section 6). - [`PDE_DETERMINING_MODES_POSTULATE1.md`](PDE_DETERMINING_MODES_POSTULATE1.md) — the C1 sidecar. The Reading-2 fiber criterion gets its continuous-measure realisation through this protocol. The strictness witness's "independent non-injectivity" requirement is upstream of this protocol and unchanged. - [`PHASE2_MDP.md`](PHASE2_MDP.md) — the finite-MDP fiber-selector Phase-2 result. This protocol is the PDE/Galerkin-instance translation, with binning replacing the finite state-action partition. - [`PDE_C2_SHELL_SIGNATURE_SCOPING.md`](PDE_C2_SHELL_SIGNATURE_SCOPING.md) — C2 scoping. The `PDE-C1-NEG-B` / `PDE-C2-NEG-B` parallelism introduced in section 7 of this protocol aligns the two candidates' post-hoc-tuning receipt labels. - [`../SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md) — the ledger. Criterion (d) closure depends on this protocol plus the cell-set v0 patch. - [`../SCIENTIFIC_CRITERIA.md`](../SCIENTIFIC_CRITERIA.md) — the parameter-pinning discipline in section 7 inherits the pre-registered audit-chain rule from this document.