Universal Rank Tests for Noncommutative Rational Formulas

Testing noncommutative rational expressions requires preserving the domains of their inverse operations while detecting their algebraic rank. We construct universal matrix evaluation lists that achieve both over every field, extending polynomial rational testing constructions over ℚ to arbitrary coefficient fields.

For n variables, total tree size s, and inversion height h, the list depends only on these bounds and the characteristic. Every family of individually admissible rational expression matrices within these bounds has a common evaluation in the list at which every original inverse is defined and every matrix attains its maximum normalized rank. All evaluation entries lie in the prime subfield.

In characteristic two, the matrix dimension is O(s² log³(s+2)), and the list length is O((h+1)s⁸ log⁶(s+2) log(n+2)). In every other characteristic, the corresponding bounds are O(s⁴ log⁴(s+2)) and O((h+1)s¹⁰ log⁷(s+2) log(n+2)). Thus matrix dimension is independent of the number of variables and inversion height, while list length depends only logarithmically on the number of variables. The construction yields coefficient-independent, nonadaptive tests for affine noncommutative rank over every field.

One deterministic Turing machine constructs the complete densely encoded lists in polynomial time, with the characteristic supplied in binary. We prove the main theorem, including serialization and the machine’s runtime bound, in Lean.

Files: the main paper (22 pages), formal supplement (24 pages), and a clean submission archive containing both PDFs, LaTeX sources, the complete Lean 4.34.1 development with pinned dependencies, proof-correspondence documentation, and reproducible verification scripts. The archive excludes working files and build caches.

Verification: all 366 local Lean files were rebuilt on macOS arm64. The audit checked 8,679 declarations using only the standard Lean foundations propext, Classical.choice, and Quot.sound. The formal supplement states the exact correspondence and limits of the formalization. Linux verification has not been performed.

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