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Numeral systems papers 24 papers
Numeral systems. Radix choice, positional notation, numeral design Tagged papers: 24.
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…
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…
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…
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…
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…
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…
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…
The Calculus of Re-Entrant Distinctions: A Unified Treatise on the Loop, the Tree, and the Constants of Self-Reference
This treatise develops a single thesis: that the primitive act of distinction — the drawing of a boundary between marked and unmarked — generates, under the discipline of re-entry and linear resource management, the constants $e$ and $\pi$, the landscape of completions of the rational numbers, the…
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…
Project Rosetta: The Approximation Entropy & The Fractal Limits of Digital Physics — v2.0
V2.0 / Rosetta v4.0 with Planck-Radix Correction + Qudit Manifesto. Corrects V1.0: transmon spectrum IS discrete (Planck quantization), binary truncation is a radix error not physics error. Transmon is a qudit (d~12), not a qubit. S_Base = ln(d/2) ~ 1.79 nats. 11-page PDF.
NUMERATA: A Multi-Axis Framework for Evaluating Numeral Systems
NUMERATA v2.0 extends the 8-axis framework with Distinction-Based Primality (WP0.3/9th axis) and meta-analysis validation (MCS=0.875, 5/5 predictions confirmed). Sunburst notation makes primality visually immediate.
Functorial Framework for Morphological Computing
This work introduces a formal framework for morphological computing that resolves the longstanding reconstruction paradox in Residue Number System (RNS) arithmetic by leveraging the physics of topological quantum matter. Conventional RNS-based accelerators are bottlenecked by the serial,…
Dual-Mode Arithmetic Representation
This presents a framework that resolves the apparent incompatibility between algebraic and geometric representations of natural numbers. We demonstrate that while static two-dimensional coordinate systems are informationally incomplete for capturing the full arithmetic structure of prime…
3 Mechanics
To fully grasp the profound consequences of implied discretization, it is essential to delve into the specific mechanics of how numbers are commonly represented and manipulated within digital computers. While various number systems exist, the overwhelming standard for scientific and engineering…
Ten-Fingered Trap
How Our Decimal Dependence Constrains Scientific Advancement Our familiar decimal system, base-10, serves as the near-universal standard for numerical representation, so deeply integrated into global commerce, education, and scientific communication that its structure often feels synonymous with…
Base-Invariant Number-Theoretic Patterns in Fundamental Constants
This presents a systematic, first‑principles investigation of the mathematical structures that underlie fundamental constants, freed from the arbitrariness of numeral‑system representation. By focusing exclusively on properties that are independent of any choice of base we uncover striking patterns…
EMERGENT NUMBER THEORY
The relationship between discrete number theory and continuous mathematical structures has been a subject of deep investigation in modern mathematics, with several well-established connections. The scholarly landscape reveals a sophisticated understanding of how discrete structures emerge from…
Ab Initio Architectonics
The current paradigm of quantum computing is increasingly constrained by a reliance on emergent quasiparticle ontologies and “epistemic patches” that obscure the underlying ab initio dynamics of quantum materials. This paper proposes a radical realignment through the “Signal-Worker” (S-W) framework,…
Math is NOT the Universe
This report re-evaluates our Euclidean understanding of reality, showing its inadequacy for modern physics. It traces the shift from Newtonian absolute spacetime to Einstein’s curved spacetime and the scale-dependent complexities of fractal geometry and quantum theory. This analysis argues for a…
Standard-Deviation Normalization Proof
Let \((\Omega, \mathcal{F}, P)\)be a probability space, and let \(X\)be a real-valued random variable defined on this space, representing a physical observable. Assume \(X\)has a well-defined mean \(\mu = \mathbb{E}[X]\)and finite, positive variance \(\sigma^2 = \mathbb{E}[(X - \mu)^2] > 0\). Let…
Silent-Radix Cryptography: Exploiting the Base Ambiguity of Positional Notation as a Cryptographic Primitive
The silent radix — a positional numeral cannot internally specify its own base — is a cryptographic primitive. Silent Radix Encryption (SRE) uses a secret base b as a trapdoor for key encapsulation. Security reduces to an unknown-weight integer knapsack problem.
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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