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612 papers from the QNFO program, filed on nine shelves by the reader each one is written for: energy per answer, error correction, physics, numbers and p-adics, logic and formalism, AI and knowledge, autonomous systems, assessments, and cross-cutting essays. Every page renders its mathematics and can be questioned in place. How a living paper works.
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Numbers & p-adics papers 14 papers
Mathematicians. Valuation theory and the p-adic and adelic programme, Bruhat-Tits trees and ultrametric geometry as mathematics, Ostrowski's theorem, numeral and radix systems, primes, arithmetic and algebraic structures, zeta and gamma functions.
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…
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…
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),…
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…
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…
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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