The Geometry of Prime Stability: From the Expanding Double-Cone to the Twin-Pinch Saddle Exclusion (A Master Synthesis of the Thirteen-Volume Semilocal Weil Spectral Positivity Series)

The Riemann Hypothesis (RH) asserts that the prime numbers are distributed as evenly as mathematically possible, with their oscillatory fluctuations governed by nontrivial zeros of the Riemann zeta function ζ(s)\zeta(s)ζ(s) lying strictly on the critical line Re⁡(s)=12\operatorname{Re}(s) = \frac{1}{2}Re(s)=21. In 1952, André Weil proved that RH is equivalent to the global positive-definiteness ("stiffness") of a bilinear quadratic form constructed from prime powers across all logarithmic window scales L→∞L \to \inftyL→∞. However, testing or proving Weil positivity directly faces three formidable obstacles: (1) an exponential O(eL)\mathcal{O}(e^L)O(eL) explosion from the average growth of primes (the pole at s=1s = 1s=1) that blinds numerical sensors at large scales; (2) the risk of methodological circularity if known Riemann zeros are inadvertently used to calibrate the prime operator; and (3) the need for a geometric principle explaining why off-critical zeros violate arithmetic stability. This master synthesis unites the findings of the thirteen-volume Semilocal Weil Spectral Positivity Series across up to 25,617,27225,617,27225,617,272 active prime powers (X=e20≈485.17 millionX = e^{20} \approx 485.17\text{ million}X=e20≈485.17 million) in certified 512-bit and 1024-bit FLINT/Arb ball arithmetic under a strict Zero-Free Firewall (ZERO_INPUT_COUNT == 0). We present the investigation through four unified pillars: The Expanding Double-Cone Geometry: Modeling the critical line as a linear double-cone rcore(t)=κ∣t∣r_{\mathrm{core}}(t) = \kappa |t|rcore(t)=κ∣t∣ joined at the narrow apex t=0t = 0t=0, where on-line zeros pinch the modulated zeta surface cleanly onto the central axis (r=0r = 0r=0). The Codimension-2 Noise-Canceling Filter: Isolating the exact two-dimensional hyperbolic subspace {cosh⁡(x/2),sinh⁡(x/2)}\{\cosh(x/2), \sinh(x/2)\}{cosh(x/2),sinh(x/2)} responsible for the 800×800\times800× Prime Number Theorem explosion at L=10.00L = 10.00L=10.00, and projecting onto its 21-dimensional pole-free, DC-free orthogonal complement H0,21super([−L,L])\mathcal{H}_{0, 21}^{\mathrm{super}}([-L, L])H0,21super([−L,L]), collapsing the prime ripple stress from 800.02800.02800.02 down to βfullsuper(10.00)=0.108918≪1.00\beta_{\mathrm{full}}^{\mathrm{super}}(10.00) = 0.108918 \ll 1.00βfullsuper(10.00)=0.108918≪1.00 (10.9%10.9\%10.9% of structural capacity) with zero artificial regularization (αBL≡0\alpha_{\mathrm{BL}} \equiv 0αBL≡0). Cramér-Littlewood Oscillatory Damping & Analytical Sub-Unity Bounds: Exploiting 6th-order contact vanishing K(m)(2)=0K^{(m)}(2) = 0K(m)(2)=0 (0≤m≤60 \le m \le 60≤m≤6) of conforming (1−ξ2)3∈H03([−1,1])(1 - \xi^2)^3 \in H_0^3([-1, 1])(1−ξ2)3∈H03([−1,1]) wavepacket correlation kernels to perform genuine numerical iterated integration across all 25.6M25.6\text{M}25.6M prime powers. Genuine oscillatory damping compresses the prime fluctuation primitives by S0(L)/S1(L)≈15.68−26.84S_0(L)/S_1(L) \approx 15.68 - 26.84S0(L)/S1(L)≈15.68−26.84 (mean ∼22.7×\sim 22.7\times∼22.7×), certifying that the 1st-order analytical Sobolev-Stieltjes bound satisfies βIBP(1)(L)≤0.9336<1.00\beta_{\mathrm{IBP}}^{(1)}(L) \le 0.9336 < 1.00βIBP(1)(L)≤0.9336<1.00 for all L≤6.00L \le 6.00L≤6.00, while the full 21-dimensional super space contracts to βfullsuper(10.00)=0.108918≪1.00\beta_{\mathrm{full}}^{\mathrm{super}}(10.00) = 0.108918 \ll 1.00βfullsuper(10.00)=0.108918≪1.00 with sectorial coercivity angle locked at θCdirect≥83.75∘\theta_C^{\mathrm{direct}} \ge 83.75^\circθCdirect≥83.75∘. The Twin-Pinch Saddle Instability, Strip Capping, & Infinity Rings: Deforming the double-cone to touch zero at an off-center quartet 12±δ±iω0\frac{1}{2} \pm \delta \pm i\omega_021±δ±iω0 forces a hyperbolic saddle stretch cosh⁡(2δL)\cosh(2\delta L)cosh(2δL) that buckles the positive cone at a finite tear horizon Ltear(δ)≈2.14/δL_{\mathrm{tear}}(\delta) \approx 2.14 / \deltaLtear(δ)≈2.14/δ. Enforcing the critical-strip boundary condition δ≤0.50\delta \le 0.50δ≤0.50, the permitted off-axis corridor contracts as δmax⁡strip(L)≈2.14/L→0\delta_{\max}^{\mathrm{strip}}(L) \approx 2.14 / L \to 0δmaxstrip(L)≈2.14/L→0 for L≥5.00L \ge 5.00L≥5.00 (δmax⁡strip(10.00)=0.213697\delta_{\max}^{\mathrm{strip}}(10.00) = 0.213697δmaxstrip(10.00)=0.213697), evaluated via transparent hybrid 64-bit quadrature assembly and certified in 512-bit Flint Arb alongside the cutoff-free Vasyunin–Möbius prime-power ring law G~pr,pr=p−(r−1)G~p,p\widetilde{G}_{p^r, p^r} = p^{-(r-1)}\widetilde{G}_{p, p}Gpr,pr=p−(r−1)Gp,p at x=∞x = \inftyx=∞.