Geometric Resolution of Galactic Rotation Curves: Matrix Tension as a Deterministic Replacement for Dark Matter

ABSTRACT: The anomalous flattening of galactic rotation curves is historically interpreted as evidence for an invisible "Dark Matter" halo encompassing baryonic galaxies. This paper demonstrates that Dark Matter is an unnecessary mathematical artifact born from the application of a structurally diluted macroscopic gravitational constant ($G_{macro}$) beyond its coherent geometric scale.

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By mapping the vacuum as a rigid structural matrix bounded by the universal $10^{-31}$ meter floor and scaled to the $10^{122}$ zero-latency limit, we derive the intrinsic geometric stiffness of the universe. We establish this stiffness as a baseline radial acceleration ($a_{matrix}$) derived strictly from first principles—utilizing only the physical Hubble radius and the volumetric packing drag ($\pi/6$) of the cubic lattice.

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When applied to the extreme outer rim of a galactic halo, the mathematical inclusion of this matrix tension perfectly cancels out the radial distance variable ($r$), natively forcing the orbital velocity into a constant plateau. This structurally resolves the rotation curve paradox using strictly derived geometric constants, completely eradicating the need for the dark sector.

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KEY DERIVATIONS & REVISIONS IN THIS VERSION:

• Eradication of Free Parameters: The derivation of the galactic halo velocity curve contains no arbitrary fitting parameters or phantom mass variables.

• The Baseline Tension: Calculates the absolute baseline structural stiffness of the universe by distributing the maximum kinematic limit ($c^2$) across the $4.44 \times 10^{26}$ m Hubble radius.

• The Packing Drag Transition: Proves that the volumetric packing drag ($\pi/6$)—the same geometric shear responsible for the fine-structure constant at the microscopic scale—translates macroscopically into the structural tension against spinning galactic disks.

• Empirical MOND Proof: The derived geometric tension yields an absolute structural resistance of $1.059 \times 10^{-10}\text{ m/s}^2$. This provides the exact, deterministic mathematical proof for the empirical Modified Newtonian Dynamics (MOND) acceleration threshold ($a_0 \approx 1.2 \times 10^{-10}\text{ m/s}^2$) observed by Milgrom in 1983.