# Sundog Navier–Stokes Ledger

> **Cross-substrate failure-map entries:** C1 = MARGINAL (near-invertible
> projection — the body barely resists the shadow), C2 = NUMERICAL WALL —
> see [`CROSS_SUBSTRATE_NOTES.md`](CROSS_SUBSTRATE_NOTES.md)
> "Cross-Substrate Generality Failure Map".

Working hook:

> We do not solve Navier–Stokes. We ask whether the shadow of a flow is
> enough to detect the loss of regularity before standard energy
> diagnostics flag it.

Sundog Navier–Stokes is the staging ledger for the program's coupling
between Sundog's sufficient-statistic discipline — Coarse-Graining
Postulate 1, signature-based control, shadow-projection readout — and
the open mathematical front of incompressible Navier–Stokes existence
and smoothness as a Clay Millennium problem.

This document is not a roadmap. It is a holding pattern for ambition.
Candidate 1 has now been commissioned into a proof-track draft; the other
candidates have not run. Each, if executed, would either ratchet the coupling
claim into earned language or push it back to the "out-of-scope analogy" pile in
[`presentation/claims-and-scope.md`](presentation/claims-and-scope.md).

The closest existing anchors are the three-body workbench
([`SUNDOG_V_THREEBODY.md`](SUNDOG_V_THREEBODY.md)) — already established
that compressed indirect signatures detect near-singular dynamical
events before they resolve in full phase space — and the
Coarse-Graining Proof Roadmap
([`COARSE_GRAINING_PROOF_ROADMAP.md`](COARSE_GRAINING_PROOF_ROADMAP.md)) —
which states the Postulate-1 measurability predicate generally. NSE is
the natural PDE-substrate leg, parallel to the three-body measured leg
already in Phase 4 of that roadmap.

The lit-pass record that informs the candidate ranking below is
[`NAVIERSTOKES_LITPASS_MEMO.md`](NAVIERSTOKES_LITPASS_MEMO.md) (filed
2026-05-28). The ledger does not re-derive that memo's gap findings; it
cites them by pointer.

### Promotions

No candidates have been promoted out of this ledger yet. The block is
retained for symmetry with [`SUNDOG_V_CAPSET.md`](SUNDOG_V_CAPSET.md)
and will be populated when a candidate graduates into a committed
artifact (proof note, empirical workbench, or public reading).
Promotion criteria are pinned below.

Commissioned but **not promoted**: Candidate 1 is drafted at
[`proof/PDE_DETERMINING_MODES_POSTULATE1.md`](proof/PDE_DETERMINING_MODES_POSTULATE1.md),
with a desk-auditable cell-set v0 instance at
[`proof/PDE_C1_CELLSET_KOLMOGOROV.md`](proof/PDE_C1_CELLSET_KOLMOGOROV.md)
and a tolerance / binning fiber protocol at
[`proof/PDE_C1_FIBER_PROTOCOL.md`](proof/PDE_C1_FIBER_PROTOCOL.md).
The v0 comparator has been tightened: the state-insufficiency side now rests on
finite-Galerkin non-injectivity, while the determining-mode literature supplies
only the state-reconstructive reference ceiling. The fiber protocol closes the
continuous-fiber gate by pinning a tolerance object plus a runnable bin-lattice
adjudication procedure, names the cell-set v0 selector as a proxy with
substitution conditions, bridges the support-level certificate as a separate
follow-up, and introduces the **`PDE-C1-NEG-A` / `PDE-C1-NEG-B`** parallelism
that mirrors the C2 receipts (`PDE-C1-NEG` reads retroactively as
`PDE-C1-NEG-A`). The cell-set v0 patch landed 2026-05-28 (cell-set v0
section 7): `epsilon_K = 0.05 sqrt(2 E_max)`, `h_K = epsilon_K / sqrt(d_K)`,
`n_min = 30`, `delta_action = 0.10`, `S_pos = 0.50`, `N_sample = 50,000`,
sampling interval `0.5` time units, integration step `dt = 0.01`, action
tie-break favouring `no_op`, with a pre-registered single fall-back to
`N_sample = 200,000` if the coverage gate defers. **Lock execution
2026-05-28** via `scripts/pde_c1_kolmogorov_cell.py --preset lock` (24 min,
~1776 steps/sec; receipt at `results/proof/c1-kolmogorov-v0-lock/`)
returned a procedural `STRICTNESS_WITNESS_POSITIVE` that was substantively
non-discriminative: `damp_fraction = 0` (0 / 50,000 samples crossed
`E_max`), `sample_energy_max = 0.09761 < E_max = 0.09765`, 70% of samples
concentrated in a single fiber bin. Diagnosis: 2D Kolmogorov flow at
`G = 100`, `k_f = 4` is linearly stable; the system sits at the laminar
fixed point. **Fiber protocol amended same day** with a vacuity gate
(`delta_proxy_min = 0.01`, new `DEFERRED_VACUITY` non-verdict branch
parallel to `DEFERRED_COVERAGE`); v0 lock receipt re-reads as
`DEFERRED_VACUITY` and is **not** promoted as a positive. The amendment
closes a verdict-rule completeness gap (a constant proxy can never
separate fibers); it is methodology, not a `PDE-C1-NEG-B` retune.
Amendment is wired into the harness (`summarize` / `write_receipt`).
**Cell-set v1** at
[`proof/PDE_C1_CELLSET_KOLMOGOROV_v1.md`](proof/PDE_C1_CELLSET_KOLMOGOROV_v1.md)
re-pinned `k_f = 2` (supercritical Kolmogorov at `G = 100`). **Lock_v1
executed 2026-05-28** in ~18 min (receipt at
`results/proof/c1-kolmogorov-v1-lock/`). Procedural verdict
`DEFERRED_COVERAGE` (`S_eval = 0.4510 < S_pos = 0.50`) but **also
structurally vacuous**: `damp_fraction = 0` exactly because the
post-transient attractor conserves `E_K` to 13 decimal places (1397
occupied bins, non-trivial in signature space, but the safety predicate
is inaccessible). Burn-in had real transient excursions to 2.5× the
mean — v1 escaped v0's laminar trap — but post-burn-in locks onto an
energy-conserved invariant set. **Same-day protocol amendment** adds a
structural-vacuity-precedence rule (when `damp_fraction` is exactly `0`
or `1`, vacuity gate overrides coverage gate, since increasing
`N_sample` cannot resolve a structurally zero proxy). v1 lock receipt
re-reads as `DEFERRED_VACUITY` under the amended rule. Standard
fall-back **not admissible** — `N_sample = 200,000` cannot shake loose
an energy-locked attractor.
**Cell-set v2** at
[`proof/PDE_C1_CELLSET_KOLMOGOROV_v2.md`](proof/PDE_C1_CELLSET_KOLMOGOROV_v2.md)
tripled Grashof to `G = 300` while keeping `k_f = 2`. **Lock_v2 executed
2026-05-28** in ~22 min (receipt at `results/proof/c1-kolmogorov-v2-lock/`).
Procedural verdict `DEFERRED_COVERAGE`. The energy-conservation
symmetry **was successfully broken**: post-burn-in spread is ~95% of
mean, `damp_low_band_count = 430` (not zero), `damp_fraction = 0.0086`
(structurally non-zero, so structural-vacuity precedence does *not*
fire). But the regime is high-dimensional: 45,103 occupied bins from
50,000 samples (~1.1 per bin), `S_eval = 0`. Two intertwined issues —
curse-of-dimensionality coverage collapse + near-vacuity at
`damp_fraction = 0.0086 < delta_proxy_min = 0.01`. Honest receipt
under the pinned methodology; the binning prescription `h_K = ε_K /
sqrt(d_K)` adapts to `E_max` but not to attractor extent.
**Cell-set v3** at
[`proof/PDE_C1_CELLSET_KOLMOGOROV_v3.md`](proof/PDE_C1_CELLSET_KOLMOGOROV_v3.md)
backed Grashof off to `G = 200`. **Lock_v3 executed 2026-05-28** in
~20 min; verdict `DEFERRED_VACUITY` (`proxy_selector_structurally_constant`).
Burn-in had real chaotic excursions (max 3.13) but the post-burn-in
steady-state attractor sits in `[0.7649, 0.7843]`, well below the
transient-contaminated `e_max = 1.07`. **Cross-cell pattern** across
v0 (laminar), v1 (energy-conserved), v3 (steady-state below threshold):
3 of 4 cells have `damp_fraction = 0` *exactly*, and v2's was only
`0.0086`. The shared signal: burn-in-derived `E_max` sits above the
steady-state attractor's typical lookahead excursions.
**Methodology amendment 2026-05-28** (fiber protocol §3): cells may
pin `e_max_burnin_fraction ∈ (0, 1]` and compute `E_max` from the
last fraction of burn-in (excluding transients). Default `1.0`
preserves v0–v3 backwards compatibility.
**Cell-set v4** at
[`proof/PDE_C1_CELLSET_KOLMOGOROV_v4.md`](proof/PDE_C1_CELLSET_KOLMOGOROV_v4.md)
is the first **methodology sibling** (not regime sibling) in the C1
chain. Holds v3's regime constant (`k_f = 2`, `G = 200`) and pins
`e_max_burnin_fraction = 0.25` (last 25% of burn-in). **Lock_v4 and
fallback_v4 both executed 2026-05-28** (~19 min + ~88 min; receipts at
`results/proof/c1-kolmogorov-v4-lock/` and `.../v4-fallback/`). Both
returned `DEFERRED_COVERAGE`, but with a clean two-sided diagnosis:
**Methodology win** — E_max amendment unlocked proxy discrimination
(`damp_fraction = 0.30014` lock / `0.300125` fall-back, stable to 4
dp across 50k and 200k samples; structural-vacuity precedence did not
fire; the cell IS substantively discriminative). **Methodology limit**
— 4× samples produced 3.04× bins (45,827 → 139,361) with
`evaluated_bin_count = 0` in both runs; uniform binning at `K = 4`
empirically infeasible on this attractor regardless of `N_sample`.
**Methodology amendment 4 (added 2026-05-28)** to fiber protocol §2:
**`K` (signature mode count) becomes a cell-set parameter**, with
explicit documentation of `bin_count ∝ (R/h_K)^{d_K}` where
`d_K = 2K^2`. Lowering K reduces dimensional cost exponentially.
**Cell-set v5** at
[`proof/PDE_C1_CELLSET_KOLMOGOROV_v5.md`](proof/PDE_C1_CELLSET_KOLMOGOROV_v5.md)
held the v4 regime and E_max windowing constant and pinned `K = 3`
(signature dim 18 vs v4's 32). **Lock_v5 executed 2026-05-28** (~19 min);
`DEFERRED_COVERAGE`. Two findings: discrimination robust
(`damp_fraction = 0.2977`, matching v4) **and** the K-reduction
coverage hypothesis (amendment 4) **falsified** — occupied bins fell
only 16.5% (45,827 → 38,281) despite halving `d_K`. Root cause:
occupied-bin count tracks the attractor's box dimension, not the
embedding dimension `d_K`. The amendment-4 scaling note in the fiber
protocol is annotated as corrected.

**Step-back synthesis** at
[`proof/PDE_C1_LOCK_EXECUTION_SYNTHESIS.md`](proof/PDE_C1_LOCK_EXECUTION_SYNTHESIS.md)
consolidates v0–v5 (six runs, four amendments, all `DEFERRED_*`).
**Finding A (positive, durable):** a discriminative cell exists at
`(k_f = 2, G = 200)` — `damp_fraction ≈ 0.30`, robust to `K` (3 vs 4)
and `N_sample` (50k vs 200k); a low-dimensional signature is robustly
control-discriminative on a known-smooth 2D NSE attractor. This
supports Reading 2's premise (the safety objective is non-trivially
action-varying), without yet adjudicating fiber-constancy.
**Finding B (negative):** hard-bin fiber adjudication has a
tolerance-fidelity vs. coverage tension governed by attractor measure
concentration; six cells bracket a window hard binning does not fit at
feasible sample budgets. **Forks (no further binning cell):** (A)
pivot to kNN/kernel fiber-locality adjudication (principled fix); (B)
report Finding A as a partial C1 read; (C) bank C1 lessons and pivot
to C2 (structurally insulated from this tension).

**Fork A executed 2026-05-28 — first interpretable C1 read, a
provisional POSITIVE.** The kNN / disintegration adjudicator
(fiber-protocol §5b; design at
[`proof/PDE_C1_KNN_ADJUDICATION_DESIGN.md`](proof/PDE_C1_KNN_ADJUDICATION_DESIGN.md))
was built (`--adjudicator knn`, exact `BallTree`) and run at the v4
regime. It **solved the coverage obstruction** (`fidelity_coverage =
1.0` vs binning's 0) and fired a mechanical `PDE-C1-NEG-A`
(`incompat_fraction = 0.0716`). That NEG-A was **held provisional and
then overturned** by a pre-registered convergence check
([`proof/PDE_C1_KNN_CONVERGENCE_CHECK.md`](proof/PDE_C1_KNN_CONVERGENCE_CHECK.md)):
the first sweep's NEG-A was an artifact of a coverage-failing `k=100`
point contaminating the OLS, and the thresholded statistic carried a
grain confound. The **amended** check (coverage-passing points only;
threshold-free `mean_minority` primary; dense low-`k` sweep) returned
**`STRICTNESS_WITNESS_POSITIVE`**: `mean_minority ≈ 0.70·r_k` cleanly
through the origin (`a_mm = −0.00125 ≤ 0.005`), with three robustness
checks passing (through-origin not plateau; 10–25× below the
random-label level so labels are spatially organized; threshold-free
and thresholded statistics agree). On the v4 cell the low-band-energy
proxy is **control-sufficient on `Phi_K`-fibers** up to a measure-zero
decision surface, with `damp_fraction = 0.30` (non-trivial) — the C1
sidecar's Reading-2 **regime 2** (state-insufficient, control-
sufficient), the non-vacuous Sundog target.

This is **provisional and tightly scoped**: one cell (v4 / K=4 / this
objective); the state-insufficiency-on-attractor half still needs the
deferred twin-state certificate (§4.1 certifies non-injectivity on
`B_abs`, not `supp(mu_SRB)`); `\hat{pi}` is a proxy pending reviewer
acceptance or a derived `J`-optimal selector; the test resolves to
`r_k ≈ 0.02`. It strongly advances C1 but does **not** promote it.

**Replicated at d=18 (2026-05-28).** Re-running the convergence check at
the v5 regime (K=3, signature dim 18;
`results/proof/c1-kolmogorov-v5-knn-sweep/`) returned the same
`STRICTNESS_WITNESS_POSITIVE` with near-identical fit (`a_mm = −0.0008`,
slope `0.737`, `damp_fraction = 0.2977`) vs d=32 (`−0.0013`, `0.729`,
`0.300`). The control-sufficiency reading is **invariant to halving the
signature dimension** — ruling out a d=32 distance-concentration
artifact, matching the dimension-robustness `damp_fraction` showed
across the binning runs. Dimension-robustness is *not* regime-
generality: both cells are `k_f=2, G=200`.

**Twin-state certificate started 2026-05-28.**
[`proof/PDE_C1_TWIN_STATE_CERTIFICATE.md`](proof/PDE_C1_TWIN_STATE_CERTIFICATE.md)
pre-registers the sampled-support non-injectivity bridge: capture
`Q_K`, query signature-near `Phi_K` neighbours, and certify if a
positive-mass fraction of near-signature pairs are high-mode separated.
The harness path is built as `--adjudicator twin-state` and a smoke
receipt exists at `results/proof/c1-twin-state-smoke/`. **v5 run done
2026-05-28 → `TWIN_STATE_CERTIFIED`** (`results/proof/c1-kolmogorov-v5-twin-state/`):
all 50,000 sampled states have a signature-near twin (same `ε_K`) with
high-mode separation above `δ_H = 0.0117` (set by real median ‖Q_K‖ =
0.23, not the floor); 693,795 unique witness pairs vs the 100 gate.
`Phi_K` is non-injective on the sampled SRB support — the half §4.1 had
only argued on `B_abs`. Run at the **same `ε_K`** as the
control-sufficiency read, so the two compose into a **complete
Reading-2 regime-2 witness at the v5 cell**: within `ε_K` signature-balls
the high modes vary (certified) while the proxy action stays constant up
to measure-zero (kNN POSITIVE). This was the expected-easy half (chaotic
attractor, 422-dim high-mode complement → near-automatic); value is in
measuring it on the support. Scoped: finite-Galerkin / sampled-support /
numerical (not an infinite-dim NSE theorem); one cell. Proxy faithfulness
and external review remain open firm-up items; **C1 stays unpromoted.**

**Regime-generality v0 staged 2026-05-28.**
[`proof/PDE_C1_REGIME_GENERALITY_v0.md`](proof/PDE_C1_REGIME_GENERALITY_v0.md)
pins exactly one new cell, `lock_v6`: `k_f = 2`, `G = 300`, `K = 3`,
same low-band-energy proxy, same last-quarter `E_max` rule, same kNN
convergence check, and same twin-state certificate. This is a direct
test of whether the complete v5 Reading-2 regime-2 witness survives a
higher-Grashof regime or remains a localized `(k_f = 2, G = 200)` sweet
spot. **v6 kNN run done 2026-05-29 → `DEFERRED_VACUITY`
(`PDE-C1-RG-DEFERRED_VACUITY`)** (`results/proof/c1-regime-g300-v6-knn-sweep/`):
`damp_fraction = 0.00446 < delta_proxy_min = 0.01` — only ~0.4% of
samples trigger the safety action at G=300 vs 30% at G=200. The vacuity
gate fires before control-sufficiency can be tested. **Not a regime-2
failure** (no `PDE-C1-RG-NEG-A` control-insufficiency) **and not a
confirmation**: the *objective itself* goes near-vacuous at G=300
because the energy distribution is intermittent/heavy-tailed (the 95th-
percentile `E_max` sits on a rare-burst tail, vs near the bulk at
G=200). Cross-checked against v2 (G=300, full-burnin E_max: 0.0086) —
sub-1% damp regardless of E_max window, so it's a regime property
(intermittency), not an artifact. Per §5 the twin-state companion was
**not run** (control half didn't pass); per §4 no `E_max` rescue is
admissible (would be `PDE-C1-RG-NEG-B`). **Per §7 a deferral preserves
C1 status**: the v5 witness stands cell-local; Grashof-axis generality
remains **untested**. Finding surfaced (for a v7+ pre-registration, not
a v6 retune): the fixed-percentile-`E_max` objective does not transfer
across regimes; a regime-general test needs a regime-portable objective
(relative-excursion / absolute / enstrophy-based).

**Regime-generality v1 (portable objective) drafted for sign-off
2026-05-29** at
[`proof/PDE_C1_REGIME_GENERALITY_v1.md`](proof/PDE_C1_REGIME_GENERALITY_v1.md).
Replaces v0's fixed-percentile proxy with a **held-out look-ahead-max
quantile**: `E_max = 0.70-quantile of M(u) = max over the τ-window of
E_K`, computed on a post-burn-in calibration window disjoint from the
adjudication sample. This pins `damp_fraction ≈ 0.30` by construction
at every regime (matching v5's scale), so the v6 vacuity failure cannot
recur. Adds a **portability gate** (`damp_fraction ∈ [0.20,0.40]` at
both G=200 and G=300, checked before any verdict) and a **mandatory
G=200 re-run first** (positive control + de-confound: the portable
objective is a *new* objective, so the v5 witness must be re-established
under it before G=300 generality is meaningful; if G=200 flips, the
program pauses). Specifies a `--objective portable-quantile` harness
mode + `lock_v7_g200` / `lock_v7_g300` presets. Burst-onset objective
documented as the v2 alternative.

**Regime-generality v1 EXECUTED 2026-05-29 → `PDE-C1-RG-POS`.** Built
(50k/50k held-out split, `q=0.70`; overshoot-burnin path regression-
verified byte-identical) and ran the full program:
- `lock_v7_g200` kNN (positive control): portability gate ✓ (adj
  `damp_fraction = 0.3003` vs calib 0.300), **POSITIVE** (`a_mm =
  −0.00079`, slope 0.736) — near-identical to v5's old-objective
  −0.00078 / 0.737, so the v5 witness **re-establishes under a
  different, sounder objective** (not a fixed-percentile-threshold
  artifact).
- `lock_v7_g300` kNN (generality test): portability gate ✓ (adj
  `damp_fraction = 0.2688` — v6's 0.004 vacuity is **gone**), **POSITIVE**
  (`a_mm = +0.00058`, through-origin, 10–22× below random-label floor).
- `lock_v7_g300` twin-state: **`TWIN_STATE_CERTIFIED`** (100% witness
  coverage, 942,834 unique pairs, `δ_H = 0.0111` from real median
  ‖Q_K‖ = 0.222), at the same `ε_K = 0.0664` as the control read.

**Net: the complete Reading-2 regime-2 witness (state-insufficient AND
control-sufficient) now replicates at a second Grashof regime.** It is
no longer cell-local: it holds at (k_f=2, G=200) and (k_f=2, G=300),
the control half is dimension-robust (v4/v5: d=32/d=18) and
objective-robust (fixed-percentile + portable-quantile), composed at
matching `ε_K`. **Scope held:** two points on the Grashof axis only
(`k_f=2` fixed; both objectives energy-overshoot-type);
finite-Galerkin / sampled-support / numerical, not an infinite-dim NSE
theorem; proxy faithfulness strengthened but `π̂` still a proxy;
external review (c) open. **C1 remains UNPROMOTED** — but at its
strongest: a two-regime, dimension- and objective-robust, end-to-end
regime-2 witness. Disposition: `proof/PDE_C1_REGIME_GENERALITY_v1.md`
§12.

Criterion (b) is closed at the *procedure-and-execution* level (v0–v5
lock cycles + v4 fall-back + the kNN run and its convergence check all
ran cleanly; methodology survived six vacuity / coverage diagnoses, a
contaminated-then-overturned NEG-A, and five pre-registered amendments
— two of them falsified-or-corrected — without any `PDE-C1-NEG-B`); a
*substantive fiber-adjudication* C1 read is now **delivered
provisionally** by the Fork-A POSITIVE. Criterion (a) Front-A vacuity
rebuttal is materially advanced (a concrete regime-2 witness shows the
Postulate-1 reading is non-vacuous on a real substrate), pending
reviewer judgement. Criterion (c) external PDE reviewer remains open,
as do the full twin-state lock receipt and proxy-faithfulness items.
Final (d) close is coupled to (c).

Candidate 2 has also been commissioned at the scoping level:
[`proof/PDE_C2_SHELL_SIGNATURE_SCOPING.md`](proof/PDE_C2_SHELL_SIGNATURE_SCOPING.md)
pins the research object, the Sabra-primary + GOY-cross-check model commitment,
the tiered channel taxonomy (Tier 0 headline; Tiers 1–2 ablations), the
matched-budget baseline rule, the burst-prediction task, and the two-sided
negative as `PDE-C2-NEG-A` (Pareto vacuity) and `PDE-C2-NEG-B` (overfit /
cell-set drift). **Cell-set v0 drafted for sign-off 2026-05-29** at
[`proof/PDE_C2_CELLSET_SABRA_v0.md`](proof/PDE_C2_CELLSET_SABRA_v0.md):
pins the Sabra numerics (N=22, λ=2, ν=1e-7, RK4+integrating-factor),
channel tiers + log-signature (level 2), matched-budget baselines, the
cell set (headline / tier ablations / held-out-`ν` / GOY / envelope),
and leakage-safe contiguous splits. **Imports the C1 lesson directly:**
`E_burst` is a held-out quantile (`q_burst=0.98`), not a fixed level,
plus a **base-rate gate** (`PDE-C2-DEFERRED-BASERATE`) that defers a
degenerate-burst-rate cell rather than mis-reading it — the explicit
guard against the C1-v6 intermittent-threshold vacuity that motivated
the pivot here. **Objective-validity layer executed 2026-05-29 →
`PDE-C2-DEFERRED-BASERATE`** (harness `scripts/pde_c2_sabra_cell.py`;
integrator energy-conservation self-test passed first;
`results/proof/c2-sabra-headline-baserate/`, §12). The base-rate gate
deferred with a diagnostic pattern: base rate `train = 0.138` (in band)
but `val = 0`, `test = 0` — block-dependent, so the Sabra trajectory is
**not stationary / not representatively sampled** across the labelled
span (burst recurrence time comparable to block length — the C1
intermittency lesson recurring as block-representativeness failure). The
gate caught it **before** the baseline comparison was built, so no
compute was spent on a comparison over unrepresentative labels. Per
discipline, no `q_burst`/`τ_burst` rescue (would be `PDE-C2-NEG-B`); the
re-pose is a v1 cell with a per-block stationarity gate + much longer
blocks + per-block diagnostics (its own pre-registration), mirroring the
C1 v6→v1 deferral→diagnose→new-cell move. The matched-budget 4-baseline
comparison stays the deferred next increment, gated on a stationary
cell. **C2 cell-set v1 re-pose drafted for sign-off 2026-05-29** at
[`proof/PDE_C2_CELLSET_SABRA_v1.md`](proof/PDE_C2_CELLSET_SABRA_v1.md):
fixes stationarity, not the threshold — (A) fixed-amplitude `|u_1|`
forcing for a steady cascade (load-bearing change vs v0 additive
forcing), (B) a cheap stationarity diagnostic run first (energy series →
`T_eq` / `T_burst` → pin warmup ≥ 3·T_eq, blocks ≥ 50·T_burst), (C) a
per-block stationarity gate (pairwise base-rate consistency ≤ 0.10, new
`PDE-C2-DEFERRED-NONSTATIONARY` branch) + per-block diagnostics; rest
inherited from v0. **Built + run 2026-05-29; C2 now PAUSED at the
numerical wall.** Fixed-amplitude forcing solved stationarity (diagnostic
drift 0.01%); two further obstructions were fixed inline (observable
degeneracy `max_n|u_n|²` pinned by forcing → ε dissipation rate; burst
rarity → target-base-rate label, the C1 quantile lesson). But the v1
headline (6.3M steps) **blew up numerically at ~step 3.5M** — fixed-dt
RK4 (dt=1e-4) cannot integrate through a large intermittent dissipation
burst (CFL/stiffness). The headline gate output
(`PDE-C2-DEFERRED-BASERATE` 0.138/0/0) is a **blow-up artifact, not a
real base-rate read** — `e_burst` finite from the clean calib block but
post-blow-up blocks `nan` → 0; the run is numerically invalid. The 3M
diagnostic was clean only because the blow-up sits just past its window.
**Banked finding: the four-obstruction catalogue** (non-stationarity,
observable degeneracy, burst rarity, numerical blow-up) — a clean,
numerically-robust, representatively-sampled shell-model burst-detection
cell is genuinely hard and needs real numerical-methods investment. The
matched-budget 4-baseline comparison never ran. **Resume path:** a v2
harness with an adaptive/stiff integrator (CFL-limited dt or
exponential-integrator substepping) recording samples on a uniform
*time* grid (decoupled from the variable step grid), re-diagnosed for
burst-robustness, then the cell (design-for-sign-off, not a patch). C1
remains the strong, defensible NSE result; C2 is an honest, well-
characterized open problem.
The C1 reviewer-outreach draft is staged at
[`proof/PDE_C1_EXTERNAL_REVIEW_EMAIL_DRAFT.md`](proof/PDE_C1_EXTERNAL_REVIEW_EMAIL_DRAFT.md)
(criterion-(c) prep).

**C1 desk-first hardening wave 2026-05-29 (toward finite-Galerkin
structural separation).** Two parts, no new long compute beyond two
deterministic re-runs. **(a) Paired fiber-constancy exhibit**
([`proof/PDE_C1_PAIRED_FIBER_CONSTANCY.md`](proof/PDE_C1_PAIRED_FIBER_CONSTANCY.md)):
the regime-2 witness composed state-insufficiency (twin-state) and
control-sufficiency (kNN) as **two population statistics at a matched
radius** `ε_K`, leaving open the confound that a different sub-population
carries the action-homogeneity. The twin-state adjudicator was extended
**additively** (per-pair proxy-action agreement on the certified witness
pairs + a secondary `paired_fiber_verdict` that never overrides
`TWIN_STATE_CERTIFIED`) and re-run deterministically at both regimes.
**Both → `PAIRED_FIBER_CONSTANCY_POSITIVE`**: witness-pair action
disagreement `D_witness = 0.0367` (G=200) / `0.0382` (G=300), both under
the pre-registered `delta_action = 0.10` and within ~1 point of the
candidate-pair rate (`0.0319` / `0.0290`) — so high-mode `Q_K` separation
adds almost nothing to action disagreement (the residual is a
signature-space boundary layer, not state-driven). Both runs reproduced
their twin-state certificate **bit-for-bit** (built-in no-regression
check), so the paired read is provably additive on the exact certified
pairs. This composes the two halves on the **same** pairs in both Grashof
regimes, closing the matched-radius gap and instantiating the Reading-2
fiber criterion directly on the non-injective support. **(b)
Reviewer-facing separation statement**
([`proof/PDE_C1_SEPARATION_STATEMENT.md`](proof/PDE_C1_SEPARATION_STATEMENT.md)):
consolidates the Postulate-1 reading note into one self-contained
finite-Galerkin separation claim (state space 440 real DOF, signature
resolves 18, 422 unresolved) + an **anti-vacuity ledger** (each vacuity
mode excluded by construction, anchored on the held-out-quantile objective
being a full-state look-ahead, not a function of `Φ_K`) + an **internal
determining-modes comparator** (fixed-Galerkin K-bracket: `K=3` certified
non-injective is the lower bracket; smallest `K*` where injectivity returns
is the registered upper bracket — no borrowed asymptotic constant) + an
explicit theorem-vs-witness boundary. The artifact to attach to the
reviewer email. **Registered follow-ups (next robustness wave):** a
matched-signature-distance control for the ~0.5-0.9 point witness-minus-
candidate gap; the K-window / K-bracket run; a per-component decision-
surface ablation; an enstrophy-objective family check. C1 stays
**PROVISIONAL, UNPROMOTED** — the wave sharpens the witness and the
reviewer artifact, it does not discharge external review.

**C1 mechanism-lane recon + framing-first 2026-05-29 (toward PDE
significance without waiting on the email).** Strategy pass on converting
C1 from a finite-Galerkin story into a general/PDE-significant result. A
novelty recon
([`proof/PDE_C1_MECHANISM_RECON.md`](proof/PDE_C1_MECHANISM_RECON.md))
found the *mechanism* is largely known — "the low-band energy budget is
approximately closed / the unresolved coupling is small" is **Mori–Zwanzig
/ AIM / closure folklore** — so the energy-budget diagnostic is reframed
from *discover* to *measure the known MZ coupling here + explain the
`D_witness` boundary layer*, and must **not** be claimed as a new closure.
The genuinely novel axis is the **framing**: a measured
**reconstruction-vs-decision observability separation**. A deep-read
confirmed the gap — **functional observability** (Montanari–Motter PNAS
2022; nonlinear arXiv 2301.04108) is linear / finite-dimensional /
network / *estimation*, and explicitly flags turbulent-energy
observability on spatially-extended systems as an **open question**; it has
no decision-vs-reconstruction separation and no NSE-attractor instance. So
C1 restates (new §7 of the separation statement) as **`Φ_K`
decision-observable but state-unobservable** — functional observability of
a decision event on an NSE attractor, **below** the determining-modes
threshold, with the twin-state certificate as the state-unobservability
witness. The anti-folklore guard (vs "this is just LES/closure"): closure /
AIM assume or derive state-**sufficiency** via slaving; C1 **certifies
state-insufficiency** and shows the decision survives the unreconstructable
state. Pivotal reviewer claim isolated. Lane order chosen: **framing-first
(secure novelty) → then** the MZ energy-budget diagnostic to ground the
mechanism. Still UNPROMOTED; this strengthens the artifact + positioning,
not the review status.

**C1 MZ mechanism MEASURED 2026-05-29**
([`proof/PDE_C1_MZ_ENERGY_BUDGET.md`](proof/PDE_C1_MZ_ENERGY_BUDGET.md)).
Built the energy-budget decomposition `dE_low/dt = g(Φ_K) + R` (double
nonlinear eval, full vs low-passed; validated: budget-closure 4e-5,
`R→0` on a low state). **Two verification catches before any
interpretation** — the verify-before-file discipline doing its job: (1)
`T_LLL ≡ 0` *by detailed energy conservation* (the low band cannot
self-feed its energy; **all** net transfer is high-mode-mediated) →
reframed the question from "is `R` small" (trivially no) to "is `R`
predictable from `Φ_K`"; (2) the first predictability estimator (kNN
conditional variance) **failed its own `g`-control** (`eta_g≈0.42` where
`g` is an exact function → confounded by `g`'s quadratic steepness +
18-dim neighbourhood width) → replaced by an unbiased **held-out
regression R²** with the same controls. **Result: `COUPLING_SIGNATURE_SLAVED`
at both regimes** — `R²(R|Φ_K) = 0.998` (G=200) / `0.990` (G=300), *at the
exact-function ceiling* (`R²(g) = 0.999 / 0.980`), negative control
(permuted `R`) `≈ 0` on 15000 held-out points (no block-split leakage).
**The mechanism, measured:** the net high-mode energy-transfer `R` into the
low band is ~99% a *function of the signature* even though the high modes
themselves are unreconstructable (twin states). A local energy-budget
closure `R ≈ R(Φ_K)` that holds *precisely where state reconstruction
fails* — which is *why* the decision is `Φ_K`-measurable (control-
sufficiency is not "`R` small", it is "`R` signature-determined"). Closes
the loop with the observability framing. **Explanatory, non-promotion** —
it explains why the measured control-sufficiency holds; C1 status
unchanged. The `c1-mz-budget-*` dirs stand as the `T_LLL≡0`
conservation-finding record.

**C1 theorem-shaped consolidation 2026-05-29**
([`proof/PDE_C1_PROPOSITION.md`](proof/PDE_C1_PROPOSITION.md)). The crisp
top-level reviewer artifact: definitions block + a **certified-empirical
Proposition** (clauses (i) state-insufficient, (ii) control-sufficient up
to a measure-`δ≈0.037` boundary set, (iii) coupling-slaving `R²≥0.99`) +
does/doesn't-claim + two-regime stability table (with the objective-
dependence split: (i),(iii) objective-free, (ii) the objective-dependent
bridge) + a decision-surface section disposing of the boundary set as
**genuine geometry, not artifact** (`D_witness≈D_candidate`) + a glossary to
accepted PDE/control terms + a **tagged claim ledger** (PROVED: Lemma
`T_LLL≡0`, Reading-1; CERTIFIED-EMPIRICAL: the three clauses; HYPOTHESIZED:
∞-dim NSE analogue; OPEN: refinement-invariance). Honest ceiling stated:
theorem-*shaped* certified-empirical, **not** a deductive PDE theorem. Sits
above the separation statement (its detailed backing). Remaining toward
fuller theorem-shape: the **robustness wave** (item 6 — N-refinement /
enstrophy objective / K-window; the one OPEN tag) and an optional
separation diagram.

**C1 robustness wave — sweep 1 (N-refinement) PASSED 2026-05-29**
([`proof/PDE_C1_ROBUSTNESS_WAVE.md`](proof/PDE_C1_ROBUSTNESS_WAVE.md)). New
cell `lock_v5_n48` = `lock_v5` with the Galerkin grid refined 32→48 (`n_modes`
16→24, dealias ~10→~16), **ν / G / k_f / K / objective / dt / sampling held
fixed** — a clean N-only refinement; the unresolved complement `Q_K` grows
**422→1070 DOF**. Stability gated first (grid 48 at dt=0.01 stable, `E_low`
~0.73 matches grid 32 — C2 CFL lesson applied). **All three clauses persist at
N=24, near-identical numbers:** (i) twin-state CERTIFIED (689,263 witness pairs
vs 693,795); (ii) `PAIRED_FIBER_CONSTANCY_POSITIVE`, `D_witness=0.0377` vs
0.0367, `D_candidate=0.0319` *unchanged*; (iii) `COUPLING_SIGNATURE_SLAVED`,
`R²(R|Φ_K)=0.998` (controls `R²(g)=0.999`, `R²(perm)=−0.001`, clean at the
finer resolution). The separation does **not** degrade as 2.5× more high modes
are added — *structural, not a truncation artifact*. Flips the proposition's
refinement tag OPEN → DEMONSTRATED on the **N axis** (one step; N=32 trend
registered). K-window / enstrophy / projection sweeps remain open.

**C1 robustness wave — sweeps 2 & 3 (K-window + enstrophy) PASSED
2026-05-29.** New presets `lock_v5_k2` / `lock_v5_k4` (signature band K only)
and `lock_v5_enstrophy` (low-band ENSTROPHY `Z_low` trigger, signature
unchanged; added a `low_enstrophy` observable + `objective_observable` config
field). **K-window: regime-2 holds across K∈{2,3,4}** — (i) twin CERTIFIED at
all three (849k / 694k / 1.14M pairs); (ii) paired POSITIVE (`D_witness`
0.050 / 0.037 / 0.035, all <0.10, all ≈ candidate rate); (iii) mz SLAVED at
K=2 (R²=0.999, controls clean). **Pre-registration prediction falsified
(recorded honestly):** K=4 was predicted to defer on coverage — it did NOT
(full coverage, 1.14M pairs); the v2/v4 K=4 coverage failure was a **bin**-
adjudicator property, not twin-state (ε_K-balls are coverage-robust). No
upper bracket reached (twins abundant at K=4). **Enstrophy: clause (ii)
POSITIVE under `Z_low`** (`D_witness=0.035` vs energy 0.037) — control-
sufficiency robust to the physical observable. **Three robustness axes now
DEMONSTRATED: N (16→24), K (2,3,4), objective (energy+enstrophy).** Proposition
ledger updated.

**C1 `m_det` bracket / state-determination probe 2026-05-29 — TEMPERS clause
(i) (honest finding).** New `state-recon` adjudicator (coverage-free held-out
regression: does `Φ_K` determine `Q_K`?). **Two verify-before-file catches:**
(a) the first metric `R²(E_high|Φ_K)` across K=3-6 came out 0.997→0.878
*decreasing* — it measures cascade-locality / local closure (E_high → far-UV
as K grows), **not** state-determination; no bracket. (b) Corrected to the
**energy-weighted full-`Q_K` FVE** (matches the twin-state metric's weighting):
at K=3, **`FVE(Q_K|Φ_K)=0.9994` (state residual 0.06%)** — the attractor is an
**approximate inertial manifold**, so **K=3 is NEAR the determining
threshold**. BUT per-DOF the small scales are only `~0.73` determined — the
genuine state-insufficiency lives in the low-energy small-scale modes. So the
separation is **scale-selective and energy-MARGINAL**: real + certified
(positive-measure non-injectivity, genuine small-scale under-determination)
but small in the decision-relevant (energy) norm; twin pairs sit at the ~2-3σ
tail of the 0.06% residual. **Honest caveat now in the proposition:** 2D NSE at
moderate G is low-dimensional, so a non-marginal regime-2 separation likely
needs a higher-dim regime (much higher G / 3D / different substrate). Clause
(i) reframed (scale-selective, energy-marginal); ledger tag MEASURED
(near-determining). The rigorous wave found the result's true size before any
reviewer did. Runs: c1-recon-k3..k6 (local-closure metric), c1-recon-k3-fullQ
(the determination number). Remaining OPEN: alt projection, N=32 trend.

**C1 non-marginal-regime hunt 2026-05-29 — cheap reframe FAILED; C1@G200 is
physically marginal (honest endpoint).**
([`proof/PDE_C1_NONMARGINAL_PROBE.md`](proof/PDE_C1_NONMARGINAL_PROBE.md)).
Goal: a *non-marginal* regime-2 (under-determined content that is physically
relevant). (1) **High-G path hit the C2 numerical wall** — `lock_hidim_g1000`
(G=1000, grid 64) blows up at dt=0.01/0.002; stable only at dt=0.001 (10×
steps, ~7h/run) AND fragile (C2-style intermittent blow-up risk over 27M
steps). Not brute-forced; robust path = adaptive integrator (shared C2 resume
build). (2) **Cheap norm-reframe at G=200 — TESTED, FAILED.** Added a
three-norm `state-recon` (energy / enstrophy-`|k|²` / equal-weight). Result:
energy `FVE=0.997`, **enstrophy `FVE=0.993` (still marginal!)**, equal-weight
per-DOF `0.71`. The enstrophy norm does NOT rescue non-marginality; the only
non-marginal number is the unphysical equal-weight per-DOF count, whose
under-determination sits in **dissipation-range noise** (negligible energy AND
enstrophy, read by no physical objective). **Honest conclusion: C1 at G=200 is
marginal in EVERY physical norm** — `Φ_K` determines all energy/enstrophy-
carrying content; only viscous dissipation-range noise is unreconstructed.
Strictly non-vacuous but a **weak** example of the regime-2 target. A genuine
non-marginal separation needs the high-dim regime → adaptive integrator; no
shortcut at G=200. Proposition clause (i) + limitation updated honestly.

## Claim Boundary

This document does **not** claim that Sundog has produced original
mathematics on Navier–Stokes existence or smoothness. It claims that:

1. there is a coherent structural argument — that Sundog's
   sufficient-statistic discipline, built to evaluate
   indirect-signal claims under partial observability, is
   well-shaped to *read* and *complement* a specific class of
   existing PDE results (determining-modes / data-assimilation,
   Onsager-type coarse-graining thresholds, shell-model
   intermittency invariants) that already share its epistemic
   structure;
2. that argument is currently defended by analogy and by the
   three-body sibling result, not yet by a Sundog-authored note,
   reading, or evaluator output on any PDE result;
3. the proof targets that would test the argument are non-trivial
   enough that they need to live in a ledger before they live in a
   roadmap.

If a candidate below is promoted, it leaves this ledger.

Avoid broader formulations such as:

- "Sundog solves Navier–Stokes."
- "Signatures replace energy methods."
- "Coarse-graining proves regularity."
- "PINNs miss what signatures see."
- "Determining modes are Sundog's idea."

## The Coupling Claim

The coupling between Sundog's apparatus and the open NSE front is
staged on two fronts, in the
[`SUNDOG_V_CAPSET.md`](SUNDOG_V_CAPSET.md) idiom.

### Front A — Reading / instrument complement (defensible)

The 2024 synchronization-framework paper
([arxiv 2408.01064](https://arxiv.org/abs/2408.01064)) — see
[`NAVIERSTOKES_LITPASS_MEMO.md`](NAVIERSTOKES_LITPASS_MEMO.md) Track B —
formalizes determining modes as *state reconstruction* via observer
synchronization, not as sufficiency for a control objective. The
Coarse-Graining Postulate 1 predicate
([`COARSE_GRAINING_PROOF_ROADMAP.md`](COARSE_GRAINING_PROOF_ROADMAP.md)
§1) names the sharper, smaller object: `𝓕_σ`-measurability of `π*`,
which is Blackwell-sufficiency for the control objective rather than
for state reconstruction.

Front A stages the claim that re-reading the determining-modes program
through Postulate 1 produces a non-vacuous distinction — there are
NSE regimes where state-reconstruction sufficiency holds but
control-sufficiency does not, or vice versa, and the gap is operational.
The product is a reading note plus a pre-registered cell-set where the
distinction is measurable, not a new theorem.

This front is defensible now in the sense that no
Sundog-original PDE mathematics is claimed.

### Front B — Empirical leg on a tractable surrogate (horizon)

A more ambitious coupling: shell models of turbulence (Sabra, GOY) are
infinite-dimensional dynamical systems that retain the NSE energy
cascade and intermittency while remaining laptop-tractable. The 2022
hidden self-similarity result
([arxiv 2201.04005](https://arxiv.org/abs/2201.04005)) gives a
deterministic invariant of cascade intermittency; instanton importance
sampling
([arxiv 2308.00687](https://arxiv.org/abs/2308.00687)) gives a labeled
rare-event dataset.

Front B stages the claim that 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, in the
[`COARSE_GRAINING_PROOF_ROADMAP.md`](COARSE_GRAINING_PROOF_ROADMAP.md)
Phase-4 idiom).

This front is *not* defensible now; the candidates below describe
what would need to happen for it to become defensible.

### Bonus precedent

[Onsager's conjecture](https://link.springer.com/article/10.1007/s00222-024-01291-z)
is the existing PDE-side precedent for "coarse-graining sets a
regularity threshold" — energy conserved at C^α for α > 1/3, dissipation
admissible below. Giri–Radu 2024 and Du–Li–Ye 2025
([arxiv 2506.15396](https://arxiv.org/abs/2506.15396)) settle pieces of
the flexible side. This is the existing pure-math precedent that makes
the Postulate-1 measurability-threshold framing not analogy but sibling.
Cited so the framing is not read as imported metaphor.

## Falsification Surface

The coupling claim can fail in five named modes (one more than the
capset ledger, reflecting that NSE has both an algebraic and an
empirical attack surface):

1. **Front-A vacuity.** The Postulate-1 reading of determining modes,
   once written, reduces to what any careful PDE analyst would already
   say. The instrument has no edge over a good review.
2. **Front-A miscalibration.** The Postulate-1 / Blackwell-sufficiency
   distinction lands on confidently wrong readings (e.g., claims a
   control-sufficient signature where the determining-modes literature
   already pins state-reconstruction sufficiency at a smaller mode count
   that suffices for control too).
3. **Front-B vacuity (signature baselines).** On shell models, a
   matched-budget DMD / critical-slowing-down / lacunarity / Rényi
   baseline detects intermittent bursts at least as well as the
   signature-based detector. Signatures add no signal.
4. **Front-B reach (PDE extrapolation).** Even a clean shell-model
   positive result silently smuggles in finite-dimensional truncation
   structure that does not survive the move to full 2D-perturbed or 3D
   small-data NSE. Cross-substrate sameness fails at the substrate
   boundary.
5. **Coupling overreach.** The ledger is published alongside a public
   workbench, reading, or claim that elevates a Front-A reading or
   Front-B shell-model result into a Clay-problem-adjacent posture.
   Detected by [`presentation/claims-and-scope.md`](presentation/claims-and-scope.md)
   audit on any deploy that touches NSE vocabulary.

Each candidate below has to declare which modes it attacks.

## Evaluation Criteria

A candidate is worth pursuing if it:

- attacks at least one failure mode above with a falsifiable artifact;
- can be written without claiming Sundog-original PDE mathematics, *or*
  has a clear external-mathematician sanity-check path if it does;
- produces a deliverable that could plausibly land on the
  application-gallery or `/geometry.html` page alongside the other
  sundog workbenches rather than a standalone diversion;
- is honest about its dependence on prior work — determining-modes
  literature, Onsager construction, shell-model machinery, PINN
  baselines — and references it explicitly.

## Promotion Criteria

A candidate leaves this ledger and becomes a committed artifact
(proof note, workbench, paper section) when **all four** are true:

- (a) the Front-A vacuity check is explicitly rebutted (the artifact
  produces a claim or reading that is *not* what a careful PDE analyst
  would already say without the Sundog apparatus);
- (b) it has a runnable Phase-0 deliverable consistent with the existing
  Sundog harness pattern (pre-registered cell set, matched-seed
  baselines, named pre-registered negative);
- (c) it has an external-mathematician sanity-check path named in the
  promotion note;
- (d) it has a pre-registered failure boundary fixed before any
  numerical reads are interpreted, per
  [AGENTS.md ▸ "Pre-register the negative"](../AGENTS.md).

## Candidate List

Candidate 1 has been commissioned as a proof-track reading draft. No empirical
candidate has run.

### Candidate 1 — Postulate-1 reading of determining modes

- **Attacks failure mode(s):** 1 (Front-A vacuity), 2 (Front-A miscalibration).
- **Cost:** Low (a short note, ≤ ~4 pages).
- **Front:** A (reading).
- **Sketch.** Re-derive the Foias–Temam–Constantin determining-modes
  / nodes / volumes result as a `𝓕_σ`-measurability statement in the
  [`COARSE_GRAINING_PROOF_ROADMAP.md`](COARSE_GRAINING_PROOF_ROADMAP.md)
  Phase-1/2 idiom. Make explicit where the literature's
  state-reconstruction sufficiency *implies* control sufficiency
  (the bulk of the regimes), and identify a regime — if any —
  where control sufficiency is *strictly weaker* than state
  sufficiency, in the sense that a smaller signature suffices for the
  control objective.
- **Pre-registered negative.** If every regime where state
  sufficiency is open also has control sufficiency strictly equivalent
  to state sufficiency under the standard data-assimilation gauge, the
  Postulate-1 reading is vacuous on NSE — record, do not rescue.
- **Draft artifact.** Filed at
  [`proof/PDE_DETERMINING_MODES_POSTULATE1.md`](proof/PDE_DETERMINING_MODES_POSTULATE1.md)
  as a C1 reading note, not yet a promotion. Cell-set v0
  ([`proof/PDE_C1_CELLSET_KOLMOGOROV.md`](proof/PDE_C1_CELLSET_KOLMOGOROV.md))
  pins the pre-registered failure boundary at a Kolmogorov-flow regime.
- **Cross-files.** Would update
  [`COARSE_GRAINING_PROOF_ROADMAP.md`](COARSE_GRAINING_PROOF_ROADMAP.md)
  Phase 2 with a PDE corollary (or a named-negative).
- **External review path.** A practicing PDE analyst familiar with
  determining-modes work (Foias, Titi, Olson collaborators). Named at
  promotion.

### Candidate 2 — Signatures on shell-model intermittency (Sabra / GOY)

- **Attacks failure mode(s):** 3 (Front-B vacuity), 4 (Front-B reach).
- **Cost:** Low-Med (laptop simulation; baselines off-the-shelf).
- **Front:** B (empirical).
- **Sketch.** Generate matched-seed Sabra / GOY trajectories with
  imminent / non-imminent intermittent bursts (labels via instanton
  importance sampling and rare-event statistics). Train a path-signature
  detector on a fixed channel set
  (e.g. shell-wise energy `|u_n|²`, transfer rates,
  hidden self-similarity coordinates per
  [arxiv 2201.04005](https://arxiv.org/abs/2201.04005)).
  Compare matched-budget against DMD / Koopman, critical-slowing-down,
  recurrence lacunarity, and Rényi-entropy detectors. Pre-register cell
  set, signature truncation depth, baseline hyperparameters, and the
  decision threshold.
- **Pre-registered negative.** 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.
- **Scoping artifact.** Filed at
  [`proof/PDE_C2_SHELL_SIGNATURE_SCOPING.md`](proof/PDE_C2_SHELL_SIGNATURE_SCOPING.md)
  2026-05-28: drafted scoping, model commitment (Sabra primary + GOY
  cross-check), tiered channel taxonomy, matched-budget baseline rule,
  burst-prediction task definition, and named negatives `PDE-C2-NEG-A`
  (Pareto vacuity) and `PDE-C2-NEG-B` (overfit / cell-set drift). Cell-set
  v0 (concrete `N_shells`, `λ`, `ν`, `τ_burst`, `E_burst`, signature
  truncation `D`, operating point) deferred to a follow-up artifact.
- **Workbench surface.** Would live next to
  [`SUNDOG_V_THREEBODY.md`](SUNDOG_V_THREEBODY.md) as the
  PDE-substrate empirical sibling. No public deployment until external
  review.
- **External review path.** A practicing shell-model researcher
  (Mailybaev / Biferale / Frisch tradition). Named at promotion.

### Candidate 3 — PINN regularity-confidence diagnostic head

- **Attacks failure mode(s):** 4 (Front-B reach, indirectly), 5
  (coupling overreach guard).
- **Cost:** Med (requires a working PINN-NSE pipeline as input).
- **Front:** B (empirical) with application surface.
- **Sketch.** Take a PINN-predicted velocity field (off-the-shelf
  trained on a controlled NSE setup), compute a signature-based
  regularity-confidence score, pre-register the operating envelope
  inside which the score predicts the PINN's actual error on held-out
  ground truth. Fail cleanly outside.
- **Pre-registered negative.** If the diagnostic score is not Granger-
  or partial-information-gain-significant over the PINN's own residual
  on the pre-registered envelope, the head adds no signal — record and
  stop.
- **Workbench surface.** Optional public diagnostic widget if it lands.
  Not deployable until Candidate 1 or 2 lands first.
- **Deferred until Candidate 1 lands.**

### Candidate 4 — Full PDE signature transit detection on 2D-perturbed NSE

- **Demoted.** Subsumed in spirit by Candidate 2 at a fraction of the
  cost. Reconsider only if Candidate 2 returns a clean positive and an
  external reviewer recommends scaling the test to full PDE.

### Candidate 5 — Vorticity-stretching structural-zero invariant

- **Deferred.** Algebraic ansatz needs clarification before this can be
  named as falsifiable. The Shadow-Faraday Branch-A receipt
  ([`faraday/SHADOW_FARADAY.md`](faraday/SHADOW_FARADAY.md)) is the
  template — but the vorticity-equation analog isn't pinned yet.
  Re-open after Candidate 1 lands.

## Shortlist Recommendation

As of 2026-05-28, the recommended sequencing is:

1. **Candidate 1** (Postulate-1 reading of determining modes) — now
   commissioned as the active Front-A draft. It remains the lowest-cost,
   highest-leverage move; next gate is external PDE sanity check plus a
   runnable cell-set artifact.
2. **Candidate 2** (signatures on shell-model intermittency) — smallest
   empirical leg with the cleanest data path; depends only on standard
   shell-model machinery and off-the-shelf baselines. Scoping is now
   drafted at
   [`proof/PDE_C2_SHELL_SIGNATURE_SCOPING.md`](proof/PDE_C2_SHELL_SIGNATURE_SCOPING.md).
   Cell-set v0 and execution should follow Candidate 1's review outcome
   (`PDE-C1-NEG` → C2 may still run but is re-framed as a turbulence-signature
   detector only, *not* as evidence for a determining-modes coupling).
3. **Candidate 3** (PINN diagnostic head) — only after 1 and 2 land.

Candidates 4 and 5 are deferred pending the outcome of 1 and 2.

## Cross-references

- [`COARSE_GRAINING_PROOF_ROADMAP.md`](COARSE_GRAINING_PROOF_ROADMAP.md)
  — the trunk roadmap. NSE is the natural PDE-substrate leg parallel to
  the three-body measured leg in Phase 4. Candidate 1 is drafted at
  [`proof/PDE_DETERMINING_MODES_POSTULATE1.md`](proof/PDE_DETERMINING_MODES_POSTULATE1.md)
  (Phase-2 PDE corollary, pending review) and Candidate 2 is scoped at
  [`proof/PDE_C2_SHELL_SIGNATURE_SCOPING.md`](proof/PDE_C2_SHELL_SIGNATURE_SCOPING.md)
  (Phase-4 PDE-substrate empirical sibling, pending cell-set v0 and review).
  Neither updates the trunk roadmap before its respective review lands.
- [`SUNDOG_V_THREEBODY.md`](SUNDOG_V_THREEBODY.md) — sibling empirical
  workbench. Different substrate, same hidden-indirect-signature-output
  pattern. The three-body Phase 13–15 results are the empirical
  precedent for "compressed signatures detect near-singular events
  before they resolve."
- [`SUNDOG_V_CAPSET.md`](SUNDOG_V_CAPSET.md) — sibling ledger and
  pattern source. This document follows its ledger structure
  deliberately so the two are auditable together.
- [`NAVIERSTOKES_LITPASS_MEMO.md`](NAVIERSTOKES_LITPASS_MEMO.md) —
  the 2026-05-28 lit-pass record that informs the candidate ranking
  here. Re-audit point: gap claims are time-stamped, not absolute.
- [`faraday/SHADOW_FARADAY.md`](faraday/SHADOW_FARADAY.md) — the
  structural-zero discipline source for the deferred Candidate 5.
- [`SCIENTIFIC_CRITERIA.md`](SCIENTIFIC_CRITERIA.md) — the research-
  object discipline this ledger inherits. Any promotion must satisfy
  the criteria there.