Subconvexity–GUE Energy Interfaces and F1-Lite Dirichlet Variance Transduction: Conditional Framework, Diagonal-Safe Bridge, and Support-Enlargement Obstructions

This record contains a three-paper manuscript set on conditional interfaces between subconvexity-type analytic savings, GUE-type pair-correlation statistics, and explicit-formula variance transduction in Dirichlet families. The set consists of: 1. **Energy Interfaces between Subconvexity Bounds and GUE-Type Pair-Correlation Statistics: A Conditional Variance–Kernel Transduction Framework**, Version v4.0r1. 2. **A Diagonal-Safe Explicit-Formula Variance-Discrepancy Bridge for Dirichlet Families**, F1-Lite Restricted Theorem Note, Version v3.7r1. 3. **Product-Congruence Obstructions to Support-Enlarged Variance Transduction in Dirichlet Families**, F1-Lite Max-Push Note, Version v3.8r1. The first paper gives the public-stabilized conditional interface architecture. It does not claim that subconvexity alone proves GUE universality, nor that GUE input alone proves subconvexity. Instead, it formulates a shared variance–kernel–energy channel through which progress on either side can constrain the other once explicit bridge hypotheses are supplied. The central interface energy has the schematic form $$ E_T[ρ_T,K_T]=KL(ρ_T\|ρ_T^)+λₛₚSpec_T(ρ_T)+λₖₑᵣ\|K_T-K_T^\|HS(w_T)^2+μregRegΔ,T(ρ_T).$$ Here \(ρ_T\) is an unfolded zero-statistic law, \(K_T\) is a windowed pair-correlation kernel, and \(K_T^\) is the GUE/sine-kernel reference. The KL–low-mode–regularity component is based on the companion foundation note “Entropy–Rigidity Energy Functionals for Arithmetic Statistics,” Version v2.0r3, DOI: 10.5281/zenodo.20132780. The second paper extracts the first restricted theorem-level F1-lite branch in a prime-modulus Dirichlet setting. For primitive nonprincipal characters modulo prime \(q Q\), heavily smoothed explicit-formula test classes, and coefficient support \(n≤ Q^σ\) with \(σ<1\), primitive-character orthogonality forces n≡ m q, n,m≤ Q^σ<q=m. After subtracting the matched diagonal target and imposing the restricted explicit-formula ledger for smoothing, gamma factors, conductor normalization, prime powers, and truncation errors, the variance-discrepancy export has the form VErrQ( FQ)(log Q)^-AEF+Q-1+σ+o(1). Thus, for fixed \(σ<1\), the diagonal-safe branch gives a restricted \(o(1)\) F1-lite variance-discrepancy bridge. The third paper records the complementary support-enlargement obstruction. Once \(σ>1\), nontrivial product-congruence pairs enter: $$n=m+kq$$ and, after Type-II decomposition, equations of the form $$ab-cd=kq$$ appear. These contributions are not controlled by the pure diagonal target alone. The support-enlarged branch is therefore reduced, in this Max-Push audit, to secondary-main-term absorption and weighted-character estimates. The refined support-enlarged target has the schematic form Dref,Q=Ddiag,Q+SMTQ, where \(SMT_Q\) is a secondary local product-congruence target. The remaining weighted-character package is summarized as WCC+WLV+GWF, where KNC denotes kernel nonconcentration, WLV denotes weighted large-value control, and GWF denotes good-character weighted fluctuation. This manuscript set does **not** claim a proof of GUE universality, subconvexity, the Riemann Hypothesis, or a new zero-density theorem. Its contribution is more specific: it provides a conditional Subconvexity–GUE interface framework, extracts a restricted diagonal-safe F1-lite variance-discrepancy theorem, and identifies the product-congruence and weighted-character obstructions that arise when one attempts support enlargement beyond \(σ=1\).