Depth Structure in Primorial Character Lattices: Invariants, an Empirical Concentration Law, and Negative Results for Zero Detection

We study how the energy of the prime-counting signal distributes across Dirichlet characters of primorial moduli Mr=p1p2⋯prM_r = p_1 p_2 \cdots p_r Mr=p1p2⋯pr, organized by the conductor lattice of squarefree divisors. We record three elementary structural results: the reduced residues modulo any primorial have center of mass exactly 1/2, forced by the involution k → M_r − k; the average normalized log-conductor at depth k in the odd-prime hypercube equals k/(r−1); and the paired term x^β + x^(1−β) arising in the explicit formula is minimized at β = 1/2 — a convexity fact that we do not claim constrains the location of any zero. We then present empirical observations on energy concentration in the conductor lattice, verified for r=4 through 8, together with a negative result: this concentration is independent of the real part σ and is equally present at non-zero heights, making it a property of the character-group geometry rather than a detector of zeros. We reproduce and mildly extend Conrad's observation that partial Euler products are small near zeros of ζ, quantifying the separation for the first 100 zeros and its breakdown above height ≈ 400. Finally, we document six approaches to the Riemann Hypothesis that failed, each with a diagnosed root cause. All code and data are public.