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[5] ai.viXra.org:2610.0053 [pdf] submitted on 2026-10-22 05:24:40
Authors: Stephen P. Smith
Comments: 12 Pages.
This paper examines whether reciprocal representation provides a mathematically coherent way to express two distinguishable but structurally equivalent sectors without privileging either as primary. The starting point is a pair of isomorphic Lie algebras and their representation within a direct-sum algebra. A natural sector-exchange map is shown to be an involutive Lie-algebra automorphism, while the paired representation forms a Lie subalgebra fixed pointwise by that exchange. The direct sum therefore supplies a precise realization of reciprocal redundancy without requiring nonzero cross-sector brackets. The paper then considers a more general ambient algebra, where closure of the paired representation constrains the cross-sector bracket to be symmetric in its corresponding arguments. The Jacobi identity imposes additional consistency requirements but does not, by itself, require cross-sector brackets to vanish. The limiting case of coincident embeddings is examined, as is an illustrative example in which paired closure coexists with a nonzero cross-sector bracket. The resulting mathematical distinctions are related cautiously to a philosophical proposal of ontological two-sidedness. The algebra does not prove that reality has a bilateral ontology; it provides a formal representation in which two distinguishable sides can be related reciprocally without assigning either side ontological priority.
Category: Algebra
[4] ai.viXra.org:2609.0044 [pdf] submitted on 2026-09-21 20:30:42
Authors: Stephen P. Smith
Comments: 18 Pages.
Lie-algebraic structure can be examined through two complementary forms of organization that will be called vertical and horizontal emergence. These terms are used here as a conceptual framework rather than as standard classifications within Lie theory. Vertical emergence refers to the construction of new algebraic objects from structures already present at a lower level—for example, the construction of curvature from a covariant derivative, or the appearance of higher multilinear operations in $L_infty$-algebras. Horizontal emergence refers to the integration of multiple algebraic structures into a larger structure in which interactions between the component algebras become essential. Semidirect products, matched pairs, and related constructions provide mathematically precise examples. The distinction is useful because the familiar derived series [ L^0 = L,qquad L^1 = [L,L],qquad L^2 = [L^1,L^1],qquad ldots ]does not by itself capture every way in which algebraic structure can become more elaborate. The derived series produces a nested sequence of subalgebras using the same underlying bracket. By contrast, extensions, deformations, higher brackets, curvature, and interacting combinations of algebras can introduce genuinely new structural relationships and new compatibility conditions. This provides a possible mathematical analogy for Arthur Koestler's concept of the holon: a structure may retain a degree of autonomy at one level while simultaneously participating in a larger integrative structure. The analogy should not be mistaken for a theorem establishing that Lie algebras are literally holarchies. Rather, it suggests a productive correspondence between a well-defined mathematical phenomenon—structural integration across levels—and a broader philosophical account of organization.
Category: Algebra
[3] ai.viXra.org:2607.0051 [pdf] submitted on 2026-07-21 17:44:48
Authors: Joerg Moesch
Comments: 27 Pages.
In this paper, the natural three-dimensional extension of the complex and dual numbers is introduced: the trinion algebras Y(k) of the form a+bi+cj with i²=-1, j²=0 and the commutative coupling ij=ji=k, where k = α+βi+γj is an element of the algebra. The family Y(k) is systematically investigated: multiplication, division and inverses are derived. Special attention is given to the phenomena induced by non-associativity, which lead to invertible zero divisors, non-unique division, and rich geometric structures. The eight fundamental algebras with α,β,γ ∈ {0,1} are examined and their zero divisor sets are visualized. Numerical examples for the algebra Y(1) illustrate the observed phenomena.
Category: Algebra
[2] ai.viXra.org:2603.0055 [pdf] submitted on 2026-03-12 18:00:46
Authors: Ahcene Ait Saadi
Comments: 6 Pages.
This paper presents an original analytical method for solving the reduced quintic equation By establishing a conformal mapping between the roots of an auxiliary quartic and those of the quintic, we reveal a Universal Scaling Law governed by a 1/8 power factor. This approach calculates all five roots (real and complex) across all topological regimes, including cases with three real roots, with a precision of . This discovery marks a new era for polynomial algebra.
Category: Algebra
[1] ai.viXra.org:2601.0030 [pdf] submitted on 2026-01-10 01:38:08
Authors: Eero Koskela
Comments: 10 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present an elementary proof of Fermat's Last Theorem for all powers n ≥ 3 using only binomial coefficients and basic algebra. The proof establishes that the binomial representation 6C(x+1,3) + x serves as a unique fingerprint for the cube x³, and demonstrates that this uniqueness property forces a secondary constraint that is incompatible with the original Fermat equation. For n=3, this leads directly to a geometric contradiction. For n=4, we obtain an algebraic impossibility. For n ≥ 5, we show that the secondary constraint forces fractional exponents that can only yield integers if a lower-dimensional Fermat equation holds—but these are already proven impossible. This proof is independent of Wiles's work and relies only on elementary techniques accessible to undergraduate students.
Category: Algebra
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