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arXiv2026

Term Coding and Dispersion: Exact and Asymptotic Decision Problems

Let t be a tuple of r terms that, under an interpretation on an n-element alphabet A, defines a map from k-tuples over A to r-tuples over A. We study the decision theory of its maximum image size, separating exact perfect dispersion from asymptotic rate. Building on the term-cut theorem of Riis and Gadouleau, we prove that every eventual threshold strictly between consecutive integer powers is decidable in polynomial time. More precisely, if a threshold is eventually greater than n to the power d and grows strictly more slowly than n to the power d plus one, then the maximum image size eventually meets that threshold exactly when the term-cut exponent is at least d plus one. For the exact problem, we introduce the perfect-alphabet spectrum and prove that it is multiplicatively closed, that a nonempty spectrum forces full rate, and that the converse fails. We completely characterize the one-output case. On square instances, perfect dispersion is precisely finite square term bijectivity. We give explicit linear-size padding reductions from three-dimensional square bijectivity to perfect dispersion for every fixed output dimension of at least three. We also characterize scalar-linear witnesses by a determinant polynomial, obtaining decidability over fixed finite fields, over extensions of a fixed characteristic, and over arbitrary finite fields. General square bijectivity remains open. The principal mathematical results have been machine-checked in Lean.

cs.IT↗

arXiv2026

Term Coding: An Entropic Framework for Extremal Combinatorics and the Guessing--Number Sandwich Theorem

Classical existence problems in extremal combinatorics ask whether finite operations can satisfy prescribed identities universally. Term Coding replaces this yes-or-no question by a graded one: for a finite system $Γ$, the maximum code size $S_n(Γ)$ is the largest number of satisfying assignments attainable on an $n$-element alphabet. We prove that normalisation and diversification associate $Γ$ with a labelled guessing game of guessing number $α$ and give finite-alphabet sandwich bounds. Consequently, $\log_n S_n(Γ)=α+o(1)$. Entropy and polymatroid inequalities provide systematic upper bounds. Examples include a five-cycle with exponent $5/2$, self-orthogonal Latin squares, and presentation-dependent exponents for universally equivalent identity systems. All theorems, lemmas and propositions in this paper have been machine-checked in the Lean 4 proof assistant; the development is available at https://github.com/SR123/term-coding-lean.

cs.IT↗