Elishia Algebra and the Seam Calculus

Elishia Algebra and the Seam Calculus 10.5281/zenodo.22553219We develop an algebraic and geometric framework for retaining transformation history that is normally lost when a system is reduced to scalar observables or projected onto a single representation. The construction combines a noncommutative matrix realization in M2(Q)M_2(\mathbb{Q}), characteristic-five Galois transport over ramified quadratic structures, quadratic Seam branch products, conjugate Fibonacci–Lucas recurrence geometry, lifted circular fibrations, and unimodular/Farey modular transformations. A central role is played by the golden quadratic pair (Φ,Ψ)(\Phi,\Psi), with Ψ=1−Φ=−Φ−1\Psi=1-\Phi=-\Phi^{-1}, whose conjugate branches encode complementary fold and phase behaviour rather than being collapsed into a single scalar approximation. The paper develops the Seam as an explicit mathematical object: a location at which distinct representations meet while retaining enough information about their respective transformation histories to reconstruct the relation between them. This provides a common language for branch orientation, fold parity, residual terms, recurrence transport, winding structure, and modular dressing. Within this framework, several numerical structures associated with physical constants are investigated as geometric transition readouts rather than inserted empirical parameters. Particular attention is given to the fine-structure constant and the appearance of the correction structure 137+π/87137+\pi/87, together with the algebraic and modular machinery from which such expressions arise. The broader aim is methodological: to construct a calculus in which discarded residuals, conjugate branches, and transformation paths remain available as mathematical data. The resulting Elishia/Seam framework provides a basis for investigating whether apparently unrelated constants, recurrence structures, and physical scaling relations can arise from a shared transformation geometry. This work should be read as a mathematical framework and research programme. Exact algebraic identities and derived constructions are distinguished from proposed physical interpretations and empirical identifications.