MachineSex/results/figS2_kernel_smooth

kernel — the learning kernel: why real learners deviate from neutral drift

(This legend covers both results/figS2_kernel_sharpen/ and results/figS2_kernel_smooth/; the figure kernel.png is written into both.)

Claim tested. Neutral Wright–Fisher drift (the histogram bridge, and the baseline of Riis 2026) is the null model of collapse. But Layer 1.5 showed real trained models deviate from it — and in opposite directions. Can a single extension of Layer 1 — a parameterized learning kernel on the refit step, p_{t+1} = T_θ(counts/n) — reproduce both deviations, and does neutral drift genuinely fail without it?

Setup. The kernel (knowledge/kernel.py) has two population-genetics knobs, both reducing to neutral drift at their defaults (so the histogram and every scientific-validation test are unchanged): reset u — mutation toward a prior (p ← (1−u)p + u·π), i.e. smoothing; and temperature τ — sharpening (p ∝ p^{1/τ}, τ<1 concentrates), i.e. mode-competition. Two matched-to-neural regimes, 24 replicates each.

The four panels (kernel.png; blue = neutral, red = kernel-on, green dashed = the real neural model)

Top row — VAE regime (n=6000, K=30), pro-collapse:

1. Heterozygosity. Neutral drift is inert — at n=6000 it barely moves (H stays at H*). Yet the real VAE (green) collapsed to H≈0. Sharpening (τ=0.8) reproduces the collapse. Neutral drift is falsified; the estimator's mode-competition is required.
2. Support. Neutral holds ~all 30 modes; sharpening → 1 mode, matching the VAE.

Bottom row — RNN regime (n=200, K=256), anti-collapse: 3. Heterozygosity. Neutral drift drives H → 0, but the real RNN (green) only partially collapses (H floors at ~0.68). Mutation u=0.006 reproduces the floor. The estimator here removes collapse pressure. 4. Forward-KL. Neutral diverges; smoothing plateaus. Honest caveat: uniform-mutation plateaus above the RNN's KL (~5 vs ~2) — evidence the RNN's smoothing target is truth-like, not uniform (a refinement for future work). The sign is unambiguous.

Takeaway

Model collapse in real learners = neutral drift ⊕ an architecture-specific estimator-bias operator that can point either way. The histogram sits at the neutral null (u=0, τ=1); the VAE sharpens (adds collapse); the RNN/MLP smooth (add a diversity floor). This mechanistically explains the Layer-1.5 architecture-generality result and the softened neural g*, and develops the exact axis Riis (2026) names as future work ("different smoothing schemes… each induce their own fixed-point geometry… a natural direction for further work"). u/τ are calibrated from a single neural diagnostic and pinned in the configs. Falsifier (not triggered): if neutral drift had already reproduced the neural curves, the estimator axis would be superfluous — instead it fails in both regimes, oppositely.