SI: adopt the clearer rewrite, with factual corrections; fix two build bugs it exposed
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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\# SI Appendix --- The evolution of sex for artificial intelligence
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\section*{Reproducibility}
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*Every experiment has a committed config (\texttt{configs/}), an artifact triple (\texttt{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. \texttt{reproduce.sh} re-runs the whole study from the master seeds; \texttt{REPRODUCING.md} maps each manuscript panel to the config and seed behind it.*
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Every experiment in this paper is defined by one committed configuration file under \texttt{configs/}. Running it produces three artifacts under \texttt{results/<name>/}: the results table (\texttt{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 \texttt{reproduce.sh} re-runs the entire study from the master seeds, and \texttt{REPRODUCING.md} maps every panel of the manuscript to the configuration and seed behind it.
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\section*{SI Text S1--S2: formal statements}
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\section*{SI Text S1. The incompatibility floor: what no alignment can remove}
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\section*{S1. The incompatibility floor: what no alignment can remove (E13c)}
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\textbf{Setting.} Two models, A and B, are trained on the same input distribution. Their label functions \texttt{f\_A} and \texttt{f\_B} agree everywhere except on a \emph{conflict set} \texttt{S}, whose size is its probability mass \texttt{\(\mu\)(S)}. In the conflict condition of the trained-network speciation experiment, \texttt{S} consists of the cyclically relabelled classes, so \texttt{\(\mu\)(S)} is approximately the configured conflict fraction, up to class-balance corrections.
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\textbf{Setting.} Models A and B are trained on the same input distribution; their target label functions \texttt{f\_A} and \texttt{f\_B} agree except on a conflict set \texttt{S} of probability mass \texttt{\(\mu\)(S)} (in E13's conflict condition, the cyclically-relabelled classes; \texttt{\(\mu\)(S) \(\approx\) conflict\_frac} up to class balance). A \emph{function-preserving transformation} \texttt{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 \texttt{T(B)(x) = B(x)} for all \texttt{x} by construction.
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A \emph{function-preserving transformation} \texttt{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 \emph{unit symmetry group} of the architecture. By construction \texttt{T(B)} computes the same function as B, that is \texttt{T(B)(x) = B(x)} for every input \texttt{x}.
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\textbf{Proposition 1 (endpoint invariance --- with the term ``chord'' defined precisely).} Here ``chord'' means the α-linear interpolation \textbf{of the endpoint loss values}, \texttt{(1−α)\(\cdot\)L(A) + α\(\cdot\)L(B)} --- the baseline in the barrier definition, a function of the endpoints only --- NOT the weight-space interpolation path. For every function-preserving \texttt{T}, the endpoint functions, hence the endpoint losses and this chord, are identical for \texttt{(A, T(B))} and \texttt{(A, B)}. The \textbf{interpolation path itself is generally NOT invariant} --- losses along \texttt{(1−α)\(\cdot\)A + α\(\cdot\)T(B)} change with \texttt{T}, which is precisely why alignment can lower a barrier. \emph{(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.
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\textbf{Proposition 1 (endpoint invariance).} Define the \emph{chord} as the straight line connecting the two endpoint loss values, \texttt{(1\(-\)\(\alpha\))\(\cdot\)L(A) + \(\alpha\)\(\cdot\)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 \texttt{T}, the pair \texttt{(A, T(B))} has the same endpoint losses as the pair \texttt{(A, B)}, and therefore the same chord. The interpolation path itself is generally not invariant: the losses along \texttt{(1\(-\)\(\alpha\))\(\cdot\)A + \(\alpha\)\(\cdot\)T(B)} change with \texttt{T}. This is exactly the room an alignment has to lower a barrier. The proof is immediate from the definition of function-preserving.
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\textbf{Proposition 2 (no merged model can serve both parents).} Let \texttt{h} be \emph{any} single classifier (in particular, any interpolated/merged model, under any alignment). On every \texttt{x ∈ S}, \texttt{f\_A(x) ≠ f\_B(x)}, so \texttt{h(x)} disagrees with at least one of them. Hence
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\emph{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.
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\texttt{ε\_A(h) + ε\_B(h) \(\geq\) \(\mu\)(S)}, and therefore \texttt{max(ε\_A(h), ε\_B(h)) \(\geq\) \(\mu\)(S)/2},
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\textbf{Proposition 2 (no merged model can serve both parents).} Let \texttt{h} be any single classifier; in particular, any interpolated or merged model, under any alignment. On every input \texttt{x \(\in\) S} the two parents disagree, \texttt{f\_A(x) \(\neq\) f\_B(x)}, so \texttt{h} must disagree with at least one of them. Writing \texttt{\(\varepsilon\)\_P(h)} for \texttt{h}'s error rate against parent \texttt{P}'s labels,
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where \texttt{ε\_P(h)} is \texttt{h}'s error against parent \texttt{P}'s labels. A hybrid of two models whose conventions conflict on mass \texttt{\(\mu\)(S)} errs at rate at least \texttt{\(\mu\)(S)/2} against at least one parent --- \textbf{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, \emph{no} recombination operator produces an offspring loyal to both lineages.
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\texttt{\(\varepsilon\)\_A(h) + \(\varepsilon\)\_B(h) \(\geq\) \(\mu\)(S)}, hence \texttt{max(\(\varepsilon\)\_A(h), \(\varepsilon\)\_B(h)) \(\geq\) \(\mu\)(S)/2}.
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\textbf{What remains empirical, and why the experiment is designed as it is.} Propositions 1--2 do \emph{not} bound the single-task path barrier (the loss along the interpolation between A and \texttt{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 (\texttt{residual}) \emph{and} by alignment modulo the full permutation \(\times\) positive-rescaling group (\texttt{residual\_scale}), while the same aligner removes \textasciitilde{}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 \emph{removable} side of the decomposition and are therefore complementary: the more barrier a larger group can remove for \emph{compatible} models, the sharper the meaning of the residual that survives for \emph{incompatible} ones --- and Proposition 2 caps what any of them could ever achieve on the conflict set.
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When two models' conventions conflict on a set of mass \texttt{\(\mu\)(S)}, any hybrid of the two is wrong on at least one parent's task at least \texttt{\(\mu\)(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.
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\textbf{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.
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\textbf{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 \texttt{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 \texttt{residual} readout) and by alignment modulo the full permutation-and-positive-rescaling group (the \texttt{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 \emph{compatible} models, the sharper the meaning of the barrier that survives for \emph{incompatible} ones. Proposition 2 caps what any of these methods could ever achieve on the conflict set.
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\section*{S2. Emergent vs imposed incompatibility (E13b framing)}
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\textbf{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.
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The conflict condition \emph{imposes} contradiction (the two label maps disagree on \texttt{S}), which pins \texttt{\(\mu\)(S) > 0} and activates Proposition 2. A true Bateson--Dobzhansky--Muller incompatibility is \emph{emergent}: each lineage's substitutions are harmless on their own background (\texttt{\(\mu\)(S) = 0} --- the training signals never contradict), and incompatibility, if any, arises only in the \emph{combination}. The \texttt{disjoint} (complementary class specialists) and \texttt{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 \(\rightarrow\) model speciation is emergent in real weights (E12's trajectory realised); residual stays at the \texttt{shared}-control level \(\rightarrow\) within this regime, trained networks are \emph{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).
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\section*{SI Text S2. Emergent versus imposed incompatibility}
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\textbf{Outcome (2026-08-11 run, 4 reps, t\_div \(\leq\) 3200): the second reading.} Residual 0.000 at every divergence in both emergent conditions, and the merge \emph{rescues} the forgetting \texttt{disjoint} specialists (parents \(\rightarrow\) 0.535/0.474 on the full task; merged \(\approx\) 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 \texttt{llm\_speciation} question (Phase 3).
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The conflict condition \emph{imposes} contradiction: the two label maps disagree on \texttt{S} by construction, which pins \texttt{\(\mu\)(S) > 0} and activates Proposition 2. A genuine Bateson--Dobzhansky--Muller incompatibility is instead \emph{emergent}. Each lineage's substitutions are harmless on their own background, so the training signals never contradict and \texttt{\(\mu\)(S) = 0}; any incompatibility appears only when the two lineages are combined.
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Two conditions realise this emergent setting. In \texttt{disjoint}, the parents are specialists on complementary classes. In \texttt{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.
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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 \texttt{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.
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\textbf{Outcome.} Four replicates, with divergence up to 3,200 steps --- up to 6.4\(\times\) the shared base training --- returned the second reading. The residual barrier was 0.000 at every divergence in both emergent conditions. Merging moreover \emph{rescued} the \texttt{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.
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\section*{SI Table S1: the claims ledger (status / assumptions / evidence / limits)}
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@ -32,19 +38,19 @@ The conflict condition \emph{imposes} contradiction (the two label maps disagree
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\begin{tabular}{p{0.184\textwidth} p{0.184\textwidth} p{0.184\textwidth} p{0.184\textwidth} p{0.184\textwidth}}
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\hline
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Claim & Status & Key assumptions & Evidence & Known limits \\ \hline
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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) \\[3pt]
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Grounding = immigration; critical real-data fraction \(\ll\) 1 & Exact + empirical sign & Fresh samples from a fixed, non-drifting truth & Exact \texttt{H\_eq}; \texttt{g*\(\approx\)0.048}; sign holds in RNN/MLP/VAE and on MNIST & Deepest tail unrescuable at feasible budgets (\texttt{m ∼ 1/p}); sharp threshold softens in trained nets \\[3pt]
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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) \\[3pt]
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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 \texttt{H\_eq}; \texttt{g*\(\approx\)0.048}; sign holds in RNN/MLP/VAE and on MNIST & Deepest tail unrescuable at feasible budgets (\texttt{m \(\sim\) 1/p}); sharp threshold softens in trained nets \\[3pt]
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``Merge, don't average'' conservation & Exact \textbf{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 \\[3pt]
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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 \\[3pt]
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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 \\[3pt]
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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 \\[3pt]
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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 \(\approx\) 0 (compatible) vs \(\approx\) naive (conflict); cliff in hybrid fitness & Scoped to aligned linear interpolation; conflict floor is information-theoretic, not genetic \\[3pt]
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Epistasis (not divergence) sets the cliff; snowball onset & Biological-model result; \textbf{hypothesis} at the neural tier & BDM incompatibility structure & E12 & Snowball count ≠ performance cliff without the effect-size link; neural test outstanding \\[3pt]
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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 \(\approx\) 0 for compatible parents versus \(\approx\) 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 \\[3pt]
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Epistasis (not divergence) sets the cliff; snowball onset & Biological-model result; \textbf{hypothesis} at the neural tier & BDM incompatibility structure & E12 & Snowball count \(\neq\) performance cliff without the effect-size link; neural test outstanding \\[3pt]
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Pre-merge functional disagreement predicts merge penalty & Empirical, within a controlled grid (0.5B, 13 conditions \(\times\) 3 seeds) & Constructed conflict/overlap/duration axes; oracle-potential outcome (pre-registered; ordering sensitive to reference) & Clustered CIs exclude 0; held-out LOCO \(\rho\)\(\approx\)0.4; selected geometry baselines \(\approx\) 0 & Head-to-head predictor differences not individually significant; only selected baselines; generalisation to real task pairs open \\[3pt]
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Confidence weighting improves rank prediction over raw disagreement & \textbf{Not supported} (pre-registered internal prediction) & --- & Paired Δ\textbackslash{} & \(\rho\)\textbackslash{} \\[3pt]
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Confidence weighting improves rank prediction over raw disagreement & Not supported (pre-registered internal prediction) & --- & Paired contrast over the same bootstrap resamples: \(\Delta\)\textbackslash{} & \(\rho\)\textbackslash{} \\[3pt]
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The predictor improves budget-matched operator choice & \textbf{Open} & --- & Soup-vs-route gap readout noise-dominated at 0.5B & The practical payoff; untested \\[3pt]
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Emergent speciation without conflict & \textbf{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 \\[3pt]
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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 \\[3pt]
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Emergent speciation without label conflict & Not observed (pre-registered) & Shared ancestry; compatible tasks; the divergences tested & Residual 0.000 to 6.4\(\times\) base training; the merge rescues the specialists & Bounds the hypothesis; longer horizons/distribution shift/capacity pressure untested \\[3pt]
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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 \\[3pt]
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\hline\end{tabular}\end{center}\medskip
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\section*{SI Table S2: headline quantitative results}
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@ -56,13 +62,13 @@ Headline quantitative results with sample sizes, uncertainty, and outcome defini
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\hline
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Result & Setting / n & Outcome definition & Headline \\ \hline
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Closed-form validation & Biological model; standing tests & Simulated vs closed-form H-decay, immigration equilibrium, multi-teacher union & Agreement < 0.5\% \\[3pt]
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Grounding retention & Minimal model; 18+ replicates per point & Fraction of equilibrium diversity retained at grounding g (operational threshold) & g \(\approx\) 0.05 retained \(\geq\)95\% (tested setting); smooth in g \\[3pt]
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Grounding retention & Biological model (E2); 100 lineages per grounding level & Fraction of equilibrium diversity retained at grounding \texttt{g} (operational threshold) & \texttt{g \(\approx\) 0.05} retains \(\geq\)95\% in the tested setting; smooth in \texttt{g} \\[3pt]
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MNIST collapse \& rescue & Conv-VAE, 4 replicates; frozen oracle (98.5\% mode acc.) & Mode support / forward-KL over generations & Dry: 30\(\rightarrow\)1 modes; 10\% grounding: 30/30 held \\[3pt]
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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 \\[3pt]
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Fisher--Muller in LLMs & 5 seeds (0.5B), fixed tests; single 7B run & Merged vs best-specialist accuracy (overall; worst family) & Ties 0.647\(\pm\)0.027 vs 0.592\(\pm\)0.009; 7B 0.87 vs 0.77 \\[3pt]
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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 \\[3pt]
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Speciation decomposition & MLPs, 3 replicates & LMC error barrier residual after permutation+rescaling alignment & Same-task 0.001; conflict 0.497 (naive 0.502) \\[3pt]
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Emergent isolation & MLPs 4 reps to 6.4\(\times\) base training; LLM 1\(\rightarrow\)12 epochs & Residual barrier; merged vs parent accuracy & 0.000 everywhere; merge rescues parents (\(\approx\)0.955 vs \(\approx\)0.50) \\[3pt]
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Predictive test & 13 conditions \(\times\) 3 seeds (0.5B) & Merge penalty vs oracle parent potential (pre-registered; ±: clustered 95\% CI) & Functional \(\rho\) +0.45/+0.46, CI excl. 0; LOCO \(\rho\) \(\approx\) 0.4; geometry n.s.; paired differences n.s. \\[3pt]
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Predictive test & 13 conditions \(\times\) 3 seeds (0.5B) & Merge penalty vs oracle parent potential (pre-registered; \(\pm\): clustered 95\% CI) & Functional \(\rho\) +0.45/+0.46, CI excl. 0; LOCO \(\rho\) \(\approx\) 0.4; geometry n.s.; paired differences n.s. \\[3pt]
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\hline\end{tabular}\end{center}\medskip
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\section*{SI Methods: experimental procedures}
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\emph{Parameter choices.} \texttt{K = 500}--\texttt{1000} with \texttt{zipf\_s = 1.1} and half the items designated tail: large enough that the rare tail contains hundreds of items (so tail statistics are not dominated by a handful of them) and small enough to sweep densely. \texttt{n = 100}--\texttt{200} sets drift strength; it is the population size in the Wright--Fisher correspondence and the distillation sample size in the AI reading. Horizons of 400--600 generations were chosen so that ungrounded lineages reach fixation and grounded ones reach stationarity within the run, which the trajectories confirm.
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\emph{Sweeps.} E2 sweeps grounding \texttt{g ∈ {0, 0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.4}}; E3 contrasts uniform against region-matched grounding allocation; E4 crosses parent count \texttt{K\_T ∈ {1,2,3,5}} with teacher correlation \texttt{\(\rho\) ∈ {0, 0.25, 0.5, 0.75, 1}} and \texttt{g ∈ {0, 0.02, 0.05}}; E5 crosses selection mode (none / greedy / quality-diversity) with novelty weight; E6 compares four re-minting arms.
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\emph{Sweeps.} E2 sweeps grounding \texttt{g \(\in\) {0, 0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.4}}; E3 contrasts uniform against region-matched grounding allocation; E4 crosses parent count \texttt{K\_T \(\in\) {1,2,3,5}} with teacher correlation \texttt{\(\rho\) \(\in\) {0, 0.25, 0.5, 0.75, 1}} and \texttt{g \(\in\) {0, 0.02, 0.05}}; E5 crosses selection mode (none / greedy / quality-diversity) with novelty weight; E6 compares four re-minting arms.
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\emph{The correlated-parent construction (E4).} Teacher correlation is constructed directly rather than obtained by tuning drift, so that \texttt{\(\rho\)} is not confounded with \texttt{n}, \texttt{m}, tail size, or generation count. For each tail item a shared switch \texttt{z \textasciitilde{} Bern(\(\rho\))}, a shared retention \texttt{s \textasciitilde{} Bern(q)}, and per-teacher \texttt{u⁽ᵏ⁾ \textasciitilde{} Bern(q)} give teacher \texttt{k} retention \texttt{s} if \texttt{z} else \texttt{u⁽ᵏ⁾}. This yields exact marginal retention \texttt{q} and exact pairwise correlation \texttt{\(\rho\)}, and is exchangeable, so \texttt{\(\rho\)} is a single scalar knob.
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\emph{The correlated-parent construction (E4).} Teacher correlation is constructed directly rather than obtained by tuning drift, so that \texttt{\(\rho\)} is not confounded with \texttt{n}, \texttt{m}, tail size, or generation count. For each tail item a shared switch \texttt{z \textasciitilde{} Bern(\(\rho\))}, a shared retention \texttt{s \textasciitilde{} Bern(q)}, and per-teacher \texttt{u\(^{(k)}\) \textasciitilde{} Bern(q)} give teacher \texttt{k} retention \texttt{s} if \texttt{z} else \texttt{u\(^{(k)}\)}. This yields exact marginal retention \texttt{q} and exact pairwise correlation \texttt{\(\rho\)}, and is exchangeable, so \texttt{\(\rho\)} is a single scalar knob.
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\emph{Multi-locus experiments (E7--E11, E14).} Genotypes are \texttt{L = 12} biallelic loci (4096 genotypes --- effectively open-ended relative to the population sizes used), with fitness either additive or a Kauffman NK landscape whose interaction count \texttt{K} tunes ruggedness from 0 to 10. E9 and E10 breed from \texttt{n\_parents = 6} local optima into populations of 200 offspring; E10 additionally screens offspring and iterates (5 rounds, keeping 8). E11 runs a population of \texttt{N = 60} agents for 80 generations at ruggedness \texttt{K = 8}, with mutation \texttt{\(\mu\) = 0.03}, 120 offspring per generation, and selection weighting true fitness against consensus conformity at \texttt{g = 0.85}. E14 sweeps mate-pool breadth on a ring of \texttt{N = 48} against ruggedness.
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\emph{Speciation (E12).} \texttt{L = 20} loci, incompatibility density \texttt{\(\rho\) ∈ {0.1, 0.25, 0.5}}, parental divergence swept 0--20 substitutions, 500 offspring per cell at recombination rate 0.5. E12\_nk repeats the question on NK landscapes (\texttt{L = 16}, \texttt{K} 0--10, 40 parent pairs, 200 offspring).
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\emph{Speciation (E12).} \texttt{L = 20} loci, incompatibility density \texttt{\(\rho\) \(\in\) {0.1, 0.25, 0.5}}, parental divergence swept 0--20 substitutions, 500 offspring per cell at recombination rate 0.5. E12\_nk repeats the question on NK landscapes (\texttt{L = 16}, \texttt{K} 0--10, 40 parent pairs, 200 offspring).
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\emph{Validation.} Three closed forms are asserted as standing tests to within 0.5\%: neutral heterozygosity decay \texttt{E[H\_t] = H\_0(1 − 1/n)\textasciicircum{}t}, the exact immigration--drift equilibrium, and the multi-parent union formula. These run in CI alongside the correctness tests. If they fail, the science is wrong rather than merely the code.
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\emph{Validation.} Three closed forms are asserted as standing tests to within 0.5\%: neutral heterozygosity decay \texttt{E[H\_t] = H\_0(1 \(-\) 1/n)\textasciicircum{}t}, the exact immigration--drift equilibrium, and the multi-parent union formula. These run in CI alongside the correctness tests. If they fail, the science is wrong rather than merely the code.
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\subsection*{M4. The trained-network tier}
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@ -142,9 +148,9 @@ Knowledge is a distribution over \texttt{K} discrete items; reality is a fixed Z
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\emph{The bridge gate.} Before any trained model is interpreted, a histogram generator is run through the identical harness; it must reproduce the biological model exactly. This separates harness bugs from model behaviour, and is why the bridge run carries 60 replicates.
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\emph{Architectures and training.} The recurrent generator is an embedding (24) \(\rightarrow\) GRU (128 hidden; 192 in the architecture-generality run) \(\rightarrow\) linear readout, trained each generation from scratch with Adam, learning rate 2\(\times\)10⁻³, batch size 256, 25 epochs, and evaluated by sampling 12,000--15,000 sequences. Feedforward and variational autoencoder generators share the harness. Retraining from scratch each generation (rather than fine-tuning) makes the generational step a clean refit, matching the biological model's operator.
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\emph{Architectures and training.} The recurrent generator is an embedding (24) \(\rightarrow\) GRU (128 hidden; 192 in the architecture-generality run) \(\rightarrow\) linear readout, trained each generation from scratch with Adam, learning rate 2\(\times\)10\(^{-3}\), batch size 256, 25 epochs, and evaluated by sampling 12,000--15,000 sequences. Feedforward and variational autoencoder generators share the harness. Retraining from scratch each generation (rather than fine-tuning) makes the generational step a clean refit, matching the biological model's operator.
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\emph{MNIST tier.} Dataset: MNIST via torchvision (60,000 training images). Modes are digit class \(\times\) stroke-thickness bin (10 \(\times\) 3 = 30 modes) with a Zipf frequency profile, so roughly eighteen modes are rare. The generator is a convolutional variational autoencoder (latent 32, β = 1), retrained from scratch each generation with Adam, learning rate 10⁻³, batch 256, 30 epochs, on 6,000 images drawn from the previous generation's own samples, for 15 generations, at \texttt{g ∈ {0, 0.1}}. The oracle is a frozen two-convolution classifier trained once (5 epochs) combined with a deterministic thickness measure; it reaches 98.5\% mode accuracy and its 30 \(\times\) 30 confusion matrix is recorded in the manifest as the measurement floor. Build gates: the oracle's accuracy, and generation-0 recovery of all 30 modes.
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\emph{MNIST tier.} Dataset: MNIST via torchvision (60,000 training images). Modes are digit class \(\times\) stroke-thickness bin (10 \(\times\) 3 = 30 modes) with a Zipf frequency profile, so roughly eighteen modes are rare. The generator is a convolutional variational autoencoder (latent 32, \(\beta\) = 1), retrained from scratch each generation with Adam, learning rate 10\(^{-3}\), batch 256, 30 epochs, on 6,000 images drawn from the previous generation's own samples, for 15 generations, at \texttt{g \(\in\) {0, 0.1}}. The oracle is a frozen two-convolution classifier trained once (5 epochs) combined with a deterministic thickness measure; it reaches 98.5\% mode accuracy and its 30 \(\times\) 30 confusion matrix is recorded in the manifest as the measurement floor. Build gates: the oracle's accuracy, and generation-0 recovery of all 30 modes.
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\emph{Speciation in trained weights.} Two multilayer perceptrons (784--512--512--10, ReLU, no batch normalisation --- batch statistics would break the permutation correspondence the analysis depends on) are forked from a shared base trained for 500 steps, then trained apart for 100--3,200 further steps (up to 6.4\(\times\) the shared base) under SGD at learning rate 0.05, batch 128. Merges are weight averages; the readout is the linear-mode-connectivity error barrier before and after alignment. Alignment composes deterministic Re-Basin permutation matching with exact per-unit scale canonicalisation --- the unit symmetry group of this architecture --- and is gated by a control that must recover a permuted-and-rescaled copy exactly. Since the search space is that group rather than all possible alignments, the removable share is a lower bound and the residual an upper bound.
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@ -156,7 +162,7 @@ Knowledge is a distribution over \texttt{K} discrete items; reality is a fixed Z
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\emph{Data splits.} Training, validation, routing-calibration, and test items are drawn from non-overlapping seed offsets by construction (test from 1000 + family index, routing from 2000 +, validation from 3000 +, training from the run seed). Test sets are fixed across seeds in the multi-seed protocols. Selection of merge weights uses validation only; the winners are then reported on the untouched test split.
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\emph{Specialisation.} Each parent is a LoRA adapter (rank 16, α = 32) on the frozen base, applied to all attention and MLP projection matrices, trained with a manual supervised fine-tuning loop: answer-only cross-entropy (prompt tokens masked out of the loss), AdamW at 2\(\times\)10⁻⁴, batch size 8, 3 epochs, bfloat16, 400--800 training items per family. Low-rank adaptation is the right instrument here for a structural reason rather than a computational one: it confines each parent's specialisation to an additive low-rank delta over an identical frozen base, which is what makes weight-space recombination between parents well defined.
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\emph{Specialisation.} Each parent is a LoRA adapter (rank 16, \(\alpha\) = 32) on the frozen base, applied to all attention and MLP projection matrices, trained with a manual supervised fine-tuning loop: answer-only cross-entropy (prompt tokens masked out of the loss), AdamW at 2\(\times\)10\(^{-4}\), batch size 8, 3 epochs, bfloat16, 400--800 training items per family. Low-rank adaptation is the right instrument here for a structural reason rather than a computational one: it confines each parent's specialisation to an additive low-rank delta over an identical frozen base, which is what makes weight-space recombination between parents well defined.
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\emph{Recombination operators.} Fusion by uniform weight averaging (soup) and by sign-reconciled, magnitude-pruned task arithmetic (TIES); union by keeping specialists intact and selecting per input (oracle routing, and a training-free nearest-centroid router over the base model's own prompt embeddings) or per module (winner-take-all by delta norm); and directed recombination, which breeds a population of Dirichlet-weighted merges, scores each on validation, and keeps the fittest.
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@ -164,7 +170,7 @@ Knowledge is a distribution over \texttt{K} discrete items; reality is a fixed Z
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\emph{The controlled predictive test.} Thirty-nine parent pairs (13 conditions \(\times\) 3 seeds) span three axes that are decorrelated by construction: conflict (contradictory conventions on shared prompts, with private training budgets held fixed), compatible overlap (the same shared prompts under the same convention --- overlap and volume without conflict), and duration (weight divergence with no conflict, 1 to 12 epochs). Six predictors are computed before any merge: confidence-weighted functional conflict, raw disagreement, gradient alignment at the shared base, LoRA-delta cosine and L2 distance, and a cross-task performance baseline. Probes are drawn blind to where the conflict lives. The outcome is the merge penalty against oracle parent potential, pre-registered, and also reported against best-parent and mean-parent references because the predictor ordering is sensitive to that choice.
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\emph{The composed society.} A population of \texttt{N} LoRA agents on a shared frozen base evolves for \texttt{G} non-overlapping generations. Each generation every agent answers a fixed validation pool (verifier scored) and a fresh conformity pool (whose modal answer defines the population consensus); selection scores agents by \texttt{g\(\cdot\)fitness + (1−g)\(\cdot\)conformity}; parents are chosen with or without a quality-diversity term over behavioural distance; offspring are bred by screened recombination; and each child is a fresh adapter distilled from its source model's own answers, which makes the inheritance channel literally self-consuming. The verifier enters the loop only where \texttt{g > 0}, but is used for reporting in every arm. The four-arm ablation removes grounded evaluation, recombination, or diversity preservation in turn.
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\emph{The composed society.} A population of \texttt{N} LoRA agents on a shared frozen base evolves for \texttt{G} non-overlapping generations. Each generation every agent answers a fixed validation pool (verifier scored) and a fresh conformity pool (whose modal answer defines the population consensus); selection scores agents by \texttt{g\(\cdot\)fitness + (1\(-\)g)\(\cdot\)conformity}; parents are chosen with or without a quality-diversity term over behavioural distance; offspring are bred by screened recombination; and each child is a fresh adapter distilled from its source model's own answers, which makes the inheritance channel literally self-consuming. The verifier enters the loop only where \texttt{g > 0}, but is used for reporting in every arm. The four-arm ablation removes grounded evaluation, recombination, or diversity preservation in turn.
|
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|
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\subsection*{M6. Negative controls}
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