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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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Tree geometry papers 16 papers
Tree geometry. Trees, depth, hierarchies and dendrograms Tagged papers: 16.
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 = ⌈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…
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…
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).
The Observer Inside the Tree: Can Self-Location in an Ultrametric Structure Resolve the Inside/Outside Schism?
The 29-schisms framework claims to resolve the central schism of quantum foundations — S10, the observer inside vs. outside — by embedding the observer as a node in an ultrametric TREE structure, eliminating the need for an external vantage point. This paper subjects that claim to independent…
Ultrametric Engine: Deploying a 20-Principle p-Adic Discovery Worker
We describe a discovery protocol operating in ultrametric spaces, formalizing 20 principles that govern knowledge graph navigation, paper discovery, and research question generation under non-Archimedean distance constraints.
Monna Map Generation and Hallucination
Every thought you have—a memory, a plan, a sudden insight—arrives not as a neat list of words but as a living structure. It is a branching hierarchy of concepts, where broad categories contain finer details, and relationships are defined by containment and connection rather than sequence. This…
Proof-of-Concept for Auditable Attention using Ultrametric Tree Distances
The dominance of high-dimensional Euclidean vector spaces in contemporary transformer architectures has precipitated an interpretability crisis, as continuous geometric embeddings can obscure the causal mechanisms of semantic attention. This paper presents a formal proof-of-concept for a…
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…
Bruhat-Tits Tree as a Unifying Geometric Object
Modern artificial intelligence, particularly in high-stakes domains, is hampered by a fundamental geometric mismatch: the reliance on continuous, “flat” Euclidean spaces to represent intrinsically hierarchical data. This incongruity leads to information loss and the proliferation of opaque “black…
Convergence Consilience and the Hierarchical Architecture of Reality
This document argues that convergence (nature independently producing similar forms across separate lineages) and consilience (knowledge from different disciplines converging on the same truths) are symmetric faces of a single deeper structure: a hierarchically organized reality shaped by attractors…
Topological Signal Processing
The inherent limitations of conventional digital signal processing—specifically the latency and energy bottlenecks imposed by analog-to-digital conversion—represent a critical barrier for next-generation systems like 6G networks. This paper introduces a novel physical computing paradigm that…
Wave-Built World
The observable universe, in all its astonishing diversity and intricate organization, is not a static collection of isolated objects but a ceaseless dance of dynamic processes. From the quantum fluctuations that shape the very fabric of spacetime itself to the rhythmic swing of pendulums, the…
Qudit Quantum Error Correction
Extends the Ultrametric Foundation thesis into quantum computing. Formalizes geometric error confinement on tree-topology quantum processors.
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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