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Results for “p-adic” 19 papers
Valuation theory. Valuations, Ostrowski's theorem, completions Tagged papers: 79.
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…
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: the Archimedean absolute value and, for each prime p, a p-adic absolute value. Their completions are the real numbers and the p-adic fields. This classification is often read as closing the question of what a…
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…
Frequency as Valuation Theory: The Rational Ratio at the Heart of Physical Law
Physics expresses frequency as a real number — 5.783 × 10^14 Hz, 1420.405751768 MHz. Yet in dimensionless Planck units (h = c = G = k_B = 1), frequency reduces to a ratio of two integers: count-of-oscillations per count-of-Planck-times. This is a rational number q = a/b in Q — not an Archimedean…
The Ostrowski Dimensionless Reformulation: A Systematic Compilation of Fundamental Physics Equations in Planck Units
We present the definitive edition of the Ostrowski Dimensionless Reformulation (ODR) — a systematic compilation, dimensionless reformulation, and detailed derivation of 53 fundamental physics equations across ten disciplines. The central claim is refined from the original v2.0 compilation:…
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…
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),…
Depth, Breadth, and Valuation: A Unified Ontology of the Physical Continuum
The standard mathematical ontology of the continuum — the real numbers the real numbers with their uncountable cardinality and Dedekind-complete order — conflates three distinct properties with different physical status. We decompose the continuum into three axes: depth (Archimedean completeness),…
The Harmonic Paradigm Under Ostrowski’s Theorem: A p-Adic/Adélic Re-Evaluation with Helical Compton Vortex Synthesis
The Harmonic Paradigm (V1.0–V4.0) proposed the harmonic oscillator as the universal IR attractor of quantum theory, spanning an 8-rung ladder from transmon anharmonicity to quantum gravity. Its bibliography cited Vladimirov-Volovich-Zelenov p-adic QFT (1994) and Ostrowski's theorem (1918), invoking…
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 — 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…
Measure-Theoretic Artifacts of the Archimedean Place: A Complete Taxonomy and the Adelic Restructuring of Fundamental Science
Ostrowski's theorem partitions all non-trivial completions of ℚ into exactly one Archimedean completion (ℝ, the ∞-place) and infinitely many non-Archimedean completions (ℚₚ for each prime p). These topologies are mutually singular — no sequence converges simultaneously in ℝ and any ℚₚ. Physics, as…
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
The Bekenstein-Hawking entropy bound $S \leq A/4\ell_P^2$ is a pillar of quantum gravity, typically expressed in Planck units with entropy measured in bits or nats. We ask: what remains of this bound when ALL anthropocentric constructions are stripped away — unit choices, logarithmic bases, and even…
The p-Adic Temperley-Lieb Parameter: Cyclotomic Units, Markov Traces, and the p-Adic Jones Polynomial
In Phase 1, we defined the p-adic braid group $B_n(\mathbb{Q}_p)$ on the Bruhat-Tits tree $\mathcal{T}_p$ and showed it satisfies the standard braid relations. Here we complete the connection to the Temperley-Lieb algebra and the Jones polynomial at non-archimedean places. The TL algebra parameter…
Adelic Quantum Error Correction: Intrinsic Qubit Protection from Ostrowski
We formalize an adelic approach to quantum error correction (QEC) based on Ostrowski's theorem: the only non-trivial completions of the rational numbers are the Archimedean (real) completion $\mathbb{R}$ and the p-adic completions $\mathbb{Q}_p$, and these topologies are mutually singular. The…
Computational Toolkit for p-Adic Spacetime
The fundamental incompatibility between the continuous manifolds of general relativity and the discrete nature of quantum mechanics remains a central tension in theoretical physics. This manuscript develops and validates a computational toolkit designed to test theories of p-adic spacetime, which…
From Force to Fractal
The transition from force-based epistemic placeholders to a purely geometric ontic bedrock represents the defining trajectory of modern theoretical physics. By synthesizing Ontic Structural Realism with contemporary developments in Invariant Set Theory, we propose a unified framework where reality…
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