SS-L5D: Clock Geometry and Consistent Five-Dimensional Extensions of Scale Space

A configuration geometry and a rule for clock readings are distinct ingredients of a physical model. This paper constructs and compares static five-coordinate clock-reading extensions of the normalized four-dimensional Scale Space geometry, with dimensionless logarithmic scale coordinate and invariant curvature radius. The extension introduces a positive clock coefficient, or lapse, and derives its clock comparisons, geodesic coupling, curvature, and required sources while keeping the configuration metric fixed. Three constructive results organize the analysis. First, a general positive lapse gives explicit source and clock formulas, allowing the consequences of a clock choice to be calculated rather than inferred from configuration geometry alone. Second, among lapses depending only on logarithmic scale, the unique Einstein-vacuum extension up to reference-clock normalization is the exponential lapse. This construction is a Poincaré patch of five-dimensional anti-de Sitter geometry, and its optical metric is Euclidean on a half-space. Third, the power-law exponential lapse family corresponds to established Lifshitz geometry, yields explicit conserved diagonal sources, and satisfies the matter null-energy condition for exponents greater than or equal to one, assuming positive gravitational coupling. For exponents greater than one over an unbounded logarithmic-scale domain, the lower-scale end exhibits a tidal singularity despite constant curvature scalars. Local satisfaction of the null-energy condition therefore does not establish global regularity. These results provide a classification of clock-reading geometries and benchmarks for further development of the Scale Space research programme. The clock-reading interpretation remains a modelling assumption, and a prescribed Einstein residual is not a derived matter theory. No derivation of the measured value of the speed of light, exterior gravitational fields, ordinary-matter clock responses, or empirical superiority is claimed. The constructions are evaluated alongside, rather than adopted in place of, the programme's leafwise dynamical architecture.