The Reduced (5k+1(3))/4 Collatz-Type Domain with Four Observed Attractors up to 10^10
Year: 2026
Type: preprint
Abstract: This work studies a residue-dependent reduced Collatz-type map on the positive odd integers, referred to as the (5k+1(3))/4 domain. For an odd state x, the affine correction is chosen from {1, 3} so that the numerator is divisible by at least 4, after which all powers of two are removed. The map exhibits four directly verified periodic attractors: the fixed point {1} and the three cycles {31, 39, 49}, {37, 47, 59}, and {61, 77, 97}. An exhaustive C computation using unsigned 128-bit arithmetic and four POSIX threads tested all 5,000,000,000 positive odd starting values below 10^10. Every tested orbit entered one of the four observed attractors, with zero unresolved cases. The work also presents basin statistics, stopping-time and peak records, an exact Diophantine identity for periodic orbits, inverse branches, the two-adic valuation law, and a probabilistic contraction analysis. These algebraic and computational results motivate a Four-Attractor Conjecture for the positive odd integers. The conjecture is explicitly distinguished from the exhaustive finite verification and is not claimed as a proof of global convergence. The accompanying computational archive contains source code, numerical results, record data, reproducibility instructions, and checksums for independent verification.
Author: Farhad, Banazadeh
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212398538
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