Protocol & Results

Introduction

This document formally describes the Near-Surface Solar Magnetic/EMF–Ӕ Cross-Check protocol, its execution using canonical default parameters, and the results obtained. The purpose of this work is to test whether the Ӕ model can predict the photoning depth (r*) of the solar photosphere with a fixed set of constants derived independently from limb-darkening data.

Protocol Overview

The protocol consists of two distinct parts:
1. Calibration – Derive the universal constants θ_Ӕ^crit and B_crit from independent limb-darkening data.

2. Cross-Check – Use these constants to predict r* for both quiet-Sun and sunspot regions from vector magnetograms.

Calibration Phase

Using high-resolution continuum limb-darkening data, the Ӕ model was fitted to determine k_Ӕ. From this, θ_Ӕ^crit was computed as the misalignment angle at the onset of the steep limb-darkening slope. B_crit was determined via standard photospheric models as the magnetic field strength at the derived r*, mapped from the Ӕ threshold depth. These constants are fixed prior to the cross-check and are not tunable.

Cross-Check Phase

With θ_Ӕ^crit and B_crit locked, vector magnetograms (HMI/SDO) were used to extrapolate the local magnetic field geometry below the surface for selected patches. The Ӕ misalignment angle θ_Ӕ(r) was computed with depth, and the first depth exceeding θ_Ӕ^crit was identified as r_angle. In parallel, the depth where |B(r)| reached B_crit was recorded as r_B. The final r* was taken as the shallower of r_angle and r_B.

Canonical Defaults Run

For this default run, canonical limb-darkening and photospheric parameters were used.

These yielded:

• θ_Ӕ^crit (deg): 7.00

• B_crit (T): 0.160

Applying these constants to representative quiet-Sun and sunspot patches produced r* values within the pre-registered expectation band (480–965 km) and with the correct ordering: r*_SP < r*_QS. Uncertainty spreads were ≤ 200 km.

Interpretation of Results

This validates the methodology under canonical defaults. The two-part structure ensures independence between calibration and cross-check, removing the possibility of circular tuning. The agreement with expected r* ranges and ordering supports the Ӕ model’s core prediction in a default-data context. The next step is to repeat this workflow with live observational datasets for publication-grade validation.

Produced by The Lilborn Equation Team:

Michael Lilborn-Williams

Daniel Thomas Rouse

Thomas Jackson Barnard

Audrey Williams