Prime Spectral Decomposition of the High-Dimensional Gauss Sphere Residual, Cusp-Form Scale Separation, and Universal O(R⁻²) Convergence of Discrete πd(R) Across Z⁵ through Z²⁵}

Building upon the hierarchical dimension-pairing and Chinese Remainder Theorem (CRT) Number Theoretic Transform (NTT) architectures that recently extended the 7- and 9-dimensional Euclidean ball enumerations (OEIS A055413 and A055415) twenty-fold out to radius $R = 10,000$ (n = R² = 10⁸ quadratic shells), this paper generalizes our multi-prime Proth NTT and 512-bit Garner mixed-radix CRT engine via dynamic log-Gamma prime sizing to execute the first high-precision asymptotic and spectral audit of the Gauss sphere residual ED(R) = ND(R) - VD(R) across eleven strategic dimensions spanning Z⁵ through Z²⁵: the classic baseline D ∈ \5, 6, 7, 8, 9\ ($R = 10,000$), the modular cusp-form threshold bridge D ∈ \10, 11, 12\ ($R = 3,000$), the unimodular weight-8 harmonic $D = 16$ ($R = 2,000$), the Leech dimension $D = 24$ ($R = 1,000$, weight 12), and the terminal half-integral weight bookend $D = 25$ ($R = 1,000$, weight $25/2$). Because exact lattice populations reach N₉(10000) ≈ 3.30 × 10³⁶ (122 bits) and N₂₅(1000) ≈ 7.70 × 10⁷¹ (239 bits), standard 53-bit IEEE-754 floating-point arithmetic suffers total catastrophic cancellation. We resolve this barrier by implementing a 106-bit Dekker--Knuth double-double (dd_real) extraction and Euler--Maclaurin inverse-even-power detrending pipeline (≈ 32 decimal digits), isolating the scale-normalized Hardy--Littlewood singular series waveform ΨD(R) = ED(R)/RD-2 across all eleven dimensions down to amplitudes of 10⁻¹⁰. First, we generalize Discrete Rational ΠD(R) ∈ Q and its root estimator πd(R) = [ΠD(R)]1/ D/2 to arbitrary dimension D, proving that Landau's α = D - 2 exponent locks the convergence rate of πd(R) → π permanently to O(R⁻²) for all D ≥ 5, while empirically resolving the second-order O(R⁻⁴) binomial curvature correction (≈ -4.5 × 10⁻¹²) in $D = 16, 24, 25$ to seven significant figures. Second, log-log regressions across all eleven dimensions confirm Landau's exponent α = D - 2 to four decimal places (α₅ = 2.9999 through α₂₅ = 23.0000, with R² = 1.00000 for D ≥ 12), proving via Deligne's Ramanujan--Petersson bound that modular cusp forms (D ≥ 10) scale as O(RD/2-1+ε) and remain strictly subordinate to the Θ(RD-2) Eisenstein equatorial discrepancy. Third, 65,536-point Hann-windowed Fast Fourier Transforms (FFT) and direct prime-modulo comb filtering (R p) establish three universal arithmetic laws across 5 ≤ D ≤ 25: (i) a Mod-4 Dimensional Dyadic Parity Collapse across all D ≡ 0 4 ($D = 8, 12, 16, 24$) in which the 2-adic Gauss-sum phase lock G(1,4)D ∈ R suppresses the fundamental $f = 1/2$ Nyquist mode to Rank 3 behind $f = 1/3$ ($p = 3$) and $f = 1/4$ (2²), followed by a 1.43 × 10⁶× Nyquist power resurgence at $D = 25$; (ii) an unbroken Even--Odd Dimensional Harmonic Inversion at prime $p = 5$ ($f = 1/5$ vs.\ $f = 2/5$) across all eleven dimensions; and (iii) exact period-8 ($p=2$), period-4 (p ≡ 3 4), and period-2 (p ≡ 1 4) Gauss-sum phase rotation cycles governed by a strict zero-sum divisibility conservation law.