On the Ergodic Geometry of Scale Generators: Unitary U(1) Representations, Winding Invariants, and Kronecker Flows of Fundamental Transcendentals

### Overview & Epistemological Formulation In this paper, we resolve the nature of imaginary powers of fundamental constants (eⁱ and πⁱ) by transforming what has historically been treated as an empirical, static subtraction of decimals into an exact, continuous dynamical system on the 2-torus T². Rather than treating complex exponentiation through disjointed trigonometric approximations, we formalize the mapping a ↦ aⁱ as a unitary Lie representation of the scale dilation semigroup (R⁺, ·) on the circle group U(1) S¹. ### Key Analytical Results: 1. **Exact Phase Linearization:** We prove that the angular separation between the fundamental bases of analysis (e) and geometry (π) is governed by the scale-invariant Modular Phase Discrepancy Invariant: Δθ = ln(π/e) ≈ 0.1447298858 rad ≈ 8.292412^∘ 2. **Invariant Winding Fraction:** Normalized to full revolutions of the circle (2π), geometric curvature (π) advances ahead of continuous semigroup dissipation (e) by exactly: Δν = ln(π/e)/2π ≈ 2.303448% of a complete turn 3. **Kronecker–Weyl Toral Dynamics:** We formulate the joint evolution X(t) = (t, t ln π) 2π as an ergodic Kronecker flow on T², demonstrating asymptotic decay of Fourier–Weyl exponential sums scaling strictly as O(N⁻¹) with zero rational resonances. 4. **Information Conservation:** We calculate that the Kolmogorov–Sinai metric entropy vanishes identically (hKS = 0), proving that the flow achieves global phase-space mixing with zero stochastic chaos or information dissipation. 5. **Ergodic Convergence & Benford Law:** The empirical occupational measure converges weakly to the uniform Haar measure, which pulls back on logarithmic quotient cells Cb = [1, b) to the continuous scale-invariant Benford density dμ_∞(u) = 1/ln bdu/u and its Fisher Information functional. ### Computational Reproducibility:Includes the autonomous Python verification script `benchmark_kronecker_weyl_scale_generators.py` validating 100,000 steps with machine-level precision (100% pass rate). ### Canonical Provenance:This work extends the foundational theorems of the Modular Rotation Algebra (ARM) originating in the Harmonic Inflection and Unfolding Algebra (AIDH, Fernandes, 2024, DOI: 10.5281/zenodo.22802518) and the Projeto Ômega Technical Corpus (DOI: 10.5281/zenodo.23019172).