A Geometric Origin for the MOND Acceleration Scale a0

Statement of Intellectual Property & Priority: This work (DOI: [10.5281/zenodo.20109310]) establishes the explicit geometric identification of the MOND acceleration scale $a_0$ and Milgrom's universal surface density $\Sigma_M$ as algebraic consequences of a geometric ansatz for gravity in the disk limit. The core originality of this Letter lies in three results derived from a single geometric ansatz $g(r) \propto r^{-(d-1)}$ under dimensional renormalization:

(i) the algebraic derivation of MOND's deep-acceleration relation $g_{\text{obs}} = \sqrt{a_0 , g_{\text{bar}}}$ from the geometric ansatz in the limit $d \to 2$, obtained without invoking any postulate of the MOND framework;

(ii) the explicit geometric identification $a_0 = GM/R_{\text{scale}}^2$, which assigns a definite geometric origin to MOND's universal acceleration scale;

(iii) the corresponding geometric identification of Milgrom's universal surface density $\Sigma_M = M/(2\pi R_{\text{scale}}^2)$, establishing both $a_0$ and $\Sigma_M$ as parametrizations of the same underlying geometric transition.

The numerical identity $a_0 = 1/(\beta G)$, while itself a known consequence of MOND combined with the baryonic Tully-Fisher relation, is here recovered through an independent geometric route, with no fit parameters and with explicit geometric content for both $a_0$ and $\Sigma_M$.

This Letter is a focused result within the broader geometric framework previously established by the author (DOI: 10.5281/zenodo.20109310), in which gravitational anomalies on galactic scales are identified as the emergence of effective dimensional deficit ($3-d$) in baryonic matter distribution.

Any subsequent theoretical models or empirical analyses employing (a) the algebraic derivation of MOND from a geometric ansatz with effective dimension $d$, (b) the geometric identification of $a_0$ as $GM/R_{\text{scale}}^2$, or (c) the geometric identification of $\Sigma_M$ as $M/(2\pi R_{\text{scale}}^2)$, are considered derivative works of this original framework. All priority rights are reserved under the time-stamped registration of this Zenodo record.