Smoothing Does Not Compress & Hard Structured Inference

Differentiable structured-inference layers replace hard $\min$ recurrences by regularized operators. We ask whether this smoothing can reduce the representation cost of the underlying hard value function. For a broad class of finite DAGs, the answer is no in the sublinear-ray regime. We pair a finite-temperature computation with a same-wiring tropical shadow: bounded min/max deformations are replaced by hard extrema, positive scalar gates are retained, and finite constants are removed. A recursive DAG budget $E(C)$ then bounds the discrepancy on every score ray. Consequently, any $o(t)$-accurate realization has the target's exact shadow and inherits its min-plus or STC lower bound without a size loss. The same budget gives finite-scale open-cone slope and relative-support floor certificates, while annealing changes only the deformation term. For $q$-ary linear-code inference, we prove $q^{\rho }\le S_{\min }\le O(nq+n{q}^{2\rho })$; for the $k\times k$ grid cycle code, $\rho =k-1=\Theta (\sqrt{n})$, giving a $2^{\Omega (\sqrt{n})}$ barrier. CUDA double-precision experiments recover the exact width threshold and code-distance slopes, and separate deformation from representation error under annealing. Signed affine preprocessing is covered by a signed shadow with an explicit affine budget.