arXiv subjects

Takao Inoué

Publications and source records attributed to Takao Inoué.

At least 19 recordsLinked to original sources

arXiv2026

Constructive Peter--Weyl Theory: What Is Known and What Remains Open

What is known about constructive Peter-Weyl theory, and what remains open? This expository article separates established theorems, comparisons between formulations, and questions requiring further investigation in the Bishop-Coquand-Spitters line. Its central finding is that constructive Peter-Weyl theory already has an established compact-group approximation core: under specified analytic and Gelfand hypotheses, finite-rank central convolution projections converge strongly in the Haar L2 norm. We make these hypotheses and the projection-kernel normalization explicit, and record the resulting spectral Hilbert completion. This answers the question of completion affirmatively for that precise core theorem. Broader claims about the whole classical package or its validity across constructive foundations require separate comparisons; uniform approximation, irreducible representation data, and point-set recovery are not inferred from strong convergence alone. The historical problems of left-right almost periodicity and universal Bohr compactification are examined with a dated follow-up literature review, distinguishing related progress from a verified resolution. A sufficient metric realization criterion, finite examples, and a separate nonassociative outlook clarify the scope of the established method. An extended appendix proves the classical Peter-Weyl theorem through Haar averaging, convolution, spectral density, point separation, and irreducible orthogonality. Stage-by-stage comparisons explain which arguments transfer constructively and which require additional data or a different proof. The article therefore identifies what is proved and what additional statements or data remain to be examined, without treating an unidentified resolution as proof that a problem is currently open.

math.FA↗

arXiv2026

TAPO-Description Logic for Information Behavior: Refined OBoxes, Inference, and Categorical Semantics

This paper develops a refined version of TAPO-description logic for the analysis of information behavior. The framework is treated not as a single homogeneous object logic, but as a layered formalism consisting of a static descriptive layer (TBox/ABox), a procedural layer (PBox), and an oracle-sensitive layer (OBox). To make this architecture mathematically explicit, we introduce a metalevel guard-judgment layer governing procedural branching and iteration. On this basis we formulate a core inference system for TAPO-description logic, covering static TBox/ABox reasoning, guarded procedural transition in the PBox, and validated external import in the OBox. We then give a categorical semantics for the resulting framework and indicate its sheaf-theoretic refinement. The theory is illustrated by examples of information-seeking behavior, including simple search behavior and review-sensitive ordering behavior in a curry restaurant. The aim is to treat not only static knowledge representation but also hesitation, external consultation, and action-guiding update within a unified logical setting.

cs.LO↗

arXiv2026

From Torsors to Topoi: An Introduction with a View Toward $Σ$-Protocols in Cryptography

This paper provides a preparatory introduction to sheaves and topoi, written as a conceptual continuation of the author's earlier introduction to torsors and as preparatory background for the author's arXiv paper \emph{Grothendieck Topologies and Sheaf-Theoretic Foundations of Cryptographic Security:\ Attacker Models and $Σ$-Protocols as the First Step}~\cite{InoueSecurity}. Rather than attempting an encyclopedic survey of all of topos theory, the exposition develops those parts of the subject that are most relevant for passing from torsor-based local-to-global reasoning to sheaf-theoretic and topos-theoretic reasoning: Grothendieck topologies, sheaves, torsors over a site, descent, sheaf topoi, elementary topoi, Cartesian closed structure, subobject classifiers, and internal logic. The goal is not merely motivational. We try to develop enough genuine topos theory that the reader can understand, not only heuristically but structurally, why the later cryptographic framework of~\cite{InoueSecurity} uses Grothendieck topologies and sheaf-theoretic language. To make the note more self-contained, we also include substantial appendices on basic category theory, Yoneda's lemma, limits and colimits, equalizers and coequalizers, Kan extensions, the relation between internal logic and intuitionistic logic, and exercises with solutions. In the final part, we explain how these ideas prepare the ground for a conceptual understanding of $Σ$-protocols, especially in connection with local consistency, simulability, and the passage from compatible local data to global structure.

math.CT↗

arXiv2026

Celebrating the Day of $π$: Joyful Variations on Euler's Identity

This short essay celebrates the mathematical meaning of Pi Day through Euler's formula \[ e^{ix}=\cos x+i\sin x, \] from which Euler's identity \[ e^{iπ}+1=0 \] follows immediately. We briefly note the historical background of the formula, usually traced to Euler's \emph{Introductio in analysin infinitorum} (1748), while also mentioning Roger Cotes's earlier precursor of 1714. We compare Euler's identity, in an explicitly analogical way, with several famous formulas in physics in order to highlight its remarkable compactness and conceptual richness. We then consider a number of joyful variations arising from the same Eulerian source, including the negative-angle case, prime-number multiples, the substitution $x=π/2$, and a functional-equation variation of the form \[ f(iπx)+1=0. \] This last variation leads naturally to a contrast between rigidity in the holomorphic setting and freedom in the discrete interpolation setting. The central aim is to organize these observations into two simple families of variations: geometric-angle variations and functional-equation variations. The earlier part of the exposition is intended to be accessible to motivated high-school students, while the later discussion points toward more advanced ideas from complex analysis.

math.HO↗

arXiv2026

Modular Cocycles and Haar-Type Measures on Topological Loops

This paper is a continuation of the author's companion work \cite{InoueQuasi} on Haar-type measures for topological quasigroups, where the quasigroup setting was analyzed in connection with Kunen's theorem. We extend that framework to locally compact topological loops and study Haar-type (quasi-invariant) Radon measures together with modular cocycles describing the distortion of such measures under translations. Unlike the classical group case, the composition of translations in a loop is affected by the failure of associativity, which produces an additional correction term governed by an associativity deviation map. We derive the resulting cocycle relation and show that loop identities, in particular Moufang- and Kunen-type identities, impose structural restrictions on the modular data. In the associative limit, the cocycle reduces to the classical modular function of a locally compact group.

math.GR↗

arXiv2026

An Introduction to Torsors in Mathematics with a View Toward $Σ$-Protocols in Cryptography

This paper provides a preparatory introduction to torsors, written with a view toward later applications in the author's work. Rather than aiming at a comprehensive survey, the exposition focuses on those aspects of torsors that are most useful for understanding torsor-based reasoning: group actions, orbits, free transitive actions, the absence of a canonically chosen origin, and the interpretation of group elements as transports between points. After developing the basic definition and several elementary examples, we emphasize a central theme: torsors are not only characterized abstractly by free transitive group actions, but also arise naturally as objects obtained by gluing local trivial pieces by means of transition data satisfying cocycle conditions. A brief optional section indicates a sheaf- and topos-theoretic perspective. In the final part, we explain how these ideas prepare the ground for later conceptual applications, including aspects of $Σ$-protocols.

math.GR↗

arXiv2026

Haar-Type Measures on Topological Quasigroups and Kunen's Theorem

Haar measure is a fundamental structure in harmonic analysis on locally compact groups. Its existence reflects the compatibility between topology and the associative algebraic structure of groups. In this paper we propose a framework for Haar-type measures on topological quasigroups. Since associativity is absent, strict translation invariance is generally too strong to expect. We therefore introduce quasi-invariant measures whose defect is measured by a modular cocycle attached to translations. We then explain, in a detailed and cautious form, how Moufang-type identities may impose strong constraints on this cocycle. In particular, under additional quasi-invariance assumptions for right translations, the Moufang-type identity $(N1)$ leads naturally to a multiplicativity relation for the cocycle. This suggests a measure-theoretic interpretation of Kunen's theorem: the emergence of loop structure may be viewed as the collapse of a modular defect in the translation geometry of a quasigroup.

math.GR↗

arXiv2026

Structural Chirality from Inverse Semigroups to Twisted Groupoid $C^*$-Algebras

We develop a structural theory of chirality for inverse semigroups and show how it propagates canonically to étale groupoids and twisted groupoid $C^*$-algebras. Starting from inverse semigroup data equipped with admissible twist information, we construct a canonical twisted universal groupoid in the sense of Paterson and introduce a mirror correspondence encoding intrinsic asymmetry. Our main result identifies a structural obstruction to mirror self-duality at the level of twisted universal groupoids and shows that this obstruction descends to an obstruction for the associated reduced twisted groupoid $C^*$-algebra to be isomorphic to its opposite. The framework is representation-independent, yet compatible with concrete germ groupoid models, and provides a unified bridge between partial symmetries, groupoid structures, and analytic invariants in noncommutative operator algebras.

math.OA↗

arXiv2026

Mayer--Vietoris and Twisted Čech Spectral Sequences for C$^*$-Algebras with Free Quantum Group Coefficients

We formulate a Mayer--Vietoris/Čech viewpoint on $K$-theory for crossed products by discrete quantum groups, emphasizing how local-to-global gluing data interacts with quantum-group coefficients. Starting from a $G$-equivariant ideal cover and the associated Mayer--Vietoris six-term exact sequence, we package the resulting $K$-theoretic computation into a Čech-type spectral sequence whose $E^1$-page is explicitly described by iterated intersections. We then introduce a minimal ``twisted'' gluing mechanism controlled by a $\mathbb Z/2$-valued Čech $2$-cocycle and an involutive automorphism of the coefficient algebra. Under a Kirchberg--UCT hypothesis on the quantum-group crossed-product coefficient, the twist produces a nontrivial differential $d_2$ identified as $φ_*-\mathrm{id}$ on coefficient $K$-theory. In a concrete regime where the coefficient $K$-groups are cyclic (e.g.\ order $3$), the differential becomes an isomorphism and forces a $K$-theoretic obstruction to Morita triviality. This yields a conceptual mechanism for producing non-Morita-trivial twisted C$^*$-algebras with quantum-group crossed-product fibers, detected purely by Mayer--Vietoris/Čech data.

math.OA↗

arXiv2026

Chirality and Racemization on Isotopy Classes of Loops: A Groupoid-Based Structural Theory

We develop a theory of chirality and racemization on isotopy classes of finite loops, formulated intrinsically within the loop isotopy groupoid understood in the categorical sense. Motivated by earlier work on quasigroups \cite{InoueQuasiChirality} and by the classical medical paradigm of mirror-related enantiomers, we restrict admissible mirror transitions to those generated by intrinsic, unit-preserving symmetries. Within this framework, racemization is modeled as a two-state dynamics on isotopy classes, with an effective rate determined by the presence of mirror-isotopisms. Our main result shows that this rate vanishes if and only if no loop isotopism exists between a loop and its opposite, providing a structural criterion for chirality. A strengthened variant based on translation-generated symmetries is discussed in the appendix.

math.GR↗

arXiv2026

A Formal Group Perspective on the Riemann Zeta Function

We develop a formal group--theoretic framework for the Riemann zeta function by treating its Euler product as an element of the multiplicative formal group $\widehat{\mathbb{G}}_m$ and its logarithm as the associated formal group logarithm. This provides a purely algebraic linearization of the multiplicative prime-wise structure of the Euler product. Starting from a finite cutoff model, we introduce a formal completion via evenization and a natural normalization procedure. The resulting logarithmic expansion exhibits a Gaussian leading term, while higher-order terms form a hierarchy of cumulants. We show that this Gaussian structure is not probabilistic in origin, but arises from the infinitesimal quadratic geometry of the multiplicative formal group after linearization. The higher cumulants encode arithmetic deviations and admit a decomposition governed by weighted integrals of the Chebyshev error function $θ(x)-x$. A systematic formal group axiomatization is then provided, placing the cutoff model and cumulant hierarchy within a coherent algebraic framework. The approach is deliberately non-spectral, focusing on a structural reorganization of the Euler product rather than on operator-theoretic realizations of zeros. We briefly indicate possible connections with absolute arithmetic and geometry over $\mathbb{F}_1$.

math.GM↗

arXiv2026

Chirality and Racemization on Isotopy Classes of Quasigroups

We develop a structural and dynamical theory of chirality for quasigroups formulated at the level of isotopy classes. Interpreting isotopy as a gauge symmetry of re-coordinatization and mirror parastrophy as handedness reversal, we introduce a gauge-invariant continuous-time two-state Markov model in which transitions occur only between a quasigroup and its mirror. We prove that this dynamics descends to the isotopy quotient, yielding a reduced generator governed by a single class-dependent rate $k([Q])$. Symmetric mirror transitions lead to convergence toward a racemic equilibrium, whereas the vanishing condition $k([Q])=0$ characterizes dynamical chiral stability. By restricting admissible transitions to those generated by intrinsic symmetries, we show that $k([Q])=0$ is equivalent to the absence of mirror-isotopisms. A concrete example of order $7$ demonstrates the existence of structurally chiral quasigroup classes.

math.DS↗

arXiv2026

On the Category-Theoretic Independence of Meaning, Object, Name and Existence

We prove a category-theoretic independence theorem for four fundamental notions: meaning, object, name, and existence. Working in a Lawvere-style categorical semantics and in particular in toposes, we show that these notions occupy distinct structural levels (object, morphism, element, and internal logical level) and are not uniformly recoverable from one another. The key separation arises between internal existence and global naming. Using a concrete example in the topos $\mathbf{Sh}(S^1)$-the sheaf of local sections of a nontrivial covering-we exhibit an object that is internally inhabited but admits no global element. These results provide a precise structural basis for treating geometric universes as foundational frameworks for information networks.

math.CT↗

arXiv2026

A Structural Fixed-Point Principle in Kunen's Theorem on Quasigroups

Kunen proved that a quasigroup satisfying a Moufang-type identity ($N1$) must be a loop. We reformulate the argument in the category $\mathbf{Set}$ as a fixed-point extraction principle. From $N1$ one canonically obtains an idempotent endomorphism $j:G\to G$. Its fixed-point object $\mathrm{Fix}(j)=\mathrm{Eq}(j,\mathrm{id}_G)$ splits off as a retract. The $N1$-symmetry forces $j$ to coequalize the (regular) translation action, hence $j$ factors through the terminal object. Thus $\mathrm{Fix}(j)\cong 1$, yielding a unique global identity element. This provides a conceptual reformulation of Kunen's original algebraic proof \cite{Kunen}.

math.GR↗

arXiv2026

Grothendieck's Geometric Universes and A Sheaf-Theoretic Foundation of Information Network

This paper proposes an interpretation of Grothendieck's geometric universes as a foundational framework for \emph{information networks}. We argue that Grothendieck topologies, sheaves, and topoi provide a sheaf-theoretic semantics in which distributed and locally held information can be integrated into globally coherent structures. In this setting, local informational states are represented by sections, while the sheaf condition governs consistency, agreement, and consensus across a network. Logical validity and mathematical existence are therefore not imposed externally but arise intrinsically from geometric and categorical conditions. From this perspective, Grothendieck's geometric universes constitute a natural foundation for information networks governed by intrinsic logical principles. Moreover, we propose that Grothendieck's geometric universes themselves concretely instantiate what the author calls \emph{intrinsic logicism}. This position is intended as a contemporary reconstruction of the classical logicist program of Frege and Russell, reformulated within the framework of category theory and topos theory, where logical structure is generated internally by geometric and categorical organization rather than presupposed as an external foundational layer.

math.CT↗

arXiv2026

TAPO-Structured Description Logic for Information Behavior: Procedural and Oracle-Based Extensions

We introduce \emph{TAPO-Structured Description Logic} (TAPO--DL), a formal extension of classical description logic designed to model \emph{information behavior} as a structured, dynamic process. TAPO--DL extends the standard T--Box/A--Box architecture with two additional layers: a \emph{Procedural Box} (P--Box), which supports concept-driven, imperative-style programs such as conditional and iterative actions, and an \emph{Oracle Box} (O--Box), which formalizes controlled interaction with external information sources. While the terminological and assertional components capture static conceptual and factual knowledge, the procedural and oracle-based components enable the explicit representation of information-generating actions and external validation. We provide a unified semantic framework for TAPO--DL based on a co-generative, sheaf-theoretic interpretation, in which local informational states are modeled as sections and informational stability corresponds to the existence of coherent global structures. Within this setting, informational truth is characterized as stability under repeated agentive interaction rather than correspondence to a fixed global state. By integrating description logic with procedural dynamics, oracle-based reasoning, and sheaf-theoretic semantics, TAPO--DL offers a principled formal framework for analyzing information behavior in contexts involving interaction, uncertainty, and contextuality.

cs.LO↗

arXiv2026

Grothendieck Topologies and Sheaf-Theoretic Foundations of Cryptographic Security: Attacker Models and $Σ$-Protocols as the First Step

Cryptographic security is traditionally formulated using game-based or simulation-based definitions. In this paper, we propose a structural reformulation of cryptographic security based on Grothendieck topologies and sheaf theory. Our key idea is to model attacker observations as a Grothendieck site, where covering families represent admissible decompositions of partial information determined by efficient simulation. Within this framework, protocol transcripts naturally form sheaves, and security properties arise as geometric conditions. As a first step, we focus on $Σ$-protocols. We show that the transcript structure of any $Σ$-protocol defines a torsor in the associated topos of sheaves. Local triviality of this torsor corresponds to zero-knowledge, while the absence of global sections reflects soundness. A concrete analysis of the Schnorr $Σ$-protocol is provided to illustrate the construction. This sheaf-theoretic perspective offers a conceptual explanation of simulation-based security and suggests a geometric foundation for further cryptographic abstractions.

cs.CR↗

arXiv2026

On Brain as a Mathematical Manifold: Neural Manifolds, Sheaf Semantics, and Leibnizian Harmony

We present a mathematical and philosophical framework in which brain function is modeled using sheaf theory over neural state spaces. Local neural or cognitive functions are represented as sections of a sheaf, while global coherence corresponds to the existence of global sections. Brain pathologies are interpreted as obstructions to such global integration and are classified using tools from sheaf cohomology. The framework builds on the neural manifold program in contemporary neuroscience and on standard results in sheaf theory, and is further interpreted through a Leibnizian lens \cite{Churchland2012, Leibniz1714, MacLaneMoerdijk, Perich2025}. This paper is intended as a conceptual and formal proposal rather than a complete empirical theory.

q-bio.NC↗