SS-MEAS: Measurement Projection and Identifiability in Scale Space: Calibrated Observables, Null Spaces, and Withheld-Probe Tests

Scale Space represents physical logarithmic scale by u = ln(ℓ/ℓ₀) on a space–scale configuration manifold with normalized metric dΣ² = e²ᵘ d𝐱² + a² du². SS-MEAS isolates the programme’s measurement layer around a categorical distinction inherited from SS-GEO and sharpened in Paper 5: u is a physical logarithmic-scale coordinate, not a fourth ordinary spatial direction. A represented object may possess finite configurational extent in u without possessing a scale current. Static scale extension therefore precedes any assumption of scale transport. For a positive configurational observable q(𝐱,u), we define the invariant space–scale configuration measure dμSS = a e³ᵘ d³x du and a normalized scale marginal P(u). Nonzero width of P(u) defines scale extension with respect to the chosen observable. Ordinary measurements are represented as calibrated probe-dependent projections: Oα(𝐱obs) = ∫ dμSS Kα(𝐱obs;𝐱,u)q(𝐱,u). This measurement map yields an admissibility-qualified identifiability framework. Two physically admissible configurations are observationally indistinguishable when their difference lies in the measurement null space. In finite-dimensional models, reconstruction depends on rank, conditioning, and admissibility. More importantly, a withheld linear observable can be uniquely predictable even when the complete latent profile is not uniquely reconstructible, provided that the observable is invariant over the admissible training equivalence class. The earlier configurational-stress mathematics is retained. For a diagonal spatially isotropic configurational stress, covariant balance remains: dPu/du + 3(Pu − P⊥) = fu. Free, fixed, elastic, and reduced periodic scale-boundary conditions remain idealized configurational constructions. They are not automatically gravitational source conditions. SS-DYN and SS-MAT control the separate gravitational interpretation. The first empirical gate is a common-profile withheld-probe test, not a transport measurement. SS-MEAS specifies the general protocol: calibrate the forward operators, fit on a declared subset of probes, predict a withheld probe, compare against conventional null models, and report whether success concerns only the measurement map or discriminates the physical-scale hypothesis. The existing dentin and correlative-microscopy proposals are retained as downstream SS-BIO applications rather than as evidence already establishing physical scale extension. Scale transport, including drift, Fickian, persistent, and jump regimes, is likewise downstream constitutive physics.