Four identities and one correction: the exact, asymptotic, and truncation structure of the truncated Weil operator

Building on the exact rank-one structure of the pole term [1] and the strict Herglotz inequality [2], we establish four identities that reduce the ground-state problem of the odd sector of the truncated Weil operator to a one-dimensional scalar equation with explicit asymptotics: (i) the exact secular equation m_o(λ_0^odd) = -1/μ_0;(ii) the strict Herglotz inequality m_o(λ_0^even) < -1/μ_0 with a quantitative mechanism (the margin is dominated by the closest pole ν_1, not by the dominant mode);(iii) the asymptotic growth M_1(c) = <û, B_odd û> ~ √c;(iv) the asymptotic identity |μ_0(c)| = M_1(c)(1+o(1)), verified to 0.06% at c=8000. The four identities are not independent: (i) + (iv) imply (iii). We further report three structural results: the concentration w_dom(c,N) of the pole direction on a single eigenvector of B_odd converges to 1 as c → ∞, reaching w_dom = 0.99993 at c=8000; the spectral alignment δ(c) ~ c^(-1.21); and the prime-sector domination (97.5%) of the resolvent at c=8000. The correction concerns the even-odd gap λ_0^odd(c,N) - λ_0^even(c,N) of the full truncated operator. At N=50 this gap oscillates in sign with at least six transitions in c ∈ [200, 32000]. We show that this oscillation is not a structural property of the operator: at c=316 and at c=16000 the sign changes from negative (ODD) at N=50 to positive (even) at N=100 and N=200, with a 6% convergence in magnitude between N=100 and N=200. The N=50 landscape is dominated by truncation error. We provide explicit thresholds: dps=50 is sufficient for c ≤ 4000; dps ≥ 75 is required for c ≥ 8000; at c=32000, dps=100 is also insufficient. We also document and fix a cache-growth bug in the connes_cvs package that prevents completion of dps ≥ 75 runs for c ≤ 4000. [1] J. A. Franco Olguín, Exact rank-one structure of the pole term in the truncated Weil operator, Zenodo, DOI:10.5281/zenodo.23228212 (2026).[2] J. A. Franco Olguín, A convexity proof of the scalar Herglotz inequality for the truncated Weil operator, Zenodo, DOI:10.5281/zenodo.23266965 (2026).