SS-DYN: A Scale-Localized Leafwise Dynamical Closure for Scale Space: Protected Tensor Gravity, a Repaired Scalar Continuum, and Constrained Collective Matte

We give the canonical low-energy successor formulation of the Scale Space gravitational dynamics, resolving scalar- and matter-sector obstructions in earlier formulations. Physical logarithmic scale remains a real coordinate, while the observable gravitational geometry is leafwise: each scale leaf carries a four-coordinate clock-reading measurement geometry g₍μν₎(x, λ). A dynamical localizer X₀(λ) = vₓ tanh[κ(λ − λc)] generates the exact positive weight F₀(λ) = sech⁴[κ(λ − λc)]. The transverse-traceless tensor operator has one normalizable even zero mode, an odd bound state at 3κ² excluded by reflection parity, and an even continuum beginning at 4κ². The zero mode is a tensor of ordinary three-space and therefore carries exactly two polarizations. Its coupling is universal for every normalized protected matter profile. The nonminimal term F(X)R[g] makes the localizer a genuine scalar–tensor degree of freedom and invalidates an earlier zero-localizer-DOF argument. In the unrepaired theory the fully constrained quadratic localizer kinetic form is rank one, while nonlinear finite regulators retain additional scalar canonical directions and a nonzero cubic interaction in the null sector. We repair that background degeneracy by adding the site-local leaf kinetic term S₄X = −(rZₓ/2) ∫dλ d⁴x √−g g⁽μν⁾ ∂μX ∂νX, r > 0. It leaves the kink and tensor spectrum unchanged, preserves sitewise lapse linearity of the regulated square-root tree, and makes the reduced scalar kinetic operator positive and full rank. In the regulator-independent limit the scalar sector is one isolated Q-like pole plus a gapped continuum governed by K̂ψ = μ[rψ + γf̂⟨f̂, ψ⟩], K̂ = −∂ᵧ² + 4 − 6 sech²y, γ = 9/(8ζₓ). The essential edge is exact: mₛ,gap = 2κ/√r. A broad low-energy interval 2.5 × 10⁻¹² ≲ r ≲ 2.57 × 10⁻⁶ survives the present benchmark constraints. At r = 10⁻⁶ and ζₓ = 1.8115 × 10⁵, the exact continuum resolvent gives mQ/κ ≃ 340.936 and, in the normalization where the unrepaired pole has αQ = 1/3, an exact repaired pole coefficient αQ ≃ 0.320825. Ordinary matter is defined at low energy as a constrained rank-one collective scale bundle. A finite-regulator Dirac reduction removes all orthogonal matter scale pairs, and the reduced path integral contains only one collective canonical pair per ordinary field component. Gauge, Higgs and chiral-fermion sectors admit protected embeddings with different raw field profiles but a common gravitational stress envelope F₀(λ)T₍μν₎⁽⁴⁾(x). Arbitrary four-coordinate EFT operators remain in this protected sector. Nonprotected scale structure is therefore an explicit SS-MAT matching correction rather than an unavoidable KK-like matter tower. This matter completion also changes the phenomenology. In the weighted tensor problem a protected source has no massive-continuum overlap at linear order, so earlier millimetre W/W tensor-continuum curves are not universal SS-DYN predictions; they survive only as matched nonprotected-profile benchmarks. Protected ordinary matter couples instead predominantly to the short-range isolated scalar pole, with only about 1.2 × 10⁻³ of its scalar spectral strength transferred to the repaired continuum at the working benchmark. The nonlinear claim is deliberately finite-regulator: an open scalar-dependent dRGT/Hassan–Rosen tree supplies the constraint-controlling parent, while a strict fully nonlinear regulator-independent continuous-scale theorem remains open. Within that boundary, SS-DYN is internally closed enough to serveas the proposed low-energy dynamical architecture.