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609 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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p-adic papers 53 papers
p-adic. p-adic numbers and valuation-driven models Tagged papers: 53.
Time as Epistemic Emergence on a $p$-Adic Bruhat–Tits Tree: Static Constraint, Counting Laws, and the Observer-Generated Arrow
The problem of time in quantum gravity blocks any straightforward union of general relativity, in which the time coordinate is dynamical, with quantum mechanics, in which evolution requires an external background parameter; until it is resolved, a non-perturbative quantum theory of gravity cannot…
Time as Bruhat-Tits Tree Depth: Discrete $p$-Adic Temporal Ordering and Tree-Geodesic Causality
We develop a conjectural but internally precise model in which physical time is not a continuous real parameter but a discrete depth coordinate on a Bruhat-Tits tree: an event is a vertex of a rooted $(p+1)$-regular tree, and its time is the integer depth $d\in\mathbb{N}$, identified conjecturally…
Ultrametric $p$-Adic Noise Structure and Heisenberg-Limited Quantum Metrology: Error Resilience by Geometry, Not Overhead
We formalize a conjecture from the ultrametric research program: if decoherence on a qubit register is modeled as a $p$-adic (ultrametric) hierarchical stochastic process, the strong triangle inequality $d(x,z) \leq \max\{d(x,y), d(y,z)\}$ forces the family of error events on nested clusters to be…
Adelic Zitterbewegung: p-Adic Regularization of Dirac Trembling Motion and the Product-Formula Cancellation at Macroscopic Scales
Zitterbewegung (ZBW) — the trembling motion of Dirac wave packets at the Compton scale — is conventionally treated as a purely Archimedean interference effect between the positive- and negative-energy branches of the free Dirac equation. Missing from the standard account is any mechanism by which…
Zitterbewegung as a $p$-Adic Ultrametric Observable: A $\mathbb{Z}_2$ Invariant on the Bruhat–Tits Tree, Its Readout, and Its Non-Additivity Signature
Zitterbewegung (ZBW) — the Compton-frequency trembling of Dirac wave packets — is conventionally modeled as an Archimedean interference oscillation between positive- and negative-energy branches. We develop a conjectural alternative in which the ZBW readout is ultrametric: the observable lives on a…
Zitterbewegung as an Archimedean–2-adic Adelic Mixing Signature: Spectral-Gap Formalization, Numerical Anchors, and a Falsifiable Simulation Program
Zitterbewegung (ZBW) — the trembling motion of a Dirac wave packet at angular frequency $\omega = 2mc^{2}/\hbar$ — is conventionally treated as an interference artifact of the Dirac Hamiltonian. We propose and partially formalize a stronger claim: ZBW is the observable trace of mixing between the…
Zitterbewegung as Archimedean–2-adic Channel Mixing: A Conjectural Adelic Framework and Its Testable Consequences
Zitterbewegung (ZBW) — the trembling motion of Dirac wave packets at the Compton scale, first identified by Breit and Schrödinger — is conventionally explained as interference between positive- and negative-energy branches on the real line. We develop a conjecture, motivated by the adelic physics…
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…
Ultrametric p-Adic Arithmetic as Energy-Proportional Precision Computing: An Energy Model for Digit-Terminable Multiply-Accumulate
Fixed-width Archimedean floating-point arithmetic pays a constant energy cost per multiply-accumulate (MAC) regardless of the precision actually required by the computation. We conjecture and formalize an alternative: p-adic (ultrametric) number representations, whose digits are hierarchically…
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…
Post-Positional Numeracy: Finite-Adele Encoding and Product-Formula-Verified Exact Rational Arithmetic
Exact computation on the rationals today inhabits a single completion. Digital arithmetic runs in the real numbers and accepts rounding; exact alternatives run in a single $p$-adic completion through Hensel codes and reconstruct through the Chinese remainder theorem with Farey bounds. This paper…
Finite-Distinction Quantum Mechanics: Unitary Evolution and Superposition as the Large-Distinction Limit of Stochastic Thermodynamics
A finite-entropy world is a finite-distinction world: uncountable precision is unphysical, while computable depth and p-adic valuation remain physically real. The state-space geometry of finite distinctions is combinatorial and ultrametric; quantum mechanics read as thermodynamics is a stochastic…
The Trapped-Ion Ultrametric Testbed: A Falsifiability Register for Testing p-Adic Structure in Quantum Dynamics
Sixteen published records from a single research program, spanning December 2025 to August 2026, are here organized into one testable claim: trapped-ion quantum simulators are the first near-term platform on which ultrametric (p-adic) structure in quantum dynamics can be accepted or rejected by…
Implications for Computing and Quantum Error Correction
The anchor paper introduced the branch-depth reading: the p-adic valuation $v_p(n)$ is a measure of depth along a prime branch of the integer factorization tree (not a measure of size), and because the tensor product multiplies Hilbert-space dimensions, $v_2(\dim H) = n$ counts tensor-branch depth…
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…
Radix-Agnostic Detection of Discrete Scale Invariance: A Certified Three-Stage Protocol and a Null Result from Planck 2018
Discrete scale invariance (DSI) — invariance under a discrete set of rescalings $x \to \lambda^n x$ — predicts log-periodic oscillations whose angular frequency in the logarithmic coordinate $u = \ln x$ is $\omega_0 = 2\pi/\ln\lambda$. When the scaling radix $\lambda$ is unknown, a radix-locked…
Searching for P-Adic Log-Periodic Signatures in the Cosmic Microwave Background Bispectrum: Upper Bounds from Planck 2018
We search for p-adic (ultrametric) log-periodic signatures in the cosmic microwave background bispectrum. Radix-locked oscillations with angular frequency omega_p = 2*pi/ln(p), p in {2,3,5,7}. Upper bounds from Planck 2018: epsilon_p < 2.5 at 95% CL, no detection; within single-modulation model this…
Zitterbewegung as a p-Adic Observable: Ultrametric Readout and Intrinsic Topological Protection for Majorana Qubits
The Zitterbewegung (ZBW) of a Dirac fermion — the rapid oscillatory motion at the Compton frequency arising from interference between positive- and negative-energy states — has resisted direct experimental observation for nearly a century. The standard interpretation attributes this to the Compton…
Passive Error Resilience Through Ultrametric Geometry: A Proposal for p-Adic Quantum Metrology
Active quantum error correction is the dominant paradigm for achieving fault-tolerant quantum computation, but its thermodynamic cost at utility scale remains unaddressed. Each round of syndrome extraction, decoding, and feedback erases information — and under Landauer's principle, that erasure…
Ultrametric Consilience Atlas: Cross-Domain Applications of p-Adic Mathematical Structure
A single mathematical structure — the p-adic ultrametric topology — appears in apparently unrelated domains: quantum error correction (stabilizer codes), protein folding (energy landscapes), neural coding (population activity clustering), financial markets (hierarchical correlation), climate…
The QWAV Decade: Enterprise p-Adic Computing 2025-2035
Computing efficiency — measured by Joules Per Computational Unit (JPCUB †) — will be the dominant competitive axis in enterprise computing by 2035. This document forecasts the JPCUB ecosystem's evolution across three eras. Era 1: Benchmark (2025-2028) — JPCUB is validated against public hardware…
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 Rate-Distortion Theory: The p-Adic Distortion Measure and the Adelic Information-Rate Function
The Adelic Shannon Theory [1] generalised channel capacity to the adele ring $\mathbb{A}_{\mathbb{Q}}$. The companion bridge paper [2] established that the entropic number $(x, S)$ from Measurement Stratigraphy [3] is precisely the adelic information vector…
Adelic Shannon Theory: From Problem Statement to Constructive Foundations
Standard Shannon theory — channel capacity, entropy, source coding, and the Gaussian noise model — is archimedean: it measures information in bits over $\mathbb{R}$. The Adelic Physics Programme's central claim — $\mathbb{Q}$, not $\mathbb{R}$, is the physically accessible base field — suggests that…
FACTORING, Adelic Complexity, and the Silent-Radix Principle
The silent radix -- the observation that no positional numeral internally specifies its own base -- was previously explored as a cryptographic primitive through Silent Radix Encryption (SRE), a key encapsulation mechanism whose security rests on the hardness of recovering a secret base from public…
The Adelic Physics Program: Epistemological Foundations and Communications Framework
The QNFO Adelic Physics Program proposes that the physically accessible base field of physics is Q (the rational numbers), not R (the real numbers). Ostrowskis theorem demands all p-adic completions of Q be physically meaningful. This paper provides the epistemological and pedagogical…
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.
P-adic Spin, Information, and Ultrametric Internal Quantum Numbers
P-adic completions of Q encode discrete quantum numbers structurally via Bruhat-Tits geometry: p=2 gives spin, p=3 gives color charge. The Z2 invariant distinguishes Dirac from Majorana.
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 --…
The Morita p-Adic Gamma Function: Computation, Adelic Structure, and the Factoring Question
<p>The Morita p-adic Gamma function $\Gamma_p(n) = (-1)^n \prod_{k=1,\,p \nmid k}^{n-1} k$ is a p-adic analytic object that interpolates factorials while explicitly skipping multiples of $p$. This document provides a self-contained exposition: the definition and functional equation, a computational…
Adelic Quantum Error Correction: A Synthesis Across All Primes v1.0.0
The $p$-adic quantum error correction program has developed three independent but connected strands: a mathematical theory of $p$-adic stabilizer codes (Phase A), an experimental protocol to test for ultrametric error structure (Phase B), and a hardware co-design proposal for tree-structured qubit…
Infomatics: An Axiomatic Framework for Information as Fundamental Substrate
Axiomatic framework treating information as fundamental substrate. 4 axioms, 4 theorems. QED from information action. Fine-structure constant alpha=1/137. p-adic correction c1=5 matches 0.0014%. Falsifiable prediction: small-integer p-adic expansion of 1/alpha.…
Zitterbewegung as the Physical Realization of p-Adic Anyon Braiding
Shows that the ZBW-induced oscillatory trajectory of a Dirac fermion in a Bruhat-Tits building generates the braid group B_n, establishing ZBW as a physical mechanism for p-adic anyon braiding in topological quantum computation.
Bruhat-Tits Readout Protocol: Measuring the ZBW Z2 Invariant in Majorana Systems
Proposes three measurement protocols for the p-adic Z_2 invariant: Direct current-correlator measurement, Hensel-lifting amplification via the Bruhat-Tits building, and Gromov delta-hyperbolicity comparison of ZBW trajectories in Dirac/Majorana systems.
Zitterbewegung as a p-Adic Observable: Ultrametric Readout and Intrinsic Topological Protection
Formulates Zitterbewegung as a p-adic topological observable. Establishes that the ZBW current J_mu_ZBW carries a Z_2 invariant encoding the Dirac/Majorana distinction, and proposes ultrametric readout protocols via Bruhat-Tits buildings.
The Adelic Physics Program: A Grand Synthesis
This paper consolidates the 6-paper adelic physics chain into a unified research program, from Zitterbewegung as a p-adic observable through Majorana correlators, Bruhat-Tits readout protocols, p-adic anyon braiding, and Ostrowski-based quantum error correction.
p-Adic Anyon Fusion and Braiding: Quantum Groups at Roots of Unity, Verma Modules, and Ultrametric Anyon Models
Constructs p-adic anyon models via restricted quantum groups at roots of unity with fusion and braiding.
p-Adic Braid Groups on Bruhat-Tits Buildings
Constructs p-adic braid groups on Bruhat-Tits buildings, establishing the geometry of discrete braiding.
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.
Computational Simulation Approach to Non-Archimedean Quantum Architectures
The prevailing quantum computing paradigm is fundamentally constrained by the continuous analog fragility inherent to Archimedean spaces, resulting in uncontrolled linear error accumulation that demands massive active correction overheads. To address this structural bottleneck, we propose a…
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…
Ratio-Based Adelic Physics
The tension between continuous Archimedean spacetime paradigms and discrete, non-Archimedean topological realities constitutes the central mathematical bottleneck in modern complexity science, hindering the unification of quantum mechanics, spin glass phase dynamics, and cognitive psychophysics.…
Unity of Ultrametric Physics
This document develops a unified framework for physics grounded in ultrametric geometry. No prior mathematical knowledge is assumed. Every concept — from the notion of a set to the adelic field equations — is defined in place, every operation explained, and key properties proved. The text can be…
ULTRAMETRIC PHYSICS
This research plan presents a framework for ultrametric physics based on fundamental scaling ratios rather than anthropocentric number representations. The core insight: physical reality is described by dimensionless scaling ratios (π, φ, e, α, etc.) that operate on hierarchical discrete structures,…
Ultrametric Physics Research Plan
This research plan presents a framework for ultrametric physics based on fundamental scaling ratios rather than anthropocentric number representations. The core insight: physical reality is described by dimensionless scaling ratios (π, φ, e, α, etc.) that operate on hierarchical discrete structures,…
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…
Spectral Analysis of Anomalous Diffusion on p-Adic Fractals
The reconcilement of continuous Riemannian geometry with the discrete arithmetic of the integers remains one of the most profound challenges in mathematical physics, particularly regarding the spectral interpretation of the Riemann zeros (Berry & Keating, 1999). While standard diffusion on…
Functional Decoupling Framework for p-adic Hamiltonian Models of Granular Field Interactions
The reconciliation of continuous temporal evolution with discrete physical granularity remains a central challenge in theoretical physics, requiring novel mathematical approaches to model localized action within unbroken fields. This study introduces a functional decoupling framework that divides…
Unified Theory of Non-Archimedean Ontology
URL: https://papers.qnfo.org/papers/unified-theory-of-non-archimedean-ontology/ Date: 2026-03-15 Version: 1.0 The foundational premise of this framework is that the universe is constructed not from matter or energy, but from pure information. In this view, physical objects and phenomena are emergent…
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