Folded-Algebraic Matroids: Characteristic Rigidity and Almost

Mathematics > Combinatorics
[Submitted on 31 Aug 2026]
Title:Folded-Algebraic Matroids: Characteristic Rigidity and Almost-Entropic Separation
View PDF HTML (experimental)Abstract:We introduce folded-algebraic matroids. In such a representation, every matroid element is replaced by a finite tuple of algebraic quantities, and transcendence degree agrees with matroid rank after one uniform scaling. The resulting class contains both algebraic and folded-linear matroids and is contained in the class of almost-entropic matroids, whose rank functions are limits of scaled entropy functions. We prove that the latter containment is proper. Our main result concerns the classical rank-three matroids $M(p)$ of Gordon. For every prime $p$, we show that $M(p)$ has a folded-algebraic representation over a field $K$ if and only if $K$ has characteristic $p$. We then use a point-identification construction that preserves almost-entropicity to obtain a $13$-element rank-three $3$-connected matroid $C_{2,3}$ that is almost entropic but not folded algebraic. Choosing a common element as dealer also yields a connected $12$-participant port with incompatible characteristic requirements. Finally, we record compact explicit witnesses and size bounds for several other separating regions among the representation classes.
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Facts Only

* Folded-algebraic matroids replace every matroid element with a finite tuple of algebraic quantities.
* Transcendence degree agrees with matroid rank after one uniform scaling.
* The class of folded-algebraic matroids contains algebraic and folded-linear matroids.
* This class is contained within the class of almost-entropic matroids, where rank functions are limits of scaled entropy functions.
* The containment between these classes is proper.
* For every prime $p$, the classical rank-three matroids $M(p)$ have a folded-algebraic representation over a field $K$ if and only if $K$ has characteristic $p$.
* A point-identification construction preserves almost-entropicity to yield a 13-element rank-three 3-connected matroid $C{2,3}$ that is almost entropic but not folded algebraic.
* Choosing a common element yields a connected 12-participant port with incompatible characteristic requirements.
* Compact explicit witnesses and size bounds are recorded for several other separating regions among the representation classes.

Executive Summary

The research introduces folded-algebraic matroids, a representation where matroid elements are replaced by tuples of algebraic quantities, with transcendence degree aligning with matroid rank after scaling. This class encompasses both algebraic and folded-linear matroids and is contained within the class of almost-entropic matroids, whose rank functions are limits of scaled entropy functions. The containment between these classes is proven to be proper. A key finding involves the classical rank-three matroids $M(p)$ of Gordon: they possess a folded-algebraic representation over a field $K$ if and only if the characteristic of $K$ is $p$. Furthermore, point-identification constructions that preserve almost-entropicity allow for the creation of specific examples, such as a 13-element rank-three 3-connected matroid $C{2,3}$ that is almost entropic but not folded algebraic. The work also details connections to constructing connected port systems and recording explicit witnesses for separating regions among representation classes.

Full Take

The framework establishes a hierarchy of structural properties—folded-algebraic, almost-entropic, and algebraic—suggesting that algebraic structure is a specific, constrained subset of broader entropic structures. The core result links this algebraic specificity directly to the characteristic of the underlying field for known combinatorial objects like Gordon's matroids, implying a deep arithmetic constraint on their geometric/combinatorial realization. The separation demonstrated by constructing $C{2,3}$ shows that almost-entropicity is a broader property than folded-algebraicity, allowing for structures that lie outside the desired algebraic regime while still respecting entropy limits. This suggests that complexity in combinatorial representations often exists in intermediate spaces, and fully characterizing these intermediate regions—the "separating regions"—is critical for understanding the full landscape of mathematical structure beyond strict definitions. The implication is that concepts related to information entropy (almost-entropic) provide a powerful, continuous space into which discrete algebraic structures can be embedded, offering new pathways to analyze structure through continuous limits rather than purely discrete categorization.