Spectral Taxonomy of Mathematical Constants: A Unified Framework for Constants as Eigenvalues of Kinetic Fractal Operators

Why do π, the golden ratio φ, and √2 appear across seemingly unrelated mathematical and physical structures? This paper proposes that these constants are not isolated numerical accidents but spectral invariants — resonance parameters of self-adjoint operators on geometric spaces — organized by a single framework of kinetic fractal operators. Three cases are developed at decreasing levels of rigour. Case I (π) is fully rigorous: Euler's Basel identity ζ (2) = π²/6 is recast as a spectral trace formula for the cyclic kinetic operator on the circle, where π emerges as √ (3·ζH (1) ). This is a reformulation of a classical result in spectral language, not a new proof. Case II (φ) combines rigorous and heuristic elements: Hurwitz's theorem (1891) characterizes φ as the optimally Diophantine number, and qualitative connections to spectral gaps of quasi-periodic operators (the almost Mathieu / Harper operator) provide a "maximal gap" interpretation. The claim that φ globally maximizes a precisely defined spectral gap measure is plausible but not rigorously proved. Case III (√2) is conjectural with numerical support: a variational energy-resonance functional F (α) associated with kinetic operators on self-similar geometries is constructed, and numerical experiments indicate its critical point lies near α = √2. This is stated as Conjecture (√2-criticality), not theorem. The paper is deliberately positioned as a hybrid: rigorous reformulations of known results, heuristic models with qualitative justification, and reproducible numerical experiments. The "spectral genesis" narrative — that fundamental constants are spectral fingerprints of geometric and arithmetic structure — is presented as a research program, not a complete theory. A unified taxonomy emerges: π governs cyclic/periodic geometry (eigenvalue traces), φ governs quasi-periodic geometry (Diophantine gap optimization), and √2 governs fractal/self-similar geometry (variational criticality of multi-scale operators). Each constant occupies a distinct niche in the spectral landscape, and the kinetic fractal operator framework provides the common language connecting them. Limitations are stated with unusual transparency in a dedicated "Honest Assessment Summary": the Basel reformulation is rigorous but not new, the φ gap maximization is plausible but unproved, the √2 variational minimum is conjecture supported by numerics in a specific model (with the companion paper on optimal spectral-fractal balance clarifying that √2 is non-universal and parameter-dependent), and the unified narrative is a research program rather than a complete theory. All numerical results are reproducible via provided code.