CROSS-ENERGY OF DIRICHLET PARTIAL SUMS AND THE ZEROS OF THE RIEMANN ZETA FUNCTION PART I: CRITICAL-LINE DETECTION, A FRESNEL TRANSITION LAW, AND STRUCTURAL LIMITS

We study the cross-energy increment associated with the Dirichlet partial sums Xₙ(s) = Σₙ≤N n⁻ˢ, where s = σ + it. The cross-energy is defined by C₂(N,s) = |Xₙ(s)|² − |Xₙ₋₁(s)|² − N⁻²σ, equivalently C₂(N,s) = 2N⁻σ Σⱼ 1/2. On the critical line, eventual monotonicity of C₂ occurs exactly at zeros of ζ. At a zero ρ = 1/2 + it, the trajectory crosses the axis at N₀ = t² + 1/3 + O(t⁻²) and satisfies N₀ lim[N→∞] C₂(N,ρ) → 1. A closed-form uniform expansion of the tail Σₙ≥N n⁻ρ gives strict monotonicity for N ≥ (1 + ε)t/(2π). A Fresnel model predicts that monotonicity begins at N = t/(2π) + κ√t, with the universal constant κ = −0.207798…, in agreement with numerical tests for zeros up to height 2 × 10⁴. We show that phase-locked C₂ recovers Hardy's function 2Z(t), while the Riemann–Siegel main sum arises as a truncated C₂ sum evaluated at x* = exp(θ(t)/t) ≈ √[t/(2πe)]. This yields a proof of Hardy's theorem within the framework. We determine the anatomy of the mollified C₂ associated with ζ(s)Mᵧ(s) and establish a structural obstruction. The Epstein zeta function associated with m² + 100n² shares every C₂ structural property used here, yet possesses an off-critical zero with real part 0.8133678… at which its C₂ collapses. Consequently, no argument based only on the C₂ structure can prove the Riemann Hypothesis. Finally, with von Mangoldt weights, RH is equivalent to the energy law Σₙ≤N C₂Λ(n, 1/2) = 4N + O(N¹ᐟ²⁺ε). These results provide a cross-energy framework for critical-line detection and prime-weighted formulations of RH. We do not prove the Riemann Hypothesis.