LOOP FOLD-IN (box homogenization) — read first. The box relaxes to a spatially near-uniform STATE at uniform Carroll params (verified: tracer CV 4e-5 @200 -> 1e-15 @6400; 2026-06-27_box_homogenization_DEFINITIVE.md). This does NOT block the map, but it names where the map's spatial signal actually comes from: NOT the box-state pattern (flat), but (1) the observation noise model Sigma — i.e. WHERE real data exists (GEOTRACES iron, Daniels/MODIS PIC, dense POSi) — and (2) the forcing-modulated sensitivity J = d(observable)/dtheta (SST/wind vary the per-cell Jacobian). The FIM = J^T Sigma^-1 J formulation below already encodes this (Sigma -> inf where no data, so the FIM is zero there), so the goal is sound. Two binding additions: (a) a Phase 0 feasibility gate — confirm the converged per-cell J has nonzero forcing-driven spatial variation before building the full map (if J is also flat, the deliverable reduces to an observation-coverage map, still useful but a weaker claim); (b) the convergence gates G1/G2/G3 below must use the sensitivity pattern, not the state pattern, since the state's variance floor is machine-epsilon. Do NOT interpret any output as box-state fidelity.

All grounding confirmed. The convergence finding note (2026-06-27_box_convergence_finding.md) is referenced by track 4 but isn't on this branch — it lives on the spine-d worktree. I'll cite it as the load-bearing finding the brief already provides, treating the slow-mode numbers (eig 0.9957) as given. I have everything needed. Here is the goal document.

Goal Document: Per-Parameter Spatial Identifiability Map for the Carroll-6 Parameters in the DarwinDiff Box¶

Status: scoping / pre-build Owner: Track 1 (parameter-learner), spine D adjacent Target artifact: scripts/identifiability_map.py + docs/findings/identifiability_map_regions.json + per-parameter global figures Frames the manuscript as: a surrogate-to-model identifiability study (where on Earth is each Carroll-6 parameter identifiable), not a 6/6 recovery claim.

1. Problem statement¶

DarwinDiff recovers the six Carroll-6 parameters (alpfe, scav_rat, Smallgrow, Biggrow, diatomgraz, R_PICPOC; registry in src/darwindiff/carroll6.py) by gradient descent through a 0-D differentiable box (carroll6_5pft_2layer, 15 state vars, 2 layers) driven by a per-cell DINN. Training currently uses three hand-picked AOIs — equatorial Pacific, N Atlantic subpolar, Southern Ocean Pacific — and a hand-ablation (v3.1.1, n=200 fresh-seed fits) attributed parameters to regions: eqpac → alpfe+diatomgraz, natl → Biggrow+R_PICPOC, SO → scav_rat (STATUS.md per-AOI gating). The hand-picked AOIs are a methodological liability: the region set is unjustified, the attribution is an empirical recovery-rate artifact rather than a structural statement, and a reviewer cannot tell whether a parameter recovered because the data there constrain it or because the loss weights were tuned to make it recover.

This document scopes a principled, information-theoretic diagnostic that answers, per grid cell, where on Earth each Carroll-6 parameter is identifiable — computed from the Fisher information / Cramér-Rao bound of the box's observables with respect to each parameter, at the converged steady state. It is a standalone forward diagnostic over a fixed global parameter vector. It is not a learned component of the recovery NN: letting the recovery DINN select its own regions from this map would reward-hack and destroy the clean per-cell structural argument. The deliverable replaces "we hand-picked three regions" with "the iron pair is identifiable in high-dust / strong-DFe₂-gradient cells; R_PICPOC in thermally-gated coccolithophore cells; the growth pair is unidentifiable everywhere."

2. Hypotheses (falsifiable)¶

H1 (steady-state, not transient). The identifiability verdict computed at the converged steady state (≥1600 forward-Euler steps) differs qualitatively from the 200-step transient verdict. Concretely: the eqpac iron sensitivity sign flips between the two (brief: box-vs-Darwin FeT pattern r = −0.96 @200 steps → +0.81 @3200 steps). Falsified if the per-cell sensitivity argmax is identical at 200 and 1600 steps for the iron cells.

H2 (validation — map reproduces the hand-ablation attribution, in rank). Aggregated over each hand-picked AOI footprint, the per-cell marginal identifiability ranks the parameters consistently with the v3.1.1 recovery ablation: eqpac ranks alpfe/diatomgraz highest, natl ranks R_PICPOC (and Biggrow) highest, SO ranks scav_rat highest. Pre-registered pass: Spearman rank correlation ρ ≥ 0.6 between the map's per-AOI parameter-stiffness ordering and the ablation's per-AOI cal-grade ordering across the 6×3 = 18 (param, AOI) cells. This is a rank test, not a value match — the ablation folds in cross-AOI coupling, seed noise, and bounds-clipping that a local FIM cannot, so exact agreement is not expected and disagreements are diagnosed, not auto-failed.

H3 (sanity — growth pair flat everywhere). The Smallgrow and Biggrow marginal-identifiability fields are low and flat over the entire ocean under a real-data noise model, and these two parameters are never the per-cell argmax. No real observable constrains growth rates, so a correct map must show them sloppy everywhere. Falsified if any cell flags either growth parameter as confidently identifiable — that is a contamination signal (growth↔grazing or growth↔iron-quota degeneracy leaking signal, or a transient-FIM leak), not a discovery.

H4 (scav_rat is contrast-driven, not single-cell-local). scav_rat is marginally identifiable only where the surface↔subsurface dissolved-iron contrast is present (the DFe₂ channel, state index I_DFE_2 = 10, breaks the alpfe↔scav_rat surface-budget degeneracy). The map shows scav_rat jointly sloppy with alpfe wherever only surface DFe is constrained, and separable only where DFe₂ is informative. Consistent-with, not falsified-by, scav_rat looking sloppy in single-cell-local FIM away from strong subsurface gradients.

H5 (R_PICPOC thermally gated). R_PICPOC marginal identifiability lights up where calcite production is both large and thermally gated (coccolithophore window), i.e. subpolar N Atlantic / bloom zones, and is sloppy in oligotrophic low-PIC cells.

3. Method — Fisher / CRLB at the converged steady state¶

The map is a per-cell, per-parameter scalar field built in five stages. Stages 2–4 are the load-bearing distinction from a naive sensitivity heatmap.

3.0 Notation¶

θ ∈ ℝ⁶ — the Carroll-6 vector, held constant across all cells at θ₀ = CARROLL_VALUES (single global θ; the diagnostic asks where the global θ is most constrained).
u_c — per-cell forcing (SST, SSS, wind, latitude, dust), from the Darwin target cache.
f(x, θ; u_c) — the per-step tendency assembled in carroll6_5pft_2layer_step (the dx/dt vector; box source src/darwindiff/carroll6_5pft_2layer.py:306).
x(θ, u_c) — the converged steady state, f(x, θ; u_c) = 0.
g(x) — the observable map: exactly what the loss compares (see §3.2).

3.1 Stage 1 — Converged steady state (non-negotiable)¶

The recovery integrates only N_STEPS = 200 forward-Euler steps at dt = 0.25 d (box_vs_darwin_fidelity.py:48-49). The linearized box has a slow dissipative mode (one-step-map eigenvalue ≈ 0.9957 ⇒ relaxation time τ = −dt / ln(0.9957) ≈ 230 steps). At 200 steps the trajectory is an under-converged transient — a sensitivity evaluated there measures dx(t)/dθ of an unconverged trajectory, a different and artifactual object than the steady-state dx*/dθ.

Requirement: integrate to ≥ 1600 steps (≥ ~7τ; residual transient ≈ e⁻⁷ ≈ 0.1%) and verify convergence per cell — a single global step count is not uniformly converged because τ varies with forcing. Per-cell convergence gate (all three must hold, else mask the cell):

G1 — steady-state residual: ‖f(x; θ)‖ / ‖x‖ < tol.
G2 — pattern stability across a doubling: the per-cell observable sensitivity at N and 2N agrees (z-scored pattern r(N, 2N) ≥ 0.99). This catches the documented FeT oscillation (the brief's convergence probes report the box-vs-Darwin FeT r alternating regimes deep in the integration, not monotone relaxation) — a single fixed step count can sit on either side.
G3 — variance floor: spatial CV = std/|mean| of the pattern stays well above machine-ε relative to Darwin's own CV, so a "converged r" is not correlating round-off.

If the iron rows show a genuine limit cycle at a cell (no fixed point), mark that cell UNDEFINED rather than reporting a steady-state FIM there.

3.2 Stage 2 — Steady-state sensitivity via the implicit function theorem¶

At the fixed point, differentiating f(x*, θ) = 0 w.r.t. θ (implicit function theorem; standard in adjoint / parametric-sensitivity literature, Dickinson & Gelinas 1976; Saltelli et al. 2008):

(∂f/∂x) · (dx/dθ) + ∂f/∂θ = 0 ⟹ dx/dθ = −(∂f/∂x)⁻¹ (∂f/∂θ)

∂f/∂x = A is the 15×15 state Jacobian (its eigenvalues are the relaxation modes; the slow mode is A's eigenvalue nearest 0).
∂f/∂θ is the 15×6 parameter-of-tendency matrix.

Both come cleanly from autograd on the single-step function at x (torch.func.jacrev / torch.autograd.functional.jacobian) — no 200-step unroll/backprop is needed once x is found**, which is both cheaper and free of the transient artifact. The observable sensitivity is then

J_c = (∂g/∂x) · (dx*/dθ), shape [N_obs × 6].

Observable map g (must be exactly what the loss reads; state indices from the box):

Observable	State index	Real anchor / noise source
surface DFe (FeT)	`I_DFE_1 = 0`	GEOTRACES `Fe_*_err` 1σ (`geotraces_loader.py:36`)
subsurface DFe₂	`I_DFE_2 = 10`	GEOTRACES subsurface (breaks iron-pair degeneracy)
PIC	`I_PIC_1 = 7`	MODIS PIC climatology spread (`modis_pic_loader.py`)
POC	`I_POC_1 = 6`	Darwin field representativeness variance
realized PIC:POC ratio	derived = PIC/POC	identifies `R_PICPOC` orthogonally to the iron pair
DIC, ALK	`I_DIC_1 = 8`, `I_ALK_1 = 9`	GLODAP / field variance
bSi (diatom)	via `diagnostic_bsi_steady(state[I_DIATOM=1], diatomgraz)` (`silica.py`)	GEOTRACES bSi (pins `diatomgraz`)
5 Chl/biomass channels	`I_DIATOM…I_PROHL` (1–5)	biology — inflate Σ; box is a surrogate

For z-scored pattern channels, fold the z-score (subtract spatial mean, divide by spatial std) into ∂g/∂x so J and the residual the loss sees are unit-consistent.

Stage 2b — Gramian/Lyapunov cross-check (justification, not primary). The dynamic counterpart of the IFT sensitivity is the variational ODE ds/dt = A s + ∂f/∂θ; the observability Gramian W_o solving the Lyapunov equation AᵀW_o + W_o A + CᵀC = 0 (C = ∂g/∂x) has near-null directions = unobservable state combinations. The finite-time Gramian → steady Gramian only after ~7τ — this is the formal justification for the ≥1600-step requirement (Kailath 1980; Stigter & Molenaar 2015). Use it to diagnose whether the growth pair is W_o-null (structurally unobservable) vs merely weakly excited.

3.3 Stage 3 — Fisher information with an explicit noise model¶

FIM_c = J_cᵀ Σ_c⁻¹ J_c (6×6 per cell)

This is the Gaussian-likelihood Fisher information and the Gauss-Newton / JᵀJ approximation to the least-squares loss Hessian (Kay, Fundamentals of Statistical Signal Processing vol. 1, ch. 3). Σ_c is the per-cell observation-noise covariance and is load-bearing, not cosmetic — each runner loss weight w_k is implicitly 1/σ_k². Report two versions:

Σ_v1 (loss-faithful): Σ⁻¹ = diag of the runner's actual weights (GEOTRACES_W, GEOTRACES_SUB_W, PIC_ABS_W, POC_ABS_W, ALK_ABS_W, RATIO_W, per-AOI AOI_W). Explains the deployed recovery.
Σ_v2 (data-faithful): real observation error (GEOTRACES _err, MODIS spread, GLODAP) plus an additive model-discrepancy covariance Σ_model inflating the biology channels by (1 − r_box-vs-Darwin) per channel (from box_vs_darwin_fidelity.py). Without Σ_model the FIM claims spurious confidence where the box cannot reproduce Darwin even at θ₀.

Where Σ_v1 and Σ_v2 disagree localizes where the hand-tuned weighting — not the data — is doing the identifying (a reviewer-grade honesty check). The growth pair has no real observable ⇒ infinite variance / zero Fisher weight in Σ_v2 by construction — exactly the flat-everywhere signature H3 predicts.

3.4 Stage 4 — Marginal identifiability = diag(FIM⁻¹), NOT 1/diag(FIM)¶

This is the single most important methodological point. diag(FIM)_jj measures sensitivity to θ_j with all other params held fixed at truth — it ignores trade-offs and overstates identifiability. The honest, other-params-unknown quantity is the Cramér-Rao bound:

Var(θ_j) ≥ (FIM⁻¹)_jj ; marginal standard error = √((FIM⁻¹)_jj).

The gap between 1/FIM_jj and (FIM⁻¹)_jj is the sloppiness/trade-off. If alpfe and scav_rat are each individually sensitive but anti-correlated (the surface iron-budget degeneracy: dust source vs scavenging sink), the FIM is near-singular in that 2×2 block and both marginal CRBs blow up even though each diagonal looks healthy — exactly the failure the project already hit. The map will show scav_rat becoming marginally identifiable only where the DFe₂ channel conditions that block (H4).

Stage 4b — structural vs practical (sloppiness; profile likelihood). Eigendecompose FIM_c = Σ_i λ_i v_i v_iᵀ. The log-uniform eigenvalue spectrum is the sloppy-model signature (Brown & Sethna 2003; Gutenkunst et al. 2007; Transtrum et al. 2015). Stiff (large λ) = tightly constrained combos; sloppy (tiny λ) = unconstrained. Structural non-identifiability = a truly zero eigenvalue / exactly-flat profile-likelihood direction (expected for the growth pair everywhere). Practical non-identifiability = small-but-nonzero λ. The FIM (a local quadratic) cannot by itself separate these — the gold-standard confirmation at selected cells is the profile likelihood (Raue et al. 2009): fix θ_j on a grid, re-optimize the other 5 through the box, trace χ²(θ_j); flat = structural, finite-well = practical, well-hits-bound = bound-limited. Caveat: sloppiness ≠ non-identifiability (Apgar/Transtrum 2014) — always report eigenvectors/combinations, confirm edge cases with profile likelihood, never label from FIM diagonals alone.

3.5 Stage 5 — Collapse to scalar fields + parameter coordinates¶

Work in bound-normalized / log parameter coordinates (params span scav_rat ~6e-7 s⁻¹ to Smallgrow ~0.66 d⁻¹; the recovery already uses bounded_params). Use the chain rule dθ_phys = (hi−lo)·sigmoid′(·)·dθ_unc with PARAM_BOUNDS from the registry, so the CRB is dimensionless and comparable across the six and the eigenvalue spectrum is a meaningful sloppiness plot.

Per parameter j, emit three scalar fields: 1. Marginal identifiability I_j(c) = −log₁₀(relative CRB) = −log₁₀(√((FIM⁻¹)_jj)/|θ₀,j|) — higher = more identifiable. 2. Trade-off / inflation factor ρ_j(c) = √((FIM⁻¹)_jj · FIM_jj) — ratio of marginal to conditional error; ρ = 1 no trade-off, ρ ≫ 1 sloppy/degenerate (the alpfe↔scav_rat tell). 3. Binary identifiable-mask at a threshold (the "place an AOI here" decision).

Plus two joint fields: (4) smallest-FIM-eigenvalue field (worst unconstrained direction), (5) participating-parameter field (which params load on the sloppiest eigenvector) — shows which pair is degenerate where.

Conditioning: FIM is deliberately near-singular in the degenerate blocks; never naive-invert. Use the eigendecomposition with an eigenvalue floor, or a Bayesian/regularized CRB (FIM + prior precision)⁻¹ with the physical bound width as prior — finite, interpretable marginal errors, the honest statement of "identifiable given a physically-bounded prior."

4. Implementation plan (phased)¶

The build lifts the existing scripts/box_vs_darwin_fidelity.py (which already runs carroll6_5pft_2layer_integrate on a full 2D [H,W] AOI field at CARROLL_VALUES, building state0 from the Darwin cache) from per-AOI → global and from 200 → ≥1600 steps, with autograd of d(observable)/d(θ) added. No new physics, no runner dependency for J.

Scoping gaps confirmed against this branch (refactor-param-registry): - scripts/box_linearization.py does not exist on any branch (verified git ls-files). The brief's "reuse box_linearization.py" instruction is not actionable — the FIM pseudo-inverse / eigenvalue-floor handling must be built fresh inside identifiability_map.py. - scripts/identifiability_sloppiness.py is not on this branch (it lives on the spine-d worktree, untracked in the snapshot). The reward-hacking-safe import pattern (NB23_N_EPOCHS=0, no-gating) and the eigh pattern must be re-implemented or cherry-picked, not assumed present. - Plan accordingly; do not budget as if either scaffold exists here.

Phase 0 — Convergence calibration (do FIRST; gates everything, ~0.5 day). Copy the state0-from-Darwin recipe from box_vs_darwin_fidelity.py for 3–5 representative cells (HNLC eqpac, dust-fed gyre, subpolar bloom, Southern Ocean, oligotrophic). Compute per-cell J = d(observables)/d(θ) at CARROLL_VALUES via torch.func.jacrev at n_steps ∈ {200, 400, 800, 1600, 3200}. Plot |J| and the iron-vs-Darwin sign vs n_steps; pick N_STEPS_CONV as the smallest count where J changes < 1–2% to 3200. Correctness gate: reproduce the eqpac iron sign flip (−0.96@200 → +0.81@3200). If it does not reproduce, STOP and reconcile before building the global map. Expected N_STEPS_CONV ≈ 1600.

Phase 1 — Global field staging (~0.5–1 day). Build a global Darwin target+forcing cache by running the IC-cache / bundle path with AOI bounds = whole globe (lat −90..90, lon −180..180) so bin_to_1deg_grid (llc270_loader.py:353) produces a [180,360] field. Stage state0 (FeT/POC/PIC/DIC/ALK from Darwin), forcing (SST/SSS/wind/pco2_atm), and the global ocean mask (~41k–46k ocean cells). Output a single global_targets.pt. Confirm against D:\ecco_darwin_v5 (DARWIN_ROOT).

Phase 2 — Per-cell Jacobian engine (~1–2 days). compute_J(theta[6], state0[15,N], forcing) → J[N_obs, 6, N] integrating to N_STEPS_CONV. Use torch.func.vmap(jacrev(f, argnums=θ), in_dims=cell-axis) where f maps θ → observable-vector at one cell (reverse-mode gives all 6 columns from one graph). Memory: a global [15, 46k]×1600-step autograd graph is ~26 GB — tile into chunks of ~5–10k cells (fresh graph per tile; embarrassingly parallel). Fits H200 (140 GB); the full sweep is an H200 / NU Explorer job, Phase 0 + tiled tests run on the 5090. Use the compiled per-step where available. Flag and mask cells whose trajectory touches the clamp(min=0.0) non-negativity floor (carroll6_5pft_2layer.py:558) — the clamp is non-differentiable at 0 and corrupts J there.

Phase 3 — FIM assembly + Σ (~1 day). Per cell FIM = Σ_obs J_obsᵀ Σ_obs⁻¹ J_obs. Build both Σ_v1 (loss weights) and Σ_v2 (real error + Σ_model inflation). Growth-rate-direct entries → +∞ variance. Handle rank-deficient/conserved modes via torch.linalg.eigh + eigenvalue floor (or Bayesian CRB), never naive inverse. Report raw eigenspectrum (sloppiness span) and the floored pseudo-inverse.

Phase 4 — Per-parameter scalar maps + region ranking (~0.5 day). Emit the six I_j, ρ_j, mask fields + two joint fields as [180,360]. Rank cells per parameter, aggregate to named regions / lat-lon boxes, emit top-K per parameter. Sanity gates that must hold or the map is wrong: (a) growth pair flat/low everywhere; (b) iron pair peaks where GEOTRACES iron + dust gradients are strong; (c) R_PICPOC peaks in coccolithophore/calcite regions.

Phase 5 — Artifacts + packaging (~0.5 day). scripts/identifiability_map.py with argparse (--n-steps default N_STEPS_CONV, --tile-cells, --theta {carroll|json}, --sigma-source {loss|data}, --out). Outputs: per-param GeoTIFF (EPSG:4326, 1°) + PNG (matplotlib, coastlines, log-precision colormap); docs/findings/identifiability_map_regions.json ({param: {best_regions:[{name,bbox,mean_precision,n_cells}], field_stats}}); a provenance block (θ, n_steps, Σ sources, Phase-0 convergence result). Add a tests/ smoke test: a 3-cell map at n_steps=50 runs end-to-end and growth-pair precision < iron-pair precision in an iron-rich cell.

5. Validation plan¶

V1 — Falsification against the hand-ablation (H2), as rank-recovery. Per hand-picked AOI, aggregate the per-cell marginal identifiability to a per-AOI score per parameter. PASS: argmax/top-rank per AOI matches the ablation carriers (eqpac→{alpfe,diatomgraz}, SO→scav_rat, natl→{Biggrow,R_PICPOC}) in rank, with Spearman ρ ≥ 0.6 across the 18 (param,AOI) cells (pre-registered). Treat scav_rat and R_PICPOC specially: both are contrast/coupling-driven (scav_rat needs surface↔subsurface DFe; R_PICPOC is per-AOI-unidentified, only joint-straddle recoverable per the ALK-anchor finding), so a purely-local map is expected to under-rank them — build that into the criterion rather than calling it a failure. The ablation is a prior to corroborate, not an oracle (it is n=200 seeds, coupling-laden; stage1_per_aoi_gating_2aoi.md shows severing cross-AOI coupling drops mean cal 2.95→1.65).

V2 — Convergence check (H1). Compute the map at 200 and at N_STEPS_CONV; show the eqpac iron cells flip sign, demonstrating that a 200-step map mis-attributes iron identifiability. Publish the fraction of cells failing each of G1/G2/G3 as a map-quality figure. Keep only cells whose per-cell argmax is stable between N_STEPS_CONV and 2×.

V3 — Growth-pair sanity (H3), automated assertion. Hard check: across all cells/AOIs the {Smallgrow, Biggrow} rows have total influence below threshold and are never the per-cell argmax under a real-data loss. If any region flags growth as identifiable, dump that cell's 6×6 normalized Hessian and find the degeneracy partner before trusting the map. Cross-check against the existing per-AOI FIM verdicts (growth pair pinned far from Carroll at low curvature).

V4 — Profile-likelihood spot-check (locality of the FIM verdict). At one representative cell per regime, run the full profile likelihood for each of the 6 params (Raue et al. 2009). Confirm the FIM/CRB ranking matches the profile-well curvature, growth-pair profiles are flat (structural), and R_PICPOC's profile is a finite well only in the coccolithophore-zone cell.

V5 — Cross-tool agreement. Run the global-θ profile-likelihood diagnostic (identifiability_sloppiness.py, in --loss realiron/realbsi/realpic) on the 3 AOIs and confirm its verdicts agree with the new per-cell map aggregated over the same footprints. Two routes to the same question; agreement is the strongest non-artifact evidence.

6. Acceptance criteria (specific, measurable)¶

scripts/identifiability_map.py exists, runs end-to-end on the global 1° field at N_STEPS_CONV ≥ 1600, tiled, and emits 6×{I_j, ρ_j, mask} + 2 joint fields as GeoTIFF+PNG, plus identifiability_map_regions.json with a provenance block.
Phase 0 correctness gate passes: the eqpac iron sign flip (≈ −0.96@200 → +0.81@3200) is reproduced.
Convergence verified per cell: G1/G2/G3 fractions reported; cells failing any gate (or showing a limit cycle in the iron rows) are masked/UNDEFINED, not mapped.
H2 passes: Spearman ρ ≥ 0.6 between per-AOI parameter-stiffness ranking and the v3.1.1 ablation cal-grade ranking; any sub-threshold (param,AOI) cell has a written diagnosis (coupling/contrast, not a silent fail).
H3 passes: growth-pair marginal identifiability below threshold in 100% of cells under Σ_v2; growth params are the argmax in 0 cells. Automated assertion in the smoke test.
Two-Σ honesty figure delivered: Σ_v1 vs Σ_v2 difference map showing where weighting (not data) identifies, with Σ-sensitivity pass (0.5×/2× Σ) confirming region rankings are robust.
Reward-hacking guard asserted in code: the θ tensor is constant across cells (broadcast); no recovery metric, gating mask, or DINN-learned param enters the map pipeline (code-level assert).
V4 profile-likelihood spot-checks at ≥4 regime cells corroborate the FIM ranking and confirm flat growth-pair profiles.

7. Risks + honesty guards¶

Diagnostic, not a recovery component. Computed at fixed CARROLL_VALUES (or a real-data optimum) over a single global θ, broadcast across cells, with no gating, no recovery loss, no seed-tuning in the loop (mirror the NB23_N_EPOCHS=0, no-gating import discipline). Code-assert θ is constant across cells so the map cannot encode the DINN's regional choices. Making the recovery NN select regions from this map would reward-hack and break the per-cell structural argument.

Transient artifact (highest risk). A 200-step map inverts the eqpac iron sign and is scientifically wrong. Mitigation: IFT on the verified fixed point at ≥1600 steps with per-cell convergence + Jacobian-stability gating; never trust a fixed step count globally.

Convergence may not be a solved "run longer" problem. The box shows a slow FeT oscillation deep in the integration (not monotone relaxation); there may be no single step count at which every cell's iron sensitivity is stable. If the iron rows are a genuine limit cycle at a cell, the steady-state FIM is ill-defined there — mark UNDEFINED. The forward-Euler scheme (dt=0.25, stability margin thin at eig 0.9957) may itself be the source of the slow mode; state that all convergence claims are conditional on this integrator.

Surrogate scope (state it in every artifact). The box is a 0-D emulator of Darwin, not Darwin; chemistry vs biology fidelity differ (Chl rows are biomass-not-chlorophyll; DIC/ALK/FeT are IC-confounded). A FIM that ignores model error claims confident identifiability where the box structurally cannot match Darwin. Mitigation: Σ_model inflation of biology channels; report the data-faithful Σ_v2 alongside Σ_v1. Title every figure "identifiability of the BOX SURROGATE at steady state" and carry the box→Darwin fidelity caveat (noting the published fidelity r's were 200-step transients). "Identifiable in the box" is necessary but not sufficient for "recoverable from real data" — cross every positive verdict against the real-anchor inventory (GEOTRACES iron/bSi, Daniels CP:PP / MODIS PIC).

FIM is a local Gaussian approximation. It captures only quadratic curvature at θ₀; it cannot represent bounded/one-sided regions, multimodality, or structural-vs-practical. Bounds make this worse near edges (R_PICPOC, scav_rat sit near bound edges). Mitigation: profile likelihood at representative cells; report when the marginal CRB exceeds the bound width (then bound-limited, not data-identified).

FIM conditioning. The point is that FIM is near-singular in degenerate blocks. Naive inversion explodes. Mitigation: pseudo-inverse with eigenvalue floor / Bayesian CRB; report directly from the eigendecomposition.

Σ = loss-weights is a choice, not ground truth. The runner weights were hand-tuned to force 6/6; adopting them makes the map explain the deployed recovery but inherits its biases. Always pair with Σ_v2.

Per-cell 0-D scope. The box is 0-D per cell with no horizontal coupling (DINN is 1×1 conv only), so the per-cell FIM is independent across cells — legitimate for this model, but it omits that real-world identifiability could borrow strength across neighbors. State this scope limit so the map is not over-read as a full spatial-field identifiability.

8. Deliverables and how it feeds the manuscript + issue tracker¶

Deliverables. - scripts/identifiability_map.py (standalone diagnostic; argparse; tiled; GeoTIFF + PNG output). - global_targets.pt (global Darwin state0 + forcing + mask cache). - docs/findings/identifiability_map_regions.json (per-parameter best-region rankings + provenance). - Six per-parameter global identifiability figures + the two joint sloppiness/participation figures + the Σ_v1-vs-Σ_v2 honesty figure + the 200-vs-1600 convergence falsification panel. - A short findings note (per repo convention: returned in PR/issue text, not a standalone hype .md) recording N_STEPS_CONV, the H2 Spearman ρ, the H3 assertion result, and the profile-likelihood spot-checks.

Manuscript (surrogate-to-model identifiability study). This is the central figure-set: it replaces "we hand-picked three AOIs" with an information-theoretic, falsifiable "where on Earth is each Carroll-6 parameter identifiable in the box, at steady state." It positions the work as the per-grid-cell, parameter-space forward-Fisher analog of domain-localized BGC parameter estimation (Mamnun et al. 2025, JGR Oceans, per-column estimation), with the differentiator that it is a no-assimilation, no-reward-hackable diagnostic; it stratifies identifiability against per-cell environment (SST/MLD/wind/lat) the way Mamnun et al. 2023 (Sobol, MLD-dependent) and Beck et al. 2017 (collinearity index) motivate; it turns the project's own slow-mode finding (eig 0.9957) into a methods-section strength (transient-FIM-as-artifact panel) that pre-empts the obvious referee attack; and it states the growth pair as the honest negative control (flat everywhere = the method is not hallucinating signal).

Issue tracker (2imi9/ECCO-DarwinDiff, spines A–D under epic #124). File a new issue under the index epic capturing this scope (acceptance criteria from §6, the two confirmed scaffolding gaps from §4, and the H200-tiling/NU-Explorer dependency), cross-referencing the convergence finding and the per-AOI gating ablation. The map is the principled replacement for the hand-picked-AOI practice that several open spine items assume; link it from those.

Cited prior art (inline above)¶

Raue et al. 2009, Bioinformatics 25:1923 (profile likelihood, structural vs practical) · Brown & Sethna 2003, PRE 68:021904; Gutenkunst et al. 2007, PLoS Comput Biol 3:e189; Transtrum et al. 2015, J Chem Phys 143:010901 (sloppiness / FIM spectrum) · Apgar/Transtrum 2014, arXiv:1403.1417 (sloppiness ≠ non-identifiability) · Kay, Fundamentals of Statistical Signal Processing vol. 1 ch. 3 (CRB, FIM=JᵀΣ⁻¹J) · Dickinson & Gelinas 1976, JCP 21:123; Saltelli et al. 2008 (IFT steady-state sensitivity) · Kailath, Linear Systems 1980 ch. 6; Stigter & Molenaar 2015, Automatica 58:118 (observability Gramian / Lyapunov) · Constantine et al. 2014, SIAM J Sci Comput 36:A1500 (active subspaces) · Palmer et al. 1998, JAS 55:633 (targeted-observation / adjoint-sensitivity design) · Mamnun et al. 2025, JGR Oceans 130:e2025JC022752 (domain-localized per-column BGC parameter estimation) · Mamnun et al. 2023, SESMO 5:18613 (1-D BGC Sobol, MLD-dependent) · Beck et al. 2017, Ecol Modelling 363:17 (collinearity index, BGC identifiability) · Atkinson, Donev & Tobias 2007 (D-optimal design over the FIM).

In-repo grounding: src/darwindiff/carroll6.py (PARAMS registry; values/bounds verified this branch) · src/darwindiff/carroll6_5pft_2layer.py (carroll6_5pft_2layer_step:306, _integrate:561, state indices I_DFE_1=0…I_ALK_2=14, clamp(min=0.0):558) · scripts/box_vs_darwin_fidelity.py (global-field template; N_STEPS=200, DT=0.25) · src/darwindiff/geotraces_loader.py:36 (iron _err Σ) · src/darwindiff/modis_pic_loader.py (PIC Σ) · src/darwindiff/silica.py (diagnostic_bsi_steady) · src/darwindiff/llc270_loader.py:353 (bin_to_1deg_grid) · docs/findings/stage1_per_aoi_gating_2aoi.md, docs/findings/alk_anchor_rpicpoc_mutex.md (coupling/contrast evidence).

Document location for the engineer: intended path docs/findings/2026-06-27_identifiability_map_goal.md (not yet written to disk — this is the synthesis return). Scaffolding caveat for whoever picks it up: on branch 2imi9/refactor-param-registry, neither box_linearization.py nor identifiability_sloppiness.py is present; budget Phase 2–3 to build the FIM eigenvalue-floor + reward-hacking-safe import from scratch (or cherry-pick identifiability_sloppiness.py from the spine-d worktree first).