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

This upload collects the RH–MMLS v4. 1ₐ 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 \ ( (, ) \) -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 4N (T) ₀^1/2uLT (u) \, du. In particular, a Selberg-type near-line profile gives the logarithmic concentration scale CT (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_ (, T) = _¹ (N (, 2T) -N (, T) ) \, d is directly dual to the dyadic horizontal entropy moment: ₀^1/2uN_ (1/2+u, T) \, du = ₁/₂^1A_ (, T) \, d. Therefore, the final Selberg–Simonič Littlewood-area inequality can be inserted directly into \ (Cₓ, ₂ₓ\), 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 profileT-bound tube entropy bound and, in the Level II-a note, Littlewood areaₓ, ₂ₓ-bound without passing through a pointwise \ (N (, T) \) -profile. This v4. 1ₐ 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