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Life sciences · Preprint
arXiv · September 4, 2026
Raises a question worth testing. It does not answer one.
This is a mathematical and computational study proving that ordered bases in finite fields belong to Galois orbits that induce identical coordinate inversion maps, with n-to-one correspondence. Three Boolean formulations of inversion show differing algebraic degrees and joint ANF leaps, and multilayer perceptron experiments confirm the predicted ordering of learning difficulty, though Galois orbit redundancy provides limited generalization benefit.
Theoretical analysis with computational verification and controlled experiments. Intervention: Three Boolean formulations of finite-field inversion: reference, mixed representation, and complete raw formulation.. Compared with: Comparison of algebraic structure (degree, ANF leap) and empirical learning difficulty across formulations..
Two ordered bases induce the same coordinate inversion map if and only if they belong to the same Galois orbit. Every Galois orbit has size n, giving exactly n-to-one correspondence between ordered bases and distinct inversion maps. Reference formulation has algebraic degree n−1 and joint ANF leap 1; mixed representation has degree 2(n−1) and leap 2; raw formulation has degree at most 3(n−1) and leap at least n.
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This is a theoretical and computational study of mathematical structure in finite-field inversion, not a clinical or health-related investigation; it raises questions about representation and learning rather than testing a clinical intervention or established therapeutic claim.
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The representation chosen for a mathematical operation can affect both its algebraic form and its empirical learning difficulty. We study this phenomenon for inversion over \(\mathbb F_{2^n}\), with field elements expressed in varying ordered \(\mathbb F_2\)-bases. We prove that two ordered bases induce the same coordinate inversion map if and only if they belong to the same Galois orbit. Since every orbit has size \(n\), the correspondence between ordered bases and distinct inversion maps is exactly \(n\)-to-one. We then analyze three Boolean formulations of inversion. The reference formulation has algebraic degree \(n-1\) and joint ANF leap \(1\), the mixed representation formulation has degree \(2(n-1)\) and joint ANF leap \(2\), and the complete raw formulation has degree at most \(3(n-1)\) and joint ANF leap at least \(n\). Exhaustive computations agree with the theoretical results and bounds in the cases considered. Controlled experiments with multilayer perceptrons show the same ordering in learning difficulty, while Galois orbit redundancy provides only a limited generalization benefit under the tested conditions. These results show that exact redundancy among representations can coexist with changes in Boolean structure and learning behavior when the representation is exposed as part of the input.
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