Three Readouts, One Surface Tile: Newton, Planck, and the Bekenstein Count as a Single Identity

One surface tile, three familiar readouts Version: 2.0 Concept DOI: 10.5281/zenodo.21249112 Author: Ali Attar Website: quantumtraction.org Main book: Quantum Traction Theory, v10.01 This paper asks whether three familiar gravitational and quantum coefficients are separate inputs, or three readings of one finite surface object. Quantum Traction Theory assigns one completed two-sided address the area quantum \[ Q_\Sigma=8\pi\ell_A^2. \] Its count face and response face are \[ \frac{N_H}{A}=\frac{2\pi}{Q_\Sigma}=\frac{1}{4\ell_A^2}, \qquad \kappa_E=\frac{8\pi G_A}{c^4}=\frac{Q_\Sigma}{\hbar c}. \] Multiplying those two readings removes the surface scale and returns one completed modular charge: \[ \boxed{ \frac{N_H}{A}\,\kappa_E\,\hbar c=2\pi }. \] The Newton and Planck faces are therefore connected by \[ \boxed{ G_A=\frac{\ell_A^2c^3}{\hbar} } \qquad\Longleftrightarrow\qquad \boxed{ \hbar=\frac{c^3Q_\Sigma}{8\pi G_A} }. \] Version 2.0 proves the dependency structure behind these equations. With fixed \(c\) and \(\hbar\), the three logarithmic readouts carry one source degree of freedom: \[ \frac{\partial(\ln(N_H/A),\ln\kappa_E,\ln G_A)} {\partial\ln Q_\Sigma} =(-1,+1,+1). \] This rank-one result is both the economy and the evidential limit of the theorem: the rows agree because they share one surface tile, so they are not counted as three independent observations. The inherited non-gravitational weak-sector ruler gives \[ G_A=6.674301404655\times10^{-11}\ { m m^3\,kg^{-1}\,s^{-2}}, \] which is \(+0.210457\) ppm, or \(+0.009566\sigma\), from the stated CODATA \(G\) comparator. The inverse Planck row is printed as a round-trip consistency transport, \[ \hbar_{\rm rt}=\frac{c^3\ell_A^2}{G_{\rm lab}} =\hbar\frac{G_A}{G_{\rm lab}}, \] because the weak-sector ruler already contains SI-exact \(\hbar\). The paper also separates the closed horizon-address count from the still-open entropy measure: \[ S_H=s_0k_BN_H, \qquad \eta_S\kappa_E\frac{\hbar c}{k_B}=2\pi s_0. \] Thus the standard Bekenstein-Hawking coefficient is the \(s_0=1\) branch; deriving that microscopic measure remains a named source theorem. A separate normalization firewall shows that bundle completion produces the physical endpoint \(m_P\), while \(2\pi m_P\) is only the ordinary-Compton convention obtained from \(h=2\pi\hbar\). Included files: PDF paper, LaTeX source, standard-library verifier, frozen verification output, release notes, render audit, and SHA-256 manifest.