FieldNet: A Source-Free Normalizer for Any Transition, and Closed-Form Under Time Homogeneity

Selectivity, bounded normalization and linear complexity are treated as a trilemma, reached in three steps: an input-dependent decay leaves the kernel lag-dependent, and a lag-dependent kernel has no closed-form mass, so the state cannot be normalized. The first two steps are right. The third holds only if the readout is unnormalized: under a posterior-mean readout (A3) the normalizer is the kernel's own mass, driven by the constant source $1$, and it exists under any causal transition, content gating included. The object is a coupled dual recursion—one operator at two sources, not two recurrences—and the field locates the absence in the readout itself: removing the denominator degrades vision accuracy and destabilises token norms (Han et al., 2024). As stated, the trilemma is a statement about unnormalized readouts; under A3 it does not hold. The paper forms a closed loop. Derived: under A3 the normaliser is forced to be the kernel's own mass, and the coupled dual recursion it forces holds under any causal transition. Measured: a companion mass and weak content dependence compose, and that composition is the best model here—the 400M dual arm leads the matched baseline on WikiText-103 ($40.22$ against $57.13$), seven-task zero-shot ($46.80$ against $38.17$) and PG19 ($26.01$ against $28.67$). Direction: the strong-gating end is supported by the same implementation; it opens the effect-size question.