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| ai.viXra.org > Functions and Analysis > 2602 |
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[2] ai.viXra.org:2602.0067 [pdf] submitted on 2026-02-14 01:33:55
Authors: Cornelius Moore
Comments: 24 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
Modern theoretical physics employs distinct mathematical formalisms—Lagrangian mechanics, Hamiltonian dynamics, quantum amplitudes, statistical ensembles, field-theoretic path integrals—that, while empirically successful, lack a unified structural foundation. We present the Universal Mathematical System (UMS), a vari-ational—probabilistic framework in which standard physical theories arise as limiting cases, projections (marginalizations), or constrained reductions of a single maximum-entropy measure over configuration spaces. The framework is built on an exponential-family measure µ[C] = Z−1 exp(−Φ[C]), where Φ is a constraint functional encoding physical laws. We show under standard regularity assumptionshow classical mechanics emerges via the Laplace principle (β →∞), statistical mechanics is directly identified with the framework, and quantum mechanics corresponds to complex-weighted measures under standard path-integral formalism. Additionally, we formalize five distinct algebraic structures—quantity (Rn), growth (semigroups), information (entropy), phase (U(1)), and ratio (R+)—clarifying thatdifferent physical questions inhabit different mathematical domains and that confusion arises from naive cross-domain interpretation. The framework is intended as structural unification of existing formalisms rather than a proposal of novel fundamental ontology or new empirical predictions. We include a proof of a coarse-graining monotonicity theorem using the data-processing inequality, provide explicitreduction pipelines, and discuss extensions to chemistry, biology, neuroscience, and computation.
Category: Functions and Analysis
[1] ai.viXra.org:2602.0010 [pdf] submitted on 2026-02-03 19:55:50
Authors: Mohammad Saeed Alnatour
Comments: 172 Pages. (Note by ai.viXra.org Admin: For the last time, author name is required in the article after article title and please cite listed scientific references)
Divergent series and singular integrals arise naturally in analysis, geometry, and theoretical physics, yet their standard treatment relies on analytic continuation or limit-based regularization. While these methods successfully assign finite values, they necessarily suppress information about how infinity is traversed. This work proposes a structural framework—Interconnected Infinities Giant Sphere Space (IIGSS), together with an intrinsic regulator, the Discrete Laplace Regulator (DLR), in which divergence is treated as a boundary phenomenon rather than a failure of summation.DLR operates directly on discrete sequences by introducing controlled exponential damping and expanding the resulting kernel at a well-defined infinite-traversal gate. Divergence appears explicitly as algebraic pole terms or logarithmic singularities in the gate expansion, encoding growth class and traversal density, while a pole-invariant constant—the Convergence Momentum (CM)—emerges as a finite structural quantity. Valuation is performed exclusively through gate expansion followed by pole removal, without index shifting, limit evaluation, or analytic continuation.Within this framework, classical zeta and Dirichlet regularizations are recovered as special projections under standard traversal, while traversal-sensitive features—such as zero insertion, spacing modulation, and phase structure—remain distinguishable. The framework accommodates finite-gate and oscillatory sequences and clarifies the limitations of reconstruction from regularized values alone. In physical applications, CM functions as a retained boundary invariant: when applied to spectral mode sums, such as those appearing in the Casimir effect, the regulator preserves observable finite quantities while rendering the underlying divergence structure explicit. DLR thus provides a higher-resolution language for infinity, preserving established results while exposing structural information necessarily omitted by classical methods.
Category: Functions and Analysis