Structural Differentiation Cosmology: A Logistic Background Model and Its Projection to Observed Expansion

This paper presents an exact analysis of a restricted background model for Structural Differentiation Cosmology (SDC), combining logistic evolution of a structural state C(t) with an exponential projection to an observable scale factor. Under a shift of the structural variable and the corresponding redefinition of the rate parameter, the model is equivalent to the earlier “Structural Differentiation Cosmology: Minimal Dynamical Core” (doi:10.5281/zenodo.18856851). The background model itself is not presented as new. Both evolution and projection laws are postulates, not deductions from a microscopic theory. For positive constant coefficients and an interior initial state, the paper derives the exact time evolution, the projected Hubble function, finite past and future limits of the scale factor, and finite total logarithmic expansion. It establishes a unique transition from accelerated to decelerated projected expansion. This branch does not produce a transition from deceleration to acceleration. With the additional identification 1+z=1/a, the normalized Hubble function is a concave quadratic in ln(1+z), defined on a finite redshift interval whose endpoints correspond to infinite-time limits. The paper gives an inverse map from the two background shape coefficients to dimensionless model parameters and identifies a structural-unit rescaling degeneracy. Two figures illustrate the mathematical behavior using arbitrary dimensionless parameters, without fitting observations. The supplementary archive includes verification scripts, numerical outputs, source photographs, explanatory records, licence documents, and checksums. Numerical checks cover the analytic solution, background identities, acceleration transition, inverse parameter map, rescaling invariance, and correspondence with the earlier model. These checks do not constitute independent verification or physical validation. No observational data are analysed. The model does not yet specify perturbation dynamics, light propagation, or an operational measurement of C. It therefore does not determine fσ₈, derive a growth-suppression amplitude, or establish an alternative to the full cosmological standard model. In particular, it does not derive the conditional 5–7 percent benchmark in the separate Euclid decision protocol (doi:10.5281/zenodo.23665310), nor reconstruct earlier SDC growth-rate figures. Third-party verification and observational verification have not been performed. The paper, figures, photographs of the author’s source document, explanatory documents, numerical outputs, and checksum lists are licensed under CC BY 4.0. The verification code is licensed under the MIT licence. These licences do not extend to third-party material. File-level assignments and licence information are provided in the supplementary archive.