Log–Scaled Axiomatic and Structural Invariance Framework for the Collatz Dynamics

A Non–Ergodic Local Obstruction Proof of the Collatz Conjecture

We develop a log–scaled axiomatic framework for the 3n + 1 (Collatz) dynamics, designed to isolate structural features of the iteration from empirical regularities observed across high bit–length ranges. The framework consists of five axioms governing local pre-decessor structure, logarithmic potential compression, excursion tightness, convergence density and a corrected log–scaled tail behaviour for the total stopping time.

These axioms are explicitly local–in–bits: they are formulated on each dyadic range [2B , 2B+1 ) in terms of (i) the odd predecessor graph, (ii) a logarithmic potential Φ(n) = log2 n − κ, (iii) excursions above the current potential, (iv) density of bounded stopping times in large samples, and (v) a range–wise log–scaled tail bound on τ (n)/ log2 n.

Together, they can be encapsulated in a single Structural Invariance principle asserting that the log–scaled statistics of the dynamics stabilise uniformly across bit ranges.
The main theorem shows that, for any function T : N → N satisfying these axioms with respect to Φ(n) = log2 n − κ, all orbits converge to the trivial cycle {1, 2}. Thus, if the Collatz map obeys the log–scaled axioms globally (equivalently, if Structural
Invariance holds for the deterministic dynamics), the 3n + 1 conjecture follows as a logical consequence. We do not claim to prove the axioms or Structural Invariance in this paper; rather, they are introduced as conjectural structural laws, motivated by
probabilistic models and by the numerical evidence described below. 

Complementing the axiomatic part, we present extensive numerical experiments on disjoint logarithmic ranges, from [210 , 2150 ) up to [25000 , 26000 ), with an extreme test on [210000 , 211000 ) and an ultra–high bit range around [230000 , 230500 ). A second ultra–high bit test on [250000 , 250500 ) confirms that the same log–scaled drift and tail behaviour persist well beyond thirty thousand bits. These experiments validate, within the tested ranges and with a single frozen choice of structural parameters, the log–scaled behaviour predicted by Axioms II–V: the tail fraction for τ (n)/ log2 n decays as the bit height
grows, and the empirical constants governing the log–scaled stopping time remain tightly concentrated around a universal value C ≈ 7.22.

Throughout the paper we write C ≈ 7.22 for the log–scaled constant extracted from our initial numerical experiments. Later high–bit locality runs, including ranges up to [251000 , 252500 ), show that the effective constant Cemp (B) stabilises into a narrow 1 plateau around this value, with small scale–dependent fluctuations (typically in the range 7.2–7.4). In particular, every occurrence of the numerical value 7.22 should be understood as a representative value of this plateau rather than as a uniquely fixed decimal constant.


Eduardo M. Dammroze
dammroze@gmail.com
Curitiba, Paraná, Brazil. 

This is the third version of the paper.