Jenni's Quell-Theorem / Jenni's Source-Theorem: A Unified Generative Framework for Linear Recursive Sequences

This paper introduces "Jenni's Quell-Theorem," a universal generative framework that unifies the theory of linear recursive sequences, such as the Fibonacci, Tribonacci, and Padovan sequences. Traditionally, these sequences are defined individually by their specific homogeneous recurrence relations and initial conditions. This work reverses this perspective, demonstrating that they all emerge naturally as difference sequences from a single, universal source: a cumulative generator defined by the simple inhomogeneous recurrence Sₙ = Sₙ₋₁ + Sₙ₋₂ + ... + Sₙ₋ₖ + 1, initialized with zeros. The familiar homogeneous sequences are derived as aₙ = Sₙ - Sₙ₋₁. The constant term +1 is not an arbitrary addition but is revealed as a fundamental "arithmetic conservation quantum"—the minimal symmetry-breaking impulse that initiates all structured growth from a universal zero state. This framework shows that the distinction between homogeneous and inhomogeneous recurrences is artificial; the inhomogeneity is the origin, and the homogeneity is its emergent image. Key contributions of the Quell-Theorem include: Unification: Providing a single generative principle for all k-step Fibonacci-like sequences. Minimalism: Replacing the need for specific initial vectors with a universal, minimal configuration. Conceptual Depth: Re-framing linear recurrences as manifestations of a deeper conservation law inherent in the discrete arithmetic of the number system itself. This work offers a new foundational perspective for number theory, combinatorics, and the study of recursive structures, suggesting that complex sequences arise from the elementary discreteness of arithmetic. Keywords: Linear Recurrence Relations, Fibonacci Sequence, Tribonacci Sequence, k-bonacci Numbers, Generative Models, Number Theory, Discrete Dynamics, Unification Theorems, Arithmetic Foundations