The Sigma–Friedmann Bridge: A Coupled Dynamics Between the Duality Signal σ(t) and Cosmological Expansion

This paper integrates the duality signal σ(t) of the Sigma–Schmidt Duality into classical Friedmann dynamics.An energy-exchange term Q=βHσ(t)ρtotQ = \beta H \sigma(t)\rho_{\text{tot}}Q=βHσ(t)ρtot is introduced, linking the matter sector with an effective dual field.In the limit β→0\beta \to 0β→0 or σ=const,wσ=−1\sigma = \text{const}, w_\sigma = -1σ=const,wσ=−1, the model reduces exactly to ΛCDM.The Sigma–Friedmann Bridge provides a dynamic coupling between cosmic expansion and local energy balance, offering a continuous transition between equilibrium and asymmetric phases.The framework remains observationally testable using SN Ia, BAO, and cosmic chronometer data, and opens a new interpretation of the Friedmann equations through dual-energy feedback between expanding and extracting singularities.