Linear Exponential Scaling in Non-Abelian Groups: A Universal Map Theorem for Non-Abelian Group Separation

zenodo.org · 2026-10-04 · by Long Nguyen ·generic
Abstract We present a comprehensive, universal, abstract group-theoretic, module-theoretic,and category-theoretic formalization of bijectively separated rank arithmetic. Given an arbitrary family of algebraic groups {Gi}ri=1 and elements Pi ∈ Gi with finite multiplicative orders oi = ord(Pi), we establish that the additive rank-coordinate space A(P) = Qri=1 Z/oiZ is canonically isomorphic to the external direct product of cyclic power orbits G(P) = Qri=1⟨Pi⟩ under the coordinate evaluation map ΦP(([ni]oi)i) = (Pnii)i. We prove that coordinatewise multiplication in the multiplicative state space corresponds strictly to coordinatewise addition of discrete expo-nent rank classes without introducing analytic logarithms, continuous curves, branchcuts, or topological ambiguities. We extend this framework to general category-theoretic limits, establishing that sub-space coordinate projections and factor permutation reindexings form natural transformations in the category of groups Grp. Furthermore, when lane outputs collapse into a single target background group H, we isolate the exact relation subgroup KP =ker(ΨP) and prove the quotient isomorphism A(P)/KP∼= ⟨P1, . . . , Pr⟩ ≤ H, measuring the precise algebraic information lost during scalar collapse. Finally, we formulate the extension to infinite complex power orbits, polynomial root reconstruction via Newton symmetrization of ranked power arrays Pr ∈ Cr×r, Chinese Remainder Theorem grid decompositions over composite moduli, native linear rank complexity bounds Θ(m), and an axiomatic minimal proof assistant kernel. Keywords: Abstract algebra, cyclic groups, direct products, exponent coordinates, re-lation lattices, projection functors, natural transformations, Newton identities, ChineseRemainder Theorem, rank complexity. Dr. Charles Tibedo Alternate Working Title: A Universal Theory of Bijectively Separated Rank Arithmetic Contents1 Introduction and Mathematical Foundations 41.1 Motivation: Analytic Logarithms vs. Discrete Exponent Representation 41.2 The Bijectively Separated Rank Arithmetic Paradigm 51.3 Paper Organization 52 Universality of Bijectively Separated Rank Spaces 62.1 General Definitions 62.2 Main Isomorphism Theorem 73 Category Theory and Natural Transformations in Grp 83.1 Coordinate Projections and Sub-Space Functors 93.2 Permutation Covariance and Factor Reindexing 104 Module Theory, Collapsed Multiplication, and Relation Lattices 114.1 The Collapsed Homomorphism and Relation Subgroup 115 Sequential Accumulation and Asynchronous Pathwise Equivalence 126 Extension to Complex Power Orbits and Polynomial Geometry 136.1 Infinite vs Finite Complex Orbits 136.2 Ranked Power Arrays and Polynomial Reconstruction 137 Chinese Remainder Factorization over Composite Moduli 148 Native Computational Complexity Bounds 149 Axiomatic Minimal Proof Kernel for Formal Verification 1510 Conclusion and Future Directions 15A Step-by-Step Derivation for Single Lane Homomorphism 16B Exact Irrational Complex Quintic Verification Audit 16C Theorem Dependency Graph 16  

Linear Exponential Scaling in Non-Abelian Groups: A Universal Map Theorem for Non-Abelian Group Separation

Published October 4, 2026 | Version v4

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Linear Exponential Scaling in Non-Abelian Groups: A Universal Map Theorem for Non-Abelian Group Separation

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Abstract

We present a comprehensive, universal, abstract group-theoretic, module-theoretic, and category-theoretic formalization of bijectively separated rank arithmetic.

Given an arbitrary family of algebraic groups {Gi}r i=1 and elements Pi ∈ Gi with finite multiplicative orders oi = ord(Pi), we establish that the additive rank-coordinate space A(P) = Qri=1 Z/oiZ is canonically isomorphic to the external direct product of cyclic power orbits G(P) = Qri=1⟨Pi⟩ under the coordinate evaluation map ΦP((\[ni\]oi)i) = (Pnii)i.

We prove that coordinatewise multiplication in the multiplicative state space corresponds strictly to coordinatewise addition of discrete expo- nent rank classes without introducing analytic logarithms, continuous curves, branch cuts, or topological ambiguities.

We extend this framework to general category- theoretic limits, establishing that sub-space coordinate projections and factor permutation reindexings form natural transformations in the category of groups Grp.

Furthermore, when lane outputs collapse into a single target background group H, we isolate the exact relation subgroup KP =ker(ΨP) and prove the quotient isomorphism A(P)/KP∼= ⟨P1, . . . , Pr⟩ ≤ H, measuring the precise algebraic information lost during scalar collapse.

Finally, we formulate the extension to infinite complex power orbits, polynomial root reconstruction via Newton symmetrization of ranked power arrays Pr ∈ Cr×r, Chinese Remainder Theorem grid decompositions over composite moduli, native linear rank complexity bounds Θ(m), and an axiomatic minimal proof assistant kernel.

Keywords: Abstract algebra, cyclic groups, direct products, exponent coordinates, re- lation lattices, projection functors, natural transformations, Newton identities, Chinese Remainder Theorem, rank complexity.

Dr. Charles Tibedo

Alternate Working Title:

A Universal Theory of Bijectively Separated Rank Arithmetic

Contents 1 Introduction and Mathematical Foundations 4 1.1 Motivation: Analytic Logarithms vs. Discrete Exponent Representation 4 1.2 The Bijectively Separated Rank Arithmetic Paradigm 5 1.3 Paper Organization 5 2 Universality of Bijectively Separated Rank Spaces 6 2.1 General Definitions 6 2.2 Main Isomorphism Theorem 7 3 Category Theory and Natural Transformations in Grp 8 3.1 Coordinate Projections and Sub-Space Functors 9 3.2 Permutation Covariance and Factor Reindexing 10 4 Module Theory, Collapsed Multiplication, and Relation Lattices 11 4.1 The Collapsed Homomorphism and Relation Subgroup 11 5 Sequential Accumulation and Asynchronous Pathwise Equivalence 12 6 Extension to Complex Power Orbits and Polynomial Geometry 13 6.1 Infinite vs Finite Complex Orbits 13 6.2 Ranked Power Arrays and Polynomial Reconstruction 13 7 Chinese Remainder Factorization over Composite Moduli 14 8 Native Computational Complexity Bounds 14 9 Axiomatic Minimal Proof Kernel for Formal Verification 15 10 Conclusion and Future Directions 15 A Step-by-Step Derivation for Single Lane Homomorphism 16 B Exact Irrational Complex Quintic Verification Audit 16 C Theorem Dependency Graph 16

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