Primal–Dual Mellin Neutralization: Exact Residue Cancellation, Sharp (T-3/2) Tail Suppression, and Finite-Rank Window Design

This paper develops a primal–dual Mellin neutralization framework for suppressing scalar and finite-rank spectral obstructions in logarithmic Mellin inversion. Centered log-Mellin Plancherel separates two complementary cancellation mechanisms: the primal condition q(0)=0, which removes the central Mellin mode, and the dual condition $q(0)=0$, which suppresses the reflected contribution. For Hölder multipliers the resulting decay is of order T-1/2-α, while the Lipschitz regime admits the sharp separated-scale law T-3/2. The construction is extended to vector-valued systems through finite-rank Gram neutralization and an explicit leakage ledger. The abstract theory is realized in automorphic models involving one-cusp scattering, boundary-heat reconstruction, and the signed Rankin–Selberg system associated with Ramanujan’s third-order mock-theta multiplier. For a fixed off-diagonal spectral pair, the original and lowered Eisenstein periods form a pole-free vector with polynomial vertical growth. After extracting the common gamma factor from the Whittaker-product Mellin matrix, the completed matrix is shown to be automatically invertible at sufficiently large height, with polynomial reciprocal growth. The remaining determinant zeros are confined to a finite-height compact region. Let Δⱼₖ(z) denote the completed determinant. If its finite zero divisor is Pⱼₖ(z) = ∏ζ (z-ζ)^ordζΔⱼₖ, then the modified sign vector Pⱼₖ(z) Zⱼₖ(z) is holomorphic on the working strip and has polynomial vertical growth. For a compact logarithmic seed hT(x) = T-1/2h(x/T), the differential window qPT,jk(x) = Pⱼₖ(-∂ₓ)hT(x) satisfies the exact bilateral Laplace identity L qPT,jk(z) = Pⱼₖ(z) L hT(z). Thus the complete finite determinant divisor is cancelled by finitely many exponential-moment jet conditions. Within this canonical spectrally neutralized window class, the individual positive and negative coefficient sums satisfy S^εjk,P(X,T;h) T1/2(Xe-RT)-η, and the differentiated windows satisfy Sε,∇jk,P(X,T;h) T-1/2(Xe-RT)-η. Under the separated-scale condition Xe-RT ≥ T1/η, this gives the sharp estimate Sε,∇jk,P(X,T;h) = O(T-3/2). The safe determinant neutralizer is further reduced to the exact minimal pole divisor. Writing Zⱼₖ(z) = adj( Mⱼₖ(z)) Fⱼₖ(z)Δⱼₖ(z), the actual pole order of each sign component at a determinant zero ζ is μ^ε_ζ = max\ 0,\, ord_ζ Δⱼₖ - ord_ζ N^εⱼₖ \. The resulting polynomial Pᵃᶜᵗⱼₖ(z) = ∏ζ (z-ζ)max_ε μ^ε_ζ is the unique minimal monic scalar polynomial that regularizes both sign components. At a simple determinant zero, removability is equivalent to the completed forcing being annihilated by a local left-kernel covector. Higher-multiplicity removability is characterized by the corresponding adjugate-numerator jets. The final part gives a root-free contour-moment certification of the actual pole divisor. For a Jordan domain U whose boundary contains no determinant zero, define νU = 1/2π i ∫∂ U Δ'ⱼₖ(z)Δⱼₖ(z)\, dz and bU,m = 1/2π i ∫∂ U zᵐ Zⱼₖ(z)\, dz. The principal-part transform RU(w) = 1/2π i ∫∂ U Zⱼₖ(z)w-z\, dz is a rational vector whose reduced monic common denominator is exactly the local minimal pole polynomial Pᵃᶜᵗjk,U. It is the unique least-degree monic polynomial annihilating the vector moment sequence through a finite Padé-type recurrence. The first 2νU vector moments determine the complete rational principal part, while bU,0 = ⋯ = bU,νU-1 = 0 is equivalent to complete removability of every singularity in U. This provides a finite certification protocol that does not require locating the determinant zeros individually. The manuscript proves unconditional individual-sign smoothing for pair-dependent spectrally neutralized windows and gives exact local and contour criteria for arbitrary unmodified windows. It does not claim universal zero-freeness of the completed determinant, removability for every fixed pair, or a completed pair-specific numerical certification. Those questions are isolated as subsequent computational-spectral and augmented-rank problems. Version v0.15r1 is a precision revision of v0.15. It preserves all mathematical results while aligning the global and local exceptional divisors, removing a matrix-notation collision, clarifying the holomorphic left-kernel compatibility condition, and tightening the zero-count and disk-holomorphy conventions.