Paper 10: Resolving the Gravitational Time Dilation Discrepancy in the Five-Dimensional Scale Space Framework: The Corrected gtt and Its Consequences

Paper 9 of this series embedded the four-dimensional Riemannian scale-space framework in a five-dimensional Lorentzian parent theory and identified a factor-of-2 discrepancy in gravitational time dilation as an open problem: the block-diagonalmetric dΣ² =−c² dt² + e^ (2s/L) dx² + α² ds² gives dτ/dt= sqrt1−4GM/ (Rc²) versus the Schwarzschild result sqrt1−2GM/ (Rc²) confirmed by GPS measurements. We resolve this discrepancy exactly. The correction requires a single modification: replacing the flat gtt =−c² with gtt =− (1 + 2/L) c², L= Rc²/GM, yielding the corrected 5D metric dΣ²corr = − (1 + 2/L) c² dt² + e^ (2s/L) (dx² + dy² + dz² + α² ds²). Within the framework’s Foundational Principle (Section 1. 2), this modification to gtt is not a correction to time but an effective metric representation of configurational change not yet captured by the (x, y, z, s) manifold. Time remains the measure of total configurational change; the model of that change is what is corrected. (1) All geodesics of Papers 1–8 are exactly preserved. Since gtt depends only on L (a body parameter), not on (x, y, z) or s, all Christoffel symbols with a t-index in the spatial sector vanish. The spatial and scale geodesic equations are algebraically identical to those of the block-diagonal metric. (2) The key algebraic identity. The resolution rests on the exact identity (1 + 2/L) −4/L= 1−2/L, where 4/L= α²˙s²/c² is the ˙s-contribution to the proper time formula. The +2/L correction to gtt precisely cancels the excess, yielding dτ/dt= sqrt1−2/L = sqrt1−2GM/ (Rc²) exactly — the Schwarzschild result, valid to all orders in GM/ (Rc²), not merely to first order. (3) SR time dilation is exact. In the flat-space limit (L→∞), the correction 2/L→0 and ˙ s→0, recovering dτ/dt= sqrt1−v²/c² with universal c. (4) The L (s) -dependent case. When L is allowed to vary with s (the full dynamical theory), a new Christoffel symbol Γˢ tt =− (c²/L³) dL/ds appears in the scale geodesic, sourcing a new s-force from the t-sector. For the current framework with L constant at a given scale position, this term vanishes and the scale geodesic is unchanged. The corrected metric is the unique minimal modification of the block-diagonal metric that satisfies the three constraints simultaneously within the current diagonal, spatially-uniform, constant-L ansatz: correct Newtonian force, exact SR time dilation, and correct gravitational time dilation. It is consistent with but not yet derived from the 5D field equations; the derivation from G^ (5) MN + Λ5g^ (5) MN = κ5T^ (5) MN remains open.