The Adrian Structure: A Heuristic Framework for Discrete Transition Points

This work presents the Adrian Structure, a heuristic framework that investigates discrete transition points—referred to as Champions—across arithmetic systems, quantum-mechanical models, spectral operators, and empirical physical data. The framework explores how additive growth interacts with multiplicative capacity limits, leading to characteristic reorganization points in both mathematical and physical structures.

The study compares three domains:

• Arithmetic and prime number structures, including record points and Euler-product–based dimensional growth

• Physical and chemical datasets, such as atomic radii, electronegativity, ionization energies, and electron shell transitions

• Spectral and quantum systems, including formal analogies to collapse thresholds in quantum-mechanical models

A central idea is that many systems—despite differing mechanisms—exhibit structurally similar points where a smoothly increasing additive index meets a quantized, multiplicative capacity constraint. These points define discrete reorganizations that the Adrian Structure describes in a unified way.

The document also develops speculative but mathematically consistent parallels to zeta-function formalism, Mellin symmetry, and the critical line ℜ(s)=1/2, interpreted as a structural equilibrium between additive and multiplicative representations. No physical claims, proofs, or solutions to the Riemann Hypothesis are asserted; the work is strictly heuristic.

Author contribution and use of AI assistance:

The conceptual ideas, structure, data selection, interpretation, and overall 
theoretical framework were developed entirely by the author. Limited AI assistance 
was used solely for LaTeX formatting support, English phrasing, and technical 
editing. The scientific content, hypotheses, interpretations, and conclusions 
originate from the author.

A key intention of this work is to highlight a new way of thinking: that additive 
mathematical processes cannot always be represented or sustained by multiplicative 
structures when the system operates within dimensions that possess strict capacity 
limits. In such settings, discrete transitions become necessary, and these 
transitions form the core of the Adrian Structure.