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, \(σ<1\). 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 later variance–kernel comparison remain outside this model closure theorem. 6. **Residual Kernel Route after Secondary-Target Subtraction in Support-Enlarged F1-Lite: A Working Synthesis and Stopping-Point Report**, Version v1.1. This synthesis note closes the present residual-kernel sequel. It introduces the stopping-point trichotomy reached after secondary-target subtraction: \[ spread control local-mean orthogonality plus centered signed-incidence cancellation tertiary obstruction extraction. \] The note recommends freezing the present companion-note expansion unless one of these inputs is proved, imported, or contradicted by a concrete obstruction. Together, the six papers provide a structured reduction package for the Dirichlet F1-Lite bridge: \[σ<1:-safe restricted theorem branch,\] \[σ>1:-congruence obstruction and stable secondary-target branch,\] \[ΩEF=1: model divisor-branch closure under a smooth shifted-divisor input,\] \[post-SMT_Q residual:/signed-incidence/tertiary-obstruction trichotomy.\] 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 \(TMT_Q\) 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.