MATHEMATICS • NUMBER THEORY
A bounded‑real Herglotz/Schur route on \(\{\Re s>\tfrac12\}\) using Schur–determinant splitting, \(\det_2\) continuity, and KYP lossless closure. Built with our axiomatic bridging method.
The proof casts RH as a bounded‑real problem on the right half‑plane. Define the prime‑diagonal operator \(A(s)e_p = p^{-s} e_p\), the completed zeta \(\xi(s)\), and the Hilbert–Schmidt regularized determinant \(\det_2\). Setting \(J(s)=\det_2(I-A(s))/\xi(s)\) and \(\Theta(s)=(2J-1)/(2J+1)\), RH follows from Schur positivity: \(|\Theta(s)|\le 1\) on \(\Re s>\tfrac12\).
Using axiomatic bridging, we translate the recognition‑ledger primitives into operator constraints, then execute a control‑theoretic program: realize prime‑grid lossless finite stages, certify passivity by KYP, control \(\det_2\) limits in the Hilbert–Schmidt topology, and conclude Schur positivity in the limit.
Block‑factorization: \(\log\det_2(I-T)=\log\det_2(I-A)+\log\det(I-S)\). Separates \(k\ge 2\) (HS) terms from the finite \(k=1\)+archimedean block.
Prime truncations \(A_N\to A\) in HS imply local‑uniform convergence of \(\det_2(I-A_N)\) on \(\{\Re s>\tfrac12\}\).
Finite‑stage passive realizations tied to primes (diagonal templates, exact/k‑fold blocks) with lossless KYP certificates.
Uniform‑in‑\(\varepsilon\) local \(L^1\) theorem via a smoothed estimate for \(\partial_\sigma\Re\log\det_2(I-A)\) and de‑smoothing. Outer neutralization → unimodular boundary values.
Finite stages align with the \(\det_2\) target (Cayley difference bound). The Schur class is closed under local‑uniform limits → \(|\Theta|\le1\).
Notation: \(\Theta=(H-1)/(H+1)\), \(H=2\,\det_2(I-A)/\xi-1\). HS = Hilbert–Schmidt, KYP = Kalman–Yakubovich–Popov.
We isolate the Euler \(k=1\) term and archimedean/pole pieces into a finite Schur complement that stays contractive on \(\{\Re s\ge \sigma_0\}\).
Prime‑grid realizations with diagonal Lyapunov certificates yield explicit lossless equalities and \(\|H_N\|_\infty\le1\).
A direct, unconditional smoothing bound on \(\partial_\sigma\Re\log\det_2(I-A)\) gives a uniform‑in‑\(\varepsilon\) local \(L^1\) limit after de‑smoothing.
With HS control and alignment, the Schur class persists in the limit — delivering \(|\Theta(s)|\le1\) for \(\Re s>\tfrac12\).
Recognition Science begins from a ledger‑based conservation and fairness calculus. The same invariances that force \(J(x)=\tfrac12(x+1/x)-1\) and \(8\)-tick completeness drive the control‑theoretic structure used here: passivity, contractivity, and Schur positivity. The bridge lets us import these invariances into analytic number theory without assuming zero‑free regions.
See the bridge design: Axiomatic Bridging. Foundations: Logical Foundations.
Prime‑diagonal operator: \(A(s)e_p=p^{-s}e_p\). Completed zeta: \(\xi(s)=\tfrac12 s(1-s)\pi^{-s/2}\,\Gamma(s/2)\,\zeta(s)\).
Regularized determinant: \(\det_2(I-K)=\det\big((I-K)\,e^{K}\big)\), continuous on HS with Carleman bound \(|\det_2(I-K)|\le e^{\|K\|_{\mathrm{HS}}^2/2}\).
Target Schur function: \(\Theta(s)=\dfrac{2\,\det_2(I-A(s))/\xi(s) - 1}{2\,\det_2(I-A(s))/\xi(s) + 1}\).
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How we translate invariances into other domains.
Core formulas and the fairness cost \(J\).
How predictions are tested.
Where this fits in the scientific story.