Off-Line Coefficient Scaling

Abstract

Off-line coefficients split into density, phase, and scaling factors.

Theorem 1.1 (Coefficients split into density, phase, and scaling factors).

$$s=\frac{1}{2}+\Delta +it,e^{-s\ell }=e^{-\frac{1}{2}\ell }\cdot e^{-it\ell }\cdot e^{-\Delta \ell },\mathrm{scalingLedger}=\Delta \ell .$$

Proof. Machine-checked in Lean as D5/S3/Midline/OffLineCoefficientScaling.off_line_coefficient_scaling_spec (✓ std3). ∎

Source. Repository-derived.

Commentary.

For a spectral parameter with real displacement delta, each labeled coefficient factors into its critical half-density, unitary phase, and real scaling terms. The scaling ledger is exactly delta * length; when delta is nonzero the existing growth theorem supplies nonvanishing, common sign, natural scaling, and unboundedness on every positive-length address.

Multiplication by any complex unit preserves the coefficient’s norm. These are coordinatewise statements only and do not assert anything about cancellation after analytic continuation.

References

• Truth anchor: D5/S3/Midline/OffLineCoefficientScaling.off_line_coefficient_scaling_spec
• Dependency: D5/S3/Midline/OffLineScaling