# PDE C2 — Shell-Model Signature Detector Scoping

> Proof-track scoping document for the PDE-substrate empirical leg of
> [`COARSE_GRAINING_PROOF_ROADMAP.md`](../COARSE_GRAINING_PROOF_ROADMAP.md)
> Phase 4 (sibling to the three-body measured leg, not a replacement).
> Status: drafted, unreviewed, unrun, 2026-05-28. Commissioned from
> [`SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md) Candidate 2.
> This document pins research object, model commitments, channel
> taxonomy, baselines, task, two-sided negative, and promotion path.
> It does **not** instantiate a cell set; the cell-set v0 is a deferred
> follow-up artifact, in parallel to the C1 staging
> ([`PDE_DETERMINING_MODES_POSTULATE1.md`](PDE_DETERMINING_MODES_POSTULATE1.md)
> → [`PDE_C1_CELLSET_KOLMOGOROV.md`](PDE_C1_CELLSET_KOLMOGOROV.md)).

## Entry And Gate

This scoping enters from the Navier–Stokes ledger's Front B:

> A Sundog-style signature derived from shell-model trajectories
> detects imminent intermittent bursts inside a pre-registered
> operating envelope, with regret → 0 against an oracle baseline and
> bounded-away-from-zero outside the envelope (matched-seed).

The pre-registered negative is **two-sided** and quoted verbatim from
[`SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md) Candidate 2:

> (a) If the signature detector's burst-detection lead-time and
> false-positive rate are not strictly inside the Pareto frontier of
> the matched-budget baselines on the pre-registered cell set,
> signatures are vacuous on shell models — record and stop.
> (b) If signatures match baselines on the registered cell set but
> the cell set was the entire span, the result is over-fit to the
> data — record as Front-B vacuity.

Negative-receipt labels: `PDE-C2-NEG-A` for (a), `PDE-C2-NEG-B` for
(b). These are introduced here so a future numerical run files them
under the same label the ledger expects.

## Claim Boundary

This scoping does **not** claim:

- a solution to the Navier–Stokes Millennium problem;
- that shell models reproduce NSE blow-up structure;
- that path signatures outperform DMD, critical-slowing-down,
  lacunarity, or Rényi-entropy detectors on any cell — that is the
  experimental question, not the position;
- that hidden self-similarity coordinates are a Sundog invention —
  they are due to Mailybaev 2022
  ([arxiv 2201.04005](https://arxiv.org/abs/2201.04005));
- that a shell-model positive result would extend to full
  2D-perturbed or 3D small-data NSE without further work — Front-B
  reach is named in the ledger as a separate failure mode.

It claims only a pin on the empirical question and the
audit surface that will adjudicate it:

1. shell models supply a laptop-tractable substrate where
   intermittency rhymes with NSE near-singular dynamics;
2. a signature detector on registered channels has a well-posed
   matched-budget comparison against four off-the-shelf baselines;
3. the comparison either lands strictly inside the Pareto frontier,
   fails as (a), or fails as (b).

All literature citations live in
[`../NAVIERSTOKES_LITPASS_MEMO.md`](../NAVIERSTOKES_LITPASS_MEMO.md);
this scoping references them by pointer.

## Research Object

> Let `u_n(t) ∈ C` for `n = 1, ..., N_shells` be Sabra shell-model
> amplitudes on a geometric wavenumber ladder `k_n = k_0 λ^n` with
> viscosity `ν`, deterministic large-scale forcing on shell 1, and
> standard nonlinear coupling. Let `Φ_S : { trajectory history } →
> R^d` be a path-signature feature derived from a fixed channel set
> of the trajectory (channel set defined below). The pre-registered
> binary task `J_burst`: at each query time `t`, predict whether an
> intermittent burst occurs in the window `(t, t + τ_burst]`, where
> a burst is defined as `max_n |u_n(s)|² > E_burst` for some
> `s ∈ (t, t + τ_burst]`. Compare a `Φ_S`-based detector against
> matched-budget DMD, critical-slowing-down, recurrence lacunarity,
> and Rényi-entropy detectors on a pre-registered cell set, with
> matched-seed evaluation, on the Pareto frontier of lead-time vs.
> false-positive rate.

The exact values of `N_shells`, `λ`, `ν`, `τ_burst`, `E_burst`, and
the signature truncation depth are **deferred to the cell-set v0**
in parallel with the C1 staging
([`PDE_C1_CELLSET_KOLMOGOROV.md`](PDE_C1_CELLSET_KOLMOGOROV.md)). The
scoping pins their *roles* in the predicate, not their values.

## Local Symbols

| Symbol | Definition |
| --- | --- |
| `u_n(t)` | Sabra shell-model amplitude at shell `n` and time `t`. |
| `N_shells` | Number of shells in the truncation. Pinned in cell-set v0. |
| `k_n = k_0 λ^n` | Geometric wavenumber ladder; canonical `λ = 2`. |
| `T_n(t)` | Inter-shell energy transfer at shell `n`. |
| `χ_n(t)` | Hidden self-similarity coordinate at shell `n` per [arxiv 2201.04005](https://arxiv.org/abs/2201.04005). |
| `Φ_S` | Signature map from a trajectory window to `R^d`; truncation depth `D` pinned in cell-set v0. |
| `F_S` | σ-algebra generated by `Φ_S` (parallel to `F_K` of C1). |
| `J_burst` | Pre-registered binary burst-prediction objective at horizon `τ_burst`. |
| `E_burst` | Pre-registered exceedance threshold on `max_n |u_n|²`. Pinned in cell-set v0. |
| `τ_burst` | Pre-registered look-ahead horizon. Pinned in cell-set v0. |
| `B(t)` | Ground-truth binary burst indicator at time `t` (computed from the simulation). |

## Model Commitments

**Sabra primary; GOY cross-check.** The headline cells run on the
Sabra shell model (Mailybaev tradition; cleaner statistical
properties; recent hidden-self-similarity literature is Sabra-anchored).
A *robustness subset* of the headline cells re-runs on the GOY shell
model with the same channel set and the same baseline configuration,
to guard against the "works on this specific model" critique. The GOY
cross-check is not required to land at the same Pareto position; it
is required to land on the same *side* of the pre-registered negative.

Both models are simulated under standard published numerical schemes
(reviewer-checkable; the schemes are not Sundog-original). All concrete
numerics (`N_shells`, `λ`, `ν`, forcing amplitude, integration step,
warm-up time) are pinned in the cell-set v0.

## Channel Taxonomy

**Tiered channel set, minimal headline, ablate up.** The headline
detector is reported on the minimal channel set; richer channel sets
are reported as explicit ablation cells.

**Tier 0 (headline).** Shell-wise log-energy:
`c⁰_n(t) = log(|u_n(t)|² + ε)` for `n = 1, ..., N_shells`.

**Tier 1 (ablation A).** Tier 0 + inter-shell energy transfers
`T_n(t)`.

**Tier 2 (ablation B).** Tier 1 + hidden self-similarity coordinates
`χ_n(t)` per Mailybaev 2022.

The signature `Φ_S` is computed on the chosen tier's channel vector
over a rolling trajectory window of fixed length (window length and
sampling rate pinned in cell-set v0). Signature truncation depth `D`
is pinned per tier in the v0; matched-budget rules below enforce that
deeper signatures do not get a free parameter-count advantage.

Each tier produces a separate ablation result. The headline claim is
Tier 0; Tier 1 and Tier 2 are interpretation cells, not headline.

## Baselines (Matched Budget)

Four off-the-shelf regime-transition / precursor-detection methods,
each in their canonical form (citations in
[`../NAVIERSTOKES_LITPASS_MEMO.md`](../NAVIERSTOKES_LITPASS_MEMO.md)
Bonus Competitor track):

1. **DMD / Koopman.** Spectral mode decomposition on the same
   trajectory window the signature uses.
2. **Critical slowing down.** Variance and autocorrelation of a
   pre-registered scalar (default: total energy `Σ_n |u_n|²`).
3. **Recurrence lacunarity.** Recurrence-plot-derived multiscale
   measure.
4. **Rényi entropy.** Order-q entropy of a coarse-grained trajectory
   embedding.

**Matched-budget rule.** Each baseline gets:

- the *same* trajectory data and matched-seed train/val/test splits as
  the signature detector;
- a hyperparameter search of *equal* compute budget (wall-clock or
  trial count, pinned in cell-set v0);
- a trainable parameter count *not strictly less than* the signature
  detector's at the chosen tier.

The matched-budget rule is enforced *before* any held-out evaluation,
not after. A baseline that under-performs at smaller parameter count
does not "lose" — the budget is normalised first.

## Task Definition

**Objective `J_burst`.** Binary classification: predict
`B̂(t) ∈ {0, 1}` where `B(t) = 1` iff
`max_n |u_n(s)|² > E_burst` for some `s ∈ (t, t + τ_burst]`.

**Evaluation surface.** The Pareto frontier of *lead-time* and
*false-positive rate*:

- **Lead-time.** Mean of `(s* − t)` over true positives, where `s*` is
  the first `s ∈ (t, t + τ_burst]` with `|u_n|² > E_burst`.
- **False-positive rate.** Fraction of `B(t) = 0` query times where
  `B̂(t) = 1`.

A detector strictly *inside* the matched-budget Pareto frontier of
the four baselines wins the comparison. A detector on the frontier
ties; outside the frontier loses, filing `PDE-C2-NEG-A`.

The decision threshold (operating point) is pinned in the cell-set v0
*before* held-out evaluation. Sliding the operating point post hoc
counts as filing `PDE-C2-NEG-B`.

## Pre-Registered Cell Set (Sketch — v0 Pending)

The cell-set v0 follow-up artifact will pin:

1. **Headline cell.** `N_shells`, `λ`, `ν`, forcing amplitude,
   `τ_burst`, `E_burst`, sampling rate, window length, train/val/test
   split (matched-seed), operating point.
2. **Tier ablation cells.** One cell each for Tier 0 (headline),
   Tier 1, Tier 2.
3. **Held-out Re extrapolation cell.** A `ν`-shifted variant the
   detectors are *not* trained on, audited to detect overfitting that
   would file `PDE-C2-NEG-B`.
4. **GOY cross-check cells.** Sabra Tier 0 headline cell re-run on
   the GOY shell model with the same channel set and matched budget.
5. **Operating envelope cells.** A small grid of `(ν, forcing
   amplitude)` cells inside and outside the envelope on which the
   detector is meant to be informative. Outside the envelope, regret
   must be bounded away from zero (Coarse-Graining Phase 4 idiom).
6. **Compute budget per cell.** Hyperparameter-search trial count and
   wall-clock cap per baseline.

Cells are *pre-registered before any held-out evaluation is
interpreted*. Adding cells post hoc that move the verdict counts as
`PDE-C2-NEG-B`. Removing cells post hoc that move the verdict counts
as `PDE-C2-NEG-B`.

## Pre-Registered Negative (Verbatim, Two-Sided)

> (a) If the signature detector's burst-detection lead-time and
> false-positive rate are not strictly inside the Pareto frontier of
> the matched-budget baselines on the pre-registered cell set,
> signatures are vacuous on shell models — record and stop.
> (b) If signatures match baselines on the registered cell set but
> the cell set was the entire span, the result is over-fit to the
> data — record as Front-B vacuity.

**File label.** `PDE-C2-NEG-A` for (a), `PDE-C2-NEG-B` for (b). Both
labels are filed in the ledger and in the cell-set v0 results once
adjudicated.

**Re-framing rule (inherited from the C1 sidecar).** If
[`PDE_DETERMINING_MODES_POSTULATE1.md`](PDE_DETERMINING_MODES_POSTULATE1.md)
returns `PDE-C1-NEG`, the C2 run may still proceed as a
*turbulence-signature detector* workbench, but its results may **not**
be framed as evidence for a determining-modes / Postulate-1 coupling.
This is the standing constraint from the C1 sidecar's
"Equivalence / Named-Negative Regimes" section.

## Promotion Path

C2 leaves the ledger when **all four** ledger promotion criteria hold:

- **(a) Front-B vacuity rebuttal.** Either the matched-budget
  comparison lands strictly inside the Pareto frontier on the
  pre-registered cell set, or the result is filed as a clean
  `PDE-C2-NEG-A` / `PDE-C2-NEG-B` (a clean negative also closes the
  candidate, just on the other side of the ledger).
- **(b) Runnable Phase-0 deliverable.** The cell-set v0 artifact is
  filed and desk-auditable; the runnable script for one headline cell
  exists and reproduces on a reviewer's machine.
- **(c) External review path.** A practicing shell-model researcher
  (Mailybaev / Biferale / Frisch tradition) is named at promotion and
  has confirmed the model setup, channel set, baseline choices, and
  matched-budget rule are not Sundog-favouring.
- **(d) Pre-registered failure boundary.** Closed by the cell-set v0
  with the two-sided negative quoted above.

Order of operations:

1. **This scoping** (closes part of (a) at the position level, names
   (c) lineage).
2. **Cell-set v0 follow-up** (closes (d); supplies concrete numerics
   for (b)).
3. **Desk audit** (reviewer reads scoping + cell-set v0; rejects,
   amends, or signs off).
4. **Numerical run** (executes baselines and the signature detector
   on the registered cells; adjudicates the negative).
5. **Promotion or named-negative filing.**

No numerical run is interpreted as a verdict before step 3 lands.

## Cross-File Consequences

If this scoping plus the eventual cell-set v0 plus the numerical run
lands a clean positive:

- update [`COARSE_GRAINING_PROOF_ROADMAP.md`](../COARSE_GRAINING_PROOF_ROADMAP.md)
  Phase 4 with a PDE-substrate sibling result (parallel to the
  three-body measured leg), pending external review;
- update [`SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md)
  Promotions with Candidate 2 as a reviewed Front-B result;
- decide whether Candidate 3 (PINN diagnostic head) is now unblocked
  and whether Candidate 4 (full PDE signatures) should be reopened.

If this scoping plus the eventual cell-set v0 plus the numerical run
lands `PDE-C2-NEG-A` or `PDE-C2-NEG-B`:

- file the named negative in the Navier–Stokes ledger;
- do not deploy a public shell-model signature workbench;
- do not use the shell-model result as evidence for Front-B reach;
- treat Candidate 4 as deferred indefinitely (the cheaper sibling
  already failed; the more expensive one inherits no head start).

## External Review Path

Promotion requires a practicing shell-model researcher familiar with
intermittency, hidden self-similarity, instanton importance sampling,
or shell-model machine-learning closures to confirm:

1. the Sabra primary + GOY cross-check commitment is faithful to the
   community's robustness expectations;
2. the channel taxonomy is not Sundog-favouring (Tier 0 minimal,
   Tier 1/2 explicit ablations);
3. the matched-budget rule is enforced before evaluation, not after;
4. the cell set (once v0 lands) is a faithful operating-envelope
   pre-registration, not a curated favourable slice.

Until that review is complete, the artifact status is **drafted
scoping**.

## Exit Status

- **Drafted.** 2026-05-28.
- **Desk-auditable at the scoping level.** A reviewer can read this
  file end-to-end and audit the *position*, the *predicate shape*,
  and the *negative-receipt structure* without running code.
- **Unreviewed.** No external shell-model reviewer has signed off.
- **Cell-set v0 deferred.** All concrete numerics (`N_shells`, `λ`,
  `ν`, `τ_burst`, `E_burst`, signature truncation depth `D`, sampling
  rate, window length, operating point, hyperparameter-search budget)
  remain to be pinned in the v0 follow-up.
- **Unrun.** No numerical execution of the baseline or signature
  detector has been attempted.

Promotion status: **blocked.** This scoping closes part of criterion
(a) at the position level and names the lineage for criterion (c).
Criterion (b) is partially staged (the runnable surface is named but
not pinned); criterion (d) waits on the cell-set v0.

## Cross-references

- [`PDE_DETERMINING_MODES_POSTULATE1.md`](PDE_DETERMINING_MODES_POSTULATE1.md)
  — the C1 sidecar. Its "Equivalence / Named-Negative Regimes" section
  imposes the re-framing rule on C2 if C1 returns `PDE-C1-NEG`.
- [`PDE_C1_CELLSET_KOLMOGOROV.md`](PDE_C1_CELLSET_KOLMOGOROV.md) —
  the C1 cell-set v0. This file's structure is mirrored by the
  forthcoming C2 cell-set v0 (cell taxonomy, deferred numerics,
  desk-auditable status).
- [`PHASE4_THREEBODY.md`](PHASE4_THREEBODY.md) — the three-body
  measured-substrate leg of Postulate 1 Phase 4. C2 is its PDE
  sibling, not its replacement.
- [`../SUNDOG_V_NAVIERSTOKES.md`](../SUNDOG_V_NAVIERSTOKES.md) —
  the ledger. Candidate 2 entry defines the promotion criteria and
  the two-sided negative this scoping pins.
- [`../NAVIERSTOKES_LITPASS_MEMO.md`](../NAVIERSTOKES_LITPASS_MEMO.md)
  — the lit-pass record. All baseline citations and the
  hidden-self-similarity / instanton importance sampling references
  live there.
- [`../SUNDOG_V_THREEBODY.md`](../SUNDOG_V_THREEBODY.md) — sibling
  empirical workbench in a different substrate. The shape of the
  hidden / indirect / transformation / output predicate is shared;
  the substrate is not.