Structural Identity Is Necessary but Not Sufficient: A Delta Taxonomy for Cross-Domain Algorithm Transfer

Published June 6, 2026 | Version v1

Preprint Open

In prior work we introduced the Principle of Structural Identity (PSI): when two domains share

identical governing equations, an algorithm mature in one is transferable to the other, and we

validated it on forest carbon-flux prediction by translating DC Optimal Power Flow (DC-OPF) from

electrical engineering (RMSE 140.8 vs naive 167.6 gC/m²/yr, Δ = +26.8, Holm-corrected p = 0.0450).

Here we ask the question that result raises but does not answer: *is structural identity

sufficient?* We argue it is not. Sharing a shape is the zeroth order of transfer; the intelligence

is the **delta** Δ between donor and recipient — `f_R(φ(x)) = f_D(x) + Δ(x)` — and naïve transfer

silently assumes Δ ≈ 0. We propose a five-axis delta taxonomy (structural form, conservation,

parametric, observational, boundary) and show on eight transfers spanning electrical, thermal,

hydraulic, ecological, epidemiological, and economic domains that the total delta predicts transfer

success near-monotonically (Spearman ρ = 0.91, p = 0.0016, n = 8), with each failure named by its

dominant axis. We then *model* the delta empirically — fitting `y ≈ a + b·f_D(x)` recovers the

forest source-phase sign flip (b = −1), a parametric scale change (b = 1.5, R² = 1), and a

structural mismatch (R² ≈ 0) directly from data — and emit `a + b·f_D` as the corrected donor.

Finally we report an honest prospective test: on the BattLeDIM leak-localization benchmark, a

delta-aware observational correction improves over copy-paste transfer (271 → 254 m median) but

does **not** beat a strong naïve baseline (203 m), because at 4% sensor coverage the delta is below

the *identifiability floor* — a scope wall, not a tuning target. In a second, field-guided

prospective test on urban heat, the method located the delta successively in the observable choice

(daytime maxima carry no urban signal), an additive elevation term, and spatial sparsity — and a

controlled dense-grid run then turned the same transfer into a near-clean copy (Δ = 2) that beats

both naïve and nearest-neighbour baselines, showing the same isomorphism can be COPY or merely-ADAPT

depending on the recipient's data realities. We package these deltas as an "isomorphic field": a

navigable, accumulating map of which transfers are cheap, which are correctable, and which should be

refused — which itself chose the urban-heat target and refused a conservation-violating alternative.

The refined thesis: **structural identity is necessary; a small *and identifiable* delta is

sufficient.**

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