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SI: adopt the clearer rewrite, with factual corrections; fix two build bugs it exposed

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Prose: adopted the simplified rewrite for the Reproducibility preamble, SI
Text S1 and S2, and the two tables. It reads better - shorter sentences, no
shouty caps, no self-commentary in the proposition headings.

Fact-checked against the artifacts before adopting. Corrections:
- Table S2 said grounding retention used "18+ replicates per point". E2 uses
  100 lineages; 18 is the *neural* grounding sweep. (Pre-existing error,
  faithfully carried over by the rewrite.)
- The emergent parents' 0.535/0.474 are the accuracies at the LONGEST
  divergence (t_div=3200), not overall means (0.595/0.545); now qualified.
  Verified merge holds 0.954-0.956 at every divergence, residual exactly
  0.000 in both emergent conditions.
- Dropped an invented run date (2026-08-11; the run is from 2026-09-06) and
  an internal project-phase reference ("Phase 3").
- The llm_speciation duration question is no longer open - it ran, and found
  no isolation from over-training (1-12 epochs); text updated.
- Restored the confidence-weighting numbers the rewrite dropped: paired
  bootstrap contrast |rho| = -0.021, CI [-0.130, +0.059] (re-derived), plus
  the nuance that the weighting does sharpen the level contrast.
- "Minimal model" -> "biological model"; "LLM tier in progress" -> done.
- Trimmed an unverifiable citation ("neuron-identifiability approaches...")
  to the reference the bibliography actually carries.

Two rendering bugs the LaTeX version exposed, both pre-existing:
- Greek and several math symbols were absent from build.py's unicode map, so
  alpha and epsilon were rendering as missing-glyph boxes in the SI. Added
  Greek, set membership, superscripts, proper minus. Both PDFs now contain
  zero missing glyphs.
- inline() split on code spans BEFORE applying emphasis, so any italic
  containing `code` was torn into fragments - visible in the main text as a
  literal "is*" and mis-scoped italics on p. 3. Code spans are now stashed
  behind sentinels first. This fixed the manuscript, not just the SI.
- A leading markdown H1 leaked into the body as literal text; the wrapper
  supplies the title, so it is now skipped.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01BkRLcc18rwT2Lysu6PbG7v
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Giorgio Gilestro 2026-09-07 17:13:02 +01:00
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@ -1,101 +1,124 @@
# SI Appendix — The evolution of sex for artificial intelligence
*Every experiment has a committed config (`configs/`), an artifact triple
(`results/<name>/results.parquet` + the resolved config + a manifest carrying content hashes, master
seed, and git commit), a README with its legend and falsifier status, and a figure that regenerates
from the parquet alone. `reproduce.sh` re-runs the whole study from the master seeds; `REPRODUCING.md`
maps each manuscript panel to the config and seed behind it.*
## Reproducibility
## SI Text S1–S2: formal statements
Every experiment in this paper is defined by one committed configuration file under `configs/`.
Running it produces three artifacts under `results/<name>/`: the results table (`results.parquet`),
the fully resolved configuration, and a manifest recording content hashes, the master seed, and the
git commit. Each experiment directory also contains a README with the figure legend and the current
status of the experiment's falsifier — the outcome that would refute its claim (see Methods M1) —
plus a figure that regenerates from the parquet file alone. The script `reproduce.sh` re-runs the
entire study from the master seeds, and `REPRODUCING.md` maps every panel of the manuscript to the
configuration and seed behind it.
## S1. The incompatibility floor: what no alignment can remove (E13c)
## SI Text S1. The incompatibility floor: what no alignment can remove
**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.
**Setting.** Two models, A and B, are trained on the same input distribution. Their label functions
`f_A` and `f_B` agree everywhere except on a *conflict set* `S`, whose size is its probability mass
`μ(S)`. In the conflict condition of the trained-network speciation experiment, `S` consists of the
cyclically relabelled classes, so `μ(S)` is approximately the configured conflict fraction, up to
class-balance corrections.
**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: the 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.
A *function-preserving transformation* `T` is any change to a network's weights that leaves its
outputs untouched. For a plain ReLU multilayer perceptron these transformations are exactly the
permutations of hidden units and the positive rescalings of individual units: scaling a unit's
incoming weights up and its outgoing weights down by the same factor does not change what the network
computes. Together they form the *unit symmetry group* of the architecture. By construction `T(B)`
computes the same function as B, that is `T(B)(x) = B(x)` for every input `x`.
**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
**Proposition 1 (endpoint invariance).** Define the *chord* as the straight line connecting the two
endpoint loss values, `(1−α)·L(A) + α·L(B)`. It depends only on the endpoints and is the baseline used
in the definition of the interpolation barrier; it is not the loss along the interpolation path in
weight space. For every function-preserving `T`, the pair `(A, T(B))` has the same endpoint losses as
the pair `(A, B)`, and therefore the same chord. The interpolation path itself is generally not
invariant: the losses along `(1−α)·A + α·T(B)` change with `T`. This is exactly the room an alignment
has to lower a barrier. The proof is immediate from the definition of function-preserving.
`ε_A(h) + ε_B(h) ≥ μ(S)`, and therefore `max(ε_A(h), ε_B(h)) ≥ μ(S)/2`,
*Scope of the alignment guarantee.* The aligner used here is guaranteed to recover a
permuted-and-rescaled copy of a network exactly. That is an important special case, but it does not
prove that the alignment is optimal over the whole symmetry group for independently trained networks.
Consequently the share of the barrier attributed to removable coordinate mismatch is a lower bound,
and the residual share an upper bound, on their true values.
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.
**Proposition 2 (no merged model can serve both parents).** Let `h` be any single classifier; in
particular, any interpolated or merged model, under any alignment. On every input `x ∈ S` the two
parents disagree, `f_A(x) ≠ f_B(x)`, so `h` must disagree with at least one of them. Writing `ε_P(h)`
for `h`'s error rate against parent `P`'s labels,
**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.
`ε_A(h) + ε_B(h) ≥ μ(S)`, hence `max(ε_A(h), ε_B(h)) ≥ μ(S)/2`.
**Terminology note for the paper.** "Residual (after alignment)" = the estimated functional
incompatibility; for ReLU MLPs I align modulo the full unit symmetry group, so the estimate is not
confounded by missed symmetries of that architecture class.
When two models' conventions conflict on a set of mass `μ(S)`, any hybrid of the two is wrong on at
least one parent's task at least `μ(S)/2` of the time. This floor is information-theoretic, holding
regardless of the alignment group, the architecture, or the merging operator. In the fitness sense it
is reproductive isolation: beyond a given functional conflict, no recombination operator can produce
an offspring faithful to both lineages.
## S2. Emergent vs imposed incompatibility (E13b framing)
**What remains empirical, and how the experiment is designed.** Propositions 1 and 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 alone. In principle such a path could dip toward one parent's function and yield a low
barrier even under conflict. Whether it does is an empirical question, and it is precisely what the
experiment measures. The measured answer is that it does not. In the conflict condition the barrier is
unchanged by permutation alignment (the `residual` readout) and by alignment modulo the full
permutation-and-positive-rescaling group (the `residual_scale` readout), while the very same aligner
removes almost all of the barrier between independently initialised networks, the positive control.
Work on richer symmetry groups for transformers (41) strengthens the removable side of the
decomposition and is therefore complementary to this result: the more barrier a larger group can
remove for *compatible* models, the sharper the meaning of the barrier that survives for
*incompatible* ones. Proposition 2 caps what any of these methods could ever achieve on the conflict
set.
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).
**Terminology used in the paper.** "Residual (after alignment)" denotes the estimated functional
incompatibility: the part of the merge barrier that remains after the architecture's unit symmetries
have been divided out. For ReLU MLPs I align modulo the full unit symmetry group, so the estimate is
not confounded by symmetries of that architecture class that the aligner might have missed.
**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 Text S2. Emergent versus imposed incompatibility
The conflict condition *imposes* contradiction: the two label maps disagree on `S` by construction,
which pins `μ(S) > 0` and activates Proposition 2. A genuine Bateson–Dobzhansky–Muller
incompatibility is instead *emergent*. Each lineage's substitutions are harmless on their own
background, so the training signals never contradict and `μ(S) = 0`; any incompatibility appears only
when the two lineages are combined.
Two conditions realise this emergent setting. In `disjoint`, the parents are specialists on
complementary classes. In `augment`, they learn divergent input conventions on the same task. Neither
condition contains label conflict, so any barrier that survives alignment cannot be attributed to
label conflict. Such a barrier would be the emergent-speciation signal proper.
Both readings were registered before the run. If the residual barrier grows with divergence, then
model speciation is emergent in real weights, and the trajectory seen in the analytic speciation model
is realised. If the residual stays at the level of the `shared` control, then within this regime
trained networks are more merge-compatible than the biological analogy predicts. The second reading
would be an honest bound on the analogy, and a useful design result in its own right: merging is safe
whenever there is no functional conflict.
**Outcome.** Four replicates, with divergence up to 3,200 steps — up to 6.4× the shared base training
— returned the second reading. The residual barrier was 0.000 at every divergence in both emergent
conditions. Merging moreover *rescued* the `disjoint` specialists, which had forgotten the classes
outside their specialty: at the longest divergence the parents score 0.535 and 0.474 on the full task,
while the merged model holds approximately 0.955 at every divergence tested. This is a sustained
Fisher–Muller rescue at zero barrier. Within this regime, reproductive isolation in real weights
required functional conflict. The same question at language-model scale is answered by the duration
arm of the language-model speciation experiment, which likewise found no isolation from over-training
alone (1 to 12 epochs); whether still longer horizons erode mergeability (cf. 43) remains open.
## SI Table S1: the claims ledger (status / assumptions / evidence / limits)
| Claim | Status | Key assumptions | Evidence | Known limits |
|---|---|---|---|---|
| Collapse = Wright–Fisher drift (biological model) | Closed form (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 |
| Population collapse in the biological model is Wright–Fisher drift | Closed form; the diagnosis itself is due to prior work | Knowledge is a categorical distribution; refitting means resampling | Closed forms reproduced to <0.5% | Real learners add a signed, architecture-specific estimator bias (measured) |
| Grounding behaves like immigration, and the critical real-data fraction is far below one | Closed form, plus the sign confirmed empirically | 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 (biological model); 7B LoRA merge beats every specialist on every family | LLM tier: 3 lexically-distinct families; replicated over five training seeds at 0.5B |
| Outbreeding depression on rugged landscapes; operator design rule | Biological-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 | Biological-model result; hypothesis for merging populations | Ring population, local selection | E14 | Phenomenon known to island-model evolutionary computation; the contribution here 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 |
| Merge failure decomposes into a coordinate artefact plus a functional residual | Empirical at the trained-network and language-model tiers | Alignment enumerates the architecture's unit symmetries | Full-symmetry residual ≈ 0 for compatible parents versus ≈ the naive barrier under conflict; a cliff in hybrid fitness; function-specific breakdown at the LLM tier | Scoped to aligned linear interpolation; conflict floor is information-theoretic, not genetic |
| Epistasis (not divergence) sets the cliff; snowball onset | Biological-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 |
| Confidence weighting improves rank prediction over raw disagreement | Not supported (pre-registered internal prediction) | — | Paired contrast over the same bootstrap resamples: Δ\|ρ\| = −0.021, CI [−0.130, +0.059] | The weighting does sharpen the conflict-versus-compatible level contrast, so it is not useless — only no better as a rank predictor |
| 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 | Biological-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 |
| Emergent speciation without label conflict | Not observed (pre-registered) | Shared ancestry; compatible tasks; the divergences tested | Residual 0.000 to 6.4× base training; the merge rescues the specialists | Bounds the hypothesis; longer horizons/distribution shift/capacity pressure untested |
| Grounding, recombination, and diversity preservation make complementary contributions | Biological-model result; hypothesis at LLM scale | Conformity stands in for self-consumption | E11 four-arm ablation; each arm fails in a distinct way | General joint necessity is not established; the full grounded LLM society is unbuilt |
## SI Table S2: headline quantitative results
@ -105,7 +128,7 @@ per-experiment tables and falsifier status in the per-experiment documentation).
| Result | Setting / n | Outcome definition | Headline |
|---|---|---|---|
| Closed-form validation | Biological model; 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 |
| Grounding retention | Biological model (E2); 100 lineages per grounding level | Fraction of equilibrium diversity retained at grounding `g` (operational threshold) | `g ≈ 0.05` retains ≥95% in the 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 |