Building on the formal definition of the Primal Manifold \(\mathcal{P}\), its Hilbert space \(L^2(\mathcal{P})\), and the decomposition via Pontryagin duality, address the second challenge: constructing the kernel \(K(x,y)\) of the non-local operator \(H\) then evaluate the trace of the associated operator using a non-perturbative adelic path integral approach, linking it to the Riemann-Weil explicit formula. This step provides the causal mechanism for the stability criterion, with the amplification and decay forces manifested in the operator's decomposition.
The operator \(H\) is a non-local integral operator on \(L^2(\mathcal{P})\), defined by \[ (H\psi)(x) = \int_{\mathcal{P}} K(x, y) \psi(y) \, d\mu(y), \] where \(d\mu\) is the normalized Haar measure. To capture the opposing forces, we decompose \(H = H_\text{amp} - H_\text{decay}\), with corresponding kernels \(K = K_\text{amp} - K_\text{decay}\).
The kernel \(K(x,y)\) ensures \(H\) is self-adjoint under the symmetry \(\mathcal{I}\), with the balance \(H_\text{amp} = H_\text{decay}\) on the critical line; enforcing stability.
To evaluate the trace \(\operatorname{Tr}(H) = \int_{\mathcal{P}} K(x, x) \, d\mu(x)\), we use a non-perturbative adelic path integral over the full adele class space, extending to the non-compact factor \(\mathbb{R}_+^\times\). This "path integral" is the integral over closed "paths" (fixed points of the scaling action), corresponding to periodic orbits labeled by primes.
For a test function \(g\) with compact support and \(g(1) = 0\), the trace of the induced operator \(U(g) = \int g(\lambda) e^{-\lambda H} \, d\lambda\) is \[ \operatorname{Tr}(U(g)) = \sum_p \sum_k \frac{\log p}{p^{k/2}} g(\log p^k), \] where the sum is over primes \(p\) and positive integers \(k\). This equals the prime-power side of the Riemann-Weil explicit formula.
Trace is computed by integrating kernel on the diagonal, extended to adele class space \(C_{\mathbb{Q}} = \mathbb{R}_+^\times \times \mathcal{P}\). The path integral representation is the sum over fixed points of the scaling act, which are the hyperplanes \(H_p = \{x \in A_{\mathbb{Q}} : x_p = 0\}\) for each prime \(p\).
Spectral side: \(\operatorname{Tr}(U(g)) = \sum_{\rho} g(\gamma)\), where \(\rho = 1/2 + i\gamma\) are the zeros, equating the two sides under RH as the stability condition.
The prime sum arises from \(H_\text{amp}\), amplification force, balanced by \(H_\text{decay}\) (from Fourier duality) so the spectral side aligns on the critical line. The reason for the zeros' location? the stability of the dynamical system. The zeros "look" random but they kind of have to be there <80)