Integral Bridge for AIT in the Weight-1 Cyclic-Defect Regime: Spectral Adjugate Identity, Normalization, and General-s Algebraic Extension

4) Description (long form, Zenodo “Description” field)

This working draft develops an integral bridge framework extending the Adjugate Identity Theorem (AIT) program after the unified spectral paper. The manuscript isolates an algebraic spectral core, identifies the transfer matrix categorically at the K_0-level, and formulates a GL(n) bridge in the weight-1 unipotent cyclic-defect regime with explicit normalization control.

Main components:

1. A t-deformed spectral Cartan framework C(t)=(1+t^2)I+t\,\mathrm{Adj}(\Gamma), with cofactor identity and uniqueness over \mathbb{Q}(t).
2. Weight-1 GL(\ell,\mathbb{F}_q) block formulas in the regime e=\mathrm{ord}_\ell(q)=\ell-1, including binomial/Pascal specialization at t=1.
3. Separation of theorem layers:
• algebraic core under explicit hypotheses,
• categorical determination of \Lambda=\mathrm{adj}(C) from K_0-level spectral identities (not merely dimension-level identities),
• GL(n) identification layer using cited block/center inputs.
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5. A normalization-explicit bridge theorem:
   exact in Hecke normalization; in p-adic normalization, agreement holds up to a blockwise \ell-adic unit.
6. General-s extension via t=q^{-s}:
   unconditional as an algebraic identity in the t-model; analytic automorphic identification at s\neq0 is stated conditionally on the same bridge inputs.
7. A finite-group analogue check outside GL(n) presented with explicitly limited scope.

This release is intended for transparent verification: theorem status is scoped and labeled, dependencies are listed, and conditional inputs are separated from unconditional algebraic statements.