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Numbers & p-adics papers 143 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.
Positional Notation as an Ultrametric Tree of Cyclic Time
Positional numeral systems are traditionally interpreted as linear encodings of magnitude. This paper shows that such systems naturally form rooted ultrametric trees whose levels correspond to nested cyclic temporal units (seconds → minutes → hours → days). The $b$‑adic valuation…
Real Numbers as Coarse‑Grained Projections of Re‑Entry‑Generated p‑Adic Trees
The foundations of mathematics rely on set‑theoretic axioms that are not directly computable, limiting the verification of real‑analysis results in constructive settings. A concrete gap is the absence of a minimal, distinction‑based formalism whose normal forms are provably isomorphic to the ring of…
The Bruhat–Tits Tree as a Unifying Computational Primitive: Confluence, Quantum Walks, and Tree‑Distance Attention
Hierarchical data, quantum‑amplitude dynamics, and proof‑normalisation each rely on distinct geometric primitives that incur sample‑inefficiency, slow convergence, or non‑confluent reductions. The $p$‑adic Bruhat–Tits tree provides a single ultrametric structure that can replace Euclidean…
Adelic Ruler‑Independence and Rationality of Standard Model Parameters
The Standard Model (SM) contains dozens of dimensionless parameters whose measured values appear finely tuned, yet the theory provides no internal mechanism to select them. A gap exists in linking these parameters to a universal mathematical principle. We adopt the hypothesis that physical law must…
Primality as a Non‑Archimedean Structure Projected onto the Real Line
Primes are defined as discrete divisibility objects in the integers, yet many analytic phenomena—Riemann zeros, cryptographic protocols, and quantum‑information patterns—depend on a continuous view of number theory. The gap lies in the lack of a real‑valued functional that captures $p$‑adic…
Discrete Bruhat–Tits Tree Analog of Kapustin–Witten S‑Duality
The Kapustin–Witten construction identifies S‑duality of four‑dimensional $\mathcal{N}=4$ supersymmetric Yang–Mills theory with the geometric Langlands correspondence on a Riemann surface $C$ [2]. This framework is limited to the Archimedean place, leaving the non‑Archimedean (p‑adic) sector…
L‑Function Zeros as Gauge‑Invariant Spectra Across Archimedean and $p$‑adic Geometries
The quest for a unified spectral invariant that simultaneously governs harmonic resonance on the hyperbolic plane and on $p$‑adic Bruhat–Tits trees remains open. Existing analogies between Maass‑form eigenvalues of the Laplacian on the modular surface and spectra of discrete Laplacians on $p$‑adic…
Anthropocentric Invariants in Measurement: A Formal Taxonomy, Cross‑Ratio Analysis, and Quantitative Bias Assessment
Conventional measurement units embed human‑specific cognitive choices—decimal numeracy, body‑scale references, and water‑based mass standards—creating anthropocentric invariants that obscure observer‑independent physics. Existing work has not formalized how such conventions transform under changes…
Ultrametric Quantum Computing as the Natural Realization of Adelic Factoring
Integer factorisation underpins modern public‑key cryptography, yet classical algorithms such as the General Number Field Sieve (GNFS) scale sub‑exponentially and quantum Shor’s algorithm requires fault‑tolerant hardware that remains unavailable. A gap therefore exists between the theoretical…
Reformulating Quantum Mechanics on Bruhat–Tits Trees: A Deterministic Boundary‑Measure Framework
Quantum foundations confront the measurement problem and the postulated Born rule $P_i=|c_i|^2$. Existing platforms—type‑I parametric‑down‑conversion sources delivering $10\;\text{kHz}$ coincidences at $200\;\text{mW}$ pump power [1] and superconducting processors achieving $99.90\%$ single‑qubit…
Mechanism‑First Pre‑Registered Testing of p‑adic Mass‑Ratio Hypotheses
The Standard Model mass spectrum lacks a mechanistic explanation, and previous adelic and p‑adic numerology has produced only look‑elsewhere false positives. We close this gap by introducing a mechanism‑first testing framework that requires any proposed p‑adic or adelic mechanism to generate…
Numeral Trees as the Native Ultrametric Geometry of Cyclic Time
Cyclic timestamps are stored linearly but intrinsically form mixed‑radix positional strings that nest hour, day, month, and finer cycles. Existing pipelines treat timestamps as points on the real line, ignoring this hierarchy and causing mis‑grouping across cycle boundaries. The gap is a lack of a…
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 Tamagawa number formula $\tau(T)=|\mathrm{H}^{1}(k,\widehat{T})|/|\Sha(T)|$ for an algebraic torus both express that freely chosen local contributions are forced by a global consistency condition to a canonical rational…
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.
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