A Finite Quantum Cell as the Source of Newton's Constant: A conditional, parameter-free MAT chain from arithmetic topology to an atomic SI readout

This paper tests whether a finite quantum cell can determine Newton’s constant rather than take it as an empirical input. Within the stated MAT architecture, a normalized finite trace determines one dimensionless response coordinate, δ*. A boundary closure then fixes the laboratory coupling α₀ and the electron action ηₑ. Dimensional transmutation generates the electron–Planck hierarchy: μₑ = exp(−ηₑ/δ*). The measured Rydberg constant R∞ provides the sole empirical dimensional anchor. The standard Rydberg identity then converts the dimensionless hierarchy into the Planck length and Newton’s constant. The result is: G(MAT) = 6.674240942595(15) × 10⁻¹¹ m³ kg⁻¹ s⁻². No direct measurement of gravity or the electron mass enters the forward calculation. Neither a Hubble scale nor a cosmological density is used. The result lies 8.85 parts per million below the CODATA 2022 central value, corresponding to 0.394 of its stated standard uncertainty. General Relativity treats G as an empirical coupling, while the Standard Model leaves the electron Yukawa coupling as an independent input. MAT instead assigns the atomic hierarchy, vacuum closure, and cosmological readouts to one common upstream response. Its central contribution is therefore a rigid and cross-sector falsifiable finite-to-continuum chain, rather than an isolated numerical agreement. The result remains conditional on the stated cell architecture. Deriving that architecture from a single master operator and establishing its universal stress–energy coupling remain the principal requirements for microphysical completion.