Structural Differentiation Cosmology: Applicability Conditions and Testability of the Logistic Background Model

This paper characterizes the expansion histories that the constant-coefficient logistic–exponential background model of Structural Differentiation Cosmology (SDC) can and cannot represent. It retains the equations analysed in version 1.0 (doi:10.5281/zenodo.23695729), without introducing a new physical model. Under the additional identification 1+z=1/a, the normalized Hubble function E=H/H₀ has the form E(L)=1+cL−bL², where L=ln(1+z), c is real, and b>0. The paper characterizes this entire family, including its finite-time domain between the two roots, and establishes a one-to-one correspondence between the shape coefficients and the dimensionless model parameters. Every member has strictly negative second derivative with respect to L. By contrast, the flat matter-plus-Λ reference model without radiation, with 0<Ωm≤1 and ΩΛ=1−Ωm, has strictly positive second derivative. Consequently, no member matches that reference through second order at any common redshift. On a finite interval 0≤L≤Lᵦ, the paper proves a parameter-independent lower bound on the maximum absolute difference between the normalized Hubble functions. For Ωm=0.3 and Lᵦ=1, this bound is 0.07171875. This is an absolute difference in E=H/H₀, not a relative percentage error or an observational discrepancy. The model also forbids a history that is accelerating today but decelerating at an earlier redshift within its domain. An exact condition is derived for simultaneous present acceleration and coverage of a prescribed finite redshift interval. Such parameter choices exist, but all remain accelerating throughout the covered past interval. The paper identifies additional assumptions required to connect the projected background to observations and outlines two future background-only tests: the curvature sign of E(L) and the sign pattern of the deceleration parameter. This is a test-design framework, not a completed preregistration. No dataset, estimator, covariance treatment, or decision threshold is fixed. Changes made after examining target data are distinguished from confirmatory testing on previously unexamined validation data. The supplementary archive provides a verification script, numerical results, explanatory records, licence information, and checksums. The numerical checks include an intentionally incorrect bound coefficient as a control. Computed maximum deviations use a 401-point grid; the general validity of the continuous-interval bound rests on the mathematical proof. These checks are not independent verification. No observational data are analysed, no parameter fit is performed, and no observational exclusion is claimed. The conclusions concern this restricted background model under its stated assumptions, not SDC as a whole. The paper does not determine fσ₈ or derive the conditional 5–7 percent suppression benchmark of the separate Euclid decision protocol. Third-party verification and observational verification have not been performed. The manuscript, 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 are provided in the supplementary archive.