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The aim is to establish a cubic energy law for digit collisions in odd prime bases and explore its implications.

June 6, 2026Open Access

Digit Collisions and Carry-Boundary Spectra

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The aim is to establish a cubic energy law for digit collisions in odd prime bases and explore its implications.
Proved a cubic energy law for the threshold-defined lag-one digit-collision energy in odd prime base.
Connected the collision/carry table to a primitive odd Dirichlet-character square mass using Parseval's theorem.
Utilized various mathematical tools, including Rademacher reciprocity and Dedekind sums, to derive results.
Established the equation E_b = b^3 + O(b^2 (log b)^2) with leading terms contributing 1/3 b^3 and 2/3 b^3.
Formulated the arithmetic boundary-defect conjecture for the secondary term.

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Preprint proves cubic energy law for digit-collision energy in odd prime base, suggesting deeper arithmetic connections.

Alexander S. Petty (2026) studied this question.

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