D ↔ k

The coupling correctly identifies the well-known structural analogy between Fick's law of diffusion (J = -D ∇C) and Fourier's law of heat conduction (q = -k ∇T). Both are linear, first-order transport laws of identical mathematical form, and the mapping D ↔ k under variable renaming is the canonical textbook analogy. The semantic descriptors honestly label distinct but structurally analogous quantities (diffusion coefficient vs. thermal conductivity) rather than claiming they are the same physical quantity; the transfer function is 'identity' in the sense of structural role equivalence, not dimensional equality, which is consistent with the dimensional mismatch correctly noted in the Tellegen check. The Onsager reciprocity check appropriately passes for a 1×1 trivially symmetric transport coefficient. The Buckingham π groups are dimensionally consistent within each domain. No spurious emergent properties are claimed. This is a legitimate tier-2 structural similarity coupling that reflects genuine physics (irreversible thermodynamic transport phenomena sharing the same linear-response framework).

D (mass diffusion coefficient) and k (thermal conductivity) are distinct transport coefficients with different physical dimensions and meanings; equating them via an identity map is not physically plausible. The semantic descriptors refer to DiffusionCoefficient vs ThermalConductivity, not the same quantity nor a named transform; the correct cross-domain analogue for Fick’s/heat equations is D ↔ thermal diffusivity α = k/(ρ c_p), not k itself. The physical-constraint filter’s Onsager pass is trivial and does not address the dimensional mismatch (indeed dim_a ≠ dim_b appears in the evidence), so it does not validate the proposed identity. The listed π groups are tautologies of the constitutive laws, not genuine emergent properties. Therefore the coupling should be rejected.

The proposal correctly identifies the canonical mathematical analogy between Fick's law of diffusion and Fourier's law of heat conduction, a classic and valid example of a cross-domain structural similarity. While the 'identity' transfer function is physically invalid due to the dimensional mismatch between the diffusion coefficient (D) and thermal conductivity (k), the discovery of this isomorphism is valuable for the ledger. The physical constraint filter correctly passes the relevant checks for such linear transport laws and its evidence logs the dimensional inequality, providing the necessary context for this Tier 2 similarity coupling.