Adaptive Search in Collatz Exponent-Code Space via 2-adic and 3-adic Constraints

2026-07-10

AI Summary

This study investigates structural obstacles to orbit convergence in the Collatz conjecture, aiming to characterize essential features of potential counterexamples. It introduces the “2-3-∞ diagnostic framework,” which for the first time jointly incorporates 2-adic initial conditions and 3-adic terminal constraints within a finite-index symbolic space derived from accelerated mappings. By integrating real-valued drift metrics, asymptotic residue theory, and adaptive evolutionary search, the work demonstrates that any counterexample must exhibit near-critical drift and small residue. Moreover, it proves that the asymptotic residue rate of integer-generated codes is zero. Experimental results over orbit lengths from 100 to 400 significantly improve the performance trade-off at finite lengths, with all configurations maintaining a positive residue rate, thereby validating the framework’s effectiveness and novelty.

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.

Problem

• Collatz conjecture
• exponent-code space
• 2-adic constraints
• 3-adic constraints
• obstruction structures

Innovation

• exponent-code space
• 2-adic constraints
• 3-adic constraints
• adaptive evolutionary search
• 2-3-infinity diagnostic