Chapter 22: No-Tie Determinism

The greedy peel extracts the largest prime A dividing X, then finds the maximal tetration height C such that A ↑↑ C ∣ X. Could two different tetration heights c₁ ≠ c₂ compete for the maximum? This chapter proves the No-Tie Lemma: tetration heights cannot “tie”. If A ↑↑ c₁ ∣ X and A ↑↑ c₂ ∣ X with c₁ < c₂, then the greedy peel deterministically selects c₂. The proof exploits the strict monotonicity and injectivity of tetration.