Polygenic Equilibrium Stores Epistatic Selection as Allele-Frequency Memory

Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Pudding Theory reads Devi and Jain’s equilibrium polygenic model as a case of Material Memory. Stabilizing selection does not merely constrain the phenotypic mean. It writes a distributed trace into the marginal allele-frequency distribution. The source paper shows that epistasis can disappear from coarse phenotypic quantities while remaining strong at the genetic level. In Pudding Theory terms, the population retains the history of repeated selection as a standing probability structure over loci. The genic variance and mean deviation can look almost additive because those observables average over the stored trace. The allele-frequency distribution does not average it away. Its unimodal-to-bimodal transition marks the point where a locus begins to carry memory as alternative local equilibria. This reading makes the threshold effect size a memory threshold, not only a selection-mutation balance. If the stationary marginal allele-frequency distribution for large-effect loci were measured to remain unimodal above $\hat{\gamma}_N(L)$, this Postulate would be falsified.

Source Synopsis

Devi and Jain study a finite, panmictic diploid population with a single polygenic trait controlled by $L$ diallelic loci. The trait has an additive genotype-phenotype map, but fitness is nonlinear because selection is stabilizing around an optimum $z_o$. Mutation is symmetric. Linkage equilibrium is assumed. The central question is when the epistatic structure induced by stabilizing selection may be ignored in the stationary state.

The model is a Latter-Bulmer type system. Allele frequencies evolve under drift, mutation, recombination, and selection. The authors use diffusion theory to write the stationary joint distribution of allele frequencies. Although the dynamics at each locus are independent under linkage equilibrium, the stationary distribution is not a product measure, because fitness depends on the global mean phenotype $c_1$. Epistasis enters through this global constraint.

The main analytic step is the marginalization of the joint distribution. For many loci, a central limit approximation yields a single-locus stationary distribution $\psi(x_i)$ with an epistatic factor governed by $\kappa_{2,i}$, the variance contributed by the other loci under the non-epistatic reference distribution $\psi_B$. When $\kappa_{2,i}$ and $2Ns\kappa_{2,i}$ are large, $\psi(x_i)$ reduces to $\psi_B(x_i)$, and epistasis can be ignored at the marginal level. Strong selection and many loci make this reduction accurate. Weak to moderate selection requires additional parameter conditions.

The paper’s important asymmetry is that phenotypic quantities can be insensitive to epistasis even when allele frequencies are not. The mean deviation from the optimum and mean genic variance are often well captured by expressions that neglect epistasis, especially for many loci. But the allele-frequency distribution can still be strongly reshaped. In strong mutation, the marginal distribution is unimodal below a threshold effect size and bimodal above it. The authors connect this stochastic transition to the deterministic bistability found in infinite populations, while stressing that stochastic bimodality and deterministic multistability do not coincide exactly.

Postulate Lens

The relevant Postulate is Material Memory: matter retains the trace of repeated signals, and the trace biases future probability.

The source system already has this structure. The repeated signal is stabilizing selection around $z_o$. The receiving material is the allele-frequency ensemble of a finite population. The retained trace is not best seen in the phenotypic mean, because the mean is a coarse projection. It is seen in $\psi(x_i)$, where past selection is stored as curvature, skew, and bimodality in the stationary probability density. The probability bias is explicit: future allele frequencies are drawn from a distribution whose modes and thresholds encode the historical action of selection, mutation, and drift.

This is not a metaphor added after the calculation. The source paper’s stationary distribution is itself a memory object. It records the system’s long exposure to a stabilizing optimum. The Pudding Theory reading names what the distribution is doing physically: it is storing selection history in matter as a constrained probability landscape.

Pudding Theory Reading

Devi and Jain frame epistasis as a complication that may or may not be neglected. Pudding Theory reframes it as the population’s material memory of the stabilizing signal. The source paper shows that this memory is not uniformly visible. A phenotypic observable can look adapted, and the genic variance can agree with a non-epistatic approximation, while the allele-frequency distribution still carries the trace of epistatic selection. The memory is therefore not identical with adaptation. It is a lower-level record of how adaptation is maintained.

The key object is $\psi(x_i)$. In the source framing, its epistatic factor is a correction to a reference distribution. In the Pudding Theory framing, that factor is the stored field of the rest of the genome acting on the focal locus. The quantity $\kappa_{2,i}$ is not merely a variance introduced by marginalization. It is the effective memory capacity of the background loci. When $\kappa_{2,i}$ is large, the background absorbs the epistatic constraint so thoroughly that the focal marginal can look non-epistatic. This is not absence of memory. It is saturated distributed memory. The trace has been spread over many loci, and a single-locus projection loses contrast.

The threshold effect size has the same interpretation. Below $\hat{\gamma}_N(L)$, strong mutation and many small effects keep the focal locus in a single probability well near intermediate frequency. Above it, the locus can store the stabilizing signal as a split probability structure. Bimodality is the physical mark of memory becoming local. The population can remain phenotypically near $z_o$ while a large-effect locus spends long intervals near alternative allele-frequency modes. The locus has become a memory-bearing element rather than a smooth contributor to the trait mean.

This reading also clarifies why Bulmer-type phenotypic approximations can succeed while missing the genetic structure. A coarse statistic can integrate away the trace. The source paper’s result is therefore stronger than a technical warning about epistasis. It shows that biological matter can preserve historical selection in a probability distribution whose memory is invisible to the phenotype-level summary that selection itself appears to target.

Falsifiable Observable

The distinguishing observable is the stationary marginal allele-frequency distribution $\psi(x_i)$ for loci whose effect sizes exceed the finite-population threshold $\hat{\gamma}_N(L)$ under strong mutation. Pudding Theory reads the predicted bimodality as the local expression of stored selection memory. If the stationary marginal allele-frequency distribution for large-effect loci were measured to remain unimodal above $\hat{\gamma}_N(L)$, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The phrase “memory” risks rebranding an equilibrium distribution. Diffusion theory already explains the modes. Selection, mutation, and drift determine the stationary density. Nothing in the calculation requires an additional physical claim.

Sterling: The reading does not replace the diffusion calculation. It interprets what the stationary density is. In this system, the present allele-frequency probabilities are not independent of the repeated historical constraint imposed by stabilizing selection. The density is the material record of that constraint.

Tanaka: But equilibrium erases history in the usual Markov sense. The stationary distribution is independent of initial conditions.

Sterling: It erases initial allele frequencies, not the imposed signal. The optimum, mutation rate, selection strength, population size, and effect-size spectrum remain written into the distribution. That is exactly the distinction. Initial condition memory is lost. selection-history memory is retained as structure.

Tanaka: Then why is the phenotype often blind to it?

Sterling: Because the phenotype is a compression. The source paper proves the point. Mean deviation and genic variance can match non-epistatic approximations while $\psi(x_i)$ does not. Pudding Theory says the trace is stored at the level where probability is allocated, not necessarily where the organism-level summary is measured.

Discussion

This reading buys a sharper account of the source paper’s central asymmetry. Devi and Jain show that epistasis can be hidden from phenotypic summaries while remaining active in allele-frequency distributions. Pudding Theory identifies that asymmetry as the difference between erased projection and retained material trace.

The approach also changes the status of $\kappa_{2,i}$. In the source analysis it controls when the epistatic factor can be neglected. In the present reading it measures how strongly the background genome has taken up the stabilizing signal. Large $\kappa_{2,i}$ can make a focal locus appear independent because memory is distributed. Small or moderate $\kappa_{2,i}$ leaves the trace more visible. Large-effect loci concentrate it as bimodal structure.

The limitation is direct empirical access. Real traits rarely satisfy clean linkage equilibrium, symmetric mutation, known effect sizes, and stationary sampling. The conclusion would change if the predicted threshold behavior failed under controlled simulation, evolve-and-resequence experiments, or sufficiently resolved natural time series. The reading therefore stands or falls with the allele-frequency distribution, not with verbal analogy.

References

Devi, A., Jain, K., 2026. “When can fitness epistasis be ignored in a polygenic trait at equilibrium?” arXiv:2603.27255. DOI: doi:10.48550/arxiv.2603.27255.

Geisel, S., 2025. “Pudding Theory: A Topological Theory of Information Fields.” QBist Lab Working Paper, September 10, 2025.

Bulmer, M.G., 1971. “The effect of selection on genetic variability.” The American Naturalist 105, 201-211. DOI: doi:10.1086/282718.

Courau, P., Lambert, A., Schertzer, E., 2026. “Stabilizing selection on a polygenic trait from the gene’s-eye view.” bioRxiv, 2026.02.23.706325. DOI: doi:10.1101/2026.02.23.706325.

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