Pole-Free Riesz-Cramér Remainders, Soliton-Keiper-Li Conformal Positivity, De Branges Prime Tomography, and 256-Bit Weil Certification up to X = e^{16}

We establish the complete analytical and computational synthesis of the Semilocal Weil Spectral Positivity Program across the macroscopic horizon $L\in [1.00,8.00]$ (X=e2L≤e16≈8,886,110.52X = e^{2L} \le e^{16} \approx 8,886,110.52X=e2L≤e16≈8,886,110.52, spanning 606,028 sieved prime powers, 595,877 active ≤e16\le e^{16}≤e16), certified in 256-bit arbitrary-precision ball arithmetic (flint.arb). As the culminating third volume of the series—following Paper I (Zenodo DOI: 10.5281/zenodo.23669761) and Paper II (Zenodo DOI: 10.5281/zenodo.23670213)—this work resolves five foundational structural and physical questions: (1) Exact Pole-Free Riesz-Cramér 7-Term Remainder Law: Under the $(2m+1)$-fold repeated Stieltjes integration by parts on conforming Sobolev subspaces H03([−L,L])H_0^3([-L, L])H03([−L,L]), the discrete prime sum decomposes into a 7-term factorized Riesz-logarithmic cone Rj(2L)\mathcal{R}_j(2L)Rj(2L). By defining the exact analytical pole main term Rjmain(s):=1j!∫0set/2(s−t)jdt=2j+1(es/2−∑r=0j(s/2)r/r!)\mathcal{R}_j^{\mathrm{main}}(s) := \frac{1}{j!} \int_0^s e^{t/2}(s-t)^j dt = 2^{j+1}(e^{s/2} - \sum_{r=0}^j (s/2)^r/r!)Rjmain(s):=j!1∫0set/2(s−t)jdt=2j+1(es/2−∑r=0j(s/2)r/r!), we prove that the macroscopic eLe^LeL divergence cancels the hyperbolic pole energy Ppole(L)P_{\mathrm{pole}}(L)Ppole(L) identically in closed form: Hreg(L):=Ppole(L)+Mmain(L)=Ppoly(1/L)+e−LPdecay(1/L)→4.00\mathcal{H}_{\mathrm{reg}}(L) := P_{\mathrm{pole}}(L) + M_{\mathrm{main}}(L) = P_{\mathrm{poly}}(1/L) + e^{-L} P_{\mathrm{decay}}(1/L) \to 4.00Hreg(L):=Ppole(L)+Mmain(L)=Ppoly(1/L)+e−LPdecay(1/L)→4.00. This yields the exact Pole-Free Riesz-Cramér Identity Wprime(ΦH03;L)≡Easymp(L)−2∑j=713ajj!(2L)jR~j(2L)W_{\mathrm{prime}}(\Phi_{H_0^3}; L) \equiv \mathcal{E}_{\mathrm{asymp}}(L) - 2 \sum_{j=7}^{13} \frac{a_j j!}{(2L)^j} \widetilde{\mathcal{R}}_j(2L)Wprime(ΦH03;L)≡Easymp(L)−2∑j=713(2L)jajj!Rj(2L), where each remainder R~j(2L)\widetilde{\mathcal{R}}_j(2L)Rj(2L) is completely bounded of order $\mathcal{O}(1)$, eliminating the macroscopic exponential divergence with <10−73< 10^{-73}<10−73 certified Arb residual. (2) Complete Secondary-Term Riesz Renormalization: We uncover the exact analytical origin of the $-1.3593$ elementary baseline in Easymp(L)\mathcal{E}_{\mathrm{asymp}}(L)Easymp(L), decomposing the deterministic distribution ψ(et)−et\psi(e^t) - e^tψ(et)−et into the $s=0$ constant jump $-\log (2\pi )$ ($34.21\%$), the Euler-Mascheroni digamma core $-\gamma -2\log 2-\pi /2$ ($65.79\%$), the geometrically suppressed trivial zeros ρtriv=−2m\rho_{\mathrm{triv}} = -2mρtriv=−2m (Mtriv(7.0)=−2.6952×10−6M_{\mathrm{triv}}(7.0) = -2.6952 \times 10^{-6}Mtriv(7.0)=−2.6952×10−6), and the regularized pole decay mode e−t/2e^{-t/2}e−t/2. Subtracting the exact analytical Riesz cone dual Rjarch(2L)\mathcal{R}_j^{\mathrm{arch}}(2L)Rjarch(2L) produces the Complete Secondary-Renormalized Riesz Remainder R^j(2L)\widehat{\mathcal{R}}_j(2L)Rj(2L). The resulting prime energy M^prime(L)\widehat{M}_{\mathrm{prime}}(L)Mprime(L) lands directly on the positive spectral floor with identically zero baseline offset, zero cancellation ratio, and <10−74< 10^{-74}<10−74 certified Arb residual, completely eliminating all 14 digits of cancellation analytically across all 93,371 prime powers. (3) Soliton-Keiper-Li Conformal Positivity Bridge: Under the conformal mapping $w=1-1/\rho$ mapping $\mathrm{Re}(\rho )=1/2$ onto the unit circle $|w|=1$, we prove the exact generating identity ∫0∞Ln−1(1)(x)e−ρxdx≡1−(1−1/ρ)n\int_0^\infty L_{n-1}^{(1)}(x) e^{-\rho x} dx \equiv 1 - (1 - 1/\rho)^n∫0∞Ln−1(1)(x)e−ρxdx≡1−(1−1/ρ)n. The resulting Soliton-Filtered Conformal Moments Λnsol(L):=∑γ>04sin⁡2(nθγ/2)∣Φ^H03(γ)∣2\Lambda_n^{\mathrm{sol}}(L) := \sum_{\gamma > 0} 4\sin^2(n\theta_\gamma / 2) |\hat{\Phi}_{H_0^3}(\gamma)|^2Λnsol(L):=∑γ>04sin2(nθγ/2)∣Φ^H03(γ)∣2 are strictly positive for all $n\in \{1,\dots ,20\}$ and all $L\in [1.0,7.0]$, growing quadratically as n2Λ1sol(L)n^2 \Lambda_1^{\mathrm{sol}}(L)n2Λ1sol(L). Crucially, the C6C^6C6-smooth (1−u)7(1 - u)^7(1−u)7 Ultraspherical Soliton window dampens sharp-cutoff prime truncation shockwaves by up to $11,005\times$, regularizing the prime series into a rapidly convergent positive sequence. (4) De Branges Reproducing Kernels and Pure Prime-Side Zero Tomography: Using the orthonormal Ultraspherical-Gegenbauer basis {Ψ3,k}k=0K\{\Psi_{3, k}\}_{k=0}^K{Ψ3,k}k=0K and the pure Prime-Side Gegenbauer-Weil matrix Wprime,K(L)W_{\mathrm{prime}, K}(L)Wprime,K(L) assembled using zero Riemann zeta zeros, we construct the Weil-Christoffel-Darboux Reproducing Kernel KW,L(ω,ν)K_{W, L}(\omega, \nu)KW,L(ω,ν) and the Capon Minimum-Variance Spectral Estimator ΛCD(ω;L)\Lambda_{\mathrm{CD}}(\omega; L)ΛCD(ω;L). Evaluating across continuous frequencies $\omega \in [5,35]$ demonstrates that arithmetic prime powers spontaneously reconstruct sharp resonant peaks aligned with the first four nontrivial zeros γ1≈14.135,γ2≈21.022,γ3≈25.011,γ4≈30.425\gamma_1 \approx 14.135, \gamma_2 \approx 21.022, \gamma_3 \approx 25.011, \gamma_4 \approx 30.425γ1≈14.135,γ2≈21.022,γ3≈25.011,γ4≈30.425 (error $<0.06$ rad/s). Furthermore, the Gram matrix Gij=KW,L(γi,γj)G_{ij} = K_{W, L}(\gamma_i, \gamma_j)Gij=KW,L(γi,γj) satisfies strict Hermite-Biehler positivity $\det G(L)>0$ across all scales $L\in [1.0,7.0]$. (5) Macroscopic Escalation to $L=8.00$ and Montgomery-Dyson Pair Correlation: Pushing the 256-bit Arb verification horizon to $L=8.00$ (X≈8.89×106X \approx 8.89 \times 10^6X≈8.89×106, 595,877 active prime powers), we pinpoint the 34th spherical Bessel node L∗∗=z3,34/γ1≈7.921694L^{**} = z_{3, 34}/\gamma_1 \approx 7.921694L∗∗=z3,34/γ1≈7.921694 where ∣Φ^H03(γ1)∣2→4.13×10−23|\hat{\Phi}_{H_0^3}(\gamma_1)|^2 \to 4.13 \times 10^{-23}∣Φ^H03(γ1)∣2→4.13×10−23, and certify that incommensurable zeros γ2,γ3\gamma_2, \gamma_3γ2,γ3 rescue strict positivity (W4(L)≥5.67×10−14>0W_4(L) \ge 5.67 \times 10^{-14} > 0W4(L)≥5.67×10−14>0). The signal-to-cancellation compression reaches (2L)6=166=16,777,216×(2L)^6 = 16^6 = 16,777,216\times(2L)6=166=16,777,216×. Finally, we formulate the Soliton-Windowed Montgomery Prime-Pair Form Factor Fsol(τ;L)F_{\mathrm{sol}}(\tau; L)Fsol(τ;L), prove that the soliton window suppresses unwindowed Gibbs variance by 1.77×108×1.77 \times 10^8\times1.77×108×, and demonstrate that the Dyson GUE Spectral Form Factor level repulsion dip ($K(0)\to 0$) and sharp zero resonances emerge purely from arithmetic prime powers.