Chemical Memory under Load: Productive Inheritance and Hidden Operating States

Can a chemical state be inherited while the species supporting its reproduction is continuously extracted, and which observations reveal the productive state under feedback? We answer these questions for two operating arrangements of one four-species resident chemistry. Under imposed extraction in $[0.009999,0.01]$, a literal finite-count compartment protocol completes two growth–division–transfer–recovery cycles with joint probability at least $24482873/24500000$. Both ancestries survive, retained cells recover distinct chemical states, high-state ancestry exceeds an exact bound of approximately $0.69304534$, and each batch exports more than five times its initial size. Extraction at least $0.031$ separately excludes the designated high stationary state. A reservoir–consumer extension has two fully attracting positive equilibria with identical consumer proportions but uptake differing by a factor greater than $1.8186$. An exact resident-field identity connects the arrangements without transferring their inheritance guarantees. We derive a sharper stationary feedback-load cap, composition dissipation, and finite-window readout bounds that recover information lost by proportions. The inheritance, attraction, balance and load-cap results have Lean proofs; additional observation consequences are proved conventionally. The conservative witness uses compartment size $6.5536\times10^{22}$ and $4\times10^9$ compartments: these are constructive mathematical guarantees, not a calibrated laboratory design.