SS-DYN: A Scale-Localized Leafwise Dynamical Closure for Scale Space: Protected Tensor Gravity, a Repaired Scalar Continuum, and Constrained Collective Matte
Abstract
Physical scale is represented by a real logarithmic coordinate, while the observable gravitational geometry is leafwise: each scale leaf carries a four-coordinate clock-reading measurement geometry g_μν(x, λ). We give the canonical low-energy successor formulation of the Scale Space gravitational dynamics after resolving the scalar and matter-sector obstructions identified in the preceding internal audit. A dynamical localizer X₀(λ) = v_X tanh[κ(λ − λ_c)] generates the exact positive weight F₀(λ) = sech⁴[κ(λ − λ_c)]. 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 = −(r Z_X/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 + γ f̂⟨f̂, ψ⟩], K̂ = −∂²/∂y² + 4 − 6 sech²y, γ = 9/(8ζ_X). The essential edge is exact: m_s,gap = 2κ/√r. An internal low-energy EFT interval 2.5 × 10⁻¹² ≲ r ≲ 2.57 × 10⁻⁶ remains under the stated theoretical conditions; this is not an experimentally surviving interval. The fixed illustrative protected-pole benchmark lies above the published single-Yukawa graphical bound under the assumed positive, universal coupling normalization. At r = 10⁻⁶ and ζ_X = 1.8115 × 10⁵, the exact continuum resolvent gives m_Q/κ ≃ 340.936 and, in the normalization where the unrepaired pole has α_Q = 1/3, an exact repaired pole coefficient α_Q ≃ 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 serve as the candidate low-energy dynamical architecture.
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BibTeX
@misc{palmer2026ssdyn,
title = {{SS-DYN: A Scale-Localized Leafwise Dynamical Closure for Scale Space: Protected Tensor Gravity, a Repaired Scalar Continuum, and Constrained Collective Matte}},
author = {Donald G. Palmer},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = sep,
doi = {10.5281/zenodo.22731549},
url = {https://doi.org/10.5281/zenodo.22731549},
note = {CC BY},
}
RIS
TY - UNPB
TI - SS-DYN: A Scale-Localized Leafwise Dynamical Closure for Scale Space: Protected Tensor Gravity, a Repaired Scalar Continuum, and Constrained Collective Matte
AU - Donald G. Palmer
PY - 2026
DA - 2026/09/13
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.22731549
UR - https://doi.org/10.5281/zenodo.22731549
AB - Physical scale is represented by a real logarithmic coordinate, while the observable gravitational geometry is leafwise: each scale leaf carries a four-coordinate clock-reading measurement geometry g_μν(x, λ). We give the canonical low-energy successor formulation of the Scale Space gravitational dynamics after resolving the scalar and matter-sector obstructions identified in the preceding internal audit. A dynamical localizer X₀(λ) = v_X tanh[κ(λ − λ_c)] generates the exact positive weight F₀(λ) = sech⁴[κ(λ − λ_c)]. 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 = −(r Z_X/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 + γ f̂⟨f̂, ψ⟩], K̂ = −∂²/∂y² + 4 − 6 sech²y, γ = 9/(8ζ_X). The essential edge is exact: m_s,gap = 2κ/√r. An internal low-energy EFT interval 2.5 × 10⁻¹² ≲ r ≲ 2.57 × 10⁻⁶ remains under the stated theoretical conditions; this is not an experimentally surviving interval. The fixed illustrative protected-pole benchmark lies above the published single-Yukawa graphical bound under the assumed positive, universal coupling normalization. At r = 10⁻⁶ and ζ_X = 1.8115 × 10⁵, the exact continuum resolvent gives m_Q/κ ≃ 340.936 and, in the normalization where the unrepaired pole has α_Q = 1/3, an exact repaired pole coefficient α_Q ≃ 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 serve as the candidate low-energy dynamical architecture.
ER -