Structural Cosmology v3.4: Observational Constraints An Illustrative Framework for Comparing Structural Cosmology with Current Cosmological Data

Structural Cosmology v3.4 introduces the observational-constraints layer of the Structural Cosmology research program. Building upon the theoretical foundations developed in v3.0–v3.3, this work presents an illustrative framework for connecting structural-field models to contemporary cosmological observations and likelihood-based inference pipelines. The framework incorporates multiple observational probes, including growth-rate measurements, matter power spectra, weak gravitational lensing, CMB lensing, and large-scale structure surveys. A unified pipeline is presented linking structural fields, projection operators, likelihood construction, Bayesian parameter inference, cross-scale consistency evaluation, and model comparison. Key components include the Structural Bandwidth Principle (SBP), the Cross-Scale Consistency Index (CSCI), the Gravitational-Geometric Projection Test (GGPT), the Gravitational-Geometric Projection Spectrum (GGPS), Bayesian evidence evaluation, and the extended structural bandwidth parameter set C₁–C₆₀. All analyses, parameter values, posterior summaries, likelihood examples, and model-comparison results are illustrative and intended to demonstrate the proposed methodology rather than provide definitive observational constraints. The framework is designed to support future evaluation using forthcoming datasets such as Euclid, LSST, DESI, and CMB-S4. This release contains: Full v3.4 manuscript (30 figures) Observational-constraints framework Likelihood and Bayesian inference architecture MCMC and posterior-analysis workflow Cross-scale consistency methodology Constraint Survival Map (C₁–C₆₀) Reproducible illustrative Python demo code