Fixed-Heat EF Infrastructure for Automorphic L-functions on GL(m) over K, Part I: Infrastructure

This paper establishes the complete, unconditional analytic and geometric infrastructure required for the fixed-heat explicit formula (EF) framework for automorphic L-functions on GL(m) over K. It rigorously proves five foundational components (A1-A5) and a ramified damping mechanism (R*), including a quantitative spectral gap on the v-odd subspace.

The foundational components are:

• A1 (Geometric Frame): A tight slice frame on the Maaß-Selberg (MS) ledger.

• A2 (Constructive Positivity): A constructive realization of the raywise half-shift, which yields a quantitative Loewner sandwich and unconditional positivity.

• A3 (Analytic Control): Rigorous, conductor-uniform bounds for the explicit formula terms on a fixed-heat window, utilizing the specialized analytic class S^2(a).

• R* (Ramified Damping): Quantitative control of the ramified block, providing exponential decay in the conductor Q_K(π) and the heat parameter t.

• A4/A5 (Amplification Engine): The second-moment amplifier and a positive, contractive, parity-preserving averaging operator, with a proven quantitative A5 spectral gap on the v-odd subspace.

This infrastructure is self-contained and makes no assumptions regarding the Generalized Riemann Hypothesis (GRH). It provides the foundation for Parts II and III of the series, which use these results to study applications to GRH certificates.