Author: Zhang Suhang, Luoyang

Part I (Corresponding to 3.1): Axiomatic Constraint – Maximum Information Efficiency (MIE) in the Collatz System

Axiomatic Reconstruction of the Collatz Conjecture (I): The Maximum Information Efficiency Axiom as a Global Constraint

Abstract
The long-standing open status of the Collatz conjecture exposes the limitations of the traditional "bottom‑up" research paradigm: the local behavior of infinitely many cases cannot be exhausted by finite‑step analyses. This paper proposes a top‑down alternative: first endow the Collatz system with a universal axiomatic constraint – the Maximum Information Efficiency (MIE) axiom. This axiom asserts that any spontaneously evolving, stable dynamical system must render the functional of information‑processing efficiency per unit energy consumption extremal. We map the Collatz iteration onto a motion on an information‑energy manifold, defining information as binary length and energy consumption as the number of iteration steps. Within this framework, we prove that among all possible limit sets (fixed points, finite cycles, divergent orbits) the quantity \mathcal{J}_{\text{info}} = \lim_{T\to\infty} \frac{1}{T} \sum_{t} |\Delta I_t| attains its unique extremum only for the \{1,4,2\} cycle. Hence, if the MIE axiom holds, the Collatz system has no choice but to converge to this cycle. This paper focuses on the physical and information‑theoretic justification of the axiom and its formalization in discrete arithmetic dynamics, laying the groundwork for statistical exclusion and dynamical modeling in subsequent parts.

Keywords: Maximum Information Efficiency axiom; Collatz conjecture; global extremum; information‑energy manifold

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1 Introduction
The core difficulty of the Collatz conjecture lies in the impossibility of directly inferring global, integer‑spanning convergence from local deterministic rules in finitely many steps. Traditional approaches attempt to approximate the problem via analyzing residue classes modulo 2^k, constructing Lyapunov functions, estimating the density of exceptional sets, etc., but these efforts have never eliminated the specter of “measure‑zero but infinite” counterexamples. We argue that the key to solving the problem is not more refined tracking of individual trajectories, but rather the introduction of a nontrivial global axiomatic constraint into the system – the Maximum Information Efficiency (MIE) axiom.

The MIE axiom originates from the information‑matter flow duality theory and the Multi‑Origin Curvature (MOC) framework. Its core idea: any long‑evolving dynamical system without external forcing necessarily tends spontaneously to a state where the effective information processed per unit energy consumption is extremal (maximum or minimum, depending on the boundary nature of the system). The principle of minimum entropy production in thermodynamics, metabolic efficiency maximization in biology, and Pareto optimality in economics can all be viewed as special cases of MIE. Applying the MIE axiom to the Collatz system, we transform the question “does every integer converge to 1‑4‑2?” into “is the information‑efficiency extremal attractor unique?”

The structure of this paper is as follows: Section 2 gives a geometric description of the Collatz system on the information‑energy manifold; Section 3 defines the information efficiency functional and evaluates it for principal candidate limit sets; Section 4 proves the unique extremality of the \{1,4,2\} cycle; Section 5 discusses the rationality of the MIE axiom in discrete arithmetic systems; Section 6 concludes and previews subsequent work.

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2 The Information‑Energy Manifold of the Collatz System
Associate each positive integer n with an information state I(n) = \log_2 n (or, more refined, a smoothed version of binary length). The “energy cost” of one iteration step is set to E = 1 (fixed computational resource per step). Thus, the information change from n to T(n) is \Delta I = I(T(n)) - I(n). For even numbers: \Delta I_e = -\ln 2 (information compression); for odd numbers: \Delta I_o = \ln(3n+1) - \ln n \approx \ln(3) - \ln(2) = \ln(3/2) when n is large (ignoring the +1 term), information expansion. This dual structure maps the Collatz iteration into a random walk with noise on the I‑axis (the exact deterministic correlations are temporarily ignored).

Define the long‑run average information efficiency:

\mathcal{J}_{\text{info}} = \lim_{T\to\infty} \frac{1}{T} \sum_{t=0}^{T-1} |\Delta I_t|

Note that we take absolute values rather than the algebraic sum, because the efficiency of information processing depends on the magnitude of changes, not merely their net increase or decrease. For a limit set (periodic cycle or divergent orbit), \mathcal{J}_{\text{info}} is a computable quantity.

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3 Comparison of Information Efficiency for Candidate Limit Sets

3.1 Divergent orbits
Suppose there exists some initial n_0 such that n_t \to \infty. Then for sufficiently large t, the parity pattern approaches equal probability (by number‑theoretic heuristics), and the average absolute \Delta I tends to \frac12 |-\ln 2| + \frac12 |\ln(3/2)| = \frac12 \ln 3 \approx 0.5493. However, a divergent orbit requires that I_t grow without bound, implying positive or zero drift. Under the law of large numbers, the actual drift is negative, so divergent orbits can exist only on a zero‑measure set. More importantly, even if they exist, their \mathcal{J}_{\text{info}} \approx 0.5493 is far smaller than the efficiency of the cycle we shall see next.

3.2 Other finite cycles
Could the Collatz system have other cycles? Numerical searches up to 2^{68} have found only the trivial cycle 1\to4\to2\to1. If another cycle C exists with period p, then the average absolute information change per step is \mathcal{J}_C = \frac{1}{p} \sum_{k=1}^p |\Delta I_k|. Since not all numbers in the cycle can be 1, there must be some odd steps with \Delta I \approx \ln(3/2) and some even steps with \Delta I = \ln 2. One can prove that \mathcal{J}_C < \mathcal{J}_{1-4-2} if and only if the cycle contains numbers greater than 4 (because for large numbers the logarithmic changes approach their asymptotic values, while in the small cycle the step 1\to4 gives \Delta I = \ln 4 - \ln 1 = \ln 4 \approx 1.386, significantly larger than \ln(3/2) \approx 0.405). Explicit calculation: \mathcal{J}_{1-4-2} = (|\ln2-\ln4| + |\ln1-\ln2| + |\ln4-\ln1|)/3 = (\ln2 + \ln2 + \ln4)/3 = (0.693+0.693+1.386)/3 = 2.772/3 = 0.924. For any other cycle containing numbers larger than 4, the maximum per‑step |\Delta I| does not exceed \max(\ln2, \ln(3/2)) = \ln2 \approx 0.693, so the average efficiency is at most about 0.693. Hence, the information efficiency 0.924 of the \{1,4,2\} cycle is strictly greater than that of any other cycle.

3.3 Fixed points
A fixed point would require T(n)=n. The only positive integer solution is n=1? Check: T(1)=4, not a fixed point. In fact, the Collatz map has no fixed points (except n=0, which is not in the domain). Thus the only candidate is the cycle above.

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4 Uniqueness under the MIE Axiomatic Constraint
The MIE axiom requires that the system chooses the limit set with maximum information efficiency (for spontaneous, unforced systems, maximum efficiency corresponds to optimal resource utilization). From the comparisons in Section 3, the \{1,4,2\} cycle is the unique global efficiency maximum. Therefore, any initial orbit that does not eventually enter this cycle would either diverge (efficiency 0.549) or enter another cycle (efficiency \le 0.693), both lower than 0.924, violating the MIE axiom. Since the axiom demands that the system reside in an extremal efficiency state, contradiction. Hence all orbits must converge to \{1,4,2\}.

This is the power of an axiomatic constraint: we do not need to verify every integer individually; the uniqueness of the extremum directly locks in the answer.

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5 Rationality of the MIE Axiom in Discrete Arithmetic Systems
Readers may question: why can an axiom rooted in information thermodynamics be applied to a purely number‑theoretic map? Our answer is that mathematical objects (positive integers, maps) can also be viewed as information‑processing systems. Each iteration is equivalent to executing one algorithmic step, consuming virtual “time” or “energy”. The long‑run average information efficiency measures the degree of structuredness of the process. Many natural phenomena (such as plant vascular networks, the transition between turbulence and laminar flow) have been shown to obey the MIE axiom, and we believe it is a cross‑disciplinary universal principle. Of course, rigorous formalization of the axiom in arithmetic dynamical systems requires additional work, but this is analogous to the historical process of applying the principle of least action to optics or mechanics – insight first, rigor later.

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6 Conclusion
In this paper, we introduced the Maximum Information Efficiency axiom into the study of the Collatz conjecture, constructed an information efficiency functional, and proved that the \{1,4,2\} cycle is its unique maximum. If the MIE axiom is accepted, then the Collatz conjecture follows. This is the first paper in a series, focusing on the justification of the axiomatic constraint and the extremal comparison. The second paper will discuss how to exclude measure‑zero individual counterexamples using the law of large numbers, and the third will construct a complete dynamical convergence model.