Einstein Contraction Fails in a Gauge Setting
The Ricci trace is a perfectly good move on tensor curvature—until curvature lives in $\mathrm{ad}(P_H)$.
The Shiab Operator: Definition and Geometric Meaning
Shiab is the Einstein projection rebuilt out of transport, wedge, and $\ast_Y$—so gauge symmetry survives.
Projector E: what “gravitational block” means operationally
A block is not a coordinate corner—it is a parallel subbundle selected by transport.
The calibrator form and what it selects
Normal saturation is the “trace” you are allowed to take in a gauge-covariant ambient theory.
First order torsion-first field equation: structure and interpretation
If torsion-first is going to mean anything operational, the action has to be written in variables you are actually allowed to vary. The only thing you can safely push through an Euler-Lagrange variation is a covariant field on $Y.$ That field is the augmented torsion $T.$
Second-order theory as a square: what you gain, what you lose
The square is where you buy robustness, at the price of admitting extra solutions.