Adaptive Search in Collatz Exponent-Code Space via 2-adic and 3-adic Constraints
2026-07-10 • Neural and Evolutionary Computing
Neural and Evolutionary ComputingInformation Theory
AI summaryⓘ
The authors study a way to analyze the Collatz conjecture using sequences that record how many times a number is divided by two after applying a 3n + 1 step. They define a diagnostic tool combining several mathematical measures to identify potential counterexamples or interesting patterns. While they prove certain long sequences have shrinking mathematical residues, their experiments show no sequences defy the expected behavior. Their approach is meant to explore structural properties symbolically, not to prove or disprove the Collatz conjecture itself.
Collatz conjecture3n + 1 map2-adic numbers3-adic numberssymbolic dynamicsexponent codesasymptotic behavioradaptive searchresidue rates
Authors
Oliver Kramer
Abstract
We study a symbolic search space for the Collatz conjecture based on finite exponent codes of the accelerated map. Each code records the number of divisions by two after every 3n + 1 step and determines three quantities: real drift, a 2-adic start representative, and a 3-adic endpoint representative. Their combination defines the 2-3-infinity diagnostic. Counterexample-like codes should exhibit near-critical drift, small 2-adic start representatives, and endpoints compatible with growth on the scale of (3/2)^k. We prove that every infinite code generated by a fixed positive integer has asymptotically vanishing 2-adic and 3-adic residue rates. Experiments with random critical codes, mechanical critical codes, and adaptive evolutionary search at lengths 100, 200, and 400 show that adaptive search improves finite-length trade-offs, while all methods retain clearly positive residue rates. The proposed framework is not a verification method for the Collatz conjecture, but a symbolic diagnostic approach for investigating obstruction structures in exponent-code space.