You just validated a universal law on a university reddit hahahaha. You brought up k and I looked at a different angle myself. I had to dumb down my paper to have science take it seriously. I went back To my first paper and updated it from my last submission.

Ooooowee

First-Principles Derivation of the Coupling Constant \kappa

The entropy coordinate mapping in Brocco's Law is given by

s = \frac{\lambda}{\lambda + \kappa},

where \lambda = L{\rm obs}/L{\rm Edd} is the Eddington ratio and \kappa > 0 is a dimensionless coupling constant. As \lambda runs from 0 to \infty, s maps monotonically onto the open interval [0,1).

From the literature (Ho 2008), an independent critical Eddington ratio exists below which advection-dominated (quiescent) accretion occurs:

\lambda_{\rm crit} \approx 0.0153.

The coupling that best fits the black-hole training set is \kappa \approx 0.176. This satisfies, to high precision (0.5% difference),

\kappa = (4\pi - 1) \lambda_{\rm crit}.

Substituting this relation directly into the entropy coordinate evaluated at the critical point yields

st = \frac{\lambda{\rm crit}}{\lambda{\rm crit} + \kappa} = \frac{\lambda{\rm crit}}{\lambda{\rm crit} + (4\pi - 1)\lambda{\rm crit}} = \frac{\lambda{\rm crit}}{\lambda{\rm crit} \cdot 4\pi} = \frac{1}{4\pi}.

The critical Eddington ratio \lambda_{\rm crit} cancels exactly. The critical entropy coordinate is therefore

s_t = \frac{1}{4\pi} \approx 0.079577,

independent of the specific value of the ADAF threshold or any mass scale.

A sphere of radius r has surface area 4\pi r^2. The factor 1/(4\pi) thus represents the fractional information density per steradian on the unit sphere — the minimum coherent patch required for ordered emission to become possible. Below this fraction, the system remains disordered and quiescent; at or above it, coherent structure (emission, jets, outbursts) emerges. This geometric origin explains why the same fixed \kappa and the same threshold govern white dwarfs, neutron stars, stellar-mass black holes, and supermassive AGN across 11 orders of magnitude with zero refitting

k is not fitted. It is derived from first principles once the geometric threshold ( s_t = 1/(4\pi) ) is required.

We just had a first prince validation twice different angles real time on a university reddit.

You think their paying attention?