Abstract
We determine a four-dimensional exact rate family on one fixed reversible reaction graph with three species, ten complexes, and ten reversible pairs. Every positive member has full stoichiometric rank and vanishes on the same compact positive algebraic ellipse. The positive rate set is a rational relatively open polyhedral cone, while a nonempty Zariski-open subset of the family has geometrically coprime coordinate polynomials. A clean primitive integer member is simultaneously optimal for maximum rate and rate sum within this fixed-support, conic-preserving family. Its steady ideal is radical. For the immutable Version 1 specialization, exact Sturm analysis divides the ellipse into a normally attracting arc and a saddle-type arc.
The fixed graph and four-parameter family
Use the ten complexes
\(0, Z, 3Z, Y+Z, 3Y, X+Z, X+Y, X+Y+Z, 2X+Y, 3X\)
and the ten undirected edges \(01,04,06,17,24,27,29,34,59,89\), with both orientations of every edge. The graph is connected, and three reaction differences have nonzero determinant, so the stoichiometric subspace is \(\mathbb R^3\). There are no conservation laws: the one positive compatibility class is the full positive orthant.
Complete family theorem. On this directed support, all rate vectors whose three coordinate fields vanish on the conic below form a four-dimensional rational linear space. With free coordinates \((a,b,c,d)=(k_{29},k_{43},k_{95},k_{98})\), the remaining sixteen rates are explicit rational linear forms given in the manuscript.
All twenty rates are positive exactly when
This relatively open rational cone is parametrized by \(a,b,h,s>0\) through
The family is exhaustive: an exact canonical \(21\times20\) remainder matrix has rank 16 and nullity 4. A certified nonzero \(16\times16\) minor proves the rank, and the four displayed free coordinates prove the matching kernel dimension.
The conic is a nonsingular, geometrically irreducible compact positive ellipse. It has the rational parametrization
All coordinates are positive for every real \(t\); on \((-1,1)\), the points are pairwise distinct.
Generic geometric gcd one
Homogeneous triples with a common factor form a projectively closed multiplication locus. Pulling its affine cone back along the linear rate map gives a closed subset of the four-dimensional parameter space, including the zero field. The clean specialization below lies outside it by exact affine and homogenized factorization. Therefore a nonempty Zariski-open subset of the constrained family has coordinate gcd one over \(\mathbb Q\), \(\mathbb R\), and \(\mathbb C\).
On that open subset, the steady ideal has height two and the conic prime is a minimal prime. Thus the ellipse is genuinely a height-two equilibrium component, not the zero set of a common scalar factor. This is a relative statement inside the conic-preserving family, not robustness under arbitrary rate perturbations.
A clean optimal integer specialization
Set \((a,b,c,d)=(653,1,70,915)\). In directed-edge order 01, 10, 04, 40, 06, 60, 17, 71, 24, 42, 27, 72, 29, 92, 34, 43, 59, 95, 89, 98, the primitive positive integer rates are
(1160, 10296, 976, 23, 560, 5977, 1800, 25, 1629, 1237,
1, 9152, 653, 1214, 5368, 1, 5368, 70, 6039, 915)The coordinate gcd is one geometrically. Its steady ideal is radical and, over \(\mathbb Q\), is the intersection of the conic prime with a disjoint degree-15 maximal ideal. Over an algebraic closure, the residual component is fifteen reduced isolated points.
Among all positive integral rate vectors on this fixed support that preserve this conic, the specialization simultaneously minimizes the maximum rate, \(10296\), and the rate sum, \(52464\). The exact certificate is a bounded finite enumeration derived from divisibility and positivity. This is not a claim of global network minimality.
Exact stability of the frozen Version 1 ellipse
Stability is analyzed for the immutable Version 1 rate vector, not for the clean specialization. Along the rational parametrization, the tangent eigenvalue is zero. The two transverse eigenvalues are real and distinct everywhere; exact Sturm counts isolate exactly two parameters at which one transverse eigenvalue crosses zero:
The ellipse is normally attracting for \(\alpha<t<\beta\), transversely saddle-type for \(t<\alpha\) or \(t>\beta\), and nonhyperbolic at \(\alpha\) and \(\beta\). The point corresponding to \(t=\infty\) is also saddle-type. Decimal approximations in the paper are for orientation only and play no role in the proof.
Minimality and scope
One or two species cannot realize the target without a common factor, so three species are globally minimal. A one-linkage three-species target must have full stoichiometric rank. Deficiency-zero weakly reversible systems have at most one positive equilibrium per compatibility class; consequently any three-species weakly reversible realization of the target has at least five active complexes. The present ten-complex network is not claimed to minimize complexes, reactions, degree, or deficiency.
Four exact replay layers
The Version 2 archive supplies a one-command replay:
./reproduce.sh- Frozen v1 verifier. Reconstructs the original network, conic identities, graph, stoichiometry, coordinate gcd, and radical decomposition.
- Family verifier. Reconstructs the \(21\times20\) remainder matrix, certified rank minor, four-dimensional kernel, exact positive cone, and homogenized gcd-one witness.
- Strengthening verifier. Reconstructs the clean field and radical decomposition, proves both bounded integer optima, derives the transverse characteristic polynomial, and performs the exact Sturm counts.
- Independent v2 audit. Separately reconstructs the family matrix, both rate specializations, both radical decompositions, both integer optima, and hand-built Sturm chains, then checks a frozen machine-readable result file.
All evidentiary polynomial and optimization computations use exact integer or rational arithmetic. The independent v2 audit is a separate implementation; independent specialist review remains outstanding.
Version 2 release and checksums
Version 2.0.0 adds the complete fixed-support family, its relatively open positive cone, generic geometric coprimality, the clean fixed-support-optimal integer vector, its exact radical decomposition, frozen-rate transverse stability, structural lower bounds, and four-layer exact replay. Publishing the tagged GitHub release triggered the automatic Zenodo archive and minted the version DOI below.
- Version 2 DOI:
10.5281/zenodo.21753997 - PDF SHA-256:
fe429cf073b30cacfe1ba75624236cda2545c44076f711d4319dcb22ff79512b - Specialist handout SHA-256:
5f7ef76a05ef3c31a693cdd8d9d5ae238edab3c04f1a56bd227c65c5bdf6b268 - Complete archive SHA-256:
c587ec4638a734cc8438e5fd2e5c3b8a489f92d6a94675c556038dd298e13707 - Source archive SHA-256:
c9f76c6edc4aa037d9e9a8320415557483a9c03f7bf0f22bb8450bf36e4239cc - Verifier archive SHA-256:
d13d0400afde27bb444133283957aa5b699928549b36b62024c2c07d3ce5e6e1
Immutable Version 1 history
Version 1.0.0 remains byte-for-byte immutable. Its PDF and all three downloadable archives are preserved below; Version 2 does not overwrite them.
Version 1 DOI correction. The automatic GitHub-Zenodo integration minted Version 1 DOI 10.5281/zenodo.21753527. A different identifier printed in the frozen v1 files had been reserved for an unpublished manual draft before the automatic integration was confirmed. The frozen bytes remain unchanged; citations to v1 should use the live version DOI. The corrected citation file is available separately.
Zenodo repository-level concept DOI 10.5281/zenodo.21753404 groups releases from the entire AlecKriebel/Math monorepo, including unrelated projects. It is not a paper-specific all-versions DOI. Cite this paper using the applicable version-specific DOI.
Priority scope
The construction and exact verifier were completed before the targeted literature audit. Boros, Craciun, and Yu already supplied a positive-dimensional fixed-support same-curve rate family, but every member retains a common scalar factor. A narrow primary-source audit through 1 August 2026 found no earlier weakly reversible fixed-support positive rate family that preserves a continuum in one compatibility class while being generically free of a coordinate common factor within that family. This is a conservative, nonexhaustive audit conclusion, not a universal priority claim. Existing generic-finiteness results concern changes in ambient parameters and do not make this fine-tuned family robust under arbitrary rate perturbations.
Suggested citation
Alec Kriebel, “A four-parameter family of reversible three-species mass-action continua without a common factor,” Version 2.0.0, 2026. https://doi.org/10.5281/zenodo.21753997.
AI-assistance disclosure
OpenAI language models were used extensively in exploration, proof development, software generation, verification design, literature organization, and drafting. Alec Kriebel directed the work, selected the released claims, and is responsible for the decision to publish these materials. Model output was not treated as evidence by itself.
Contact
Alec Kriebel
Independent Researcher
Technical correspondence: me@aleckriebel.com