Paper 11: The Five-Dimensional Einstein Tensor of the Corrected Scale-Space Metric: Vacuum Structure, Required Stress-Energy, and the Non-Vacuum Nature of the Framework

Paper 10 derived the corrected 5D metric for the scale-space framework, replacing gtt =−c2 with gtt =− (1+2/L) c² to resolve the factor-of-2 gravitational time dilation discrepancy, and asked whether this metric satisfies the 5D Einstein equations G^ (5) MN + Λ5g^ (5) MN = κ5T^ (5) MN. We answer this by computing the full 5D Einstein tensor symbolically (SymPy-verified). Four results are established. (1) The Ricci scalar is unchanged. Both the corrected metric and the block-diagonal metric of Paper 9 give R=−12/L³, identical to the AdS4 result scaled to five dimensions. (2) The spatial and scale Einstein components are identical. Gxx = Gyy = Gzz = 3e^ (2s/L) /L³ and Gss = 3/L² for both metrics. The correction to gtt leaves the (x, y, z, s) sector of the Einstein tensor completely unchanged. (3) Neither metric is a 5D vacuum solution. The spatial sector requires Λ5 =−3/L³, while the tt sector requires Λ5 =−6/L³. These differ by 3/L³ and cannot be reconciled by a single cosmological constant. This structural inconsistency is present in both the block-diagonal (Paper 9) and corrected (Paper 10) metrics; it is a property of the entire metric class, not of the correction specifically. The framework requires a non-vacuum 5D theory with non-zero T^ (5) MN. (4) The correction is self-consistent: ∆Λ5 = 0. The gtt correction does not shift the cosmological constant. With Λ5 =−3/L³ (from the spatial sector), the required tt stress-energy changes from κ5Ttt =−3c²/L³ (block-diagonal) to κ5Ttt =−3c² (L+ 2) /L⁴ (corrected), a ratio of (L+ 2) /L= 1 + 2/L — exactly the gtt correction factor. The correction modifies gtt and Ttt in the same proportion. These results sharpen the physical interpretation of Paper 10: the gtt correction is not an independent geometric addition but a self-consistent modification requiring a corresponding non-vacuum energy component in the t-sector. This Ttt is the field-equation signature of the missing configurational change identified in Paper 10. Its physical origin — whether an additional dimension, additional coupling, or sourced matter field — remains the central open question.