RH–MMLS v4.1_a: Two-Dimensional Zero Measures, Large-Value Entropy Certificates, and Direct Littlewood-Area Extraction

This upload collects the RH–MMLS v4.1_a three-paper package on entropy–zero-density diagnostics for zeta zeros and related L-function zero sets. The package consists of: 1. **Regularized Two-Dimensional Zero Measures, Gaussian Tube Entropy, and Zero-Density Profiles** This parent manuscript develops a regularized two-dimensional zero-measure framework on the \((β,γ)\)-plane. Its purpose is to separate horizontal critical-line concentration from vertical spacing statistics. A one-dimensional height-density diagnostic cannot distinguish critical-line zeros from off-critical zeros with the same ordinates, so the paper replaces height-only diagnostics by a two-dimensional smoothed zero measure. 2. **Large-Value Estimates and Entropy Moments of Two-Dimensional Zero Measures** This companion note develops the large-value output layer of the RH–MMLS framework. It records how a classical zero-density or Dirichlet-polynomial large-value input of the form N(1/2+u,T)≤ LT(u) exports to the horizontal entropy moment CT≤ 4/N(T)∫₀1/2uLT(u)\,du. In particular, a Selberg-type near-line profile gives the logarithmic concentration scale CT(log T)⁻². The note also records Ingham/Kadiri–Lumley–Ng-type windowed inputs and log-free far-edge inputs as entropy-tail ledgers. 3. **Direct Littlewood-Area Entropy Extraction and a Leading-Constant Saving in Selberg-Type Zero-Density Output** This Level II-a companion note specializes the framework to the Selberg–Simonič Littlewood-area method. Its central observation is that the Littlewood area A_Δ(σ,T) = ∫_σ¹ (N(τ,2T)-N(τ,T))\,dτ is directly dual to the dyadic horizontal entropy moment: ∫₀1/2uN_Δ(1/2+u,T)\,du = ∫1/2¹A_Δ(σ,T)\,dσ. Therefore, the final Selberg–Simonič Littlewood-area inequality can be inserted directly into \(C[T,2T]\), without first converting it into a pointwise zero-density profile. For the same imported area inequality and the same dyadic normalization, this direct area-to-entropy route avoids the additional smoothing factor \(e\) that appears in the pointwise-profile export. The package does **not** claim a proof of the Riemann Hypothesis. It does **not** claim a new pointwise zero-density theorem. Its contribution is a mathematically explicit entropy–zero-density diagnostic framework, together with a large-value output layer and a sharper entropy extraction from an existing Littlewood-area estimate. The main conceptual chain is: two-dimensional zero measure⇒ entropy moment CT zero-density profile⇒T-bound⇒ tube entropy bound and, in the Level II-a note, Littlewood area⇒[T,2T]-bound without passing through a pointwise \(N(σ,T)\)-profile. This v4.1_a package should be read as an entropy-moment and diagnostic reformulation of horizontal zero-density information, with a concrete leading-constant saving at the entropy-output layer for the Selberg–Simonič area input. Email: leeclinic@protonmail.com