Integer Dynamics × Statistical Mechanics: Toward “Macro Number Theory” (with Collatz as a Test Case)[A historical perspective: how number theory has been “going macro,” and why I believe a structural approach to Collatz deserves attention.]

0. A brief note from the author

I’m an independent researcher in Japan. I’m currently living with depression, and I’m not fluent in English. That’s the honest reason why I haven’t produced a full English version of my “Integer Dynamics × Statistical Mechanics” (Macro Number Theory) series yet. I’m writing this Medium post anyway, because I want the core idea to be accessible and discussable. I hope to translate more later.

1. The big idea, in plain language

Number theory is often imagined as a world of “tiny” questions:

• Is this number prime?
• Does this equation have an integer solution?
• What happens to this specific sequence?

But historically, number theory has also been moving in the opposite direction: from the microscopic to the macroscopic.

• From individual integers to asymptotic laws (analytic number theory)
• From exact patterns to distributions and fluctuations (probabilistic viewpoints)
• From static properties to dynamical behavior (ergodic/dynamical number theory)

In physics, we do not track every molecule to understand a gas; we introduce macroscopic variables (temperature, entropy, free energy) that capture typical behavior.

My proposal is that we can do something analogous for integers:

Treat an integer process as a kind of integer dynamics, then introduce a principled coarse-graining that produces macroscopic “structural variables,” and study typicality, stability, and “phases” at the macro level.

Collatz is a perfect test case because it is fully deterministic at the micro level, yet its long-time behavior often feels dissipative or statistical.

2. Why I think Collatz needs a structural — not just computational — attack

Most people meet Collatz as a rule:

• if n is even: n↦n/2
• if n is odd: n↦3n+1

Then the conjecture says: everything eventually reaches 1.

Computational verification is huge, but it does not automatically explain why the dynamics seems to collapse into a small attractor. The key issue is not “checking more numbers.” It is finding the right invariants / monotone quantities / coarse variables.

My approach is to replace “raw integers” with a structural representation that makes the rewrite-like nature of Collatz steps explicit. That is what I call the Collatz Phase Expression (CPE).

3. A claim about historical positioning

I want to make a strong claim — carefully stated:

The Collatz Phase Expression (CPE) is not “yet another computation trick.” It is an attempt to place Collatz into the historical trajectory of number theory: coarse-graining, structure extraction, and macro-level reasoning.

In other words, CPE is meant as a proof-oriented representation: a language in which one can try to build inequalities, monotonicity constraints, and stability arguments without chasing individual trajectories directly.

This is the sense in which I believe CPE can be historically positioned as a meaningful “proof approach,” rather than a numerical exploration.

(If you are a mathematician: I’m explicitly claiming that the representation aims to play the role of a “good coordinate system” for the dynamics — something like what normal forms do in dynamical systems, but adapted to integer rewriting.)

4. References: my Collatz CPE papers

I wrote two versions of my CPE-based Collatz work:

• English: Structural Analysis of the Collatz Conjecture… via the Collatz Phase Expression (CPE) https://doi.org/10.5281/zenodo.18025034
• Japanese: 整数全体におけるコラッツ予想の構造的解決…（交代二進表記とコラッツ相表現） https://doi.org/10.5281/zenodo.18025050

5. From here on: a more technical section (for researchers & serious amateurs)

This section is not necessary to enjoy the main post. I’m switching gears: I’ll speak more directly in the language of statistical mechanics / combinatorics / dynamical thinking. If you’re here for the big picture only, you can stop at the end of Section 4.

5.1 Integer dynamics and coarse-graining

A core methodological point:

1. Start with a deterministic integer map (Collatz is one instance).
2. Build a representation that expresses each state as a composition of local structural units.
3. Define macro variables that forget microscopic details but preserve structural constraints.
4. Study:

• state counting at fixed macro variables
• typical macro configurations
• fluctuations under canonical/grand-canonical weighting
• phase stability (attractors / loop-structures)
• possible critical behavior under parameter changes

In my work, the ABN/CPE framework provides such a structural factorization, from which I extract a set of “structural quantities” that act like macro variables.

5.2 Why “statistical mechanics” appears without probability assumptions

A frequent misunderstanding is: “If you use entropy or ensembles, you must be assuming randomness.”

Not necessarily.

In combinatorics and information theory, entropy can arise as:

• Ω = number of micro-configurations compatible with a macro constraint
• S=log⁡Ω

This is counting, not stochastic modeling. You can define microcanonical shells purely combinatorially. Probability enters later only as a convenience (e.g., choosing a uniform measure on the shell, or adding weights to form a canonical ensemble). The conceptual foundation can remain deterministic.

That matters for Collatz because the underlying system is deterministic; the statistical layer should be a derived description, not an assumption.

5.3 “Phases” as attractor-classes of integer dynamics

A “phase,” in my intended sense, is not “convergence to 1” per se. It is more like:

integers whose dynamics share the same asymptotic destination structure (same attractor / same loop system) belong to the same phase.

For positive integers, the Collatz conjecture asserts a single dominant phase (the 1-attractor). For negative integers under the corresponding dynamics, one can discuss multiple loops, which suggests a richer “phase diagram.”

This viewpoint is meant to generalize: Collatz is only the most famous example.

6. Why I’m writing this on Medium

I’m writing here because I want feedback from both communities:

• people who like big ideas and historical framing
• serious amateurs and researchers who can critique definitions, invariants, and proof structure

If you read this and feel anything — interest, confusion, skepticism — comments help. Even one line like:

• “This part is unclear.”
• “This resembles X in dynamical systems.”
• “Why should your coarse variables be complete?”
• “Where is the key monotonicity?”

…is valuable fuel for improving the work.

7. Links: “Integer Dynamics × Statistical Mechanics” series (Japanese/Zenodo)

This is the series where I develop the “Macro Number Theory” viewpoint using Collatz as an example:

• I — Structural variables and a dictionary https://doi.org/10.5281/zenodo.18054427
• II — Microcanonical shells and exact state counting https://doi.org/10.5281/zenodo.18054455
• III — Canonical picture and fluctuations https://doi.org/10.5281/zenodo.18054493
• IV — Occupation-number representation and multi-parameter canonical structure https://doi.org/10.5281/zenodo.18054529
• V — True grand-canonical picture and critical structure https://doi.org/10.5281/zenodo.18054550

(Full English versions are not written yet, for the reasons stated at the beginning.)

Closing line (short, Medium-friendly)

Collatz is often presented as a “hard problem.” I think it is also a perspective problem: how to coarse-grain integers in a way that exposes the right invariants and stability structure. CPE is my attempt at such a perspective.