Existence of non-smooth processes satisfying only one smoothness assumption

About 4 years old · traced to 2022

Let $w$ be a process and let the three assumptions in Definition be the smooth-expectation condition, the relative-fitness-density condition, and the local-change-density condition. A process is non-smooth if it does not satisfy all three assumptions. Non-smooth-process conjecture. There exist non-smooth processes satisfying any one of these assumptions without satisfying the other two. For such processes, the Price equation

would not hold. The preceding theorem shows that satisfying any two assumptions forces the third and hence implies the Price equation, so this conjecture concerns the possible failure of the converse when only one assumption is imposed.

References

Primary source

Tom LaGatta, “The Mathematics of Evolution: The Price Equation, Natural Selection, and Environmental Change”, arXiv:2202.10289 (2022).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader-written argument claims the conjecture is false by showing one assumption is impossible, but no independent verification has been found.

The conjecture asks whether processes can satisfy exactly one of the three smoothness assumptions while failing the other two. The source also states that any two assumptions imply the third and hence the Price equation.

Posted attempt

An unverified argument claims the relative-fitness-density assumption is impossible: since $\mathbb{E}_t[U_{t,t+h}]=1$ , the norm of $U_{t,t+h}/h$ diverges as $h\downarrow0$ . It therefore claims the conjecture is false, and gives a one-point example with assumptions (1) and (3) but not (2), contradicting the preceding two-imply-third theorem. The complete refutation has not been independently verified.

Current status (as of August 2026): The published conjecture and the two-assumptions theorem remain unverified here, while a reader-written argument claims both are false; no independently confirmed resolution is recorded.

Sources

arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

The conjecture is false: its second smoothness assumption cannot hold for any evolutionary process.

Fix a time $t$ , and write $N_t=\mu_t(I_t)\in(0,\infty)$ . For every $h>0$ , the relative fitness is defined by

U_{t,t+h}=\frac{W_{t,t+h}}{\mathbb E_t[W_{t,t+h}]}, \qquad \mathbb E_t[f]=\frac1{N_t}\int_{I_t}f\,d\mu_t.

Consequently $\mathbb E_t[U_{t,t+h}]=1$ , or equivalently

\int_{I_t}U_{t,t+h}\,d\mu_t=N_t.

However, the relative-fitness-density assumption in Definition 8.1(2), equation (8.4), requires the existence of the uncentered limit

\Upsilon_t=L^2(\mu_t)\!\operatorname{-}\lim_{h\downarrow0}\frac{U_{t,t+h}}h.

Whenever $U_{t,t+h}\in L^2(\mu_t)$ , Cauchy–Schwarz gives

\left\|\frac{U_{t,t+h}}h\right\|_{L^2(\mu_t)} \ge \frac{\left|\int_{I_t}U_{t,t+h}\,d\mu_t\right|}{h\sqrt{\mu_t(I_t)}} =\frac{\sqrt{N_t}}h\longrightarrow\infty.

If $L^2$ is instead normalized by $\mathbb E_t$ , the corresponding lower bound is $1/h$ , with the same conclusion. If $U_{t,t+h}\notin L^2$ , the assumption already fails. Thus condition (2) is impossible for every admissible process. In particular, there cannot exist a process satisfying condition (2) without satisfying the other two conditions, contrary to Conjecture 8.3.

There is also an explicit counterexample to the preceding claim that any two conditions imply the third. On the one-point probability space take the identity transition kernel and the constant observable $X_t=0$ . Then the expectation derivative and local-change density both exist and equal zero, so conditions (1) and (3) hold, whereas $U_{t,t+h}=1$ and condition (2) demands the nonexistent limit $\lim_{h\downarrow0}1/h$ . The centered expression $(U_{t,t+h}-1)/h$ would avoid this obstruction, but it is not the density specified in equation (8.4).