Reliable copying of chemical states: finite-molecule guarantees and molecular redundancy

Chemical multistability supplies candidate memory states, but a list of stable states does not say whether a finite population of molecules copies a state reliably through growth, division and recovery. We treat copying as a complete stochastic cycle: an admitted parent must reach division, its molecules must be allocated to two daughters, and both daughters must return to the admitted region of the same label, from which the same guarantee applies again. We prove three separate results and keep their scopes distinct. First, a constructed pair of interacting reversible modules stores four concentration-state labels with at most 874 resident molecules per newborn module; under ideal external pool and volume control, every admitted parent produces two correctly returned daughters at its first division within 20 seconds with probability at least $0.9901$, uniformly over a positive interaction and growth-reversal interval. Second, for $k$ interacting modules satisfying a local quadratic recovery inequality, bounded reaction activity and bounded total incident exchange, the one-generation failure probability is at most $kC(\gamma)e^{-cN}$ with an exponent and an interaction allowance independent of $k$; for an explicit four-species module this gives sufficient copy numbers $N=O(\log(kG/\eta))$ for a designated lineage of $G$ divisions and $O(G+\log(k/\eta))$ for a complete binary family, with matching necessary orders for immediate return of both daughters. Third, the nominal eight-species kinetic model of Semenov et al. supports a prescribed split, refill and 500-second recovery protocol at $39.853\,\mu$L with joint daughter return probability at least $0.992$. Uniform return implies repeated copying without any independence assumption between siblings or generations. The proofs combine finite-state generator certificates, exact rational arithmetic, the actual complementary allocation law and state-dependent offspring kernels. The finite-model theorems and the certified-generation kernel are checked in Lean 4; the remaining conventional steps are identified explicitly.