Prime Spectral Decomposition of the High-Dimensional Gauss Sphere Residual, Cusp-Form Scale Separation, and Universal $\mathcal{O}(R^{-2})$ Convergence of Discrete $\pi_d(R)$ Across $\mathbb{Z}^5$ through $\mathbb{Z}^{25}$}

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 $R = 10,000$ ($n = R^2 = 10^8$ 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 $E_D(R) = N_D(R) - V_D(R)$ across eleven strategic dimensions spanning $\mathbb{Z}^5$ through $\mathbb{Z}^{25}$: the classic baseline $D \in \{5, 6, 7, 8, 9\}$ ($R = 10,000$), the modular cusp-form threshold bridge $D \in \{10, 11, 12\}$ ($R = 3,000$), 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 $N_9(10000) \approx 3.30 \times 10^{36}$ ($122\text{ bits}$) and $N_{25}(1000) \approx 7.70 \times 10^{71}$ ($239\text{ bits}$), 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 (\texttt{dd\_real}) extraction and Euler--Maclaurin inverse-even-power detrending pipeline ($\approx 32$ decimal digits), isolating the scale-normalized Hardy--Littlewood singular series waveform $\Psi_D(R) = E_D(R)/R^{D-2}$ across all eleven dimensions down to amplitudes of $10^{-10}$. First, we generalize Discrete Rational $\Pi_D(R) \in \mathbb{Q}$ and its root estimator $\pi_d(R) = [\Pi_D(R)]^{1/\lfloor D/2 \rfloor}$ to arbitrary dimension $D$, proving that Landau's $\alpha = D - 2$ exponent locks the convergence rate of $\pi_d(R) \to \pi$ permanently to $\mathcal{O}(R^{-2})$ for all $D \ge 5$, while empirically resolving the second-order $\mathcal{O}(R^{-4})$ binomial curvature correction ($\approx -4.5 \times 10^{-12}$) in $D = 16, 24, 25$ to seven significant figures. Second, log-log regressions across all eleven dimensions confirm Landau's exponent $\alpha = D - 2$ to four decimal places ($\alpha_5 = 2.9999$ through $\alpha_{25} = 23.0000$, with $R^2 = 1.00000$ for $D \ge 12$), proving via Deligne's Ramanujan--Petersson bound that modular cusp forms ($D \ge 10$) scale as $\mathcal{O}(R^{D/2-1+\epsilon})$ and remain strictly subordinate to the $\Theta(R^{D-2})$ Eisenstein equatorial discrepancy. Third, 65,536-point Hann-windowed Fast Fourier Transforms (FFT) and direct prime-modulo comb filtering ($R \bmod p$) establish three universal arithmetic laws across $5 \le D \le 25$: (i) a \emph{Mod-4 Dimensional Dyadic Parity Collapse} across all $D \equiv 0 \pmod 4$ ($D = 8, 12, 16, 24$) in which the 2-adic Gauss-sum phase lock $G(1,4)^D \in \mathbb{R}$ suppresses the fundamental $f = 1/2$ Nyquist mode to Rank 3 behind $f = 1/3$ ($p = 3$) and $f = 1/4$ ($2^2$), followed by a $1.43 \times 10^6\times$ Nyquist power resurgence at $D = 25$; (ii) an unbroken \emph{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 ($p \equiv 3 \pmod 4$), and period-2 ($p \equiv 1 \pmod 4$) Gauss-sum phase rotation cycles governed by a strict zero-sum divisibility conservation law.