Control-Limited Transient Accessibility in Near-Caustic Spacetimes

This paper develops an operational, waveform-agnostic criterion for transient accessibility in strongly curved spacetime regions. Null congruence evolution is modeled as a controlled Riccati system, and an integrated focusing surplus functional is introduced as a conservative sufficient diagnostic for avoiding caustic formation over a finite affine interval. A constant-load comparison system yields a closed-form marginal threshold Φ₍crit₎ = π² / (2 Δλ) for initially marginal congruences. The analysis shows that longer affine accessibility windows require proportionally tighter surplus control. Near-horizon gravitational redshift is then shown to convert the affine-window constraint into a geometric “dilation tax,” tightening the allowable surplus budget for fixed exterior observable duration. This establishes that sustained accessibility in strongly redshifted geometries is control-limited rather than energy-limited. The results are further interpreted within the Projection Gate framework as an operational classification of sealed, marginal, and open accessibility regimes.