H–MMLS Entropy–Zero-Density Framework: Two-Dimensional Zero Measures, Large-Value Certificates, and Direct Littlewood-Area Entropy Extraction, Version v4.2

This record contains Version v4.2 of the RH–MMLS entropy–zero-density framework, together with two companion notes on large-value certificates and direct Littlewood-area entropy extraction. The parent manuscript, `RH_MMLS_v4.2`, develops a regularized two-dimensional zero-measure framework for studying horizontal critical-line concentration of zeta and automorphic \(L\)-function zeros. The starting point is a non-identifiability observation: a one-dimensional height-density diagnostic depending only on the ordinates \(γ_j\) cannot distinguish a critical-line zero configuration from an off-critical configuration with the same ordinates. The framework therefore replaces the height density by a mollified two-dimensional zero measure\[μ_T^ε(β,γ)=1/N(T)∑0<γ_j<Tφ_ε(β-β_j)φ_ε(γ-γ_j)\,dβ\,dγ,\]so that horizontal critical-line deviation and vertical spacing behavior are measured by distinct diagnostic channels. The parent paper proves finite-dimensional diagnostic identities, including the Gaussian tube decomposition\[Klab,Tε={Cᵣₑₙ(μ_T^ε)}{2ε^2},\]together with a posterior label-loss decomposition relating labeled tube entropy to observable mixture KL. It also records the zero-density/entropy bridge\[C_T=2/N(T)∫_01/2 u\,Nₒᵤₜ(u,T)\,du\]and the converse bound showing that control of the horizontal moment \(C_T\) implies fixed-strip zero-density bounds. Version v4.2 adds a conditional Level III appendix. Under a source-normalized Weil–Guinand explicit-formula package, the appendix shows that a dyadic horizontal moment estimate of order\[C[T,2T](log T)⁻²\]would imply a dyadic fixed-strip zero-density consequence. This appendix is conditional: it isolates a sufficient explicit-formula source package and proves the entropy/zero-density consequence of that package; it does not prove the source package itself. The companion note `RH_MMLS_LargeValue_EntropyMoment_v1_0` formulates the large-value output layer of the framework. It shows how a zero-detecting large-value or Dirichlet-polynomial certificate exporting a profile\[N(1/2+u,T)≤ L_T(u)\]immediately yields a horizontal entropy-moment bound\[C_T≤4/N(T)∫_01/2uL_T(u)\,du,\]and hence a Gaussian tube entropy bound. This note does not prove a new zero-density estimate; it identifies the certificate format by which classical or future large-value estimates feed the RH–MMLS framework. The companion note `RH_MMLS_Direct_Littlewood_Area_Entropy_LevelIIa_v1_1` gives a stabilized Level II-a output. It observes that the Littlewood-area quantity\[A_Δ(σ,T)=∫_σ^1(N(τ,2T)-N(τ,T))\,dτ\]is directly dual to the dyadic entropy moment \(C[T,2T]\). For the same imported Selberg–Simonič Littlewood-area input and the same dyadic normalization, direct area-to-entropy extraction avoids the extra leading factor incurred by first converting the area estimate into a pointwise zero-density profile. This is an entropy-output saving, not a new pointwise zero-density estimate. ### Files in this record 1. `RH_MMLS_v4.2.pdf` Parent manuscript. Develops the two-dimensional zero-measure framework, Gaussian tube entropy, observable-KL/posterior-loss decomposition, zero-density-to-entropy bridge, entropy-to-density converse, and conditional Level III explicit-formula route. 2. `RH_MMLS_LargeValue_EntropyMoment_v1_0.pdf` Companion note on large-value estimates and entropy moments. Records the large-value certificate language needed to turn zero-detecting detector profiles into \(C_T\)-bounds and Gaussian tube entropy bounds. 3. `RH_MMLS_Direct_Littlewood_Area_Entropy_LevelIIa_v1_1.pdf` Stabilized Level II-a companion note. Exports an imported Selberg–Simonič Littlewood-area estimate directly into the dyadic entropy moment \(C[T,2T]\) and records the leading entropy-output coefficient saving relative to pointwise-profile export. ### Scope and claim boundary This record does not claim a proof of the Riemann Hypothesis. It does not claim that random-matrix spacing statistics imply critical-line alignment, and it does not claim a new zero-density estimate. Its purpose is to provide a mathematically explicit two-dimensional entropy diagnostic for horizontal zero displacement, to formulate the horizontal entropy moment \(C_T\) as a moment-compressed zero-density input, and to isolate Level I/II/III analytic routes by which classical or future zero-detection inputs can be exported into entropy-moment and fixed-strip zero-density consequences. Email: leeclinic@protonmail.com