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Results for “foundations” 8 papers
Non‑Archimedean Statistical Foundations for a Fully Connected Reality
Standard parametric statistics assume independent, identically distributed (i.i.d.) observations, an assumption that fails when all events share a common causal ancestry encoded by an ultrametric (p‑adic) structure. This paper addresses the gap between classical significance testing and a fully…
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: the Archimedean absolute value and, for each prime p, a p-adic absolute value. Their completions are the real numbers and the p-adic fields. This classification is often read as closing the question of what a…
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…
Adelic Shannon Theory: From Problem Statement to Constructive Foundations
Standard Shannon theory — channel capacity, entropy, source coding, and the Gaussian noise model — is archimedean: it measures information in bits over $\mathbb{R}$. The Adelic Physics Programme's central claim — $\mathbb{Q}$, not $\mathbb{R}$, is the physically accessible base field — suggests that…
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…
The Decryption Key: Cognitive Architecture as the Hidden Radix of Physical Measurement
The evolved cognitive architecture of Homo sapiens — decimal numeracy from ten fingers, water-based biochemistry, Euclidean spatial intuition from visual cortex processing, and body-scale affordances including stride length, hand-heft, and heartbeat-period temporal chunking — functions as a…
Number-Theoretic Ultrametric Foundations: A Unified p-adic Framework for Error-Correcting Code Classification
We present a unified framework connecting deep number-theoretic structures — $p$-adic valuation theory, Mahler spectral expansions, Kodaira-Néron fiber classification, and the Amice transform — to the classification of quantum error-correcting codes. The framework is organized into seven…
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…
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