How do the great “millennium themes” change when you insist on finite structure, explicit cutoffs, and spectral control? Book III offers a τ-effective spectral reading of seven major mathematical themes—framed as eight “lenses” that keep scope and claims disciplined. Rather than presenting sweeping one-line solutions, the book builds a shared dictionarythat translates problems into operator-theoretic and spectral statements on a minimal two-loop carrier: the lemniscate(a figure-eight) with its correct duality between the character torus and the integer lattice. The aim is not metaphor, but a reusable technical framework: compare sectors, impose finite windows, and state precisely what is established, what is τ-effective, and what remains conjectural. At the center is a cascade schema: (×, ∧) → ℤ² → (∞, 𝕃) → H∞ → eight lenses Here, two independent modes of composition (multiplicative structure and iterative growth) generate a bookkeeping lattice; the lattice is carried by the lemniscate 𝕃 = S¹ ∨ S¹; and an operator-theoretic object H∞ becomes the common stage on which the lenses are read. Throughout, the text enforces a strict status discipline—Established, τ-effective, Conjectural, and Metaphor—so readers can track exactly what kind of statement is being made. The eight lenses are: Finite (P vs NP): a τ-internal analysis of tractability for τ-admissible instances (bounded interface width), including a spectral–interface correspondence inside the framework—explicitly not a blanket ZFC claim about P vs NP. Spatial (Poincaré): anchored in Perelman’s theorem, reinterpreted through categorical/spectral structure and internal reconstructions. Temporal (Riemann): zeta phenomena reformulated in operator language, emphasizing finite cutoffs and regularized determinants. Eternal (Hodge): Hodge-theoretic constraints related to categorical holomorphy and finite spectral support mechanisms. Rational (BSD): rank and L-data compared via spectral multiplicities and finite-window determinants. Existential (Yang–Mills): existence and mass-gap themes explored through spectral gaps and compactness constraints, with physics analogies clearly marked. Regular (Navier–Stokes): regularity reframed as spectral control of cascades, conditional on explicit sector bounds. Spectral prism (Langlands): a finite-window comparison principle between arithmetic and automorphic sectors (schematically H_{\rho,\le N} \simeq H_{\pi,\le N}), stated as a controlled comparison—not an unqualified identity. Book III concludes with a synthesis of the dictionary, a list of open bridges, and a clear ledger of limitations and next steps. It is designed as a navigation instrument for deep problems: a way to make statements precise, finite, and comparable across domains—before claiming victory. “Prefer finite-window statements; keep scope labels explicit.”