A Predicate Zoo in the PSD Laplacian World: Integer-Gap Gadgets and Composition Rules

We collect small, fixed-arity PSD Laplacian gadget cores whose minimized Dirichlet energies, after a simple shift--clip transform, produce a fixed integer "pulse'' on precisely the violating boundary assignments. These integer-gap residents form a composable library under an explicit local pulse evaluation model: in composites, each constituent gadget energy is shift--clipped separately and the results are aggregated by a final cap--sum. The paper is organized as a "predicate zoo'': each entry records (i) the graph and weights, (ii) its energy profile on Boolean boundary labelings, (iii) a shift--clip map that yields a clean $0/g$ signal, and (iv) alignment and closure rules. We also explain how the shift--clip--cap #P-hardness reduction plugs into this library as a direct application of the OR$_3$ resident together with gap alignment and cap--sum closure. As a byproduct, the finite-state synthesis template shows that weighted resistor networks can encode any fixed regular-language predicate (DFA membership at length $k$) under the same $0/g$ pulse readout. This regular-language encoding claim is under the local $0/g$ pulse interface (per-gadget shift--clip with final cap--sum), not a claim that a single total Dirichlet energy of a glued graph directly computes the predicate. Here, "encoding/verifying under the $0/g$ pulse interface'' means: acceptance is equivalent to the existence of auxiliary one-hot state labels yielding post-processed energy $0$ (otherwise the energy equals $g$).