Effective Sourcing Epochs and the Geometry of Reheating-Selected Axion Contours

This paper develops an effective‑sourcing framework for reheating‑selected axion contours, clarifying how microscopic sourcing histories reduce to geometric data that control contour orientation in the (log⁡ma,log⁡fa) plane. Instead of relying on a detailed model‑dependent sourcing kernel, we classify sourcing histories into three effective classes—peak‑type, threshold‑type, and multi‑stage—and show that each class admits a reduction to an effective sourcing epoch a∗, an associated Hubble scale H∗=H(a∗), and an effective source amplitude Y1.

At leading order, these effective quantities may be represented by the exponent map

which directly determines the contour slope through the generalized formula

This framework separates genuine geometric effects—tilt deformations and departures from a single effective slope—from pure normalization or position shifts. It also identifies multi‑stage sourcing as a natural origin of curvature or piecewise‑tilt behavior in reheating‑selected axion contours.

The result is a model‑independent geometric layer between microscopic reheating dynamics and observable contour structure, clarifying which features of the sourcing history survive coarse‑graining and which do not.