The Clock [3600,60,1] and the (25,12)-System: A Structural Comparison

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-06-26
DOI
https://doi.org/10.5281/zenodo.20943109
Primary Topic
Cryptography and Residue Arithmetic
Type
preprint
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preprint

The Clock [3600,60,1] and the (25,12)-System: A Structural Comparison

Bilal El Issaoui
Zenodo (CERN European Organization for Nuclear Research)
Cryptography and Residue Arithmetic
preprint

The Clock [3600,60,1] and the (25,12)-System: A Structural Comparison

Bilal El Issaoui
preprint en

Abstract

The clock [3600,60,1] and the (25,12)-system are not "faulty versions" of each other; they are fundamentally different designs with fundamentally different purposes. One wants multiple representations (to count, optimize, and generate them); the other wants exactly the opposite (to register time without error and without ambiguity). This paper refutes the assumption that the clock [3600,60,1] and Diophantine systems such as (25,12) are comparable architectures. They are not. The (25,12)-system is designed for multiple representations: its coefficients satisfy Foundation (25 ≡ 1 mod 12) to generate a deterministic ladder of solutions. The clock [3600,60,1] is designed for the exact opposite: one unique (H,M,S)-triple per time value. Its coefficients satisfy 3600 ≡ 0 (mod 60), not a flaw, but the correct requirement for positional carry/overflow. We prove that these two properties (Foundation and positional divisibility) are structurally incompatible. The clock does not fail to be Diophantine; it is designed not to be, because its purpose requires the opposite architecture.

Zenodo (CERN European Organization for Nuclear Research)
Cryptography and Residue Arithmetic
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The Clock [3600,60,1] and the (25,12)-System: A Structural Comparison — Bilal El Issaoui · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS