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Odd/Odd Records for a Relative Approximation Error: Three Quadratic Classifications and an Exact Certificate

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.CA — Classical Analysis and ODEs

Contributed by Elias De Jesús ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

For a quadratic irrational α > 1 and positive odd integers m, k, set H = mα + k and ε = |mα − k|/(mα + k). A state (m, k) is a strict chronological record if ε is smaller than at every admissible state of smaller H. We classify the strict records completely for α = φ, α = 1 + √3 and α = √3. In each case they are a finite explicit union of orbits of a single unimodular matrix, with signed defect cycles (−1, +1, −5), (−3, +1, −11) and (−2) respectively. The proofs rest on two exact identities that make ε a function of the defect N and the conjugate factor η alone, on a norm cutoff |N| ≤ 18, 204, 27 derived from heights, and on a finite exact certificate that decides every remaining comparison. Two phenomena require care and are isolated: at φ the competition between the Fibonacci and Lucas ladders is a genuine tie at leading order, decided by an integer identity N_B = −5N_A; and at √3 a companion family has exactly equal error at larger height, so it is excluded by equality rather than by loss. The results resolve two conjectures of the author's earlier note and correct one assertion there. Nothing here is formally verified or refereed, and the uniform generalisation is left open.

Author comments: First version, 17 pages. Supporting materials—including the exact-arithmetic certificate, verification scripts, complete orbit tables, and supplementary documentation—are available at: https://github.com/innerlightr-wq/two-mode-oscillator-record-chronologies/tree/main/congruence-records

Provenance statement

This work builds on several months of my earlier research and conceptual development. Under my direction, Claude and ChatGPT contributed to mathematical development, computational checks, critical review, and manuscript preparation.

References

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Version history

  1. v1 · 17 pagesSubmitted by Elias De JesúsFirst version, 17 pages. Supporting materials—including the exact-arithmetic certificate, verification scripts, complete orbit tables, and supplementary documentation—are available at: https://github.com/innerlightr-wq/two-mode-oscillator-record-chronologies/tree/main/congruence-recordsCurrentNov 08, 2026