Structural Differentiation Cosmology: Operational Assumptions Linking Structural Projection to Redshift and Distance

This paper examines the additional assumptions required to connect the logistic–exponential background model of Structural Differentiation Cosmology (SDC) to redshift and distance observables. It retains the equations analysed in the preceding papers, versions 1.0 and 1.1, without introducing a new physical model. The evolution of a structural state C(t) and its projection to a scale factor do not, by themselves, define redshift, light propagation, clocks, or observational distances. An explicit family of auxiliary metric choices demonstrates this underdetermination. The paper then adopts one conditional observational correspondence: an effective spatially flat metric, comoving observers whose proper time is the model time, null-geodesic light propagation, photon-number conservation, and locally defined luminosities and rod lengths. Under these assumptions it derives the redshift relation, comoving distance, angular-diameter distance, luminosity distance, and distance duality, identifying the assumptions used at each step. The standard distance relations are not claimed as new. For the normalized background function E(L) = 1 + c₁L − bL², where L = ln(1+z) and b > 0, the paper provides an integral representation and a closed form involving exponential integrals, a low-redshift expansion, and the asymptotic behaviour near the finite upper redshift bound. This bound corresponds to an infinite-past limit, where the comoving, angular-diameter, and luminosity distances diverge logarithmically. The paper also identifies scale and structural-variable degeneracies and gives a conditional classification of the stationary points of the angular-diameter distance. The central result is observational equivalence within the stated scope: a spatially flat FLRW description with the same scale-factor history and the same clock, light-propagation, and conservation assumptions gives the same redshift and distance relations. This comparison concerns the same background history, not a particular ΛCDM solution. The structural-projection interpretation alone therefore adds no distinct prediction for these observables, and the construction does not by itself establish a physically non-expanding universe. The supplementary archive contains numerical consistency checks on fixed finite parameter sets, code, configurations, results, and correction records. Two implementation mismatches in the initial run were corrected, and checks N5 and N6 were rerun; both runs are retained. In one deliberately altered rescaling control, 89 of 100 evaluation points were analytically unreachable, while 11 remained undetermined (OPEN) because the integration’s terminal event was not reached. Those 11 points are counted neither as detections nor as passes. The numerical checks are not independent verification or mathematical proofs. In particular, the sampled stationary-point examples do not provide rigorous sign certificates or a classification over the full admissible parameter space. No observational data are analysed, no parameter fit is performed, and no observational exclusion or confirmation is claimed. The paper does not specify perturbation dynamics, determine fσ₈, or derive the conditional 5–7 percent growth-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. Licence assignments and supporting records are provided in the supplementary archive.