The material and kinetic cost of productive chemical memory: exact frontiers for support and proportion encodings

A chemical system that is to be called heritable must do two things on one probability law: make fresh product before a deadline, and hand its own identity to both of the compartments it divides into. We fix a single finite protocol — run a batch of $K$ material units to time $T$, harvest all product, allocate every intact molecule by one complementary coin, refill both daughters — and ask what that joint event costs in material, for two ways of encoding the stored identity.

For support memory, where the identity is which of two species is present, we study an explicit reversible binding network with elementary bimolecular steps. The least material budget that guarantees a quota of four fresh product units and two memory-bearing daughters with probability $0.999$, uniformly over all $287$ admitted preparations and over an entire factor-two band of release rates, is exactly $32$. The certificate is a rowwise minimum of endpoint transition kernels evaluated in exact integer arithmetic; it needs no monotonicity in budget, deadline or rate. We then enlarge the uncertainty: the same minimum $32$ survives allocation bias $\theta\in[0.39,0.61]$ and a simultaneous independent multiplicative perturbation of every one of the six rate constants by $\pm1/300$. We also exhibit a direction in which it genuinely breaks. Untying the two exit channels of the bound complex — so that dissociation and product release no longer share one rate — excludes $K=32$ by an exact upper certificate and raises the exact minimum to $42$. The guarantee is not monotone in the product-release rate: doubling release alone also breaks it, because the dissociation channel is what returns food to the pool that replication needs to build carriers.

For proportion memory, where both species are always present and the identity is the majority, no elementary network is known. We give a cooperative gate of reaction order $229$ using $415$ units, and show that the gate’s reachable class collapses to two states, which makes the whole architecture class exactly optimisable. Sharpening the partition estimate from a union bound to an exact endpoint minimum raises its guarantee from $0.9995$ to $0.9997$ per cycle and from $0.995$ to $0.997$ over ten. More importantly, we compute the class’s exact frontier: $253$ core units suffice at the same specification and no member of the class does better, so that construction is not optimal. At the support-memory specification the frontier is $230$ core units. Because a formal gate may use unbounded reaction order and unbounded rate constants, this is a lower bound on every such realisation: proportion memory costs at least $231$ units where support memory costs $32$. Across five decades the frontier obeys $N\approx33.3\ln(1/\delta)$; we prove $N=q+\Theta(\log(1/\delta))$ and identify the balance law behind the constant, in which the corridors of $m$ ordered levels have costs $4k^2z^2$ and the total grows cubically in $m$.

Concrete minima are conventional exact computer-assisted theorems; the kernel envelope, source conservation, rate bounds, the complementary-binomial event and the exact tail arithmetic are verified in Lean 4 against Mathlib. This is not an end-to-end formalisation of either probability theorem, and neither network is an experimentally calibrated chemistry.