Here is the English translation of your third paper:

Author: Zhang Suhang, Luoyang

Second Paper (corresponding to 3.2): Statistical Exclusion – Handling of the Law of Large Numbers and Parity Correlations

Axiomatic Reconstruction of the Collatz Conjecture (II): Large Numbers Law and Statistical Exclusion of Exceptional Cases

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Abstract

After the MIE axiom provides a global extremal constraint, a key problem remains: could there exist zero‑measure yet infinitely many “exceptional cases” (e.g., divergent trajectories or other cycles) that carry no probabilistic weight but might still exist in principle? Traditional methods struggle to completely rule out such null‑set counterexamples. This paper employs a strong form of the law of large numbers—combining the Wiener–Khinchin theorem and ergodic theorems—to prove that, under Collatz iteration, the random walk of the logarithmic variable has negative drift with probability one, and hence almost all trajectories must enter a bounded region. For possible zero‑measure counterexamples, we further argue that if they existed, they would have to exhibit abnormal parity correlations, which would appear as saddle points deviating from the extremum in the information efficiency functional, contradicting the MIE axiom. Thus, exceptional cases are automatically excluded within the axiomatic framework. This paper deals in detail with the issue of non‑independence of parity sequences and provides the rigorous conditions under which the law of large numbers holds (within the heuristic model).

Keywords: Law of large numbers; zero‑measure counterexamples; parity correlations; ergodic theorem; Collatz conjecture

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1 Introduction

In the first paper, we established the MIE axiom and proved that the \{1,4,2\} cycle is the unique information‑efficiency extremal attractor. However, that argument assumed that the system eventually enters some limit set (i.e., does not spontaneously escape to infinity). Trajectories that escape to infinity (divergent trajectories) are precisely the core objects that the Collatz conjecture must exclude. Moreover, there might be some sparse initial values that neither converge to that cycle nor diverge, but instead enter other cycles—though numerical searches have found none, they cannot be completely ruled out theoretically. These are collectively referred to as “exceptional cases.”

The purpose of this paper is to use the law of large numbers to prove that, in the sense of natural density, almost all integers converge (i.e., the set of initial values that diverge or enter other cycles has density zero). This is not a new result (similar conclusions exist in the literature), but we will reinterpret it and emphasize its collaborative relationship with the MIE axiom: the law of large numbers eliminates possibilities of non‑zero measure, while the MIE axiom eliminates zero‑measure possibilities—the two complement each other to form a complete exclusion.

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2 Stochastic Model and Negative Drift

Classical heuristic model: For large random integers, parity can be treated as independent Bernoulli trials with probability 1/2 each. Define X_t = \ln n_t . Then the one‑step change is:

· If odd: X_{t+1} - X_t = \ln(3n_t+1) - \ln n_t = \ln 3 + \ln(1 + 1/(3n_t)) - \ln 2 \approx \ln(3/2) (neglecting small terms)
· If even: X_{t+1} - X_t = -\ln 2

Expected drift:

\mu = \frac12 \ln(3/2) + \frac12 (-\ln 2) = \frac12 \ln\left(\frac{3}{4}\right) = -\frac12 \ln(4/3) \approx -0.1438

By the law of large numbers, \lim_{t\to\infty} X_t/t = \mu < 0 almost surely. Hence X_t \to -\infty , i.e., n_t \to 1 (strictly speaking, enters a low‑number region). This means that starting from almost every initial value, the trajectory almost surely declines.

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3 Handling Parity Correlations: From Independence to Weak Dependence

In the actual sequence, parities are not independent: if n is odd, then 3n+1 is even, so there is a deterministic pairwise pattern of odd‑even pairs. However, research indicates that such correlations decay sufficiently fast over long timescales that a generalization of the strong law of large numbers—for instance, applicable to stationary ergodic processes—still holds. Specifically, define an indicator variable \xi_t = 0 if n_t is even, \xi_t = 1 if odd. Then \{n_t\} is a deterministic dynamical system, but it can be regarded as a sampling under some ergodic measure. It can be shown (see Appendix) that this dynamical system possesses a unique absolutely continuous invariant measure (conjectured), under which \xi_t is strongly mixing, thereby satisfying the law of large numbers.

Consequently, for almost every initial value (with respect to that invariant measure, and also with respect to natural density), the sample mean converges to the expectation. Therefore, the set of divergent trajectories (i.e., those with X_t \to \infty or that do not decline) has zero natural density.

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4 Further Exclusion of Zero‑Measure Counterexamples: The Coup de Grâce of the MIE Axiom

The law of large numbers cannot rule out counterexamples on a zero‑measure set—this is precisely the real difficulty of the Collatz conjecture. Suppose there exists a zero‑measure set A such that for n_0 \in A , the trajectory diverges or enters another cycle. For these individual trajectories, the law of large numbers does not constrain their parity patterns; they can exhibit extremely singular correlations (e.g., long blocks of pure odd numbers, etc.).

However, the MIE axiom provides the final line of defense: even if such trajectories existed, they would still have to satisfy the information efficiency extremal condition. But we have already shown that the asymptotic average efficiency of divergent trajectories is approximately 0.549 , and that of other cycles is \le 0.693 , both lower than 0.924 of the \{1,4,2\} cycle. Hence, if these trajectories existed, they would violate the MIE axiom (the system would be selecting a non‑extremal state). Since the Collatz system is deterministic and no external force compels it to remain in an inefficient state, such counterexamples cannot stably exist. Therefore, zero‑measure counterexamples are also excluded.

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5 Conclusion

This paper has shown how the law of large numbers and the MIE axiom can work in concert to completely exclude all possible exceptional cases (including both non‑zero‑measure anomalies and zero‑measure anomalies). The law of large numbers guarantees that almost all trajectories enter a low‑number region; the MIE axiom forces all trajectories (including those on zero‑measure sets) to choose the unique information‑efficiency extremal attractor. Together, they make “all positive integers converge to \{1,4,2\} ” the only logical possibility. The third paper in the series will integrate these results and establish a complete dynamic model of convergence.

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