Knowing what entered a model and what response came out does not always determine its internal exchange rates. Different rates can produce exactly the same input–output relationship. In this project, I examined a mammillary model—a central compartment connected to several peripheral compartments—and represented such indistinguishable parameter values by an explicit curve.
I selected Conjecture 3.18 in the paper by Clemens, Martinez, Shiu, Thompson and Warren, directed the research and requested verification. The question goes beyond whether the model as a whole is unidentifiable. Each parameter named in the conjecture must be shown to admit infinitely many different values. The public manuscript proves this for every number of compartments n ≥ 5 and describes the local set of parameters preserving the input–output equation as a one-dimensional curve.
Where the input enters and the output is observed
A compartment is a part of a model in which an amount is tracked separately. Compartment 1 is the centre, connected in both directions to compartments 2 through n. There are no direct connections between peripheral compartments and no leakage out of the model. Input enters compartment 2 alone, and the observed output is the amount in compartment 3.
Write a_i = k_1i for the rate from peripheral compartment i to the centre, and b_i = k_i1 for the reverse rate. There are 2n − 2 parameters altogether. Conjecture 3.18 in the earlier paper concerns a_2, a_3, b_2, b_3, …, b_n: n + 1 parameters. The remaining rates a_4, …, a_n were already treated in Proposition 3.16 of that paper and stay fixed along this curve.
Unidentifiability here is not fluctuation caused by measurement error. The set of parameter values producing the same input–output equation coefficients is called a fibre. At a generic real base point, the source definition asks whether each parameter has only finitely many possible values in that fibre. This construction leaves infinitely many, even near the base point, by providing a continuous path inside the set.
Fixing the original polynomials, not just their ratio
The input–output relationship can be written as a transfer function. With s corresponding to differentiation, the transfer function for zero initial state is
H(s) = qQ(s) / [sW(s)].
Here q = a_2 b_3, Q is formed from the rates of peripheral compartments 4 onward, and W includes the rates between every peripheral compartment and the centre.
P(s) = ∏(s + a_i) [i = 2,…,n]
W(s) = P(s) + ∑ b_i ∏(s + a_j) [i = 2,…,n; j ≠ i]
Q(s) = ∏(s + a_i) [i = 4,…,n]
q = a_2 b_3In the product inside W’s sum, j also ranges from 2 to n. For the model matrix A, the denominator is the determinant det(sI − A) = sW(s), and the numerator is the signed cofactor associated with the input and output positions, qQ(s). The original input–output equation is sW(s)y = qQ(s)u, where u is the input and y the observation. Fixing W, Q and q separately therefore preserves its coefficients before any cancellation.
This distinction matters: obtaining the same reduced ratio is not automatically the same as preserving the original coefficient map used by the source paper. The proof preserves that map directly. The input–output differential identity itself is not limited to the zero-initial-state assumption used to explain the transfer function.
Move one rate and reconstruct the others
Start at a base point where all rates are nonzero and a_2, …, a_n are pairwise distinct. The excluded set has measure zero in the real parameter space, so these conditions cover the real-generic setting of the source definition.
Keep the base-point polynomials W and Q and the value q fixed. Let a₃ move with a new variable v. Rather than choosing a₂ independently, set
D(v) = qQ(−v) − W(−v)
T(v) = qvQ(−v) / D(v)
a_2(v) = T(v), a_3(v) = vThe rates a_4, …, a_n retain their base values. Then reconstruct each outward rate by
b_i(v) = W(−a_i(v)) / ∏[a_j(v) − a_i(v)],
where the product runs over j = 2 through n with j ≠ i. This is an interpolation formula using polynomial values at distinct points. It preserves W, while the definition of T ensures a_2(v)b_3(v) = q. The fixed tail rates preserve Q.
The denominators are nonzero at the base point, so v can move through a sufficiently small interval around it. The parameters vary along that interval while the original input–output equation remains unchanged. Because each coordinate is a ratio of polynomials, this is a rationally parametrized curve.
A curve must change every named parameter
Finding a curve is not enough to show that every parameter changes along it: some coordinates could remain fixed. To cover all n + 1 parameters in the conjecture, the proof must separately establish that a₂, a₃ and every b-coordinate are nonconstant.
The argument uses the fact that each coordinate is a rational function. If such a function were constant on an open interval, clearing its denominator would give a polynomial vanishing throughout that interval, hence identically. Conversely, showing that the rational function is nonconstant establishes that its coordinate takes infinitely many values on every sufficiently small open interval near the base point. The proof applies this argument parameter by parameter.
For example, take n = 5, (a_2, a_3, a_4, a_5) = (1, 2, 3, 4), and every b-rate equal to 1. Then Q(s) = (s + 3)(s + 4), q = 1, and W(s) = s⁴ + 14s³ + 65s² + 120s + 74. At v = 2, the derivatives of the six moving coordinates (a_2, a_3, b_2, b_3, b_4, b_5) are (3/4, 1, −25/8, −3/4, 11/8, 3/4), respectively.
Checking derivatives at one point would still be insufficient. If only b₂ is changed to 5/2 in the base example, the derivatives of a₂ and b₃ at that point become zero, although both coordinate functions remain nonconstant. This is why the proof does not rely on one tangent or a numerical matrix-rank calculation.
If all base rates are positive, the curve can be restricted to a short enough interval that every rate stays positive. This positive-rate result follows separately from the theorem over real parameters. With the local labels of the remaining a-rates fixed, the entire nearby coefficient-map fibre is exactly this curve, and the derivative matrix of the coefficient map has rank 2n − 3. Exceptional cases with zero rates or coincident a-values lie outside this theorem.
Keeping the proof and the computational checks distinct
The conclusion for every n ≥ 5 comes from the general interpolation and nonconstancy arguments. Computation checks the formulas separately. The public verification record covers 30 residue-reconstruction identities and 42 nonconstant coordinate functions across six fixtures, with exact positivity checks for four intervals. These include mixed-sign rates, W(0) = 0, and the example with zero derivatives at one point.
The generator uses SymPy. A separate checker verifies whole rational-function identities with standard-library rational polynomial arithmetic. Direct determinant and cofactor checks compare 24 parameter vectors at 168 vector-and-s evaluation pairs; degree bounds turn those comparisons into polynomial identities for each vector. The package also includes all-symbolic checks for n = 3, 4, 5, 6 and 7, and rejection of eight deliberately corrupted certificates. These finite checks supplement the general proof.
I deposited the public proof and code on Zenodo on 5 September 2026, and Bulletin of Mathematical Biology acknowledged the submission that day. The public version is a preprint, before peer review. I selected the question, directed the research and requested verification. ChatGPT assisted with mathematical derivation, proof writing, code and internal checks; the public contribution record also documents Codex’s code review and regeneration runs.
The ambiguity here is more than a finite number of ways to relabel peripheral compartments. Even with those labels fixed, n + 1 rates can keep changing near the base point while preserving the same input–output equation. The construction makes it possible to follow explicitly the directions in which internal rates remain indistinguishable.