Vassena and Stadler conjectured that instability must be localized in a D-unstable core; earlier counterexamples refuted this only for linearized equilibria, leaving open whether the phenomenon can support a genuine periodic orbit. This paper shows it can: a four-species, five-reaction classical mass-action network with no D-unstable child selection undergoes a certified supercritical Hopf bifurcation and has an attracting periodic orbit on an open set of rate constants. The network is minimal under reaction deletion, every proper subnetwork losing even the capacity for a positive equilibrium, and the oscillation is produced entirely by kinetic order. No laboratory realization is asserted.
Steel's question whether some reaction always lies in at least half of a system's RAFs, the RAF form of Frankl's union-closed conjecture, was previously settled for systems with an elementary core and shown to be as hard as Frankl in the coreless case; this paper extends the solved region considerably. If the system contains a locally ranked complete-supplier core, a set of reactions each with one internal supplier for its non-food reactants and an acyclic internal production digraph, some core reaction appears in at least half of all RAFs, whatever cycles, alternative producers and feedback surround the core. Every elementary core qualifies. The unrestricted question remains open.
Nandan, Nghe and Unterberger's classification and the companion unistationarity results give a rigid picture: a minimal autocatalytic core has at most one positive steady state, stable for small enough degradation. This paper shows stability need not persist: a source-minimal seven-species Type II core with rational rates and strictly positive degradation of every species has a unique positive steady state whose Jacobian has exact eigenvalues 1 plus or minus 8i, so it is oscillatory unstable on an open set of parameters. The mechanism is a phase lag from an intermediate on a return path. Six-species cores without such intermediates remain unsettled; no periodic orbit is asserted.
Steel (2023) asked which union-closed families arise as the supported subsets of a directed graph on the same ground set, equivalently which RAF families come from catalysis alone; this paper gives an exact intrinsic answer. Starting from the full power set, repeatedly pick a maximal non-member, choose an unused root element inside it that lies in no member below it, and delete the interval of sets containing the root within that non-member; a family is realizable if and only if some such sequence ends exactly at the family, and a successful sequence reconstructs the digraph. With the antimatroid characterization of general RAF families this completes Steel's same-ground programme.
Whether the sequential distributive phosphorylation cycle with one kinase and one phosphatase can oscillate for three or more sites had been open since Vassena excluded Hopf bifurcation for two sites; this paper proves that it can, with attracting oscillations. A certified supercritical Hopf bifurcation at three sites extends by site addition to every larger site count, making three the exact Hopf threshold. There is no feedback reaction: the feedback is dynamic sequestration of the shared enzymes in intermediate complexes, and since the equilibrium, fluxes and quasi-steady-state vector field stay constant while stability reverses, no static reduction can see the instability.
A reporter that consumes a regenerated cofactor also stores it in a reporter complex, where it serves neither the native reaction nor the regeneration that would replenish it. This paper shows the resulting native-function loss per reporter product exceeds the instantaneous-turnover prediction by an exact factor, one plus the ratio of regeneration to release coefficients, and that no dosing schedule removes this cost at fixed final product. It then constructs a finite readout suppressing native flux by under 4.73% while separating two functional classes, and proves that an unobserved background consumer makes native function unidentifiable without native-specific calibration.
A chemical module computes reliably only if the state encoding its input survives being used: making output consumes the same finite material that maintains the state, and division splits it. This paper constructs a seven-species reversible count network that corrects wrong-label molecules, delivers at least four product molecules by a deadline, and returns both complementary daughters to the region encoding the same value: an 81-unit preparation succeeds with probability at least 0.9997 over a wide kinetic and operational uncertainty box. The correction chain is exactly the Mabinogion urn, showing why faster chemistry cannot replace resident inventory.
Applied to a pooled prokaryotic metabolic network with 6,039 reaction actions, this computer-assisted study finds and certifies an inclusion-minimal three-action intervention that blocks structural L-valine production while preserving L-methionine production; an 18-species siphon makes the obstruction independent of catalyst annotations. A ten-reaction valine subsystem adds a cautionary lesson: two ambient networks can share the same catalogue of irreducible RAFs yet differ in what they can produce, and one food-enabled pooling operation changes the closed-RAF catalogue. The conclusions concern structural generation under an explicit medium, not concentrations or kinetics.
Steel, Hordijk and Smith (2013) gave a procedure that checks whether a supplied list of k irreducible RAFs is complete, with running time exponential in k, and asked whether the problem is fixed-parameter tractable in k; this paper shows it is not, unless FPT = W[P]. Completeness certification is co-W[P]-complete when parameterized by the number of listed irreducible RAFs and coNP-complete otherwise, already for systems with one food molecule whose listed RAFs are disjoint catalytic cycles. Under the Exponential Time Hypothesis the exponent of the known algorithm cannot be made sublinear in k, although tractability returns when the size of the largest listed RAF is also bounded.
A finite record of product collected from a microbial preparation cannot by itself distinguish fresh conversion from release of material already present. In a two-pool model with a mobile product pool, a convertible reserve, uptake, losses and pool-specific recovery, this paper proves that the maximum of five affine expressions in the two observed collections is the exact minimum fresh inventory consistent with the calibration bounds, attained by an explicit material schedule. An initial reserve ceiling cannot replace a pre-wash ceiling, but adding a cumulative uptake cap makes it valid. Washing helps only by attenuating first-window measurement uncertainty. All inputs are synthetic.
Can a chemical state be inherited while the very species supporting its reproduction is extracted, and does composition alone reveal the productive state under feedback? For one four-species resident chemistry, a finite-count compartment protocol under sustained extraction completes two growth, division, transfer and recovery cycles with both inherited states surviving, with probability above 0.9993. A reservoir-consumer extension has two attracting equilibria with identical consumer proportions but uptake differing by a factor above 1.8, so proportion-only observation loses information that finite-window readouts recover. The witness scales are constructive, not a lab design.
Vassena and Stadler found exactly three unstable cores in the dual futile cycle, all unstable-positive with determinant of modulus one, and conjectured the same picture for the n-site sequential distributive cycle; this paper settles the conjecture by splitting it in two. The determinant claim holds far more generally: every child-selection matrix of any elementary enzyme-conversion system has determinant 0, 1 or -1. The structural claim fails: every cycle with at least three sites contains a fixed six-species unstable-negative core, and for every n there is a minimal unstable-positive core of dimension 2n+1, so core sizes are unbounded.
After Kosc et al. found that individually realizable autocatalytic cores may fail to be thermodynamically realizable together, this paper asks how large a minimal conflict can be, and shows there is no bound. For every n of at least three there is a family of n two-reaction cores with no common productive state, although every proper subfamily has one and the full family passes both standard necessary tests, reaction-direction and independent-complex feasibility. Hence no test on subfamilies of bounded size, even with those global relaxations, can decide compatibility; a polynomial-time certificate is given for graded assemblies where it holds.
Stationary compatibility of two NADPH-consuming modules, a glutathione peroxidase branch and a peroxiredoxin-thioredoxin branch, says nothing about recovery speed or how long joint service lasts. This paper separates capacity, recovery and duration: from an explicit preparation region every solution meets both service quotas permanently after four milliseconds, while a matched family with identical stationary states has arbitrarily long recovery delays, so no deadline can be read from stationary data. Finite-donor certificates show stored carrier can sustain a one-second mission even while the source sits below the stationary threshold most of the time.
Deciding whether a mass-action system is disguised toric, meaning some other network with the same vector field is complex balanced and so has a unique globally attracting equilibrium, is a hard problem with few exact solutions; this paper solves it completely for a six-parameter family of four-species systems formed by coupling two minimal autocatalytic cores. Two explicit flux budgets at a steady state are necessary and sufficient, even when the competing realization may use arbitrarily many auxiliary complexes, and eliminating the state gives a criterion in the rate constants alone. Every positive trajectory is permanent, and the balancing state is globally stable throughout the locus.
Once a single reactor is known to support repeated harvesting, does the guarantee survive connection to other reactors, given that exchange redistributes catalyst in all its forms during recovery? This paper proves that it does for any finite number of six-species reactors joined by an arbitrary symmetric exchange graph transporting all species with one common coefficient. Every synchronized harvest cycle returns the coupled state to its operating region and collects certified template output at every node, with constants independent of the number of reactors, the graph and the exchange strength. A two-node witness shows why species-selective exchange falls outside the argument.
When a dividing cell splits a conserved set of molecular marks between its daughters the sisters' states are dependent, yet models often draw each daughter independently from the same marginal. For a finite reader-writer chromatin motif with explicit division and a monotone state-dependent death hazard, this paper proves that exact complementary allocation always gives smaller extinction probability and smaller population variance than the independent simplification, although both models have identical type-resolved means. The gap is the propagated conditional covariance of daughter risks; certified two-site examples show regrowth differences above 0.098.
The public generator for the finite-core theory of RAF emergence identifies the two split descriptions of one reaction, whereas the earlier exact critical-window theorem used the ordered split-position catalogue; a companion paper isolated the resulting unknown into a bulk factor and left its limit open. This paper determines the quotient-model limit exactly: when the mean number of channels catalysed per molecule grows linearly in the maximum polymer length, the RAF probability converges to a value strictly between zero and one. Hordijk and Steel's finite-threshold conjecture therefore fails in this reaction convention too; the entire linear regime is a nontrivial critical window.
How many stable states can n modification sites hold when they share one kinase and one phosphatase? For the unordered phosphoform cube with retained enzyme-substrate complexes, this paper proves the stable-state capacity is exponential, of order 2^n, in contrast to the linear capacity of a sequential chain. If effective modification ratios balance around every square of the cube, the capacity drops to exactly n, so exceeding the chain requires a nonzero effective cycle affinity. A finite-molecule theorem converts hyperbolic sinks into labels with explicit operating guarantees. The large resource bounds diagnose the example construction, not phosphorylation memory in general.
The companion separation theorem left open whether the rare productive reactors in the capped-Zipf polymer model run on a diffuse collective RAF or on a single one-channel RAF; this paper proves the latter. Conditional on productive output, the minimum RAF size converges to one: productive operation without a productive one-channel incidence has vanishing relative probability, and that nucleus supplies more than three quarters of the measured export. Conditioning reverses the structural picture, since the minimum RAF size diverges given RAF existence alone. This is statistical selection in this model, not an impossibility of diffuse autocatalysis.
The critical-window law names the limiting RAF probability in the reversible polymer model through a survival function but says nothing effective about its value, its shape near zero, or what finite networks do; this paper answers all three questions. A terminating algorithm returns certified rational enclosures of the survival function to any tolerance. Quantitative low-intensity bounds show the profile is flatter than any power at zero and non-analytic there. At fixed polymer length the probability of a singleton RAF is exactly linear in the catalysis probability, so finite systems and the infinite limit behave differently near zero: a quantified order-of-limits effect.
Steel asked whether every catalytic reaction system has a reaction lying in at least half of its RAFs, a RAF analogue of Frankl's union-closed sets conjecture; this paper proves it for every system containing a viable elementary core, a RAF whose reactants are all food, with no restriction on the remaining reactions or on catalytic feedback into the core. It then shows the remaining case is exactly as hard as the original: a producer-gate construction realizes every finite union-closed family as a RAF family while preserving frequencies, so a half-frequency theorem for systems without an elementary core would be equivalent to Frankl's unrestricted conjecture, which remains open.
Can stored red cells sustain ATP-dependent pumping while consuming an oxidative challenge? This paper asks precisely what a flux envelope decides for the proteome-constrained RBC-GEM model. It proves an exact inventory inequality linking Na/K-ATPase service to peroxide turnover, certifies that optimal turnover is essentially flat across almost the entire attainable service range, and shows that closing external sulfur supply cannot bound repeated turnover. It then shows what the envelope cannot decide: returned optima leave most imported peroxide unhandled, and only finite carrier pools supply the missing kinetic constraint. A theorem of stored-cell oxidative tolerance remains open.
Earlier papers decided shared-activity compatibility of autocatalytic cores only for junction-path architectures; this paper handles weighted two-reaction cores on an arbitrary finite oriented graph with rational activity boxes. The constraints are monotone two-variable implications with a least feasible element, and two exact algorithms compute it and return a rational productive state. Cycle acceleration replaces converging propagation by forced fixed-point jumps; variable elimination along a depth-first forest then gives fixed-parameter tractability in the length of the longest simple path. The result is static compatibility, not long-time operation.
Three open directions are settled: how the maximum amplification factor composes under network operations (Gagrani et al.), whether catalyst-aware child-selection cores can be recovered from ordinary cores (Golnik et al.), and a general criterion for autocatalysis under species-specific degradation (Nandan, Nghe and Unterberger). The scalar amplification factor cannot compose and is replaced by a threshold-indexed family of price certificates that does; catalyst-aware cores are recovered exactly by a matching-exchange decomposition; and growth and extinction regions under arbitrary degradation are characterized. Exact boundary traces then say when a module summary is safe.
Shared catalyst assignments change the exact chance of activation. With a fixed food set, just 34 entry channels bound the probability that a RAF exists as longer molecules are allowed.
What prevents eradication of a finite population whose cells inherit a molecular state, even when the intervention may respond to the whole population history? This paper proves two independent obstructions for branching populations under bounded feedback: an exposure floor, giving founder-dependent survival under any limited budget, and an amplitude floor that no longer course or larger budget can overcome. In a molecular inheritance source the amplitude threshold rises with memory size, so an actuator that eradicates at two sites fails at five to eight sites under every policy; an added death actuator on protected states repairs this. Extinction here is not a clinical endpoint.
Two coupled autocatalytic cores, each with a unique steady state on its own, can be made bistable by rerouting feedback at fixed total capacity; this paper certifies that transition and settles the global dynamics that follow. Shifting the fork current from a buffered channel to a dynamic one feeding the second core turns one attracting state into two through a certified saddle-node bifurcation. In the main family every trajectory converges to one of exactly three equilibria and the basin boundary is a smooth hypersurface of measure zero. Both attracting states show the same catalytic activity pattern despite different compositions, so activity alone cannot identify the selected state.
A chemical state that lets a compartment grow faster does not automatically leave more descendant compartments: with asynchronous division and a shared, depleting resource, accumulated material and offspring number are out of phase. This paper proves a finite-batch enrichment theorem for an explicit four-species count network in which high- and low-state newborns compete for one finite precursor pool: the high-state cell-count odds increase by a certified factor by a deadline, except on an explicitly bounded failure event, and every descendant's chemical identity is proved rather than assumed. No fitness parameter is imposed. The copy numbers needed are astronomically large.
Can a specified small number of molecules carry several distinguishable chemical states through a full reaction, division and refill cycle with a rigorous probability? This paper constructs a finite stochastic system of two modules with reversible cross-catalysis and bounded food that transmits four labels: with eighteen resident molecules per compartment, both complementary daughters return to the parent's label by time twenty with probability above 0.9919, robustly near nominal rates. Within the stated pure-species, fair-allocation architecture, eighteen residents are necessary as well as sufficient for a 99% guarantee. The mechanism is amplification and partition, not multistability.
An enzyme assay reports a rate, a recovery claim concerns a trajectory, and a population guarantee concerns how many trajectories finish a task; this paper computes what each observation certifies about the next for a G6PD-inspired kinetic family. Rate bands with a dual certificate yield a uniform recovery deadline, while a parameter sequence shows eventual recovery with no common deadline. Populations differing only in catalytic capacity can share the entire pooled initial-rate response surface yet have recovery fractions anywhere from about 49% to 100%; a calibrated classification readout narrows this to about 67% to 80%. Parameters are synthetic; no clinical predictor is supplied.
The companion optimal-affinity paper identified the response capacity L(T,w) as the lower bound on driving force at local production maxima, but left open whether those local maxima are global over the whole positive stationary set; this paper closes that gap. For square reversible mass-action sources with a nonnegative response matrix, every positive forward-response profile can be realized at a state maximizing production globally, so the response-level, local and global affinity sets coincide and L(T,w) is the sharp global capacity. A subcriticality certificate makes the maximizer unique; a worked source with two global maximizers shows that rootedness alone does not.
An accurately measured signal can still leave the biological question unresolved when sampling, population heterogeneity, stored material or the measurement itself obscures the quantity of interest. This perspective draws eight case studies into a design workflow that starts from the intended claim, identifies the competing explanations for a result, and selects the observation or intervention that distinguishes them. Three worked examples anchor it: pooled enzyme activity versus population recovery, dilution under the immunoassay hook effect, and fresh microbial conversion versus release of stored inventory. Every example remains a model construction requiring biological validation.
Compositional heredity, compartment competition, finite-molecule copying and repeated harvesting had each been treated separately; missing was one probability theorem following a finite population through competition, random transfer of intact compartments, recovery, refill and another competition, with selection measured in compartment counts. This paper supplies it: for a balanced population carrying one of two inherited chemical states, after two full serial-transfer cycles both states remain present and the high-to-low compartment-count log odds increase by more than 0.81, with certified probability above 0.99199. The scale is an existence witness, not a laboratory prediction.
Preserving every type-resolved population mean is not enough to preserve a treatment decision. Two seven-type branching models with identical rates and mean dynamics, differing only in whether sisters receive complementary or independent draws of the mother's epigenetic marks, prefer opposite orders for two equal-exposure interventions when the objective is eventual extinction, a probability, not a mean. Exact interval arithmetic certifies the reversal and traces it to sister covariance, whose sign makes the independent approximation optimistic. The effect is a deliberate near tie of about 1.7 times 10^-6 in extinction probability: information sensitivity, not clinical significance.
Hordijk and Steel (2016) showed that sparse random catalysis in the binary polymer model yields RAFs of quadratic size and asked whether linear-size RAFs, or even a single self-constructing molecule, could be expected; this paper answers both questions negatively. For any fixed catalysis intensity and constant C, the probability of a nonempty RAF with at most C times n reactions tends to zero as the maximum polymer length n grows, as does the probability that some molecule catalyses a set of reactions that builds itself, even with cleavage allowed. The smallest RAF is therefore superlinear, lying between linear and the known quadratic bound.
Avram, Adenane, Basnarkov and Horvath observed that strain supports in coinfection models are minimal siphons and left the interpretation of the extra modes created by overlapping siphons for future work; this paper supplies it. At a common boundary state the normal Jacobian is block triangular by siphon membership, giving an exact factorization of characteristic polynomials in which a shared invading species is counted once, not twice. For a two-strain, fourteen-reaction coinfection network, away from threshold, every compartment persists uniformly if and only if each absent strain can strictly invade the other's resident equilibrium.
Competitive exclusion says several consumers of one limiting resource generally cannot all persist, while chemostat models show that intraspecific self-limitation can rescue coexistence. This paper proves the rescue survives when the resource is produced by a nonlinear four-variable reaction network that the consumers load dynamically, need not approach equilibrium, and may draw on a replenished reservoir. Any number of self-limited consumers persists uniformly from every positive initial state, robustly under independent perturbation of every reaction rate, and the self-limitation coefficients prescribe the asymptotic composition. The autocatalytic-module generalization is left open.
Low-count proliferation assays are often summarized by a scalar birth-death model fitted to the expected phenotypic composition. This paper shows such a mean-composition closure can match the exact expected count and birth and death fluxes and still assign wrong probabilities: in an explicit two-type branching process with inherited states, the scalar model gives the counts 0, 1 and 2 probability above 0.9679 while the true probability is below 0.95, so a nominal 95% prediction interval fails. Widening to 0 through 3 repairs coverage, and the defect persists with many founders. A covariance identity and backward equations compute corrected thresholds without solving the joint law.
The same resistant outgrowth can reflect pre-existing state, selective survival, within-cell switching or altered daughter production. This paper shows which observations separate them in a finite-state branching model with correlated sister states: an independently randomized exponential deadline, stopped at the founder's first division or death, turns the observation law into a resolvent, so calibrated markers and joint daughter records identify the switching generator, demographic rates and daughter kernel without matrix logarithms. Sharper bounds cut a prospective founder budget from 2 million to 300,000. This is a model-conditional design result, not a validated assay.
Vassena and Stadler conjectured that an unstable equilibrium must contain a D-unstable core; earlier counterexamples needed parameter-rich kinetics or high-molecularity padding, leaving open the physically natural case where every reaction consumes at most two molecules. This paper settles that case negatively with a four-species, six-reaction mass-action network in which no species appears on both sides of any reaction, the equilibrium is unstable, yet all 25 child-selection matrices are D-nonunstable under every positive scaling. Unimolecular reactions can never produce such instability, and four species is the minimum. The case with bimolecular products as well remains open.
Earlier harvesting theorems kept the driving fuel and waste at externally maintained activities; this paper makes them a finite, counted inventory whose changing composition alters every subsequent cycle, and proves the operating guarantee for that changed law. For a six-species reactor coupled to a conserved fuel-waste bath, all m cycles return to the restart set and meet every output, food, service and bath-composition budget with probability at least 1 minus m times an exponentially small error. Two refinements roughly halve the sufficient copy scale and reservoir, so a hundred-cycle mission needs half the fuel. Reliable production does not imply net fuel consumption.
The companion finite-copy reactor theorem left three gaps: one copy scale, one finite observation window, and no competing destruction pathway or long-horizon supply ledger; this paper closes all three. With a maintained-drive cleavage channel that breaks product back into food, the six-species continuous-flow reactor still starts from food alone and exports a prescribed amount of product in every unit window over a horizon exponential in the copy scale, with exponentially small failure probability and explicit food and driving-service budgets. Removing only templated ligation makes the same output exponentially unlikely, so the long-lived production is due to autocatalysis.
A RAF certifies that catalysts could be regenerated from food, but gives no deadline, export quota or reliability for a finite reactor started from food, and the surrounding chemistry that might destroy a chosen autocatalyst is unknown. This paper proves an output certificate that survives that ignorance: a one-sided retained reward means either the selected autocatalyst grows or the interfering nonfood material becomes collected output. In a fed, diluted binary-polymer reactor the witness meets export quotas, a mass corridor and a food budget with probability above 1 minus 10^-10, whatever the nonfood catalysis. The sufficient scale is conservative; the output is aggregate material.
A count model validated on single-founder experiments may not transfer to a different founder preparation. This paper constructs two preparations of inherited-state branching populations with identical laws for every single-founder observation yet different minimal 95% two-founder prediction cutoffs, six versus seven. It gives calibration procedures that keep marginal coverage without independence between calibration and future data, shows that independent cell detection at half efficiency can erase or reverse a low-count signal, and proves uniform recorded-count bounds under rate and founder uncertainty. The lineage-data analysis does not establish biological coverage.
Existing refinement theory preserves steady states or bifurcations when an effective reaction is replaced by an explicit intermediate, but not a finite-mission guarantee with withdrawal, recovery, product collection and resource accounting on the refined stochastic network. This paper proves such a guarantee for a seven-species refinement of a reversible autocatalytic exporter coupled across any number of reactors, with failure probability exponentially small in the count scale and linear in the number of reactors and cycles. A witness shows why simply deleting the intermediate is not an exact reduction, and every successful mission of 55 or more cycles produces net new material.
The deterministic harvesting theorem left obligations at finite molecule number: random thinning of individual molecules, post-withdrawal concentration tails, and iteration through actual random endpoints without assuming independent cycles; this paper discharges them. For an explicit six-species stochastic reactor at copy scale V of at least 2 times 10^11, m consecutive harvest cycles with history-dependent withdrawals all succeed with probability at least 1 minus m times an exponentially small error, so the copy scale needed grows only logarithmically in the number of cycles: 48,000 cycles succeed jointly with 99% probability. The full chain is verified in Lean 4.
One-batch and two-cycle selection between inherited chemical states left open how to reach an arbitrary prescribed number of serial transfers without losing the rarer state to sampling or restarting the population. For a fully specified four-species compartment model, this paper follows the retained compartments through growth, uniform intact sampling, recovery and refill and proves that, for every horizon, both states remain present at every census while the compartment-count log odds grow by a certified amount per cycle; the attainable horizon has order log M in the transfer size M. Improved bounds cut the population sufficient for a ten-cycle mission from 10^13 to 4 times 10^9.
Eradicating an inherited-state target population with an intervention that also damages a finite regenerative reserve requires the reserve to stay above its failure threshold throughout treatment, not merely at the end. The main tool is an excursion bound anchored at a chosen return level, yielding a reserve-loss certificate that is asymptotically exact at high renewal. Combined with a six-state inheritance source and uncertain clearance, a single course gives target survival below 0.9% and reserve loss below one in 100,000 from a 400-unit reserve filled to 278, and still meets both 1% tolerances from 209 units, nine above threshold. The source is synthetic, not a calibrated regimen.
After the companion paper showed the three-site distributive phosphorylation cycle can oscillate, it remained open whether the same mechanism can hold a stable rest state and a stable oscillation at the same rates and totals; this paper proves it can. A certified generalized Hopf (Bautin) point yields an open wedge of parameters with a sink coexisting with an attracting cycle, and an explicit witness is validated by two independent computer-assisted proofs. Near the Bautin point there are baselines at which smoothly modulating a single rate constant, with all other rates and totals fixed, switches rest to rhythm and back. Switching at the finite witness is simulated, not proved.
Competition for NADPH is well known; what was missing is a criterion for when the glutathione and thioredoxin branches can meet their service quotas while sharing one regeneration system with finite enzyme and carrier inventories. This paper derives it: eliminating each branch's private states gives service currents increasing in shared NADPH, which against a decreasing regeneration current yields a necessary and sufficient condition, an attained minimum capacity and a unique joint steady state. For the declared parameters both branches succeed alone but fail together; a certified 13.71% capacity increase or a 3.37% enzyme redesign repairs it. The model is kinetic, not a whole-cell claim.
Wang and Sontag's bound of 2n-1 positive equilibria for the sequential distributive n-site phosphorylation cycle had been attained only for small n, and the best general stability result (Feliu, Rendall and Wiuf) gave about n/2 stable states; this paper determines both capacities exactly for every n. One list of rate constants and one compatibility class can have at most 2n-1 equilibria and at most n locally asymptotically stable ones, and rational data attain both simultaneously with n sinks and n-1 saddles, robustly on an open parameter set. Stability comes from separating equilibrium geometry from kinetic time scales, which also quantifies the slow recovery caused by crowded states.
RAF theory certifies that a network can regenerate its own catalysts from food, but not that a reactor started from food will actually produce anything; this paper puts both questions on one random network and proves an exact separation. In the reversible binary polymer model with Hordijk and Steel's capped-Zipf catalysis at its critical exponent, a RAF exists with limiting probability strictly between zero and one and the smallest RAF is typically small, yet the probability that a fed, diluted stochastic reactor exports a fixed amount of nonfood polymer is exponentially small in polymer length, even given that a RAF exists. Structural autocatalysis is common; productive operation is rare.
Wang and Sontag proved the sequential distributive n-site phosphorylation cycle has at most 2n-1 positive steady states, and Flockerzi, Holstein and Conradi attained the bound for three and four sites; whether it is attained for every n had remained open, and this paper proves that it is. For any 2n-1 distinct positive numbers, explicit algebraic formulas give rate constants and totals with exactly 2n-1 nondegenerate positive steady states whose free-kinase to free-phosphatase ratios are those numbers, so the maximal count persists on an open parameter set. A determinant formula shows n-1 of these states are always unstable; exact arithmetic certifies the other n stable through n = 10.
Whether every irreducible RAF of a catalytic reaction system can be listed in time polynomial in the input plus the output had remained open after the completeness-testing question was settled; this paper closes it exactly. Such an output-polynomial enumeration algorithm exists if and only if P = NP. The lower bound comes from a SAT encoding whose irreducible RAFs are exactly n conflict pairs plus the satisfying assignments of the formula, so an output-sensitive time bound would decide satisfiability. The same construction yields coNP-completeness of certifying a complete list, #P-hardness of counting, and the same equivalence for polynomial-delay enumeration.
What is the least material that pays both for fresh product and for handing the same chemical identity to both daughters in one batch, harvest, partition and refill protocol? For a support encoding, where identity is which of two species is present, an elementary reversible binding network needs exactly 32 units to guarantee four product units and two memory-bearing daughters with probability 0.999, robustly over rate bands and allocation bias; untying the complex's two exit channels raises the minimum to 42. For proportion encodings, where identity is the majority species, the cooperative-gate class needs at least 231 units. These are architecture-specific costs, not universal bounds.
The companion paper showed some finite-resource amplification assays have no usable readout deadline; this one asks whether a smarter classifier using the same crossing-time information could succeed, and proves it cannot. For any fixed sequential source a deadline is optimal among all randomized crossing-time classifiers, so every deadline exclusion becomes an all-classifier exclusion: at threshold five, any rule with blank error at most 1% misses more than 6.9% of loaded reactions, and the obstruction persists for every threshold above 10^6. Adding an identity-sensitive measurement or doubling the effective capacity repairs it; changing readout time or improving inference cannot.
Earlier results separated RAF existence from one window of productive output; this paper asks whether one food-only trajectory, without reset or resampling, can maintain stock and export material in two consecutive windows, and proves matching lower and upper bounds. Along the critical Zipf sequence the probability is of order 2^-n, and conditioning on success selects a productive one-channel incidence with probability tending to one. A necessary startup scale near 1.4 times 10^7 copies contrasts with a constructive sufficient scale near 5 times 10^49. The result separates availability of autocatalytic organization, creation of the first catalyst, and execution of an operating task.
Nandan, Nghe and Unterberger asked whether their uniqueness theorem for positive steady states of minimal autocatalytic cores survives mixed degradation, where some species are lost and others are not, leaving the Type V cores and the Type II cores with three or more forks open; this paper answers both affirmatively. For every reversible mass-action extension of a source-minimal Type V or Type II core, with arbitrary rate constants and any nonnegative degradation vector, including none at all, there is at most one strictly positive steady state. With the published cases this completes unistationarity for the whole classification over the closed degradation orthant.
A recovery experiment that records no event is compatible both with no recoverable units and with a failure to observe their recovery. This paper derives the sharp condition under which an all-negative paired assay can legitimately exclude a recoverable subpopulation: if two assay conditions together cover each stage of a two-stage recovery path and the within-condition stage mismatch is bounded, the complete-path response has an exact attained floor, maximized by equal allocation; without the mismatch bound, full stage coverage can leave zero chance of completing either path. The worst-case all-negative probability is exact, giving a sharp finite-sample design criterion.
A low sandwich-immunoassay signal can mean little analyte or so much that the hook effect suppresses the signal; dilution is the classical remedy, but what a neat-and-diluted pair actually certifies when capacities, affinities, dilution, gain drift and accessibility are known only within intervals had not been worked out. This paper describes the ambiguity exactly (every sub-maximal signal has two preimages; the diluted-to-neat ratio is strictly increasing), proves robust two-reading certificates over interval-valued calibration boxes, and shows exactly what neat readings, spikes and finite dilution panels cannot identify. These are conditional certificates, not clinical claims.
A negative readout need not exclude targets in the specimen when extraction failures are shared across targets and splitting consumes material. This paper connects material conservation to a sharp moment bound and a native-reference reporting rule, giving the exact worst-case probability that a count-exclusion certificate is false. It then quantifies what such a rule must survive: reusing one calibration across specimens has an irreducible familywise-error floor, so a 5% familywise level is unreachable for two specimens; heavy Poisson reference loading destroys the guarantee; and spurious control positives become the binding constraint. Biological transport remains an experimental premise.
Complete observations of one retained cell lineage need not determine which order of two interventions best suppresses the whole family. This paper constructs branching sources whose retained-branch records, including event times and censoring, have identical laws across a family of sister couplings, yet whose preferred intervention orders are opposite, and the reversal survives descendants dividing as fast as the founder. A paired sister-agreement measurement recovers the missing information: forty independent division records give a 95% correct choice on a separated constant-treatment class. The source is synthetic; the conclusion concerns an observation protocol, not a regimen.
García Puente, Gross, Harrington, Johnston, Meshkat, Pérez Millán and Shiu asked whether a unique positive zero-divisor candidate characterizes absolute concentration robustness in bimolecular networks and conjectured that their Gröbner-basis candidate algorithm is complete; this paper resolves both. Uniqueness is neither necessary nor sufficient, and the algorithm fails for an elimination order their definition admits but is complete for block orders, establishing the conjecture in that form. A multiplier floor then turns the algebra into a usable concentration bound, without which a small residual can signal extinction rather than accuracy, as EnvZ/OmpR and reactor examples show.
Deleting a reaction from an autocatalytic network can collapse far more than the reaction itself, and recomputing the maximum RAF from scratch after every deletion is wasteful. This paper introduces ranked support witnesses, stored certificates that localize exactly which part of a RAF a deletion can affect, so the new maximum RAF is recovered by one residual computation inside a bounded region. It then asks which witness to store (perfect for any single deletion, NP-complete to optimize in general, repaired by portfolios) and derives exact first- and second-order laws for expected surviving size under random deletions. The results concern structural survival, not kinetics.
After the companion papers proved that listing every irreducible RAF is intractable in general, this paper identifies input structure that makes it feasible. The supplier excess counts how many alternative producers and effective catalysts exist beyond one in the pruned system; the complete family of irreducible RAFs can then be enumerated exactly in time exponential only in that excess and polynomial in everything else, and the exponential dependence is essentially unavoidable. The catalogue converts into intervention guarantees: reaction sets whose deletion destroys every RAF are exactly its transversals, giving a greedy approximation and exact reliability formulas.
Kosc et al. showed that autocatalytic cores realizable in isolation can be incompatible when they share species; earlier papers gave two-core phase diagrams and unbounded-order conflicts. This paper identifies an infinite class where compatibility is decided exactly: assemblies of two-reaction cores arranged along private directed paths that meet at shared junction species. The interior of every path is eliminated without relaxing any current law, leaving a necessary and sufficient inequality on the junction activities alone, and for rational data a complete decision procedure returns a rational common state. Tolerance radii make it an operating certificate; general graphs remain open.
Structural criteria such as RAFs or autocatalytic cores say a network can amplify its own catalysts in principle, not that a reactor started from food alone will ignite, keep its catalyst against washout and sequestration, and export product on time. This paper proves such an end-to-end guarantee for an explicit six-species stochastic template-ligation network: at copy scale 10^8 the reactor establishes and retains a catalytic population and exports at least 10^7 covalent mass units in the specified window with probability at least 0.9, uniformly over a parameter box. Removing only the two catalytic ligation channels drops the probability of an easier comparison event below 1/5000.
Amplification assays face a timing conflict: a reaction may become sensitive only after blanks have already begun turning positive. This paper asks whether any readout deadline meeting both a 1% blank-positive and a 5% missed-call limit exists at all, for an explicit finite-resource birth-immigration source. At detection threshold five, resource capacities five through seven admit no acceptable deadline, eight admits one but none on a 0.1-minute observation grid, and ten is feasible with margin. For arbitrarily large thresholds, fixed headroom of at most two units leaves no window while three or more restore one. No clinical calibration is claimed.
Steel (2023) asked for a set-theoretic characterization of the set systems that can occur as the RAFs of a catalytic reaction system, calling it the harder of his two characterization problems; this paper answers it when the reaction set is the given ground set. A family of reaction subsets is such a RAF family exactly when it is the intersection of an antimatroid, which records how reactants are built up from food, with the supported sets of a directed graph, which records catalysis. The result also sharpens Steel's obstruction bound of twelve to the exact threshold of four, and shows that every interior operator becomes realizable if two reactions per element are allowed.
Vassena and Stadler's D-unstable cores localize instability to small square submatrices built from stoichiometry and reactant incidence alone; this paper shows that under classical mass-action kinetics such cores can miss instability entirely, because they ignore kinetic order, the number of copies of a reactant a reaction consumes. Adding the same molecules to both sides of reactions leaves every child selection unchanged, yet a four-species, five-reaction network that is stable at every equilibrium acquires an unstable one after such padding, with no D-unstable core anywhere. Any corrected localization theorem for mass action must see kinetic order.
Steel, Hordijk and Smith (2013) proved that finding a smallest RAF is NP-hard and asked whether the smallest RAF can at least be approximated within a constant factor; this paper answers no. Unless P = NP, no polynomial-time algorithm can guarantee a RAF within any fixed constant factor of the minimum size, even for systems with one food molecule and reactions with at most two reactants. The proof translates set-cover instances exactly into catalytic reaction systems whose RAFs correspond to set covers, which also shows that merely estimating the minimum size is hard and that two irreducible RAFs of one system can differ in size by an arbitrarily large factor.
Autocatalytic structure does not show that a reactor can be harvested again and again: withdrawal changes the state, catalyst is lost unevenly across its free and bound forms, and refilling food does not replace catalyst. This paper proves a uniform operating theorem for a materially balanced six-species reversible reactor: after one conditioning period, every cycle of withdrawal, recovery and collection returns the state to the operating region and exports a certified minimum of template product within explicit food and fuel budgets, for arbitrarily many cycles and state-dependent interventions. Net synthesis beyond any initial stock is certified from the thirty-first cycle.
Banaji, Boros and Hofbauer proved that no planar four-reaction mass-action network has a cusp of equilibria unfolded by its rate constants, leaving five reactions as the first unresolved case; this paper classifies it exhaustively. Among two-species, five-reaction, rank-two bimolecular networks, exactly 52 mechanism classes admit a positive cusp transversely unfolded by two rate constants, so five reactions is the minimum for such a cusp in planar bimolecular kinetics. All 9,999 candidate classes are decided by exact rational certificates verified in Lean 4. A Banaji preprint announcing an independent classification of the same class appeared concurrently; no priority claim is made.
A list of stable chemical states does not show that a finite population of molecules can copy a state through growth, division and recovery; this paper proves that it can, with explicit redundancy requirements. A constructed pair of interacting modules stores four labels with at most 874 molecules per newborn and returns both daughters to the correct label at first division with probability above 0.99. General error bounds show the molecule count needed grows only logarithmically in labels and generations for a single lineage and linearly in generations for a full binary family, with matching necessary orders; the published Semenov et al. kinetics also supports a certified protocol.
The companion operating theorems concerned an abstract labelled count process; whether it could arise from a materially balanced reversible chemistry with one consistent set of chemical potentials was a separate question, answered here. Six reversible pairs on eight species reproduce every propensity and output mark of the certified source across the whole parameter rectangle, so all inherited guarantees transfer unchanged, with success probabilities above 0.92 to 0.9995 for one hundred windows at three copy scales. The completed chemistry has one emergent fuel-to-waste cycle, obeys local detailed balance, and forces a necessary initiation scale near 4.6 million copies for 99% reliability.
Kosc, Kuperberg, Rajon and Charlat (2025) proved that any single autocatalytic core can be realized under thermodynamically consistent kinetics but that overlapping cores may not be realizable together; this paper asks when interacting cores can share one consistent set of concentrations. It refutes the natural strengthening that a common productive current plus compatible reaction directions suffice, and proves an exact phase diagram for two autocatalytic triangles sharing an edge: compatible exactly when their private rate factors are within a factor of two. Three distinct failure mechanisms are separated: direction, magnitude, and the lift to a common activity vector.
Kosc and colleagues' complexity result for detecting productive autocatalytic cores prescribed a target species and an allowed food set, leaving open the unconstrained question: does a finite reversible network contain any productive core at all? This paper proves that problem NP-complete for binary-encoded reactant and product matrices with signed reaction flows. Existence is first characterized by a nonsingular square restriction with two-sided participation in every selected reaction; a weighted incidence construction then reduces directed two-path linkage, and hence SAT, to the unrestricted search. It concerns stoichiometric cores, not the polynomial-time maximum-RAF problem.
With six food molecules held fixed, the probability of a RAF approaches a continuous value strictly between zero and one at every positive, finite ratio of average catalysis to maximum molecule length.
This paper constructs a four-species reaction network with growing oscillations near equilibrium, despite having no D-unstable core. All 24 selected matrices pass the test: none has an exponentially growing mode under any positive column scaling. The full network combines multiple species’ effects on the same reaction, revealing that instability can arise collectively even when every selection passes.
The numbers of molecules consumed and produced do not, by themselves, give the conjectured lower bound on the thermodynamic driving force at maximum production. This paper supplies explicit counterexamples and a replacement calculation based on how reaction rates respond to concentration changes; under stated conditions, it also constructs reaction rates that realize every allowed response at a strict local production maximum.
For the Type II family of minimal autocatalytic cores, the paper proves that fixed positive forward and reverse reaction rate constants, together with positive degradation of every species, allow at most one steady state in which every species is present. It also gives two exact examples outside the minimal class where the same rate constants within each example support two different positive steady states, showing why the requirement of minimality matters.