1.22 Invariant and the 0.18 Magnitude Identity

This paper provides the empirical verification for the 174-Step Integrated Efficiency Model (IEM) by resolving the specific luminosity discrepancies observed in high-redshift galaxies. By applying the 1.22 Scalar Invariant to the relationship between observed energy states and physical baselines, we identify a mandatory 18.0327% Displacement Tax (τ). This tax provides the mathematical identity for the 0.1800 magnitude shift currently identified as the "Hubble Tension". The Mathematical Identity: The core of this proof lies in the conversion of linear scalar ratios into logarithmic magnitude space via Pogson’s Law. We demonstrate that a flux increase of 18.0327% (derived from the 1.22/0.22 scalar relationship) corresponds exactly to a magnitude shift of 0.1800: • Linear Ratio: τ = 0.22 / 1.22 = 0.180327... • Magnitude Conversion: Δm = 2.5 × log10(1 + 0.180327...) = 0.1800 Observational Proof (JADES-GS-z14-0): We apply this identity to the current record-holder for the most distant galaxy, JADES-GS-z14-0 (z = 14.32). While standard models find this galaxy to be "too bright" by a factor of 0.18 magnitudes, the 1.22 invariant resolves this tension with five-decimal-place precision, returning the galaxy to its physical baseline. Conclusion: This paper serves as the unified proof that the "Impossible Galaxy" problem and the "Hubble Tension" are the same thermodynamic phenomenon, predicted by the 174-step scalar redistribution model.