Surrogate gap quantified — box vs Darwin at the TRUE Carroll params (2026-06-26)

SUPERSEDED (2026-06-27) by 2026-06-27_box_homogenization_DEFINITIVE.md. The box-vs-Darwin pattern-r table below is not a well-defined fidelity metric. Verified (CV computation + adversarial review extending to 12800 steps): at uniform Carroll params the box tracer fields are nearly spatially flat (CV ~4e-5 at 200 steps → ~1e-15 by 6400) while Darwin's fields are O(1) (CV ~0.6–2.4). So every r below correlates a near-flat field against a structured one; the values and their sign-flips (incl. eqpac FeT −0.96→+0.81→−0.79) are vanishing-structure/IC artifacts. Both this note's "good chemistry proxy r~1 DIC/ALK; eqpac iron a fundamental gap" AND the intermediate box_convergence_finding correction are wrong. What stands: ALK (an IC self-match, not fidelity) and PIC:POC = 0.0424 (IC-independent). The real surrogate gap is dimensional — a 0-D box cannot generate Darwin's circulation-driven spatial structure. The downstream conclusion (recovery leans on real absolute anchors, not the Darwin pattern) is stronger under the correct reading. Read the definitive note; treat the table below as historical.

scripts/box_vs_darwin_fidelity.py run locally (5090). No fitting: the 2-layer box is handed Carroll's published parameters and integrated to quasi-steady state; we report, over ocean cells, the z-scored spatial-pattern correlation r (box vs Darwin) per field, and the surface PIC:POC ratio-of-means. This isolates the surrogate-fidelity component of the recovery error from identifiability and optimization (the self-twin isolates the method; this isolates the proxy).

Result — z-scored spatial-pattern r (box at Carroll vs Darwin v05, 1°)

field	eqpac	natlsubpolar	southernoceanpac
Chl1	−0.26	0.07	−0.34
Chl2	0.54	−0.26	0.11
Chl3	0.11	−0.01	−0.03
Chl4	0.11	nan	nan
Chl5	0.37	0.64	−0.03
POC	−0.04	0.05	−0.33
PIC	0.44	1.00	−0.51
FeT	−0.96	0.89	0.80
DIC	1.00	0.99	1.00
ALK	1.00	1.00	1.00

(nan = the field has ~no variance in that AOI, so correlation is undefined.)

Result — surface PIC:POC ratio (box at Carroll vs Darwin)

AOI	box	Darwin	box/Darwin
eqpac	0.0424	0.0332	1.28×	OK
natlsubpolar	0.0425	0.6758	0.06×	MISMATCH
southernoceanpac	0.0424	0.0067	6.35×	MISMATCH

What this refines (numbers, not narrative)

1. The surrogate gap is large and field-specific. Carbonate chemistry (DIC, ALK) is reproduced almost exactly (r ≈ 1.0); the phytoplankton (Chl) and POC patterns are essentially uncorrelated even at the true parameters (r in −0.34…0.64). The 0-D box is a good chemistry proxy and a weak biology proxy — this is the irreducible surrogate gap, independent of any fitting or resolution.

2. It explains the loss-weighting finding mechanistically. eqpac FeT r = −0.96 — the box's equatorial iron pattern is anti-correlated with Darwin's. So a z-scored Darwin-FeT loss term actively pulls the iron parameters the wrong way, and the only valid iron anchor is real GEOTRACES iron (absolute units), not Darwin's pattern. This is the physical cause behind the FIM/profile "loss-weighting, not structural" result (full-loss alpfe → 0.103; real-iron → 0.9997).

3. It re-derives the R_PICPOC regional finding from Darwin's own field. Darwin's realized PIC:POC spans 0.0067 (SO) → 0.68 (natl), ~100×, while a single global R_PICPOC = 0.042 is flat by construction. A single global scalar structurally cannot match the spread — independent of the estimator. This is the "R_PICPOC should be regional" point, measured.

Skeptical check it forces (passes)

We recover the iron pair in eqpac despite the box's eqpac iron pattern being anti-correlated with Darwin's (r = −0.96). That is only consistent because the iron-pair claim rests on absolute-units GEOTRACES iron, not the Darwin FeT pattern. Had we claimed recovery from the Darwin pattern, r = −0.96 would falsify it. It does not — because we do not. Stated as a guard against over-reading the Darwin-graded numbers.

For the manuscript

This is the measured magnitude the 4-cause error decomposition needed: surrogate fidelity is not a label, it is r ≈ 1.0 for chemistry and r ≈ 0 (or negative) for biology, per AOI. It also strengthens the "use real, Darwin-independent data" thesis: where the proxy can't reproduce Darwin's pattern (iron in eqpac; the PIC:POC spread), recovery must and does come from real observations.