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Category theory papers 32 papers
Category theory. Categories, functors and coalgebra Tagged papers: 32.
Functoriality as a Transform: Register and Gate-Cost Analysis of a Langlands-Structured Adelic Quantum Register
Computing with automorphic representations is bottlenecked by the absence of a fast algorithm for non-abelian harmonic analysis: for the abelian group $\mathrm{GL}(1)$ the Fast Fourier Transform decomposes functions into irreducible characters in $O(n \log n)$ operations, while for…
Functorial Invariance Across Presentations of the Prime Numbers, with Context-Relative Quantitative Illustrations
The prime numbers admit many simultaneous presentations: as the atomic subset of the natural numbers, as the support of the von Mangoldt function, as the zero data of the Riemann zeta function, as the output of sieve procedures, as primes inside Beatty-type sequences, and as the output of concrete…
Automorphisms and Scaling Transformations: Conjugation Action, Parameter Rescaling, and Growth of Associative Structure
Automorphism groups encode the symmetries of algebraic varieties, non-associative algebras, parabolic geometries, group splittings, and holomorphic dynamical systems, but each setting is treated with its own tools, and practitioners lack a uniform test for when a rescaling of coordinates,…
Number Systems as Stratified Distinction Operations: An Enclosure Hierarchy over the Mark
We develop a formal sketch of the conjecture that the major number systems arise from successive primitive acts of distinction applied to a Laws-of-Form-like mark. We model each historical-cognitive stage — tallying ($\mathbb{N}$), ratio comparison ($\mathbb{Q}$), convergent sequences (computable…
Braid Group Representations and the Uniqueness of Modular Data: The Ising Test Case and General Limitations
We investigate whether the unitary braid group representation carried by a collection of non‑Abelian anyons uniquely determines the modular data—namely the $S$ and $T$ matrices—of the underlying modular tensor category (MTC). Using the Ising anyon theory as a concrete laboratory, we reconstruct the…
A Two-Index Framework for the Bulk-Boundary Correspondence of Anyon Condensation and Boundary Majorana Statistics
The bulk-boundary correspondence asserts that topological data of a (2+1)-dimensional phase determine the physics of its boundary, but in the setting of anyon condensation the correspondence is usually stated structurally rather than quantitatively. We propose a set of computable indices that…
The Boson/Fermion Distinction: Spin-Statistics as Structural Invariant
The textbook dichotomy between bosons and fermions is commonly presented as a primitive classification of nature. This paper argues that the dichotomy is a derived, dimension-dependent shadow of a single structural relation: the exchange phase of identical particles equals their topological spin, R…
The Void Is Not False: Recovering the Unmarked State in Logic from the Calculus of Indications
A critique-first paper: Boolean logic and set theory reify the void (the unmarked state) into false and the empty set, committing a category error; false is a mark, the void is not a value. A reconstruction program (distinction-logic with inside/outside/oscillating states) is labeled conjecture with…
The Universe Category: A Single Functor Encoding Quantization, Stability, and Factorization
This paper proposes a single categorical object — a functor from the divisibility category of the positive integers to the category of smooth compact manifolds — and investigates whether its image simultaneously encodes three structures that have historically been treated as separate: quantization…
Valuation Without R: A Category-Theoretic Foundation for Finite Measurement
Measurement in physics is always finite and approximate — yet its mathematical foundations are embedded in the Archimedean continuum of real numbers and the cumulative hierarchy of set theory. We invert this dependency: the act of measurement is axiomatized as a graded distinguishability map…
The Ontology of Boundary-Crossing: Spencer-Brown Cancellation Rule Defended
Systematic defense of Spencer-Brown Cancellation rule: establishes structural isomorphism with quantum measurement collapse via three-outcome dynamics (Varela 1975, corroborated by 29-schisms numerical search 2026). 4-level M-property justification framework. Boson-fermion observer analysis. 14…
Quantum Laws of Form
George Spencer-Brown's Laws of Form (1969) proposed the act of drawing a distinction as the primitive operation from which logic, mathematics, and physics can be constructed. This consolidation synthesizes known quantum correspondences, identifies 7 open problem categories, and proposes 4…
Geometric Unity of Computation
This work establishes a fundamental unification of computational state spaces, navigation systems, and holographic principles through rigorous symplectic and bundle-theoretic isomorphisms. We demonstrate that these apparently disparate domains share identical mathematical structures that can be…
Functorial Characterization of the Prime Numbers
This paper presents a complete theoretical characterization of the set of prime numbers, \(\mathcal{P}\), by demonstrating that it is the geometric support of a canonical object constructed via a three-stage functorial lift. This process begins in the logical stratum of Homotopy Type Theory (HoTT),…
Functorial Framework for Morphological Computing
This work introduces a formal framework for morphological computing that resolves the longstanding reconstruction paradox in Residue Number System (RNS) arithmetic by leveraging the physics of topological quantum matter. Conventional RNS-based accelerators are bottlenecked by the serial,…
Biological and Engineered Computational Systems
This work establishes the categorical equivalence between biological and engineered computational systems through an explicit mathematical formalism. The central thesis posits that these seemingly disparate domains implement equivalent computational architectures, an equivalence that manifests…
Monistic Reality
This paper presents a monistic reality framework that unifies fundamental physics, consciousness, and collective intelligence within a single mathematical architecture. This framework posits that reality is fundamentally a pre-geometric informational substrate structured as a Hamiltonian…
Category Error of the Ego
Epistemology represents the formal philosophical study of how humans acquire knowledge. This field investigates the fundamental differences between justified belief and mere subjective opinion. Applying this discipline to human identity reveals significant gaps in how we understand ourselves. The…
Stability Compiler
The modeling of fundamental entities as zero-dimensional points creates singularities across physics and network engineering. Current regularization methods like renormalization are procedural patches that fail to explain the origin of finite scales. This paper introduces the topological extension…
Topological Extension Hypothesis
Scientific modeling often relies on zero-dimensional point approximations, leading to mathematical singularities and empirical anomalies when resolution increases. Existing solutions like renormalization are procedural patches that fail to address the underlying ontological error across disparate…
Universal Computational Topos
This work presents a comprehensive mathematical framework for understanding computational self-reference through the unification of hierarchical computational theories and meta-computational approaches. We establish that the Computational Trinity Principle—distinguishing Object, Model, and Meta…
General Theory of Process
Modern science is currently fragmented into siloed disciplines—quantum mechanics, artificial intelligence, network science, and signal processing—each describing complex systems using disparate vocabularies. This manuscript proposes a general theory of process, asserting that these fields are not…
Process Ontology and Hydrodynamic Vacuum
Standard cosmological models face a foundational epistemic crisis at the Big Bang singularity, where the geometric description of General Relativity breaks down. This paper proposes a unified framework integrating Whiteheadian Process Philosophy with the hydrodynamics of a “stiff” superfluid vacuum.…
Universal Hamiltonian as Process Verb
The ontological status of quantum mechanics remains suspended between two mutually exclusive grammatical categories: the continuous, reversible “Verb” of unitary evolution and the discrete, irreversible “Noun” of the measured state. This paper reconciles this tension by mapping the physics of…
Chemical Category
This report posits that the major conceptual challenges of nineteenth-century chemistry—the crisis of atomic weights, the classification of an exponentially growing chemical space, and the paradoxes of isomerism and allotropy—collectively constituted a demand for a new form of structural and…
Computo Ergo Sum
This paper resolves Hilbert’s sixth problem by introducing the Self-Computing Universe Framework (𝒞), a meta-axiomatic system positing that the universe is not merely described by mathematics but is a self-executing mathematical structure. While its six foundational axioms are expressed with the…
Derivation of SM Gauge Couplings
Objective: To derive the relationship between the Standard Model’s gauge couplings and the Kappa information field (\(\kappa(x)\)) using a self-contained, category-theoretic approach based on the principle of scale-invariant information thermodynamics. This derivation establishes that effective…
Derivation of Einstein from Thermodynamics
Objective: To derive the Einstein field equations of general relativity from thermodynamic principles applied to local causal horizons, framed within the category-theoretic language of scale-invariant relationships. This derivation formalizes the physical argument of Jacobson (1995) and demonstrates…
Mathematical Structures Underlying Physical Laws
This work addresses the foundational schism in modern physics, which arises from the perceived incompatibility between classical continuous, deterministic reality and quantum discrete, probabilistic phenomena. The Post-Quantum Synthesis (PQS) framework resolves this by asserting that the universe is…
NEW LANGUAGE FOR MATH
This report presents a fundamental reframing of Causal Set Theory (CST) within the language of category theory, establishing a relational process ontology (RPO) for fundamental physics. By recasting causal sets as small, thin categories, this framework achieves a categorical reification of…
Meta-Analysis of Categorical Equivalence and Scale Invariance
The primary research question for this deliverable investigates the precise mathematical conditions under which the two documented categorical approaches to renormalization group (RG) flow—Sharpe’s model categories versus the Harm framework’s natural transformations—are either equivalent or…
Prime Numbers as Universal Optimization Primitives
The scholarly understanding of prime numbers is structured around four paradigms: quantum chaos, classical optimization, algorithmic information theory, and deterministic law. However, these paradigms are marked by unresolved tensions—dynamical, hierarchical, and methodological—that have hindered a…
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