F1-Lite Dirichlet Variance Transduction and the Residual Kernel Route after Secondary-Target Subtraction: Diagonal-Safe Theorem, Support-Enlarged Obstruction, Stable Secondary Targets, Model Divisor-Branch Closure, and Final Working Synthesis

This record contains a six-paper synthesis 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 narrower and more structural: it isolates a diagonal-safe restricted theorem branch for Dirichlet F1-Lite, identifies the support-enlarged product-congruence obstruction, constructs a stable secondary target, proves conditional model divisor-branch closure under a smooth shifted-divisor input, and records the final stopping point of the residual-kernel route after secondary-target subtraction. 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, variance–kernel comparability, reverse domination, threshold matching, and loss-compatibility gates. It also records the no-free-bridge principle: subconvexity-type savings do not become GUE statistics without explicit bridge theorems, and GUE-type kernel control does not become subconvexity without reverse-domination and threshold ledgers. 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, \ (1: -congruence obstruction and stable secondary-target branch, \] \^ EF=1: model divisor-branch closure under a smooth shifted-divisor input, \ -SMTQ residual: /signed-incidence/tertiary-obstruction trichotomy. \ 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; establishing local-mean orthogonality or centered signed-incidence cancellation for the residual kernel; deciding whether a stable tertiary main term \ (TMTQ\) is required; and proving the separate variance–kernel comparability gate needed for GUE-facing kernel-energy conclusions. This release should be read as a conditional/reduction/synthesis package, not as an endpoint theorem package.