MachineSex/paper/pnas/si.md

# SI Appendix — The evolution of sex for artificial intelligence

*Skeleton assembled at Phase 4; finalised at submission. Every numbered experiment has a committed
config (`configs/`), artifact triple (`results/<name>/results.parquet` + resolved config + manifest
with content hashes and git commit), a README with its legend and falsifier status, and a figure that
regenerates from the parquet alone. `reproduce.sh` re-runs everything from the master seeds.*

## SI Text S1–S2: formal statements

## S1. The incompatibility floor: what no alignment can remove (E13c)

**Setting.** Models A and B are trained on the same input distribution; their target label functions
`f_A` and `f_B` agree except on a conflict set `S` of probability mass `μ(S)` (in E13's conflict
condition, the cyclically-relabelled classes; `μ(S) ≈ conflict_frac` up to class balance). A
*function-preserving transformation* `T` (any composition of hidden-unit permutations and, for ReLU
networks, positive per-unit rescalings — the full unit symmetry group of a plain ReLU MLP) satisfies
`T(B)(x) = B(x)` for all `x` by construction.

**Proposition 1 (endpoint invariance — with the term "chord" defined precisely).** Here "chord"
means the α-linear interpolation **of the endpoint loss values**, `(1−α)·L(A) + α·L(B)` — the
baseline in the barrier definition, a function of the endpoints only — NOT the weight-space
interpolation path. For every function-preserving `T`, the endpoint functions, hence the endpoint
losses and this chord, are identical for `(A, T(B))` and `(A, B)`. The **interpolation path itself is
generally NOT invariant** — losses along `(1−α)·A + α·T(B)` change with `T`, which is precisely why
alignment can lower a barrier. *(Immediate from the definition of function-preserving.)* Scope
caveat: our aligner provably recovers a permuted-and-rescaled copy exactly — an important special
case — but this does not establish global optimality of the alignment over the symmetry group for
independently trained networks; the decomposition's "removable" share is therefore a lower bound, and
the "residual" an upper bound, on their true values.

**Proposition 2 (no merged model can serve both parents).** Let `h` be *any* single classifier (in
particular, any interpolated/merged model, under any alignment). On every `x ∈ S`, `f_A(x) ≠ f_B(x)`,
so `h(x)` disagrees with at least one of them. Hence

  `ε_A(h) + ε_B(h) ≥ μ(S)`,  and therefore  `max(ε_A(h), ε_B(h)) ≥ μ(S)/2`,

where `ε_P(h)` is `h`'s error against parent `P`'s labels. A hybrid of two models whose conventions
conflict on mass `μ(S)` errs at rate at least `μ(S)/2` against at least one parent — **hybrid
disadvantage with an information-theoretic floor, independent of the alignment group, the
architecture, and the merging operator.** This is reproductive isolation in the fitness sense: past a
given functional conflict, *no* recombination operator produces an offspring loyal to both lineages.

**What remains empirical, and why the experiment is designed as it is.** Propositions 1–2 do *not*
bound the single-task path barrier (the loss along the interpolation between A and `T(B)` evaluated
on one parent's task): in principle a path could dip toward one parent's function. Whether it does is
exactly what E13 measures — and the measured answer is that it does not: the conflict-condition
barrier is unchanged by permutation alignment (`residual`) *and* by alignment modulo the full
permutation × positive-rescaling group (`residual_scale`), while the same aligner removes ~all of the
independent-init barrier (the positive control). Richer-symmetry results for transformers
(arXiv:2606.23607; neuron-identifiability approaches to linear mode connectivity, 2026) strengthen
the *removable* side of the decomposition and are therefore complementary: the more barrier a larger
group can remove for *compatible* models, the sharper the meaning of the residual that survives for
*incompatible* ones — and Proposition 2 caps what any of them could ever achieve on the conflict set.

**Terminology note for the paper.** "Residual (after alignment)" = the estimated functional
incompatibility; for ReLU MLPs we align modulo the full unit symmetry group, so the estimate is not
confounded by missed symmetries of that architecture class.

## S2. Emergent vs imposed incompatibility (E13b framing)

The conflict condition *imposes* contradiction (the two label maps disagree on `S`), which pins
`μ(S) > 0` and activates Proposition 2. A true Bateson–Dobzhansky–Muller incompatibility is
*emergent*: each lineage's substitutions are harmless on their own background (`μ(S) = 0` — the
training signals never contradict), and incompatibility, if any, arises only in the *combination*.
The `disjoint` (complementary class specialists) and `augment` (divergent input conventions)
conditions realise this: any residual barrier they develop cannot be attributed to label conflict and
is the emergent-speciation signal proper. Pre-registered readings: residual grows with divergence →
model speciation is emergent in real weights (E12's trajectory realised); residual stays at the
`shared`-control level → within this regime, trained networks are *more* merge-compatible than the
biological analogy predicts — an honest bound on the analogy, and itself a design-relevant result
(merging is safe absent functional conflict).

**Outcome (2026-08-11 run, 4 reps, t_div ≤ 3200): the second reading.** Residual 0.000 at every
divergence in both emergent conditions, and the merge *rescues* the forgetting `disjoint` specialists
(parents → 0.535/0.474 on the full task; merged ≈ 0.955 throughout — a sustained Fisher–Muller rescue
at zero barrier). Isolation in real weights required functional conflict in this regime; whether
long-horizon over-specialisation erodes mergeability at LLM scale (cf. arXiv:2607.11997) is the
`llm_speciation` question (Phase 3).


## SI Table S1: the claims ledger (status / assumptions / evidence / limits)

| Claim | Status | Key assumptions | Evidence | Known limits |
|---|---|---|---|---|
| Collapse = Wright–Fisher drift (minimal model) | Exact (diagnosis conceded to prior work) | Knowledge = categorical distribution; refit = resample | Closed forms reproduced to <0.5% | Real learners add a signed, architecture-specific estimator bias (measured) |
| Grounding = immigration; critical real-data fraction ≪ 1 | Exact + empirical sign | Fresh samples from a fixed, non-drifting truth | Exact `H_eq`; `g*≈0.048`; sign holds in RNN/MLP/VAE and on MNIST | Deepest tail unrescuable at feasible budgets (`m ∼ 1/p`); sharp threshold softens in trained nets |
| "Merge, don't average" conservation | Exact **for the output-mean operator** | Rare-item regime; an oracle/verifier identifies the strongest source | E4 closed form + simulation; neural reproduction | Weight-averaging and routing are empirical cousins, not instances; budgets differ; bridge = the headroom rule |
| Offspring exceed every parent (Fisher–Muller) | Interpretation + empirical | Complementary (decorrelated) parents; verifiable fitness | E8 analytic; 7B LoRA merge beats every specialist on every family | LLM tier: 3 lexically-distinct families; multi-seed replication in progress |
| Outbreeding depression on rugged landscapes; operator design rule | Exact-model result; hypothesis at LLM scale | NK epistasis stands in for skill entanglement | E9–E10; directed selection rescues | Not yet mapped onto a real task-entanglement measure |
| Optimal mate-pool breadth shrinks with ruggedness | Exact-model result; hypothesis for merging populations | Ring population, local selection | E14 | Phenomenon known to island-model evolutionary computation; our contribution is the mapping and the diversity/mean decomposition |
| Merge failure decomposes into coordinate artefact + functional residual | Empirical (MLP tier; LLM tier in progress) | Alignment enumerates the architecture's unit symmetries | Full-symmetry residual ≈ 0 (compatible) vs ≈ naive (conflict); cliff in hybrid fitness | Scoped to aligned linear interpolation; conflict floor is information-theoretic, not genetic |
| Epistasis (not divergence) sets the cliff; snowball onset | Exact-model result; **hypothesis** at the neural tier | BDM incompatibility structure | E12 | Snowball count ≠ performance cliff without the effect-size link; neural test outstanding |
| Pre-merge functional disagreement predicts merge penalty | Empirical, within a controlled grid (0.5B, 13 conditions × 3 seeds) | Constructed conflict/overlap/duration axes; oracle-potential outcome (pre-registered; ordering sensitive to reference) | Clustered CIs exclude 0; held-out LOCO ρ≈0.4; selected geometry baselines ≈ 0 | Head-to-head predictor differences not individually significant; only selected baselines; generalisation to real task pairs open |
| Confidence weighting improves rank prediction over raw disagreement | **Not supported** (pre-registered internal prediction) | — | Paired Δ\|ρ\| ≈ −0.02, CI [−0.13, +0.06] | Weighting does double the conflict-vs-compat level contrast |
| The predictor improves budget-matched operator choice | **Open** | — | Soup-vs-route gap readout noise-dominated at 0.5B | The practical payoff; untested |
| Emergent speciation without conflict | **Not observed** (pre-registered) | Shared ancestry, compatible tasks, tested divergences | E13b: residual 0.000; merge rescues specialists | Bounds the hypothesis; longer horizons/distribution shift/capacity pressure untested |
| Grounding + sex + diversity jointly necessary | Exact-model result; hypothesis at LLM scale | Conformity stands in for self-consumption | E11 four-arm ablation, each arm failing distinctly | The full grounded LLM society is unbuilt |

## SI Table S2: headline quantitative results

Headline quantitative results with sample sizes, uncertainty, and outcome definitions (full
per-experiment tables and falsifier status in the per-experiment documentation).

| Result | Setting / n | Outcome definition | Headline |
|---|---|---|---|
| Closed-form validation | Analytic tier; standing tests | Simulated vs closed-form H-decay, immigration equilibrium, multi-teacher union | Agreement < 0.5% |
| Grounding retention | Minimal model; 18+ replicates per point | Fraction of equilibrium diversity retained at grounding g (operational threshold) | g ≈ 0.05 retained ≥95% (tested setting); smooth in g |
| MNIST collapse & rescue | Conv-VAE, 4 replicates; frozen oracle (98.5% mode acc.) | Mode support / forward-KL over generations | Dry: 30→1 modes; 10% grounding: 30/30 held |
| Fisher–Muller in LLMs | 5 seeds (0.5B), fixed tests; single 7B run | Merged vs best-specialist accuracy (overall; worst family) | Ties 0.647±0.027 vs 0.592±0.009; 7B 0.87 vs 0.77 |
| Union vs blend (headroom) | 3 seeds (0.5B hard); single 7B-hard run | Paired per-seed ordering, routing vs weight-average | Routing > blend in 3/3 seeds; one catastrophic blend failure avoided |
| Speciation decomposition | MLPs, 3 replicates | LMC error barrier residual after permutation+rescaling alignment | Same-task 0.001; conflict 0.497 (naive 0.502) |
| Emergent isolation | MLPs 4 reps to 6.4× base training; LLM 1→12 epochs | Residual barrier; merged vs parent accuracy | 0.000 everywhere; merge rescues parents (≈0.955 vs ≈0.50) |
| Predictive test | 13 conditions × 3 seeds (0.5B) | Merge penalty vs oracle parent potential (pre-registered; ±: clustered 95% CI) | Functional ρ +0.45/+0.46, CI excl. 0; LOCO ρ ≈ 0.4; geometry n.s.; paired differences n.s. |

## SI Methods (per tier — full details in the per-experiment READMEs and configs)

**Analytic tier (E1–E14).** Wright–Fisher simulator over K-item distributions; closed-form validation
suite (`tests/test_scientific_validation.py`, <0.5% tolerances); learning kernel; multi-locus
genotypes, NK landscapes, n-parent crossover (E7–E11); BDM speciation model (E12); mating structure
(E14). Bitwise reproducible from master seeds.

**Neural tier.** Histogram bridge (exact reduction to the analytic tier — the harness gate);
RNN/MLP/VAE collapse+grounding on a synthetic mode universe with an exact oracle; conv-VAE on MNIST
with a frozen CNN oracle (98.5% mode accuracy; 30x30 confusion matrix recorded as the measurement
floor); E13 speciation: no-BatchNorm MLPs, weight-average merges, LMC error barriers before/after
alignment under the complete unit symmetry group (deterministic Re-Basin permutation matching composed
with exact scale canonicalisation; sanity gate recovers a permuted-and-rescaled copy exactly);
pre-registered emergent conditions (disjoint classes; shifted-view conventions).

**Language-model tier.** LoRA rank-16 specialists on procedural task families with an exact-match
verifier; Qwen2.5-Instruct 0.5B/7B; operators: soup/TIES adapter arithmetic, per-input routing,
Dirichlet offspring populations screened on held-out validation; multi-seed protocol (fixed tests,
varied training seed); speciation knobs (conflicting conventions on ambiguous prompts; duration);
the controlled predictive test (six pre-merge predictors; three decorrelated axes; robust statistics
via `figures/stats_llm_epistasis.py`: condition-clustered bootstrap, paired contrasts,
leave-one-condition-out prediction, three outcome references). Statistical (per-seed)
reproducibility documented for GPU tiers.

## SI Statistics

Output of `figures/stats_llm_epistasis.py` (clustered CIs, paired predictor contrasts, LOCO held-out
prediction, outcome-reference sensitivity, within/between-axis decomposition) — reproduced verbatim at
submission. Chronology of the predictive test (prospective / adaptive / post-hoc) as disclosed in
`results/llm_epistasis/README.md`.

## SI Figures

One per experiment, regenerated from committed artifacts: E1–E14, bridge/collapse/grounding/
architectures/recombination, kernel (sharpen/smooth), mnist_collapse (+ montage), speciation_real
(decomposition/cliff/emergent), llm_merge(_hpc/_seeds), llm_moe(_hpc/_hard_hpc/_hard_seeds),
llm_directed(_hpc/_hard_hpc/_hard_seeds), llm_speciation(_add), llm_epistasis(_compat).