Collatz Normal Form: Time as Degree-of-Freedom Elimination and the Trace-Compressed Engine

This paper introduces a trace-compressed normal form for the Collatz (3x+1) dynamics.

The main contribution is structural rather than declarative:

the Collatz iteration is rewritten via an exact change of variables that separates

(i) multiplicative drift induced by odd steps and

(ii) dyadic compression induced by 2-adic valuations.

In the resulting normal-form coordinate, the evolution admits an exact affine increment identity driven by valuation data and a multiplicative correction cocycle.

This representation does not modify the Collatz map, introduce auxiliary dynamics, or rely on probabilistic assumptions.

All constructions are observational and derived directly from the original iteration.

Using this framework, the paper derives an exact cocycle equation that any exact periodic numeric orbit must satisfy.

This equation provides a necessary structural constraint linking valuation sums, odd-step counts, and an in-orbit correction product.

The result clarifies precisely which compatibility conditions would be required for a hypothetical nontrivial cycle.

Importantly, no convergence or termination claim is made.

The paper does not assert that the cocycle constraint is unattainable, nor does it claim to resolve the Collatz conjecture.

Rather, it isolates a compact normal-form structure and an exact constraint that any periodic scenario must obey.

The trace-compressed normal form introduced here is intended as a reproducible coordinate system for analyzing long Collatz trajectories on equal footing, and as a preparatory framework for further work on cycle obstruction or non-attainability questions.

A complete deterministic implementation for generating the normal-form trajectories and figures is provided in the appendix.

Notes

This upload establishes a permanent, citable record of a structural normalization of the Collatz dynamics.

It is intended as a foundational reference rather than a proof of the conjecture.

This v1.0.1 version corrects and clarifies the roles of block drift and cocycle terms,
and refines the cycle constraint as a necessary condition only.
The core normal-form construction and exact identities are unchanged.