infsup

by Zinan Huang 🌸

One Marking, One Failure: The Method of Unknowns

2026-03-25

We tried a new way to do balancing dilation. It didn’t work. Here’s why.

This is a sequel to Five Ways to Fail at One-Marking. If you haven’t read that, start there for context on the one-marking problem and the $\frac{1}{2}$-trick.

The Setup: G1, B1, B2

In the previous post, we showed that the $\frac{1}{2}$-trick (Attempt 5) is our current best result. The variable system from $\frac{1}{2}(S + S^2)$ has one beautiful property and two defects:

Question: Can we keep G1 while eliminating B1 and B2?

The Idea: Asymmetric Scaling

The $\frac{1}{2}$-trick scales everything by $\frac{1}{2}$, including the marking variable. What if we break the symmetry?

Proposal: Keep $x_1$ unscaled (so $x_1 \to x_1^*$ directly) and scale everything else by a small $\lambda > 0$. Introduce an independent variable $r$ to absorb conservation.

A Different Kind of $r$

This is where the construction departs from everything we’ve done before. In the $\frac{1}{2}$-trick, $r = 1 - \sum x_i$ is a derived quantity: an algebraic function of the PP species, completely determined by them. Its derivative $\dot{r} = -\sum \dot{x}_i$ follows by differentiation.

Here, $r$ is fundamentally different. It is an independent unknown β€” a CRN species in its own right. We don’t define what $r$ “is” in terms of the original PP variables. We only require three things:

  1. Conservation: $r + \sum_{\alpha \neq r} z_\alpha = 1$.
  2. Dilation: $\dot{x}_i = r \cdot P_i(x_1, \ldots, x_n)$.
  3. Absorber: $\dot{r} = -\sum_{\alpha \neq r} \dot{z}_\alpha$.

Condition 3 follows from differentiating Condition 1. We don’t prescribe $r$’s identity β€” only the constraints it must satisfy. We call this the method of unknowns.

The Variable System

For small $\lambda > 0$:

Variable Role
$x_1$ marking (unscaled β€” direct readout)
$\lambda x_i$, $i \geq 2$ other species, scaled
$\lambda r x_i$, $i = 1, \ldots, n$ degree-2 cross terms
$\lambda r x_i x_j$, $i \leq j$ degree-3 cross terms
$\lambda r^2$ degree-2 pure-$r$
$r$ conservation absorber

This is structurally the same monomial family as $S + S^2$ β€” just with asymmetric coefficients instead of uniform $\frac{1}{2}$. The hope: changing the scaling preserves the factorization structure (G1), since scaling only changes reaction rates, not the existence of valid factorizations.

B1 eliminated: $x_1$ has coefficient 1. Direct readout.

B2 eliminated: Total mass $= 1$ by construction (Condition 1).

Why It Fails

The Chain-Rule Trap

Consider the variable $z = \lambda r x_1$. For it to faithfully represent the product of species $r$ and $x_1$, the chain rule requires:

$$\dot{z} = \lambda(\dot{r} \cdot x_1 + r \cdot \dot{x}_1).$$

But $\dot{r}$ is the conservation absorber:

$$\dot{r} = -\sum_{\alpha \neq r} \dot{z}_\alpha.$$

And $\dot{z}$ (the derivative of $\lambda r x_1$) is one of those terms.

So $\dot{r}$ depends on $\dot{z}$, and $\dot{z}$ depends on $\dot{r}$. Circular.

Resolving the Circularity

We can solve for $\dot{r}$ explicitly. For $n = 1$ with variables $\{x, \lambda rx, \lambda r^2, r\}$:

$$\dot{r} = -\bigl[\dot{x} + \lambda(\dot{r} \cdot x + r \cdot \dot{x}) + 2\lambda r \cdot \dot{r}\bigr].$$

Collecting $\dot{r}$ terms:

$$\dot{r}(1 + \lambda x + 2\lambda r) = -\dot{x}(1 + \lambda r).$$

Substituting $\dot{x} = rP$ (after $r$-dilation):

$$\boxed{\dot{r} = \frac{-rP(1 + \lambda r)}{1 + \lambda x + 2\lambda r}.}$$

This is a rational function of $r$ and $x$.

Why This Is Fatal

Mass-action chemical reaction networks produce polynomial ODEs. Each bimolecular reaction $A + B \to C + D$ at rate $k$ contributes $\pm k[A][B]$ to the derivatives β€” always a product of at most two concentrations. The resulting ODE for each species is a polynomial.

The denominator $1 + \lambda x + 2\lambda r$ depends on species concentrations. It does not simplify to a constant, not even on the conservation simplex. No set of bimolecular reactions can produce a rational-function ODE.

Why the $\frac{1}{2}$-Trick Avoids This

In the $\frac{1}{2}$-trick, $r = 1 - x$ is an algebraic identity. So $\dot{r} = -\dot{x}$ β€” a polynomial. There is no circularity because $r$ is not an independent variable; its dynamics are completely determined by $x$.

The chain rule for $\frac{1}{2}rx$ involves $\dot{r} = -\dot{x}$, which is known and polynomial. Everything stays in the polynomial world.

The price we pay: uniform $\frac{1}{2}$ scaling (B1 and B2). The method of unknowns tried to eliminate this price by making $r$ independent. But independence creates circularity, and circularity creates rational functions.

The Structural Lesson

The failure reflects a fundamental tension. Three things cannot coexist:

  1. Independent $r$ β€” needed for asymmetric scaling (eliminating B1 and B2).
  2. Chain-rule consistency β€” needed for compound variables ($\lambda rx$, $\lambda r^2$) to track their intended quantities.
  3. Polynomial ODEs β€” required by mass-action kinetics.

The $\frac{1}{2}$-trick resolves this by sacrificing (1): $r$ is dependent, forcing uniform scaling but keeping everything polynomial. The method of unknowns sacrifices (3), which is non-negotiable.

There is no free lunch. The uniform $\frac{1}{2}$ in the $\frac{1}{2}$-trick is not a cosmetic choice to be optimized away β€” it is the price of polynomiality.

Updated Scorecard

Construction One marking Formal cons. Simplex Issue
Self-product ($z_{ij} = x_i x_j$) No (two) Yes Yes $O(n^2)$ markings
$\frac{1}{2}$-trick (Note 17) Yes No Yes No formal conservation
$\sqrt{\cdot}$ method Yes Yes No $\sqrt{x} > x$ for $x < 1$
$\frac{1}{3}$-trick (Note 23 v1) Yes Yes Yes Scale: $O(n^3)$ vars
$\frac{1}{2}$-trick v2 (Note 23) Yes Yes Yes Tracks $\frac{1}{2}x^*$
Method of unknowns Yes Yes Yes Rational $\dot{r}$

Six approaches. Six obstructions. The $\frac{1}{2}$-trick v2 remains the best.

Filed under “negative results worth documenting.”