Regularized Two-Dimensional Zero Measures, Gaussian Tube Entropy, and Zero-Density Profiles: Explicit Entropy Bounds from Classical Zero-Density Inputs

This manuscript develops a regularized two-dimensional zero-measure framework for studying horizontal critical-line concentration of zeta and automorphic \(L\)-function zeros, and updates the framework by inserting classical zero-density inputs into the entropy bridge. The starting point is a non-identifiability observation: a one-dimensional height density ρ^ε(t)=∑ⱼ φ_ε(t-γⱼ) cannot distinguish a critical-line zero configuration from an off-critical configuration with the same ordinates \(γ_j\). To retain the horizontal information relevant to the Riemann Hypothesis, the paper replaces the height density by a mollified two-dimensional zero measure μT^ε(β,γ)=1/N(T)∑0<γⱼ<Tφ_ε(β-βⱼ)φ_ε(γ-γⱼ)\,dβ\,dγ . The framework separates two diagnostic channels: - the horizontal critical-line concentration channel, measuring displacement in the \(β\)-direction;- the vertical spacing channel, measuring ordinate statistics in the \(γ\)-direction, including random-matrix or GUE-type spacing behavior. The first quantitative result is a Gaussian tube decomposition theorem. In the Gaussian model, the labeled horizontal KL cost is exactly the renormalized critical-line concentration divided by \(2ε^2\): Klab,Tε= Cᵣₑₙ(μT^ε)2ε². The observable mixture KL is the data-processing projection of this labeled cost, with the difference recorded by a posterior label-loss term: Klab,Tε= Kobs,Tε+ LT^ε . The second quantitative result is a zero-density-to-entropy bridge. The outside-strip zero-density profile controls the mean-square horizontal concentration through the tail-integral identity C(μT)=2/N(T)∫₀1/2u\,Nₒᵤₜ(u,T)\,du . Conversely, bounds for the horizontal concentration or the labeled Gaussian tube entropy imply zero-density upper bounds outside any prescribed critical-line strip. Version v4.1 adds explicit imported zero-density substitutions. In particular, a Selberg–Simonič near-critical-line profile of the form N(1/2+u,T)S T1-u/4log T yields, after normalization, Cᵣₑₙ(μT^ε)≤64AS/(log T)²[1-(1+log T/8)T-1/8]. Consequently, the Gaussian labeled tube entropy satisfies Klab,Tε≤32AS/ε²(log T)²[1-(1+log T/8)T-1/8]. Thus, for a logarithmically shrinking tube ε(T)=(log T)-α,0<α<1, the framework gives simultaneous localization and vanishing labeled tube entropy: ε(T)→0, Klab,Tε(T)→0. The manuscript also records how Ingham/Kadiri–Lumley–Ng type estimates give windowed entropy-tail bounds, and how Bellotti-type log-free estimates give far-right entropy-tail bounds near the edge of the critical strip. This manuscript does not claim a proof of the Riemann Hypothesis and does not claim a new zero-density estimate by itself. Its purpose is to provide a two-dimensional entropy-moment formulation of horizontal zero-density profiles, together with explicit entropy/KL consequences of imported classical zero-density estimates and an information-loss ledger for observable mixture diagnostics. Email: leeclinic@protonmail.com