Multiplicative Logarithmic Geometry and Prime‑Zero Spectral Theory Helical Embeddings, Explicit Formulae, and Logarithmic‑Geometric Residual Spectrum

Inspired by the geometric intuition of vector‑speed helical spacetime, this paper systematically establishes a spectral representation of prime‑number distribution within the framework of multiplicative logarithmic geometry, structured in four layers of rigor: strict theorems, geometric representations, conditional criteria, and open problems. On the strict mathematical level, we define a continuous family of unitary mul- tiplicative characters on the positive‑integer semigroup, and prove the algebraic condition under which the mapping is injective. We construct the logarithmic prime‑power measure and rigorously prove that its Laplace transform equals the logarithmic derivative of the Riemann zeta function. Based on the Riemann ex- plicit formula, we derive the spectral expansion of the centered residual on the logarithmic scale, and correct the weight of zero modes as well as the growth order of the residual under the Riemann Hypothesis. On the geometric level, we construct a family of continuous‑frequency helical orbits and define the pushforward measure to embed the discrete prime struc- ture into the three‑dimensional helical manifold. We clarify that orbital tangential velocity is not constant under the Euclidean metric, and distinguish three indepen- dent concepts: geometric‑parameter velocity, physical speed‑of‑light, and spectral frequency. On the spectral‑dynamics level, we define the autocorrelation function of the truncated residual, strictly expand its diagonal and cross terms, and clarify the mechanism of zero‑difference interference. The scope of Montgomery’s pair‑correlation theorem is strictly delineated. On the open‑problem level, we present three construction routes for the Hilbert– Pólya spectral operator and propose a candidate critical‑growth criterion for the Riemann Hypothesis. We emphasize that the fine‑structure constant only has a gauge‑bundle extension interface and no quantitative derivation is currently avail- able. This paper strictly delineates boundaries between classical results, extended constructions and open problems. The value of this work lies in providing a uni- fied geometric‑spectral language for prime‑number distribution while adhering to mathematical rigor.