Author: Fumio Miyata
ORCID: https://orcid.org/0009-0008-8797-5578
Email: ai.kenkyu.001@gmail.com
Date: June 2026
DOI: https://doi.org/10.5281/zenodo.XXXXXXXX
Abstract
This paper develops a formal semantic framework for modeling the irreversible structural transformations that occur in the inter-universal communication processes of Mochizuki’s Inter-Universal Teichmüller Theory (IUT). These transformations—implemented through the $\Theta$-links and log-links—produce three types of structural indeterminacy ($\mathrm{Ind}$ I, II, III). We reconstruct these phenomena using transformation monoids and the axiomatic notion of elementary irreversible morphisms $\mathcal{I}(S)$, introduced in Miyata (2026).
The aim of this research program is not to adjudicate the correctness of IUT’s proof of the ABC conjecture. Rather, it is to articulate a mathematically rigorous and philosophically transparent semantics for irreversible structural change. This framework is situated within several active movements in Philosophia Mathematica: Potentialism (Hamkins 2026; Hellman 2026), Structuralism (Assadian 2026), the conceptual foundations of computation (Quinon 2026; Burgess 2026), and the epistemology of mathematical trust (Mangraviti 2026). We argue that irreversibility provides a new lens for analyzing modal mathematical universes, cross-structural identity, deviant encodings, and the stability of computational meaning.
0. Positioning of This Paper
Recent discourse in the philosophy of mathematics has emphasized four major themes:
- Potentialism — mathematical universes as modal, non-fixed, and dynamically expandable (Hamkins 2026; Hellman 2026).
- Structuralism — the fragility of identity across distinct structural contexts (Assadian 2026).
- Computation — the conceptual stability or instability of computational encodings (Quinon 2026; Burgess 2026).
- Mathematical Practice — the epistemic norms governing mathematical trust and inconsistency (Mangraviti 2026).
IUT’s inter-universal communication—where highly structured objects are transported across non-isomorphic universes—constitutes a concrete mathematical arena where all four themes intersect. This paper does not attempt to resolve the technical debate surrounding the ABC conjecture. Instead, it provides a formal semantic grammar for describing structural degradation across universes.
Philosophical Contribution
The central philosophical contribution of this paper is the introduction of irreversibility as a foundational semantic primitive for mathematical ontology. Irreversibility marks the boundary between structural preservation and structural transformation. It provides a criterion for when identity, meaning, or computational representation ceases to be invariant across mathematical universes. This directly addresses current debates on modal ontology, structural identity, and conceptual fixed points in computation.
1. State Spaces and Transformation Monoids
Assumption 1.1 (Abstract State Space)
Let $S$ be an abstract representation of the geometric and algebraic data manipulated within IUT (e.g., elliptic curves, Frobenioids, log-structures, theta-capsules). Depending on the chosen level of abstraction, $S$ may be formalized as:
- A non-empty set or topological space,
- An algebraic variety or scheme,
- An object within an appropriately constructed category.
Remark 1.2 (Abstraction Safety)
We do not manipulate IUT’s concrete objects directly. Instead, we assume they admit a structural projection into an abstract state space $S$. This aligns with Potentialism: mathematical universes are not fixed totalities but can be dynamically recontextualized across strata of abstraction (Hamkins 2026; Hellman 2026).
Definition 1.3 (Transformation Monoid)
Let $\mathrm{End}(S)$ denote the monoid of endomorphisms of $S$.
If $S$ is modeled as a category, $\mathrm{End}(S)$ is interpreted as the monoid of endofunctors.
Note 1.4 (Representability of Inter-Universal Communication)
We hypothesize that the $\Theta$-links and log-links can be fully embedded as elements into $\mathrm{End}(S)$. This mirrors the structuralist project of analyzing how mathematical objects adapt when mapped between distinct structural contexts (Assadian 2026).
2. Axiomatic Framework for Irreversible Morphisms
Definition 2.1 (Reversible and Irreversible Morphisms)
Let $\mathrm{Rev}(S) = \mathrm{Iso}(S)$ denote the group of all reversible, structure-preserving automorphisms (or bijections) on $S$. We define the set of irreversible morphisms as the set-theoretic complement:
$$\mathrm{Irr}(S) = \mathrm{End}(S) \setminus \mathrm{Rev}(S)$$
Axiom 2.2 (Elementary Irreversibility)
For any given state space $S$, there exists a designated, non-empty set of elementary irreversible morphisms:
$$\mathcal{I}(S) \subseteq \mathrm{Irr}(S)$$
representing minimal, irreducible units of structural or informational loss.
Axiom 2.3 (Minimal Decomposition)
Every non-trivial irreversible morphism $f \in \mathrm{Irr}(S)$ decomposes as a finite composition:
$$f = R_k \circ I_n \circ \cdots \circ I_1 \circ R_1$$
with $R_i \in \mathrm{Rev}(S)$ and $I_j \in \mathcal{I}(S)$.
This decomposition parallels Burgess (2026) on deviant encodings: complex structural distortions can be systematically broken down into minimal, transparent encoding disruptions.
3. Structural Analogy with IUT Indeterminacies
Proposition 3.1 (Analogy Between Indeterminacies and Canonical Irreversible Forms)
The three canonical forms of elementary irreversible morphisms (Miyata 2026) correspond structurally to the three IUT Indeterminacies ($\mathrm{Ind}$ I, II, III):
| Canonical Form (Miyata) | IUT Indeterminacy | Core Structural Mechanism |
|---|---|---|
| Compression | $\mathrm{Ind}$ I | The rigid degradation of additive ring structures, bounding structural degrees of freedom. |
| Forgetting | $\mathrm{Ind}$ II | The deliberate omission or masking of environmental data (e.g., action of roots of unity). |
| Branching | $\mathrm{Ind}$ III | The controlled copying and expansion of data structures into non-isomorphic parallel graphs (multiverse replication). |
This aligns with Assadian (2026): cross-structural identity breaks down when objects cross domain boundaries, and the forms above classify the exact grammar of this breakdown.
–>
Figure 1: Classification of Elementary Irreversible Morphisms (Compression, Forgetting, and Branching)
4. Irreversibility Degree and Distortion Measurement
Definition 4.1 (Irreversibility Degree)
For any morphism $f \in \mathrm{End}(S)$, its metric of structural degradation—the irreversibility degree $\mathrm{irr}(f)$—is defined as:
$$\mathrm{irr}(f) = \min \{ n \mid f = R_k \circ I_n \circ \cdots \circ I_1 \circ R_1 \}$$
–>
Figure 2: Decomposition and Metric of Irreversibility Degree for Morphisms
Proposition 4.2 (Comparability with IUT Distortion Measures)
IUT’s continuous “log-volume distortion” bounded during inter-universal passage can be compared—at an abstract level—to the cumulative step-count metric of elementary irreversible steps:
$$\sum_{j} \Delta_{I_j}$$
where $\Delta_{I_j}$ is the unit of distortion assigned to each irreversible step.
This addresses the question raised by Quinon (2026): computation is not a conceptual fixed point; irreversible steps represent non-fixed transformations of mathematical information.
5. Philosophical Status of the Program
This framework contributes to current debates by:
- Treating universes as modal and non-fixed (Potentialism).
- Analyzing identity under structural change (Structuralism).
- Modeling irreversible information flow (Computation).
- Clarifying epistemic norms of mathematical trust (Mangraviti 2026).
Irreversibility becomes a foundational semantic primitive for evaluating what occurs when mathematical objects transcend their native structures.
6. Toy Model Demonstration
To illustrate the framework concretely, consider the base state space:
$$S = M_2(\mathbb{R})$$
the space of $2 \times 2$ real matrices. We define the reversible morphisms as the invertible linear maps $R \in GL_2(\mathbb{R})$, and irreversible morphisms as rank-decreasing linear or structural projections.
Example: Compression Type
Let $I \in \mathrm{End}(S)$ act via matrix multiplication:
$$I(x) = P x, \quad P = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$$
Then $\mathrm{rank}(I(x)) \le 1$. Information is rigidly compressed (loss of independent dimensional freedom), and $\mathrm{irr}(I) = 1$.
Example: Forgetting Type
Let $I \in \mathrm{End}(S)$ be the coordinate-collapsing projection:
$$I \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a & b \\ c & 0 \end{pmatrix}$$
This operator is strictly idempotent ($I^2 = I$) and non-invertible. It models the semantic “forgetting” of the parameter space associated with the $(2,2)$-entry, effectively reducing the informational dimension of the state without introducing affine shifts.
Example: Branching Type
To rigorously model branching as an endomorphism, we extend the toy state space to a finite multiverse envelope $\mathcal{S} = \bigsqcup_{k=1}^N S^k$. The branching morphism $I_{\mathrm{branch}} \in \mathrm{End}(\mathcal{S})$ acting on a single-universe state $x \in S^1$ is defined as:
$$I_{\mathrm{branch}}(x) = (x, Bx) \in S^2$$
This maps a state in universe-stratum $1$ to a correlated state in stratum $2$, strictly preserving the endomorphism property on the envelope $\mathcal{S}$ while modeling controlled multiverse branching.
These toy models demonstrate that irreversibility degrees and structural degenerations can be formulated and computed cleanly without violating standard categorical definitions.
7. Future Research Questions
A. Representability
Can the actual category-theoretic mechanisms of the $\Theta$-link and log-link be rigorously embedded into the endomorphism monoid $\mathrm{End}(S)$ of an appropriately constructed category without critical semantic remainder?
B. Categorical Formalization
Can the analogy between $\mathrm{Ind}$ I–III and canonical irreversible forms be lifted into a strict, verifiable category-theoretic equivalence or adjunction?
C. Concrete Toy Models
Can we construct richer, categorical toy models (e.g., using finite linear topoi) that exhibit non-trivial irreversibility degrees while mirroring the behavior of geometric log-volume variations?
8. Conclusion
Elementary irreversible morphisms provide a mathematically rigorous and philosophically meaningful vocabulary for describing inter-universal communication. By embedding this framework within current debates in Potentialism, Structuralism, computation theory, and mathematical practice, we offer a neutral, highly structured path toward understanding irreversible information flow across the boundaries of mathematical worlds.
References
- Assadian, Bahram. (2026). Cross-structural Identity Statements. Philosophia Mathematica.
- Burgess, John P. (2026). Is There a Problem about Deviant Encodings? Philosophia Mathematica.
- De Benedetto, Matteo & Rossi, Lorenzo. (2026). Cognitive Modelism. Philosophia Mathematica.
- Hamkins, Joel David. (2026). A Potentialist Conception of Ultrafinitism. Philosophia Mathematica.
- Hellman, Geoffrey. (2026). Introduction to Special Issue on Potentialism in the Philosophy of Mathematics. Philosophia Mathematica.
- Mangraviti, Franci. (2026). Inconsistency is in the Eye of the Mathematician. Philosophia Mathematica.
- Miyata, Fumio. (2026). A Minimal Axiomatic Framework for Elementary Irreversible Morphisms in Transformation Monoids. DOI: https://doi.org/10.5281/zenodo.20984264
- Mochizuki, Shinichi. (2021). Inter-universal Teichmüller Theory I–IV. Publications of the Research Institute for Mathematical Sciences.
- Quinon, Paula. (2026). Is the Concept of Computation a Conceptual Fixed Point? Philosophia Mathematica.
- Scholze, Peter & Stix, Jakob. (2018). Why ABC is still a conjecture.