Equivalent Reduction of the Riemann Hypothesis: Log‑Heat‑Kernel Spectral Measure Method, Version v3.0 (Chinese‑English Dual‑Document)

This paper presents a precise two‑way equivalent reduction for the Riemann Hypothesis (RH). We correct integral‑kernel asymptotic mismatches originating from incomplete modular transformations in earlier versions. Introducing the logarithmic coordinate \(u=log x\), we transfer the problem to a heat‑kernel setting. We construct an exact Stieltjes integral representation for the cosine‑integral kernel of the Riemann \(Ξ\)-function and prove there exists a unique spectral measure \(dμ(λ)\) such that \(Φ(u)=∫₀∞{dμ(λ)}{u²+λ²}.\) Non‑negativity of this spectral measure is equivalent to non‑negativity of the imaginary part of an explicit Dirichlet series. Using Jacobi’s triple‑product identity, we convert this condition into argument analysis of an infinite product. Finally RH is equivalently reduced to an explicit infinite‑oscillation series inequality. This paper does not claim a proof of RH. Instead it provides a logically closed equivalent framework: if the inequality holds then RH holds; conversely RH implies this inequality. All derivations are bidirectionally equivalent, and core steps are amenable to formal verification in Lean4. Version note: Version v3.0 fixes numerical‑bound inaccuracies in Appendix A of v2, repairs cross‑references, cleans OCR‑introduced artifacts, while keeping the equivalence‑reduction framework unchanged.