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The box homogenizes — the surrogate gap is dimensional, not per-field (DEFINITIVE, 2026-06-27)

Supersedes the box-vs-Darwin pattern-r tables in 2026-06-26_surrogate_gap_quantified.md and 2026-06-27_box_convergence_finding.md. Both reported a quantity that is not well-defined. This note is the verified, doubly-vetted conclusion (my own CV computation box_cv_check.py + adversarial review wf_3e9befaf, which independently extended integration to 12800 steps).

Verified finding — the box relaxes to a spatially near-uniform state

At the published Carroll-6 parameters (uniform across cells) under real per-cell eqpac forcing (SST CV 0.04, wind CV 0.12), the box tracer fields have almost no spatial structure at any integration length, and what little they have collapses to round-off:

steps FeT POC PIC phyto ALK
200 4.0e-5 4.4e-5 6.7e-5 3.5e-5 1.7e-2
3200 9.3e-12 1.0e-11 1.0e-11 4.5e-12 1.8e-2
6400–12800 1.2e-15 1.6e-15 1.1e-15 2.0e-16 1.9e-2

(spatial CV = std/mean over ocean cells). Darwin's fields have CV FeT 2.39, POC 0.60, PIC 0.92 — four to fifteen orders of magnitude more spatial structure than the box produces.

What this means

  1. The box-vs-Darwin spatial-pattern r for FeT/POC/PIC/DIC is not a meaningful fidelity metric. The z-score amplifies a ~1e-5 (eventually ~1e-15) residual to unit variance and correlates it against Darwin's O(1) structure; the result — including the much-discussed eqpac FeT "r" that wanders −0.96 (@200) → +0.81 (@3200) → −0.79 (@6400+) — is set by vanishing/IC residual and has no converged value. Do not report these as fidelity. Both prior notes did, in opposite directions, and both are corrected here.

  2. What is robust: ALK retains ~2% CV (it is a near-conserved mode pinned to the Darwin IC — a near-tautological self-match, not evidence of chemistry fidelity), and surface PIC:POC = 0.0424, constant 200→12800 steps and IC-independent — the one load-bearing chemistry number. The R_PICPOC regional finding (Darwin's realized PIC:POC spans ~100× across AOIs) is integration- and homogenization-independent and stands.

  3. The real surrogate gap is dimensional. A 0-D box, applied per-cell at uniform parameters, has no mechanism to generate large-scale spatial structure — Darwin's structure comes from ocean circulation/advection the box does not represent. So the box cannot reproduce Darwin's spatial pattern; not for any per-field-r reason, but by construction. This is the honest "surrogate gap."

Why this strengthens the project's thesis (not weakens it)

  • It is the strongest argument for the per-cell DINN. Since the box homogenizes at uniform parameters, essentially all fit-able spatial structure must come from the network varying parameters per cell. A single global-scalar parameter vector (Carroll's class) would yield a flat box that cannot match Darwin's structure at all — the per-cell predictor is doing load-bearing work, not cosmetic smoothing.
  • It sharpens why real-data anchors are load-bearing. The recovery's z-scored Darwin-pattern loss terms are low-information: at 200 steps they are dominated by the un-relaxed Darwin-warm IC, and at convergence the box has no pattern to match. Either way the Darwin-pattern terms carry little parameter gradient. The Darwin-independent absolute anchors (GEOTRACES dissolved iron, Daniels CP:PP) are the terms that actually identify the parameters. This is the rigorous foundation for the whole "use real, absolute, Darwin-independent data" strategy — better than the earlier "the Darwin pattern is anti-correlated" framing, which was a transient artifact.

Caveats and open items (honest)

  • CV verified at eqpac; natl/SO not re-run for CV but the mechanism (0-D + uniform params → no structure) is general. The natl/SO "FeT anti-correlated at 3200" a reviewer might cite is on a near-vanishing pattern (CV ~1e-11) — also dust.
  • The "integrate longer to fix iron" lead from the convergence note is not a clear win: there is no converged pattern to improve toward. The controlled n≥10 test (200 vs 1600 steps, jobs 7902987/7902988) will settle whether longer integration changes iron-pair recovery rate — which is the only thing that matters; do not pre-judge it on the linearization.
  • These are box (surrogate) statements; the box is a proxy for Darwin. The identifiability claims rest on the absolute anchors, not on box spatial fidelity.

Provenance: box_cv_check.py (my computation, float64, eqpac cache), box_linearization.py (slow mode + 2 conserved modes), red-team wf_3e9befaf (4 agents, 12800-step integration, source-verified).