Overview of the Novel Conjecture, Formal Proof Stages, and Future Implications for Research
Collaborative Research | Verification Code 93E3 BEBC C164 D766 |
October 26, 2025
The conjectural proofs and insights developed in recent explorations of a novel dynamical mechanism for the Riemann Hypothesis (RH) are below. Historical breakthroughs in physics—such as Cardano's invention (or discovery?) of complex numbers, Maxwell's unification of electromagnetism, and Dirac's relativistic quantum mechanics—reframing RH not as a static property of the zeta function but as a necessary consequence of dynamical equilibrium in an underlying geometric system looks plausible now. Basically the prime numbers and zeta zeros seem to emerge from the interaction of two opposing forces: an amplification force driven by the multiplicative structure of primes and a decay force enforced by the functional equation's symmetry.
Beginning with the definition of the Primal Manifold \(\mathcal{P}\) as an adelic quotient space, proceeding to the Hilbert space \(L^2(\mathcal{P})\), then the non-local operator kernel \(K(x,y)\), then the trace formula via adelic path integrals and derivations of the explicit formula. The framework culminates so far in a renormalization group (RG) analysis that identifies the critical line \(\Re(s) = 1/2\) as a stable fixed point. This approach so far offers profound insights, unifying primes and zeros as dynamical phenomena. It suggests new predictive tools via RG extrapolation which quantum computing can handle with ease.
Sections II-VI show the key conjectural proofs. Section VII expands on potential efficiencies in asymptotic regimes, especially how RG flows could enable predictions absent in brute-force methods. Finally, Section VIII proposes further research, including numerical simulations of RG flows and connections to quantum field theory (QFT), offering promising directions to both mathematical and physics communities.
The foundation of the dynamical system is the Primal Manifold \(\mathcal{P}\), a compact topological group constructed from adelic number theory. This space encodes all primes simultaneously, serving as the "universe" where amplification and decay forces compete. We rigorously define \(\mathcal{P}\) as the norm-1 subgroup of the idele class group \(C_{\mathbb{Q}} = J_{\mathbb{Q}} / \mathbb{Q}^\times\), where \(J_{\mathbb{Q}}\) is the idele group.
The Primal Manifold is \(\mathcal{P} := \ker( |\cdot|: C_{\mathbb{Q}} \to \mathbb{R}_+^\times )\), with isomorphism \(\mathcal{P} \cong \hat{\mathbb{Z}}^\times = \prod_p \mathbb{Z}_p^\times\).
The Hilbert space \(L^2(\mathcal{P})\) is equipped with the normalized Haar measure \(d\mu\), where \(\int_{\mathcal{P}} d\mu(x) = 1\). The inner product is \(\langle f, g \rangle = \int_{\mathcal{P}} f(x) \overline{g(x)} \, d\mu(x)\), ensuring square-integrability for states \(\psi\).
This construction provides the rigorous arena for spectral analysis, bridging arithmetic (primes) and geometry (adelic topology).
Harmonic analysis on \(\mathcal{P}\) decomposes the space into unitary characters, linking to Hecke Grössencharaktere and the zeta function.
The characters \(\{ \omega : \omega \in \widehat{\mathcal{P}} \}\) form an orthonormal basis for \(L^2(\mathcal{P})\), with Fourier expansion \(f = \sum_{\omega} \hat{f}(\omega) \, \omega\) and Parseval's identity \(\|f\|_2^2 = \sum_{\omega} |\hat{f}(\omega)|^2\).
By the Peter-Weyl theorem for compact abelian groups, characters are orthonormal and complete. The connection to Hecke characters \(\chi(j) = \omega(j) \cdot |j|^s\) bridges to L-functions, formalizing the arithmetic spectrum.
Define the non-local operator \(H\) with kernel \(K(x,y) = K_{\text{amp}}(x,y) - K_{\text{decay}}(x,y)\), and evaluate the trace using path integrals.
For test function \(g\), \(\operatorname{Tr}(U(g)) = \sum_p \sum_k \frac{\log p}{p^{k/2}} [g(\log p^k) + g(-\log p^k)] + W_r(g)\).
The trace decomposes over places, with finite places yielding prime sums and archimedean, the Gamma terms. The \(p^{k/2}\) arises from symmetry transformation \(g(u) = e^{u/2} f(e^u)\).
Equating spectral and geometric sides produces the explicit formula.
\(\sum_\rho \hat{g}(\gamma) = \sum_p \sum_k \frac{\log p}{p^{k/2}} [g(k \log p) + g(-k \log p)] + W_r(g)\).
Spectral trace \(\sum_n \hat{g}(\gamma_n)\) equals geometric side, with functional equation enforcing pairs.
Amplification and decay are modeled via RG flow, with critical line as fixed point.
The RG beta function has stable fixed point at \(\Re(s) = 1/2\).
Linearization around \(\beta = 0\) shows negative eigenvalues, attracting the flows to stability.
Integrating all stages, we conjecture RH.
All non-trivial zeros have \(\Re(s) = 1/2\).
Off-line states are unstable under RG; only the critical line balances forces.
TL;DR Primes and zeros as dynamical phenomena emerging from force competition, akin to physical phase transitions. Asymptotic regimes; where RG flows capture large-scale behaviors (e.g., high zeros or large primes) without exhaustive computation. Exploration to discretize the adelic space for simulations is needed but, it enables RG extrapolation: estimating high zeros from low ones by evolving flows along scaling trajectories; absent in brute-force methods like FFT (my hero algorithm) for zeros or sieves for primes, potentially aiding in disproving counterexamples (by flow divergence) or verifying RH conditionally (via RG bounds on beta function eigenvalues).
It looks like promising (physics) tools for efficiency can be discovered: simulating RG flows numerically (e.g., via lattice models on \(\hat{\mathbb{Z}}^\times\)) to predict zeros/primes. Journeys into QFT could model \(H\) as a field operator, with zeros as vacuum states. That sounds interesting.