Certified one-bit chemical computation: finite-fuel correction, productive readout, and inheritance

A chemical module computes reliably only if the state that encodes its input survives the act of using it: making the output consumes the same finite material that maintains the state, and dividing the compartment splits that state at random. We construct a seven-species reversible count reaction network, with one fuel species, whose compartment realises a complete one-bit contract. Two preparations with identical molecular support encode the two input values; the network repairs a bounded number of wrong-label molecules, delivers at least four molecules of the corresponding product by a fixed deadline, and returns both complementary daughters to the region that encodes the same value. A single probability bound covers all of these events on one trajectory. For an $81$-unit preparation we prove a uniform joint guarantee of at least $0.9997$ over a parameter box that simultaneously spans a hundredfold range in the correction rate constant, admits product recovery as low as $90\%$, and admits division bias anywhere in $[0.48,0.52]$; under ideal operations the same construction gives $0.9998$. The proof combines a fixed-time reaction-clock comparison with a downward-rounded integer certificate on a $3\,240$-state core, so that no floating-point matrix exponential enters the argument. Correction consumes fuel, and its embedded jump chain is exactly solvable: it is the Mabinogion urn, whose wrong-consensus probability $2^{-(r-3)}$ from two minority molecules among $r$ residents is independent of how fast the correcting reaction is. Weighting the exact absorption law by the fuel actually spent yields a $144$-unit, two-minority construction with joint probability at least $0.9916$, within $5.5\cdot10^{-4}$ of the ceiling that the consensus law imposes on its worst admitted state. The same law makes heritable variation quantitative: a specified mixed preparation produces a productive opposite-program daughter pair with probability above $0.0074$, whereas a pure newborn does so with probability at most $8.1\cdot10^{-6}$, because on a pure face the only channel that can create a minority molecule is first-order leakage. No single label-dependent switching rate can therefore describe both classes of newborn. An external product-dependent retention rule converts the output distinction into an explicit finite-population selection bound. The probability theorems are conventional arguments supported by exact finite computation; selected source, partition, rate and arithmetic lemmas are checked in Lean 4. High-order reactions, severe kinetic separation and idealised inventory operations remain assumptions: this is a certified mathematical model of one-bit chemical information processing, not a calibrated biochemical implementation.