Tarski Coherence
Order-Theoretic Coherence for Agent–Tool Composition, and the Resolution of the Degree-1 Hodge–Tarski Conjecture
The degree-1 Hodge–Tarski characterization: on a self-loop-free quiver, the Helmholtzian’s prefix points equal the coequalizer realization of H¹ for all complete-lattice stalks and all Galois connections if and only if every non-isolated vertex has in-degree = out-degree = 1 (machine-checked as an iff; the class direction requires no cycle decomposition)
Any Tarski sheaf on an in-out-one quiver with Pre(ℌ) ≠ Q, or an out-of-class orientation admitting equality for all stalks and connections; at the paper level, an orientation-free functor restricting to the thesis’s H¹ realization on all orientations (proven impossible via the C₃ cardinality argument)
Abstract
The agent-as-convention-translator finding relocates compositional coherence: the failure unit is caller–tool, conventions are refinement lattices rather than vector spaces, and cure protocols are monotone fixed-point dynamics. This paper builds that order-theoretic layer — star-complement localization (obstructions live entirely on opaque passthrough edges), cure-loop convergence to the least fixed point, and a nucleus-theoretic account of disclosure (minimal cures are matroid bases; a least cure exists iff every element is a loop or coloop) — and then completely resolves an open conjecture of Riess’s lattice-sheaf program: the degree-1 Hodge–Tarski conjecture is refuted as stated on a pendant edge, shown ill-posed for arbitrary orientations, repaired and proven on cyclically-oriented cycles for arbitrary complete-lattice stalks and Galois connections, defended against the natural balanced-orientation repair by a bowtie counterexample, and finally characterized exactly: equality for all sheaves holds precisely on disjoint unions of directed cycles.