Vanishing-Order Ladder for Swallowtail Walls in Reduced Inverse Geometry

“We compute the internal-coordinate vanishing order of the swallowtail (A4) wall and place it in the A2→A3→A4 ladder.” “The universal content is the internal root scaling pk = 1/k; the observable exponent l/k is diagnostic‑dependent and measured.” This work establishes the vanishing-order ladder for the A-series (A2–A4) in reduced inverse geometry, proving the universal internal scaling pk=1/k and identifying the swallowtail (A4) deepest-axis exponent 1/4. The result quantitatively links structural walls, Fisher-spectrum softness, and noise-calibrated horizons, completing the measurement‑cost hierarchy of the reduced‑observability program.