Open research library
Papers you can read, cite and question
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.
Looking for an answer rather than a paper? Ask the whole corpus: answers cite the papers they come from.
Numbers & p-adics papers 131 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.
Reflection Closure of Concurrent Line Sets: Exact Dihedral Emergence, Grid Frequency Laws, and the Finite/Infinite Dichotomy
Model builders in physics and machine learning must choose a symmetry group before fitting begins, and a wrong or hidden choice degrades predictions and obscures observables; the choice is normally an unchecked input. The specific gap addressed here is whether a minimal generative rule—draw a finite…
Ultrametric Quantum Computation for Direct Integer Factorization: A Resource Analysis with a Largely Negative Verdict
Factoring large integers underpins RSA, and the best-known polynomial-time quantum method, Shor's algorithm, extracts the period of modular exponentiation through a quantum phase-estimation detour that dominates circuit depth. A conjecture in the ultrametric research program holds that a quantum…
Joint Bruhat–Tits Tree Product as a Candidate Universal Structure for Standard Model Mass Ratios, Quantum Error-Correction Lattices, and Efimov Scaling: Formalization, Arithmetic Consequences, and a Partial Falsification
The Standard Model (SM) treats particle masses as free parameters, quantum-error-correction (QEC) lattice designs are chosen by heuristic optimization, and the Efimov discrete scaling factor lacks a structural explanation; a single mathematical object generating all three would change how each…
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…
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…
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…
The Silent Radix Is Not a Key: A Security Analysis of Unknown-Radix Encoding as a Cryptographic Primitive
Positional numerals do not internally specify their own base, and a recent informal proposal — Silent Radix Encryption (SRE) — conjectures that this "silent radix" can serve as a cryptographic secret: plaintext symbols are mapped to digits by a secret map and read as an integer in a secret radix…
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…
Non-Computable Reals as Projection Artifacts: A Reconciled Framework for Tree-to-Manifold Construction of the Continuum
We develop a formal framework for the conjecture that the continuum is constructed rather than primitive: natural numbers, rationals, and computable reals arise from a single distinction operator via enclosure operations, while the full real line emerges only through a lossy projection — the…
Factoring Hardness as a Bayesian Prior: A Decision-Theoretic Model of Post-Quantum Cryptography Migration Timing
The urgency of post-quantum cryptography (PQC) migration is usually justified by the anticipated capability of quantum computers to factor integers, yet that threat rests on an unproven complexity-theoretic conjecture: that integer factoring is not in BPP, the class of decision problems solvable in…
Frequencies of Events in the Actual World: A Reconciled Cross-Domain Framework with Explicit Arithmetic
Every empirical discipline counts events, but the practice of measuring event frequencies is fragmented across fields that rarely compare methods. This paper develops a cross-domain framework for event-frequency observation in the actual world, organized around three questions: what counts as an…
Flag Structures and the Flagging Act: A Cross-Domain Survey with Explicit Counting Derivations and a Posed Re-Entry Problem
The word "flag" names a family of mathematically precise objects — chains of nested subspaces (flag varieties), nested set systems with exchange axioms (flag matroids), asymptotic-inequality certificates (flag algebras), and directional curvature invariants of Finsler spaces — and, in ordinary…
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,…
Vertex-Anchored Geometry: Discrete Extremal Points as the Organizing Primitive of Non-Archimedean Space
Non-Archimedean (ultrametric) geometry, in which the strong triangle inequality $d(x,z) \le \max\{d(x,y),d(y,z)\}$ replaces the Euclidean one, lacks the smooth infinitesimal structure on which most physical and geometric intuition rests. We argue that in this setting the correct organizing primitive…
Optimal Radix $q$ under Resource Constraints: A Reconciled Derivation Framework for Ultrametric Hierarchies
Hierarchical (ultrametric) representations organize $N$ items in a $q$-ary tree of depth $n = \lceil \ln N / \ln q \rceil$, and the branching factor $q$ determines how storage, interconnection, and traversal resources scale. Prior ultrametric modeling has largely treated the radix as a fixed…
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…
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…
Numeral Systems as Evolving Cognitive Technologies: Error Propagation, Compactness, and the Non-Optimality of Positional Notation
Positional (Arabic-Indic) notation is conventionally treated as the endpoint of numeral-system evolution, but this paper argues that it is a historically contingent survivor whose dominance reflects fitness for particular task environments rather than global optimality. We formalize the tradeoff…
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…
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…
Spectral Decomposition of the Bruhat–Tits Tree as a Candidate Origin of Particle Mass Hierarchies: Exact Radial Spectrum, Finite Truncations, Semigroup Tests, and a Falsification Analysis
The Bruhat–Tits tree $\mathcal{T}_p$ of $\mathbb{Q}_p$ is the infinite $(p+1)$-regular tree that serves as the bulk of $p$-adic AdS/CFT. We test the conjecture that a natural Laplacian on $\mathcal{T}_p$ — or the prime-indexed arithmetic of the tree — encodes Standard Model mass ratios. We compute,…
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…
Emergent Universality from Symmetric Rules: Rule-110 Emulation by Alternating Trivial Cellular Automaton Rules
Elementary cellular automaton (ECA) rule 110 is Turing complete, yet its local rule is chiral: it treats its left and right neighbours differently, and this handedness underlies the glider dynamics that carry its computation. A recent construction shows that rule 110 can nevertheless be emulated by…
The Number Builder: Enclosure Sequences in the Laws of Form Calculus as a Constructive Route to Computable Reals
We propose and partially develop the Number Builder conjecture: every computable real can be generated by a finite specification of a re-entry structure in Spencer-Brown's Laws of Form (LoF) calculus of distinctions, in which each re-entry step appends a nested rational enclosure whose interval…
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…
Optimal Radix q under Resource Constraints: A Reconciled Derivation Framework for Ultrametric Hierarchies
Hierarchical (ultrametric) representations organize N items in a q-ary tree of depth n = ⌈ln N / ln q⌉, and the choice of branching factor q determines how storage, interconnection, and traversal resources scale. Prior ultrametric modeling has treated the radix as a fixed structural parameter…
Geometric Ratios as Generators of Hierarchical Structure: An Arithmetic Audit of π, φ, and e
A recurring conjecture in the ultrametric-physics program holds that hierarchical—i.e., ultrametric—structure "emerges" when a scale ratio q is set equal to a fundamental geometric constant: π, the golden ratio φ = (1+√5)/2, or Euler's number e. This paper converts that conjecture into falsifiable…
Why Vertices, Not Points? Vertex-Anchored Braiding, Stabilizer Arithmetic, and Displacement Gaps on Bruhat–Tits Buildings
Bruhat–Tits buildings are the canonical combinatorial-geometric objects attached to reductive groups over non-Archimedean local fields, and the recent QNFO constructions of p-adic braid groups [12], p-adic anyon models [13], and a p-adic Temperley–Lieb parameter [14] anchor their discrete braiding…
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 Corrected Primon-Gas Dictionary: Zeta Partition Functions and the Discipline of Arithmetic Interpretations
The primon gas has connected quantum statistical mechanics with multiplicative number theory since the early 1990s and remains in active use. This paper consolidates the correspondence into a single audited reference: a corrected, verified dictionary; a five-level ladder separating mathematical…
Discriminating the Arithmetic Cut: Matched-Level-Density Null Models for the Primon-Gas Specific Heat and Spectral Correlations
The arithmetic-statistics program reads the two exchange statistics of identical particles as two multiplicity rules on one integer lattice, and its published observable is the specific heat of the primon gas: a computable deviation from the smooth ideal gas at every sampled temperature. 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…
Adelic Quantum Arithmetic: Particles as Prime Factors
Standard quantum mechanics is written in the Archimedean completion of the rational numbers: Hilbert space over the complex numbers uses only the real place, one among all completions classified by Ostrowski's theorem. The primes are the other places, and their structure is multiplicative — unique…
Quantum Statistics from the Adelic Product Formula: The Squarefree Origin of the Fermi-Dirac/Bose-Einstein Distinction
The Bose-Einstein and Fermi-Dirac occupation distributions are read as the maximum-entropy solutions of one lattice with two multiplicity rules: the unrestricted integer lattice (Dirichlet series the Riemann zeta function) and the squarefree restriction (a ratio of two zeta values). The paper…
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…
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.
The Ultrametric Program: One Structural Object Across Seven Research Domains, and Its Falsifiable Tests
Seven research domains are claimed to be seven vocabularies for one structural object: nested hierarchical partition logic defined by the ultrametric inequality; the p-adic/adelic arithmetic is one realization, the hierarchy is the invariant. Three falsifiable hypotheses: H1 compression prior, H2…
The Self-Referential Scalar Family: e, the Half-Turn, and the Unification of Information Theory, Statistics, Thermodynamics, and QND Measurement in the Adelic Picture
The p-adic maximum-entropy distribution of Adelic Shannon Theory is exactly the Bose-Einstein occupation distribution at fugacity 1/p, i.e. at inverse temperature ln p at the p-adic place; the squarefree restriction of the integers is its Fermi-Dirac counterpart, with occupation probability 1/(p+1)…
One Table, Two Regimes: Standard-Model Particles and Condensed-Matter Excitations as Patterns on the Bruhat-Tits Tree
Physics maintains two particle catalogs that never meet in one framework: the Standard Model's finite table of elementary fermions and bosons, and the open-ended zoo of quasiparticles and anyons discovered in condensed matter. This paper assembles both catalogs into a single pattern table on the…
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…
Non-Archimedean Projective Perspective: The Monna Map as a Visual Rendering Interface
Things farther away appear smaller, and two-point linear perspective is taken to approximate the eye. That phenomenology is Archimedean geometry. This paper asks what changes if the metric of physical space were instead non-Archimedean and ultrametric. Four moves: (1) the Archimedean anatomy of…
The Exchange Phase as a Logical Scalar: R = e^(2 pi i s) from the Re-Entrant Calculus
The boson/fermion dichotomy is conventionally presented as a primitive classification of nature. Recent work established that the underlying invariant is the relation between the exchange phase of identical particles and their topological spin, R = e^{2πis}, with the dichotomy as its shadow in three…
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 Idempotent Core: Quantity as Broken Distinction and the Hidden Assumptions of Arithmetic and Algebra
A critique-first paper on arithmetic's hidden premise: 1+1=2 presupposes two distinguishable marks; idempotence (AA=A) is the primitive law; quantity is created distinction, not repetition. Reconstruction program (Z from ! modality) labeled conjecture with falsifiability conditions.
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…
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…
Boundary Ultrametricity: The Tree vs. ∂∞𝒯 Distinction, Applied to the ZBW Transition Graph
ACRP-02 formal note. Boundary Gromov metric on dT_p is ultrametric and recovers |.|_p; tree vertex metric is NOT ultrametric. ZBW P1 Claim C2 corrected: 0-hyperbolic core, not ultrametric core (147/500 violations per paper data).
New papers by email
One short weekly digest: titles and links. No tracking; unsubscribe any time.