Mass Geometry of Spacetime: Intertwinning Time and Gravity from Planck to General Relativity

Mass Geometry of Spacetime: The Intertwining of Time & Gravity from Planck to General Relativity "Resolve Singularity & Reframing Dark Matter” Abstract Mass remains the most fundamental yet least understood quantity in physics. Four frameworks—the Higgs mechanism, QCD, general relativity (GR), and cosmological dark matter—each provide a partial account of mass, yet none answers the deeper question: what is mass? We demonstrate that these disconnected descriptions share a single geometric root: mass has been misidentified as substance rather than as a geometric object with intrinsic spacetime extent. Three independent physical constraints—the reality of proper time, the Heisenberg uncertainty principle, and the universal linear density c 2/G—establish that every mass must occupy a finite spacetime territory. The Schwarzschild solution yields the mass-independent GR invariant gBH · R = c 2/2 and its temporal form g · t = c/2, the Weaving Condition. The same time-dilation equation defines, at the cosmic scale, a coherence horizon Rmax where proper time again vanishes, giving g∗ = c/(2Tmax). Since Tmax is directly encoded in the arrow of time, g∗ is universal, yielding Q ≡ g∗/G = 42.86 ± 0.02 kg m−2 (from Arabic Al-Qadar, ‘the precise measure’) and the Mass–Geometry Law M = Qr 2 . Reversing the Schwarzschild boundary from the horizon to the natural pre-collapse radius recovers Newton’s law as GR at a different radius of the same spacetime. The central geometric result r 2 = RsRmax places every territory radius at the geometric mean of collapse and coherence boundaries, structurally excluding singularities. Two exact algebraic identities close the framework: 2QℓPRmax = mP and 2QPℓPRmax = Muniv. The framework—formulated entirely within standard GR—is verified across 60 orders of magnitude (slope 2.0001 ± 0.003, R 2 = 0.9997), reproduces galactic rotation curves without dark matter, predicts the CMB lensing amplitude AL = 1.067 (Planck 2018: 1.073 ± 0.091, 0.6σ), elevates the equivalence principle from axiom to theorem, and yields six falsifiable predictions testable within five years. Keywords: mass geometry; universal surface density; Al-Qadar; Weaving Condition; cosmic coherence horizon; arrow of time; natural gravitational territory; inverse Schwarzschild mapping; Planck–cosmic duality; singularity elimination; Cosmic Energy Conservation Law