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 = 1, where D is the probability of a contractive step and 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 = Ed (t+1) /d (t) governs a trikotomy: convergence (1). We prove three theorems. Theorem~I: In Go-model protein folding, non-productive moves satisfy ₍₎₍₏ₑ₎₃ = 1 + O (k/M e^-/kBT), so the martingale parameter reduces to the thermodynamic contraction product = D. 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 = 1 exactly---it sits on the martingale boundary. Theorem~III: The entire qn+1 family is classified by q = q/3 (arithmetic mean) and q = q/4 (geometric mean) ; for q 3 both criteria agree, but for q = 3 they split: the Collatz map is an arithmetic martingale (= 1) while being geometrically subcritical (= 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.