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Valuation theory papers 73 papers
Valuation theory. Valuations, Ostrowski's theorem, completions Tagged papers: 73.
Singularity as an Epistemic Artifact: Finite Cyclic Cosmology without Actual Infinities
Cosmological singularities — states in which density, curvature and temperature diverge — mark the breakdown of predictive physics at the Big Bang and inside black holes, and they block any quantitative continuation of collapse or of the early universe. Whether these infinities are features of…
Ostrowski's Classification as a Selection Rule: A Falsifiable Program for the Adelic Shadow of the Standard Model Gauge Group
Ostrowski's theorem classifies the completions of the rational numbers $\mathbb{Q}$: exactly one Archimedean completion, $\mathbb{R}$, and one $p$-adic completion $\mathbb{Q}_p$ for each prime. Theoretical physics uses only the first. A conjecture in the adelic physics program holds that the…
Physical Irrelevance of Uncountability: Computable Reals Suffice for Continuum Physics
Continuum physics is formulated over the uncountable real line $\mathbb{R}$, yet every measurement, record, and computation a laboratory can produce is finite. This paper asks whether the uncountability of $\mathbb{R}$ — as distinct from its Archimedean completeness — has any physical consequence,…
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,…
Adelic Factorization of Quantum Dynamics Across All Completions of ℚ
We explore the conjecture that the conventional choice of the Archimedean completion ℝ as the underlying number field of quantum theory is a matter of convention rather than necessity. By invoking Ostrowski’s theorem, which partitions the completions of ℚ into the unique Archimedean place ∞ and the…
Critical Values on a Non-Uniform Number Line: Bifurcation Markers, Normalization Dependence, and the Conjectured Specialness of Physical Constants
We examine the conjecture that the real number line, as a model of physical state space, is misleadingly uniform: that certain parameter values act as topological phase boundaries rather than smooth increments, and that fundamental constants such as the fine-structure constant $\alpha \approx 1/137$…
Adelic Consistency Conditions Across Primes as a Candidate Selection Principle for Physical Constants: Formulation, Negative Control, and Toy Dynamics
The three constants problem—why the fine-structure constant takes its observed value, why charged-lepton masses are hierarchically spaced, and what fixes the absolute mass scale—has no accepted resolution within the Standard Model. We develop a research program, stated as a testable conjecture…
Master Calibration for Non-Overlapping Physical Frameworks: Composition, Closure, and the Limits of Partial Translation
Physics presents itself as convergent, yet systematic inventories of its foundations list dozens of unresolved bifurcations—general relativity versus quantum mechanics, continuum versus discrete description, the Heisenberg cut, the status of the observer. We propose that such schisms are symptoms of…
Completion-Invariance of Physical Ontology: The Rational Base Field in Quantum Foundations
We propose and formalize a principle of completion-invariance for physical ontology: since all recorded measurement outcomes are rational numbers, the archimedean completion $\mathbb{R}$ and the non-archimedean completions $\mathbb{Q}_p$ of the rational base field $\mathbb{Q}$ are equally…
Is There a $p$-Adic $\pi$? Haar-Measure Circle Constants over $\mathbb{Q}_p$ and an Adelic Product Formula
The real number $\pi \approx 3.14159$ is defined as the ratio of a circle's circumference to its diameter. Since Ostrowski's theorem embeds $\mathbb{Q}$ diagonally into every completion $\mathbb{R}$ and $\mathbb{Q}_p$, one may ask whether this ratio admits a well-defined analogue $\pi_p$ at each…
Measurement as Place-Crossing: Completion-Selective Adelic Projection
We propose a structural reframing of the quantum measurement problem in number-theoretic terms. By Ostrowski's theorem, the completions of $\mathbb{Q}$ consist of a single Archimedean place ($\mathbb{R}$) and countably many $p$-adic places ($\mathbb{Q}_p$); standard quantum mechanics is formulated…
The Zero-Point Phase as an Archimedean Factor of Weil-Index Reciprocity: A Formal Verification and Its Limits
The vacuum phase of the harmonic oscillator — the Archimedean Maslov (metaplectic) phase $e^{i\pi/4}$ accompanying the zero-point energy $\tfrac{1}{2}\hbar\omega$ — is conventionally treated as a purely real-place, analytic structure. We investigate the conjecture that this phase is instead one…
A Universal Adelic Normalization Principle: Ostrowski's Product Formula and Ono's Tamagawa Numbers as Specializations of One Local–Global Schema
Ostrowski's product formula $\prod_{v}|x|_{v}=1$ for a global field and Ono's formula $\tau(T)=|\mathrm{H}^{1}(k,\widehat{T})|/|\Sha(T)|$ for the Tamagawa number of an algebraic torus both assert that a family of individually free local contributions aggregates, under a global consistency condition,…
Archimedean Constants as Measure-Theoretic Artifacts of the Infinite Place: A Reconciled Derivation of the Local Status of $\pi$
Ostrowski's theorem partitions the non-trivial completions of $\mathbb{Q}$ into exactly one Archimedean place ($\mathbb{R}$) and countably many non-Archimedean places ($\mathbb{Q}_p$). Mainstream physics is formulated exclusively at the Archimedean place, and its most ubiquitous transcendental…
Ensemble Dependence of Critical Exponents at Quantum Error Correction Thresholds: Analytic Mechanisms and Statistical Consequences
Ensemble equivalence — the expectation that microcanonical, canonical, and grand-canonical descriptions of a system agree in the thermodynamic limit — is a cornerstone of statistical mechanics, yet it can fail for observables that probe exponentially rare events. Recent work on a simplified quantum…
The p-Adic Temperley–Lieb Parameter: Valuation-Theoretic Constraints and an Adelic Accuracy Framework for Quantum Predictions
The Temperley–Lieb (TL) algebra governs a broad class of exactly solvable quantum systems, from anyonic chains to braid-based teleportation circuits, through its scalar loop parameter δ. We investigate the proposal that a p-adic Temperley–Lieb parameter — a number-theoretic functional of δ — can…
Distinction, Number, and the Empirical Filter: The Pre-Arithmetic Research Framework
The QNFO arithmetic-physics program connects number-theoretic structure to statistical mechanics: prime-indexed modes with logarithmic energies yield partition functions that are exactly zeta objects, and hierarchy distance is a realization-independent ultrametric. This record states the…
The Distinction-Based Ultrametric: A Hierarchy Distance Without Primes, and the Statistical Test of Arithmetic Structure in Physical Spectra
Quantum mechanics is written in one completion of the rational numbers, the real line, though Ostrowski's theorem gives one completion for every prime. A line of work reads physical structure through those other places, but its empirical tests of prime-specific geometry in physical systems have…
Completeness Senses and the Levi-Civita Field: Ordered Non-Archimedean Number Systems Beyond Ostrowski's Classification
Ostrowski's theorem classifies every nontrivial rank-1 absolute value on the rationals; this paper distinguishes three senses of completeness and exhibits the Levi-Civita field as a Cauchy-complete, ordered, real-closed, non-Archimedean counterexample.
Five Objections, One Standard: An Evidence-Graded Adjudication of a Critique of Post-Quantum Synthesis
A five-point red-team critique of the Post-Quantum Synthesis (PQS) research program was examined point by point against the primary sources it targets and the external literature it invokes. Each objection was graded against a symmetric evidence standard: the objection's own factual premises were…
Invariant Structural Value: Fundamental Constants and Formulas as Invariant Relations
A measured physical quantity is a number that survives all arbitrary choices. The unit system, the coordinate system, the gauge choice, the Hilbert-space basis, and the energy scale are all choices; what is invariant under them is what is physical. This essay develops the structuralist reading that…
Prime Valuation Depth: Multiplication as Branching, the Calculus of Indications, and the Structural No-Cloning Reading
This paper develops a structural reading of the p-adic valuation as a measure of depth along a prime branch of the integer factorization tree, rather than a measure of size. The reading, grounded in Ostrowski's classification of absolute values on the rationals, is proposed as a bridge between the…
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…
The Consilience Framework: From Valuation Theory to the Void — A Cross-Domain Synthesis
Synthesis of valuation theory + foundational hierarchy (void -> distinction -> ZFC -> valuation) with Universal Consilience Prompt and autonomous 4-phase LLM research workflow.
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…
Killing the Framework: Seven Criteria for the Falsification of Adelic Physics
A framework that accommodates any observation has zero empirical content. This paper proposes seven kill criteria for the adelic framework, audits each for escape hatches, grades it against the incumbents (GR composite C, SM C, LCDM B, string D), tests all against existing data, and applies the…
Frequency as Valuation Theory: The Rational Ratio at the Heart of Physical Law
Frequency in dimensionless Planck units is a rational ratio a/b in Q. The Compton frequency of a fundamental particle is its invariant identity — its prime spectrum.
ODR Thesis: The Compton Count as the Only Primitive
ODR framework: Compton count as the only primitive physical invariant.
ODR Thesis: The Compton Count as the Only Primitive — A Five-Question Synthesis
The Ontological Distribution of Reality (ODR) framework identifies a particle with its Compton count \(N_C = m/m_P \in \mathbb{Q}^+\), a dimensionless rational number whose prime factorization assigns it a fixed vertex on the Bruhat–Tits tree. This paper synthesises five research questions into a…
The Ostrowski Dimensionless Reformulation: A Systematic Compilation of Fundamental Physics Equations in Planck Units
Systematic compilation and dimensionless reformulation of 53 fundamental physics equations across ten disciplines, motivated by the Ostrowski Dimensionless Mandate. Dimensional formulations implicitly privilege the Archimedean completion of the rationals per Ostrowski theorem (1916). All quantities…
When Will Non-Archimedean Geometry Displace the Real Numbers? A Structured Assessment of the Adelic Substrate Thesis
The real numbers are the default continuum for physical modeling — every quantum state lives in a Hilbert space over $\mathbb{C}$, every neural network optimizes over $\mathbb{R}^n$, every error-correcting code is designed with Euclidean distance. Yet Ostrowski's theorem (1916) demonstrates that the…
QNFO Consilient Synthesis v2.0: The Shared Adelic Kernel
QNFO Consilient Synthesis v2.0: the shared adelic kernel (valuation → tree boundary → Ostrowski → adeles). Applies all 7 corrections (C1–C7) from the 2026-07-25 red-team audit, restructured kernel-first with honest kernel-membership table and mandatory symmetry sections.
Counterfactual Physics: What If p-Adic Methods Had Won?
The p-adic approach to physics was nearly mainstream. Vladimirov and Volovich developed p-adic quantum mechanics in the 1980s. Freund and Witten connected p-adic string amplitudes to the adelic product formula in 1987. Dragovich extended p-adic methods to cosmology and the Standard Model. Yet by…
Adelic Entropic Numbers: When the Adelic Information Vector Becomes the Entropic Number
Two papers, written independently on the same day, converge on the same structure from opposite directions. Adelic Shannon Theory [1] generalises Shannon information theory to the adele ring $\mathbb{A}_{\mathbb{Q}}$, defining the *adelic information vector…
The History and Future of Measurement Stratigraphy, Number Theory, and Valuation Theory
[We propose a unified framework — the Stratigraphy of Measurement — that reinterprets the history of number systems as a sequence of expanding distinction operations. Each era adds a new enclosure: marking ($\\mathbb{N}$), comparing ratios ($\\mathbb{Q}$), convergent sequences ($\\mathbb{R}$_comp),…
The Continuum Critique Trilogy
Three papers, one thesis: the real-number continuum is a map, not the territory — and many "unsolved problems" in mathematics and physics are artifacts of representational choices rather than genuine ignorance. This monograph unifies three companion papers into a single narrative. Paper I (The…
Depth, Breadth, and Valuation: A Unified Ontology of the Physical Continuum
Three-axis framework: Depth (computable reals), Breadth (non-computable reals — physically vacuous), Valuation (p-adic completions as information carriers). Falsifiable predications for QEC, gauge structure.
The ℚ-vs-ℝ Question: Why Physical Law Requires Only the Rational Numbers
The QNFO physics programme — spanning the Adelic Physics Program (7 papers), the Ostrowski Programme (30+ artifacts), Hensel Codes (3 versions), and the ultrametric foundation — rests on an assumption that has never been systematically defended: that the rational numbers (ℚ) are the physically…
The Harmonic Paradigm Under Ostrowski’s Theorem: A p-Adic/Adélic Re-Evaluation with Helical Compton Vortex Synthesis
Systematic Ostrowski-completion-theoretic audit of the Harmonic Paradigm V1.0-V4.0. Five findings: (1) Beta-function mechanism is infinity-place-specific -- Missarov (1989) confirms different universality class. (2) Q_p not ordered -- eliminates S-matrix, Noether, measurement. (3) pi IS adelic --…
Adelic Constraints on Quantum Field Theory—Phase 2 Synthesis
<p>Phase 2 subjected the Phase 1 findings to deeper epistemic scrutiny, guided by the “constants critique”: that no physical quantity can be natively transcendental, and that true physical observables must be rational cross-ratios. This critique proved correct. The Phase 1…
HM-LWE v3.0
Hidden-modulus LWE: novel hardness assumption
Measure-Theoretic Artifacts of the Archimedean Place — v2.0: The Completion Problem, the Langlands Connection, and the Adelic Restructuring of Fundamental Physics
Ostrowski's theorem partitions all non-trivial completions of $\mathbb{Q}$ into exactly one Archimedean completion ($\mathbb{R}$, the $\infty$-place) and infinitely many non-Archimedean completions ($\mathbb{Q}_p$ for each prime $p$). These topologies are mutually singular — no sequence converges…
Consilience Between Physics and Number Theory: Convergent Theses from Ostrowski's Theorem to Adelic Quantum Mechanics
We synthesize convergent evidence from physics (quantum mechanics, quantum field theory, zitterbewegung, quantum error correction, the Standard Model mass spectrum) and number theory (Ostrowski's theorem, Tate's thesis, the Riemann zeta function, Bruhat-Tits buildings, the adele ring) into a unified…
Measure-Theoretic Artifacts of the Archimedean Place: A Complete Taxonomy and the Adelic Restructuring of Fundamental Science
Ostrowski theorem partitions all non-trivial completions of Q into exactly one Archimedean completion and infinitely many non-Archimedean completions. Physics operates exclusively at the Archimedean place. Catalogues measure-theoretic artifacts across 5 categories with 16 completion failures…
α-π-Helix: Geometric Unification of Fundamental Constants
Geometric unification of fundamental constants using α (fine-structure) and π (geometric) helix structures, classifying Standard Model particles as topological vortex/helicon states. v2.1 — Phase 10 complete with 6-domain synthesis including experimental roadmap, cosmology, mathematical rigor, QFT…
The Adelic Completion of the Harmonic Paradigm: A Five-Pillar Red-Team Assessment
The Harmonic Paradigm (V1.0-V4.0, 2026) proposed the harmonic oscillator as the universal IR attractor of quantum theory, with an 8-rung ladder spanning from transmon anharmonicity to quantum gravity. Its bibliography invoked Ostrowski's theorem and p-adic structures, yet its core mechanisms were…
Non-Anthropocentric Natural Units: From the Bekenstein Bound to Ostrowski's Theorem
We strip away all anthropocentric constructions from the Bekenstein-Hawking entropy bound. Area as dimensionless geometric invariant; exponential bound defined without logarithmic base; integer Hilbert space dimension as universal invariant. Ostrowski's theorem reveals the Archimedean completion is…
The Adelic Constraints Project — A Complete Account
<p></p> <p>This document is a complete, self-contained account of a research project conducted in May 2026. The project asked whether a specific piece of pure mathematics — the fact that the rational numbers can be "completed" in multiple incompatible ways, and that a single identity links all…
Adelic Quantum Error Correction: Intrinsic Qubit Protection from Ostrowski
We propose a quantum error-correcting code whose protection derives from Ostrowski's theorem. The code leverages the classification of non-trivial absolute values on Q to achieve intrinsic fault tolerance without active syndrome extraction.
The p-Adic Temperley-Lieb Parameter: Cyclotomic Units, Markov Traces, and the p-Adic Jones Polynomial
Establishes the Temperley-Lieb parameter as p-adic cyclotomic units and constructs the p-adic Jones polynomial.
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