Balanced on one: contraction–expansion equilibrium in protein folding, the Collatz iteration, and semantic change - Three theorems and an observation

Three superficially unrelated dynamical systems---protein folding on an energy landscape, the Collatz iteration on the integers, and semantic change in natural language---share a common critical signature: the product D · γ = 1, where D is the probability of a contractive step and γ is its magnitude. We argue that this is not coincidence but structure. We define a class of contraction--expansion systems in which a deterministic contraction competes with stochastic expansion, and show that the martingale parameter λ = E[d(t+1)/d(t)] governs a trikotomy: convergence (λ < 1), criticality (λ = 1), or divergence (λ > 1). We prove three theorems. Theorem~I: In Go-model protein folding, non-productive moves satisfy γnonprod = 1 + O(k/M · e-ε/kBT), so the martingale parameter λ reduces to the thermodynamic contraction product σ = D · γ. Theorem~II: The single-step reset distribution v₂(qn+1) ~ Geo(1/2) is an exact arithmetic fact for all odd q; under the remaining assumption of orbit equidistribution, the Collatz iteration on embedding depth has λ = 1 exactly---it sits on the martingale boundary. Theorem~III: The entire qn+1 family is classified by λq = q/3 (arithmetic mean) and μq = q/4 (geometric mean); for q ≠ 3 both criteria agree, but for $q = 3$ they split: the Collatz map is an arithmetic martingale (λ = 1) while being geometrically subcritical (μ = 3/4 < 1). This tension---the arithmetic mean cannot guarantee what the geometric mean predicts---is a precise characterization of the conjecture's difficulty. We complement the theorems with an empirical observation: semantic change rates in natural language correlate with etymological depth ($r = 0.528$, p < 10⁻⁶), consistent with a contraction--expansion balance. The Collatz conjecture remains open---and now we know precisely where it sits: the unique point where arithmetic and geometric criticality diverge.