III.D10 — Ladder Checker

Lean-grade proof harness for verifying ladder properties: existence_checker(k) verifies non-emptiness at level k, stability_checker(k) verifies template preservation, strictness_checker(k) verifies E_k \ E_{k-1} ≠ ∅, saturation_checker(k_max) verifies [E_{k_max}^op, E_{k_max}] ⊆ E_{k_max}.