The Einstein coupling from surface addresses: 8πG/c⁴ = QΣ/ℏc, and why the 1/4 and the 8π are the same counting

One surface count, two famous coefficients This paper isolates a compact claim in Quantum Traction Theory (QTT): the coefficient \(1/4\) in the horizon entropy law and the coefficient \(8π\) in Einstein's field-equation coupling are two readouts of the same surface address quantum, \[ Q_Σ = 8π _A^2. \] The source-side entropy count is \[ S/k_B = 2π A/Q_Σ = A/4_A^2, η = 2π k_B/Q_Σ = k_B/4_A^2. \] The local horizon thermodynamic route then fixes the field-equation coupling as \[ { κQTT = Q_Σ/ c = 8π _A^2/ c = 8π G_A/c^4 }. \] The same-counting closure is the short invariant: \[ { η\,κQTT\, c/k_B = 2π }. \] In plain language: divide one completed surface bundle by its area and one obtains the entropy density; divide the same surface bundle by \( c\) and one obtains the gravitational coupling. The \(1/4\) and the \(8π\) are not separate normalizations in this framework. They are the same address count read through two different laboratory windows. The numerical value leg is inherited from the companion non-gravitational Planck-length construction: \[ G_A=_A^2c^3/, 8π G_A/c^4 = 2.0766478812×10⁻⁴³\ { s^2\,kg⁻¹\,m⁻¹}, \] with \(G_A\) landing \(+0.211\) ppm from the CODATA Newton constant, or \(+0.0096σ\) on the CODATA uncertainty. The included standard-library verification script regenerates the displayed numerical identities. Related QTT anchors: Main book: Quantum Traction Theory: Main Book v10.01 Non-gravitational Planck-length construction: 10.5281/zenodo.21190193 Newton's constant in QTT: 10.5281/zenodo.20057430 A2 endurance to Einstein-field dynamics: 10.5281/zenodo.20763263 Einstein-Hilbert coefficient source paper: 10.5281/zenodo.20042843 Website: quantumtraction.org Corpus DOI map: quantumtraction.org/doi-map/ Included files: PDF paper, Version 1.1.1 Complete reconstruction ZIP containing the LaTeX source, verification script, verifier output, change note, and SHA-256 manifest