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611 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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Numbers & p-adics papers 27 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.
Ultrametric. Non-Archimedean geometry: tree metrics, strong triangle inequality, hierarchical distance Tagged papers: 69.
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…
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…
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…
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…
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…
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…
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 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…
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).
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…
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.
Harmonic Analysis is Langlands — The Fast Fourier Transform as the Computational Langlands Correspondence for GL(1)
We identify the Fast Fourier Transform (FFT) as the computational Langlands correspondence for the general linear group GL(1). Harmonic analysis on the finite cyclic group Z/nZ decomposes functions into irreducible characters -- the FFT -- and this structure generalises directly to harmonic analysis…
Five Pillars, One Structure: Consilient Convergence in QNFO Research
Five independent QNFO research programs converge on a single structural insight: ultrametric (non-Archimedean) mathematics provides the correct state-space geometry for fundamental physics, quantum computation, and optimization.
Embodied Mathematics After Lakoff & Nunez: A QNFO Perspective
This review examines Where Mathematics Comes From (Lakoff & Núñez, 2000) from the QNFO perspective, identifying convergence on anti-Platonism, the symbol-concept distinction, and critique of formalism. Extensions include ultrametric formalization of the four grounding metaphors and a cognitive…
COMMUTATIVE PHYSICS
Mathematics serves as the fundamental language through which the intricate laws and diverse phenomena of the physical universe are articulated, modeled, and predicted. The selection of mathematical tools is not a trivial matter; it profoundly shapes our comprehension of reality. Modern physics…
ultrametric-paradigm
Intelligence does not float free of its substrate. Every computational system makes an implicit commitment to a geometry—a structure of space that determines what can be represented, how information propagates, and what it means to learn. The history of artificial intelligence can be read as a…
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…
Computational Topology and Formal Proofs of Discrete Non-Archimedean Relational Networks
The fundamental reconciliation of quantum mechanics with macroscopic geometry requires a departure from continuous Archimedean manifolds, substituting them with discrete non-Archimedean topologies at the Planck scale. This foundational shift is contextualized by analyzing the Wheeler-DeWitt…
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…
Prime Numbers as Spectral Artifacts
This establishes a mathematical framework demonstrating that prime numbers emerge as spectral artifacts from a continuous quantum geometric substrate, rather than representing fundamental discrete entities. Through formal deduction from five established mathematical facts—base-independence of…
Continuous Topology and Discrete Arithmetic
The tension between the discrete cardinality of prime numbers and the continuous geometry of the complex plane is reconciled by treating primes as spectral residues governed by the explicit formulae of the Riemann zeta function. Sequential enumeration provides a linear coordinate that obscures the…
Radix to Ultrametrics to Page-Wootters to Wheeler-DeWitt to Bruhat-Tits: A Convergent Synthesis
Convergent synthesis bridging discrete p-adic ultrametrics with continuous quantum geometry via Page-Wootters, Wheeler-DeWitt, and Bruhat-Tits buildings.
The Ultrametric Foundation: A Unified Thesis on Number, Time, Knowledge, and Computation
Positional notation is inherently an ultrametric tree of nested cycles. The Archimedean line is a derived abstraction. Recovering the native ultrametric resolves cascading errors across mathematics, physics, epistemology, education, quantum computing, and interface design.
Nonlinear Tree-Based Numeration Systems: A Consolidated Synthesis
Positional notation is natively an ultrametric tree. This synthesis covers 5 research threads with extensions into quantum computing and interface design. https://papers.qnfo.org/papers. the QNFO archive: https://papers.qnfo.org/papers. Published:…
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