F1-Lite Dirichlet Variance Transduction and the Subconvexity–GUE Interface: Diagonal-Safe Theorem, Support-Enlarged Obstructions, Stable Secondary Targets, and Model Divisor-Branch Closure

This record contains a five-paper core set for the F1-Lite Dirichlet variance-transduction branch of a conditional Subconvexity–GUE interface framework. The package does not claim a proof of GUE universality, subconvexity, the Riemann Hypothesis, or a zero-density theorem. Its purpose is more specific: it isolates the Dirichlet-family F1-Lite bridge into a restricted diagonal-safe theorem branch, a support-enlarged product-congruence obstruction branch, a stable secondary-target package, and a conditional model divisor-branch closure under a smooth shifted-divisor input. The uploaded PDFs are: 1. **Energy Interfaces between Subconvexity Bounds and GUE-Type Pair-Correlation Statistics: A Conditional Variance–Kernel Transduction Framework**, Version v4.0r2. This parent manuscript formulates the conditional variance–kernel–energy interface. It separates the forward explicit-formula variance bridge, the variance–kernel comparability gate, the reverse obstruction package, threshold matching, and loss compatibility. 2. **A Diagonal-Safe Explicit-Formula Variance-Discrepancy Bridge for Dirichlet Families**, F1-Lite Restricted Theorem Note, Version v3.7r2. This note extracts the restricted theorem-level branch for coefficient support below the prime modulus threshold, \(σ<1\). In this range, primitive-character orthogonality forces \(n≡ m q\) with \(n,m<q\) to imply \(n=m\), eliminating product-congruence off-diagonal terms. Under a restricted explicit-formula ledger, it gives a diagonal-safe variance-discrepancy export. 3. **Product-Congruence Obstructions to Support-Enlarged Variance Transduction in Dirichlet Families**, F1-Lite Max-Push Note, Version v3.8r2. This note records what changes once the support crosses the \(σ=1\) threshold. For \(1<σ<1+η_0\), nontrivial congruence pairs \(n=m+kq\), and after Type-II decomposition equations of the form \(ab-cd=kq\), appear. The note formulates the no-free-support-enlargement principle and reduces the support-enlarged branch to secondary-main-term absorption and weighted-character residual gates. 4. **Secondary Product-Congruence Main Terms in Support-Enlarged Dirichlet Variance Transduction**, F1-Lite Companion Note, Version v0.5r6. This companion note constructs the stable secondary product-congruence target \(SMT_Q\). It fixes the exact primitive-character convention, separates actual shifted sums from singular factors, introduces the divisor-type singular factor, formulates coefficient-local correction factors, and gives boundedness and dyadic-refinement stability conditions for the secondary target. 5. **Model Divisor-Branch Closure for Support-Enlarged F1-Lite under a Smooth Shifted-Divisor Input**, Version v1.0r1. This note compresses the divisor-type model branch into a conditional closure theorem. In the locally unbiased divisor branch, where \(ΩEF=1\), a normalized smooth shifted-divisor input with summable block loss implies that the deterministic shifted-product component of the balanced product-congruence package is absorbed by the stable secondary target. The centered residual, endpoint/Type-I ledgers, the full \(ΩEF≠1\) coefficient-local case, and the later variance–kernel comparison remain outside this model closure theorem. Together, the five papers provide a structured reduction package for the Dirichlet F1-Lite bridge: \[σ<1:-safe restricted theorem branch,\] \[σ>1:-congruence obstruction and stable secondary-target branch,\] \[ΩEF=1: model divisor-branch closure under a smooth shifted-divisor input.\] The main remaining mathematical tasks after this release are: proving or importing sufficiently uniform shifted-product asymptotics for broader coefficient classes, proving centered product-congruence residual saving after secondary-target subtraction, handling endpoint/Type-I ledgers, extending the model divisor branch to the full EF/VHB coefficient-local setting, and establishing the separate variance–kernel comparability gate needed for GUE-facing kernel-energy conclusions.