We give a fully machine-checked exact dichotomy governing when a deterministic carrier can faithfully retrieve an unbounded committed history. Model a retrieval system as a tuple (τ, c₀, κ, ρ): a state transition τ : S → S, an initial carrier state c₀ ∈ S, an unbounded content stream κ : ℕ → α, and a readout ρ : S → α, related by the faithfulness law ρ(τⁿc₀) = κ(n) for all n. The system distinguishes all entries when κ is injective. Our central result is a dichotomy with no third option: a carrier admits a distinguishing bridge if and only if its forward orbit n → τⁿc₀ is an injective copy of (ℕ, succ). Both directions are proved; the forcing direction (a faithful bridge embeds an unbounded register into the carrier) discharges with zero axioms. The classification is sharper than the elementary counting facts it recovers, and its routing is the point: the no-go runs through finiteness, never through a fixed period, so it survives tensor products, registers, and many-site assemblies whose orders grow without bound. An odometer over (ℤ/8ℤ)^K has order 8^K, yet every such carrier is blocked verbatim; small order is not the obstruction, finiteness is. Two universal corollaries kill whole classes rather than examples: any finite-order carrier and any carrier over a finite state space cannot distinguish an unbounded history. We also record a sharp periodic instance (an order-8 cyclic carrier resolves at most eight entries) and an independent orbit-blindness obstruction (a shift-invariant readout retrieves flat content). Every numbered theorem is formalized in Lean 4 over Mathlib and is axiom-clean in the technical sense defined below. A reversibility fork (whether the re-addressing step is a one-way monoid jump or an invertible group action) is stated as a labeled open question.