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

paper/pnas/body.tex

@@ -26,7 +26,7 @@ An operator of a model population faces recurring decisions for which there is n
 
 \section*{The biological model, and where trained learners depart from it}
 
-Knowledge is modelled as a distribution \texttt{p\_t} over \texttt{K} discrete items (capabilities, facts, modes of behaviour), with a fixed true distribution \texttt{p*} whose rare tail carries the knowledge most at risk. One generation is: *draw \texttt{n} samples from the parent's distribution, optionally mix in \texttt{m} verified real samples (``grounding'', \texttt{g = m/(n+m)}), and refit the child\emph{. This construction is the paper's }biological model\emph{, deliberately minimal, and its resampling step }is* the Wright--Fisher process, population genetics' canonical model of neutral evolution, in which each new generation is a random sample of size \texttt{n} from the previous one and every statistical property of drift follows from that one step. Diversity throughout this paper is \emph{heterozygosity}, \texttt{H = 1 − Σ p\_i²}: the probability that two random draws differ (one minus a collision probability), high when many items share the mass, zero at total collapse. The identity is exploited as an engineering gate: the simulator reproduces the classical closed forms (heterozygosity decay \texttt{E[H\_t] = H\_0(1 − 1/n)\textasciicircum{}t}; the exact immigration--drift equilibrium; the closed-form multi-teacher union) to within 0.5\%, and these are standing tests in the codebase, not one-off checks.
+Knowledge is modelled as a distribution \texttt{p\_t} over \texttt{K} discrete items (capabilities, facts, modes of behaviour), with a fixed true distribution \texttt{p*} whose rare tail carries the knowledge most at risk. One generation is: \emph{draw \texttt{n} samples from the parent's distribution, optionally mix in \texttt{m} verified real samples (``grounding'', \texttt{g = m/(n+m)}), and refit the child}. This construction is the paper's \emph{biological model}, deliberately minimal, and its resampling step \emph{is} the Wright--Fisher process, population genetics' canonical model of neutral evolution, in which each new generation is a random sample of size \texttt{n} from the previous one and every statistical property of drift follows from that one step. Diversity throughout this paper is \emph{heterozygosity}, \texttt{H = 1 \(-\) \(\Sigma\) p\_i\(^{2}\)}: the probability that two random draws differ (one minus a collision probability), high when many items share the mass, zero at total collapse. The identity is exploited as an engineering gate: the simulator reproduces the classical closed forms (heterozygosity decay \texttt{E[H\_t] = H\_0(1 \(-\) 1/n)\textasciicircum{}t}; the exact immigration--drift equilibrium; the closed-form multi-teacher union) to within 0.5\%, and these are standing tests in the codebase, not one-off checks.
 
 Where a real learner departs from this model matters, and I measured that boundary rather than assumed it. Real training adds approximation, optimisation noise, and inductive bias, and when trained networks are fit against the exact drift null they deviate in \emph{opposite, architecture-specific} directions: a smoothing recurrent network resists collapse (keeping spurious variants alive), while a sharpening image generator accelerates it. A one-parameter \emph{learning kernel} (a smoothing knob and a sharpening knob on the refit) reproduces both. Throughout, a real learner is therefore treated as Wright--Fisher \emph{plus a signed, measurable estimator bias}, and the drift signs (rare-first loss; the grounding response) survived that bias in every architecture I tested, including a convolutional VAE retrained on its own generated digits, where the dry lineage collapses to a single blurred digit class while 10\% grounding holds all thirty modes (Fig. 2). Retraining on a single parent is \emph{asexual reproduction}, and sustained loss under it carries the defining consequence of \emph{Muller's ratchet} (28), the mechanism by which lineages that never recombine decay irreversibly --- the reason non-recombining genomes such as the Y chromosome have shed most of their ancestral genes. Once every copy of a rare capability is gone from all parents and sources, no recombination can rebuild it: each such loss is a click of the ratchet, and remedies must act while copies still survive somewhere (a consequence-level correspondence; the biological model lacks the ratchet's recurrent-mutation driver).
 
@@ -51,7 +51,7 @@ Selection on a fitness function & Verifier-anchored selection (``reality that ca
 
 \subsection*{Grounding is immigration: cheap, with a floor}
 
-In the biological model, grounding from a fixed real source is \emph{immigration} into a drifting population (29--31). Immigration is what conservation managers prescribe when a fragmented reserve loses diversity, and its striking property there is how little is needed --- the field's rule of thumb is that one migrant per generation holds an isolated population's diversity (32). The same economy appears here: the equilibrium diversity has a closed form the simulator matches exactly. That equilibrium is \emph{smooth} in the grounding fraction (there is no phase transition in aggregate diversity), so the practical number is an operational threshold, and I define it as such: under the tested population size and Zipf source distribution, \texttt{g \(\approx\) 0.05} retained most (\(\geq\)95\%) of equilibrium diversity indefinitely, with the required fraction depending on sample size, source distribution, and the chosen retention target (dependencies in SI). Verified real data remains, on any of these definitions, cheap insurance at fractions far below one. But the same analysis yields a floor the field's average-loss framing misses: under unstratified sampling from the source, a capability of rarity \texttt{p} appears in a real-data batch of size \texttt{m} with probability \texttt{1 − e\textasciicircum{}{−m\(\cdot\)p}}, so \texttt{m\(\cdot\)p \(\approx\) 1} marks roughly a 63\% chance of one example per batch: a soft observation floor, with higher confidence priced accordingly, and with distinct consequences for continuous retention, stationary occupancy, and reintroduction after loss (immigration can restore an absent item; SI separates these). Protecting the rarest knowledge under unstratified grounding is therefore priced per item at cost \texttt{\(\propto\) 1/p}; targeted or stratified sampling changes that cost, and recombination can recover rare capabilities \emph{that are still retained across complementary parents} (next section). In trained networks the \emph{sign} of the grounding response transfers everywhere I looked, with two deviations, both traced to the estimator bias above: sharp thresholds soften, and support-counting metrics decouple from truth (forward-KL is the operative collapse metric for a smoothing learner). On real images (Fig. 2B), dry self-training collapses a convolutional VAE to one mode while \textasciitilde{}10\% grounding holds all thirty (the trained model needs roughly twice the biological model's fraction, the measured price of the estimator bias).
+In the biological model, grounding from a fixed real source is \emph{immigration} into a drifting population (29--31). Immigration is what conservation managers prescribe when a fragmented reserve loses diversity, and its striking property there is how little is needed --- the field's rule of thumb is that one migrant per generation holds an isolated population's diversity (32). The same economy appears here: the equilibrium diversity has a closed form the simulator matches exactly. That equilibrium is \emph{smooth} in the grounding fraction (there is no phase transition in aggregate diversity), so the practical number is an operational threshold, and I define it as such: under the tested population size and Zipf source distribution, \texttt{g \(\approx\) 0.05} retained most (\(\geq\)95\%) of equilibrium diversity indefinitely, with the required fraction depending on sample size, source distribution, and the chosen retention target (dependencies in SI). Verified real data remains, on any of these definitions, cheap insurance at fractions far below one. But the same analysis yields a floor the field's average-loss framing misses: under unstratified sampling from the source, a capability of rarity \texttt{p} appears in a real-data batch of size \texttt{m} with probability \texttt{1 \(-\) e\textasciicircum{}{\(-\)m\(\cdot\)p}}, so \texttt{m\(\cdot\)p \(\approx\) 1} marks roughly a 63\% chance of one example per batch: a soft observation floor, with higher confidence priced accordingly, and with distinct consequences for continuous retention, stationary occupancy, and reintroduction after loss (immigration can restore an absent item; SI separates these). Protecting the rarest knowledge under unstratified grounding is therefore priced per item at cost \texttt{\(\propto\) 1/p}; targeted or stratified sampling changes that cost, and recombination can recover rare capabilities \emph{that are still retained across complementary parents} (next section). In trained networks the \emph{sign} of the grounding response transfers everywhere I looked, with two deviations, both traced to the estimator bias above: sharp thresholds soften, and support-counting metrics decouple from truth (forward-KL is the operative collapse metric for a smoothing learner). On real images (Fig. 2B), dry self-training collapses a convolutional VAE to one mode while \textasciitilde{}10\% grounding holds all thirty (the trained model needs roughly twice the biological model's fraction, the measured price of the estimator bias).
 
 \begin{figure*}[p]\centering  % fig2
 \includegraphics[width=\textwidth]{figs/fig2.pdf}
@@ -83,7 +83,7 @@ Sex has risks and, for AI, an unfair advantage, both quantified on Kauffman's NK
 
 \subsection*{The society: grounding, recombination, and diversity make complementary contributions}
 
-Composing the operators (Fig. 6) requires one definitional distinction first. In the inheritance model, grounding is \emph{grounded inheritance}: external samples added to the reproduction process (the data channel). In the society model, grounding is \emph{grounded evaluation}: selection weights true fitness against conformity to the population's own consensus, \texttt{g}\(\cdot\)true-fitness + (1−g)\(\cdot\)conformity, the analogue of scoring models by the crowd's approval (the fitness channel). These are related design ideas, since both couple the lineage to a non-drifting external signal, but they are different operators, and I name them separately. In the tested society (a finite agent population on a rugged NK landscape), a four-arm ablation separates the failure modes: the full system (grounded evaluation + directed recombination + diversity-preserving selection (40)) climbs to near the global optimum while keeping its specialists; removing grounded evaluation converges the population confidently on an unfit consensus (self-consumption); removing recombination strands it on local optima; removing diversity converges it prematurely to a worse answer. Each removal fails differently; the three implementations make complementary contributions \emph{under the tested conditions}; general joint necessity is not established (alternative mutation, restart, archive, or selection schemes could alter the picture). At language-model scale this composed loop remains unbuilt; it is the paper's largest stated gap.
+Composing the operators (Fig. 6) requires one definitional distinction first. In the inheritance model, grounding is \emph{grounded inheritance}: external samples added to the reproduction process (the data channel). In the society model, grounding is \emph{grounded evaluation}: selection weights true fitness against conformity to the population's own consensus, \texttt{g}\(\cdot\)true-fitness + (1\(-\)g)\(\cdot\)conformity, the analogue of scoring models by the crowd's approval (the fitness channel). These are related design ideas, since both couple the lineage to a non-drifting external signal, but they are different operators, and I name them separately. In the tested society (a finite agent population on a rugged NK landscape), a four-arm ablation separates the failure modes: the full system (grounded evaluation + directed recombination + diversity-preserving selection (40)) climbs to near the global optimum while keeping its specialists; removing grounded evaluation converges the population confidently on an unfit consensus (self-consumption); removing recombination strands it on local optima; removing diversity converges it prematurely to a worse answer. Each removal fails differently; the three implementations make complementary contributions \emph{under the tested conditions}; general joint necessity is not established (alternative mutation, restart, archive, or selection schemes could alter the picture). At language-model scale this composed loop remains unbuilt; it is the paper's largest stated gap.
 
 \begin{figure*}[p]\centering  % fig6
 \includegraphics[width=\textwidth]{figs/fig6.pdf}
@@ -107,15 +107,15 @@ The pre-registered \emph{emergent test} constrains the claim most: true BDM inco
 
 The framework's prediction-level claim was put to a designed test (Fig. 3C). Thirty-nine parent pairs (13 conditions \(\times\) 3 seeds; rows are not independent --- parents share task-data seeds across conditions, so inference is condition-clustered, and because shared seeds also couple rows \emph{across} conditions I report per-seed and leave-one-seed-out sensitivity alongside) span three axes decorrelated by construction: \emph{conflict} (contradictory conventions on shared prompts, private budgets fixed), \emph{compatible overlap} (the same shared prompts under the same convention --- overlap and volume without conflict), and \emph{duration} (weight divergence with zero conflict). Before merging, six predictors are computed: \emph{confidence-weighted functional conflict} (bilateral confident disagreement on probes drawn blind to where conflict lives --- a proposed proxy for merge-relevant interactions, motivated by the observation that raw disagreement counts harmless complementation, one parent merely ignorant, as conflict), raw disagreement, gradient alignment at the shared base (47), LoRA-delta cosine and distance, and a cross-task performance baseline. The pre-registered outcome is the merge penalty against oracle parent potential (the analogue of \emph{hybrid load}, the fitness a hybrid loses relative to what its parents' genes could jointly supply), also reported against best- and mean-parent references because the predictor ordering is sensitive to that choice.
 
-Across this controlled grid, pre-merge functional disagreement predicted merge penalties (clustered bootstrap CIs excluding zero; held-out leave-one-condition-out \(\rho\) \(\approx\) 0.35--0.40), whereas LoRA-delta cosine and L2 showed no statistically detectable association; gradient alignment carried intermediate signal. Head-to-head predictor differences are not individually significant at this sample size; only these baselines were tested; and with three seeds, uncertainty about seed generalisation remains substantial --- though the seed sensitivity favours the functional measures (per-seed \(\rho\) stable at +0.37 to +0.53 in each seed alone, geometry \(\approx\) 0 in every seed, gradient alignment seed-unstable at −0.11 to −0.55). Two further results bound the claim: the initial two-axis grid's best predictor was delta-cosine (\(\rho\) = +0.60) --- an overlap artefact that the compatible-overlap control was added to expose, and did (collapse to +0.03); and the pre-registered internal prediction that confidence weighting would beat raw disagreement \emph{failed} (they are statistically indistinguishable as rank predictors), so the present evidence favours functional disagreement generally, not the DMI-specific refinement. The framework motivated the measurement and the controls; their success does not validate the specifically population-genetic mechanism. Whether the prediction improves a budget-matched operator choice, and whether it generalises to unfamiliar conflict structures and real task pairs, are the experiment's open front.
+Across this controlled grid, pre-merge functional disagreement predicted merge penalties (clustered bootstrap CIs excluding zero; held-out leave-one-condition-out \(\rho\) \(\approx\) 0.35--0.40), whereas LoRA-delta cosine and L2 showed no statistically detectable association; gradient alignment carried intermediate signal. Head-to-head predictor differences are not individually significant at this sample size; only these baselines were tested; and with three seeds, uncertainty about seed generalisation remains substantial --- though the seed sensitivity favours the functional measures (per-seed \(\rho\) stable at +0.37 to +0.53 in each seed alone, geometry \(\approx\) 0 in every seed, gradient alignment seed-unstable at \(-\)0.11 to \(-\)0.55). Two further results bound the claim: the initial two-axis grid's best predictor was delta-cosine (\(\rho\) = +0.60) --- an overlap artefact that the compatible-overlap control was added to expose, and did (collapse to +0.03); and the pre-registered internal prediction that confidence weighting would beat raw disagreement \emph{failed} (they are statistically indistinguishable as rank predictors), so the present evidence favours functional disagreement generally, not the DMI-specific refinement. The framework motivated the measurement and the controls; their success does not validate the specifically population-genetic mechanism. Whether the prediction improves a budget-matched operator choice, and whether it generalises to unfamiliar conflict structures and real task pairs, are the experiment's open front.
 
 Headline quantitative results, with sample sizes, uncertainty, and outcome definitions, are collected in SI Appendix, Table S2.
 
 \section*{Discussion}
 
-\textbf{Design rules.} As engineering guidance, the results reduce to rules that an operator of a model population can apply, answering the four decisions posed in the Introduction. \emph{Ground every generation} in verified reality --- a few percent retained most diversity in the tested settings --- but price the rarest capabilities individually (observation probability \texttt{1 − e\textasciicircum{}{−m\(\cdot\)p}} per batch under unstratified sampling), consider targeted sampling for the deep tail, and use recombination to recover rare capabilities still retained across complementary parents. \emph{Merge, don't blend, when there is headroom}: keep specialists intact and route, or breed-and-screen candidate merges, whenever the naive average is far from ceiling; plain averaging is adequate only where a strong base has already composed the skills. \emph{Match the operator to entanglement}: merge freely when skills are additive; sparingly, with offspring selection, when they entangle; and expect the champion-optimal mating breadth to narrow as landscapes roughen. \emph{Preserve diversity as a first-class objective}, because selection can only preserve variety that exists, and in the tested society its removal produced a distinct failure mode. \emph{Before merging, measure functional conflict} --- cheap, pre-merge, and in the controlled setting predictive where the tested weight-distance baselines were not; and \emph{do not treat divergence or specialisation alone as evidence of incompatibility} --- in every regime tested here, what broke merging was conflicting conventions on shared circuitry, which is the thing to detect.
+\textbf{Design rules.} As engineering guidance, the results reduce to rules that an operator of a model population can apply, answering the four decisions posed in the Introduction. \emph{Ground every generation} in verified reality --- a few percent retained most diversity in the tested settings --- but price the rarest capabilities individually (observation probability \texttt{1 \(-\) e\textasciicircum{}{\(-\)m\(\cdot\)p}} per batch under unstratified sampling), consider targeted sampling for the deep tail, and use recombination to recover rare capabilities still retained across complementary parents. \emph{Merge, don't blend, when there is headroom}: keep specialists intact and route, or breed-and-screen candidate merges, whenever the naive average is far from ceiling; plain averaging is adequate only where a strong base has already composed the skills. \emph{Match the operator to entanglement}: merge freely when skills are additive; sparingly, with offspring selection, when they entangle; and expect the champion-optimal mating breadth to narrow as landscapes roughen. \emph{Preserve diversity as a first-class objective}, because selection can only preserve variety that exists, and in the tested society its removal produced a distinct failure mode. \emph{Before merging, measure functional conflict} --- cheap, pre-merge, and in the controlled setting predictive where the tested weight-distance baselines were not; and \emph{do not treat divergence or specialisation alone as evidence of incompatibility} --- in every regime tested here, what broke merging was conflicting conventions on shared circuitry, which is the thing to detect.
 
-\textbf{Continual learning at the population scale.} Within a single network, the discipline's remedies for forgetting are this framework's operators writ small. Rehearsal and replay of stored data (26, 27) is grounded inheritance within one lineage, and the replay fractions the field settled on empirically, on the order of 1\% for instruction tuning (48) and 5\% to 25\% by distribution-shift strength in continual pretraining (49), sit where the biological model's operational threshold lies. \emph{Pseudo-rehearsal}, the replay of a network's own generated samples, proposed as a cure in 1995 (50) and revived as generative replay (51), is precisely the ungrounded null studied here: immigration from a drifting source, benign for one hop, compounding over generations, with verifier-filtering (29, 52) converting it back into grounding. Parameter isolation (53), including frozen-base adapters, which forget far less (54), is engineered decorrelation; complementary-learning-systems consolidation (55--57) is the periodic adapter-into-base merge; the recent turn to merging as a continual-learning mechanism (58--61) applies recombination within one lineage over time, where this paper applies it across lineages; and the observation that rare examples and long-tail knowledge are forgotten first (62--64) is tail extinction seen one model at a time. The mechanisms differ (forgetting is largely deterministic interference, collapse is sampling drift) but the victims and the remedies coincide, and to my knowledge no prior work carries population-genetic formalism into continual learning. Read into that field, the results offer: (i) an equilibrium theory for the replay ratio, with the sharper prediction that the required fraction is set by the rarest capability one refuses to lose (the \texttt{1 − e\textasciicircum{}{−m\(\cdot\)p}} law) rather than by average loss, testable against published replay sweeps; (ii) a \emph{failure theory for generative replay}: self-generated rehearsal is safe for short horizons and compounds into collapse across generations unless verifier-filtered back into grounding (29, 50--52); (iii) \emph{pre-merge interference prediction with a mechanism}: where the current state of the art fits regressions over candidate metrics (47), the functional-conflict measure arrives at a convergent signal from principle and comes with an operator prescription --- when conflict is high, do not average; route or breed-and-screen; (iv) a candidate \emph{decision rule for the consolidate-versus-stay-modular question} that currently splits the field's practice (keep adapters separate vs merge them; 57--61): union-preserving operators where headroom exists, fusion where the base composes, consolidation as the slow-store step; and (v) \emph{tail monitoring as the leading indicator}: continual-learning evaluation that averages over capabilities hides exactly the losses that drift theory says come first and, past a threshold, become irreversible. On that last point I note the standing objection that apparent forgetting can be skewed task-inference over latent capability rather than erasure (65); the irreversibility results here concern oracle-measured behavioural distributions, and distinguishing latent from extinct capability at language-model scale is an open experiment whose outcome would be decisive for both readings.
+\textbf{Continual learning at the population scale.} Within a single network, the discipline's remedies for forgetting are this framework's operators writ small. Rehearsal and replay of stored data (26, 27) is grounded inheritance within one lineage, and the replay fractions the field settled on empirically, on the order of 1\% for instruction tuning (48) and 5\% to 25\% by distribution-shift strength in continual pretraining (49), sit where the biological model's operational threshold lies. \emph{Pseudo-rehearsal}, the replay of a network's own generated samples, proposed as a cure in 1995 (50) and revived as generative replay (51), is precisely the ungrounded null studied here: immigration from a drifting source, benign for one hop, compounding over generations, with verifier-filtering (29, 52) converting it back into grounding. Parameter isolation (53), including frozen-base adapters, which forget far less (54), is engineered decorrelation; complementary-learning-systems consolidation (55--57) is the periodic adapter-into-base merge; the recent turn to merging as a continual-learning mechanism (58--61) applies recombination within one lineage over time, where this paper applies it across lineages; and the observation that rare examples and long-tail knowledge are forgotten first (62--64) is tail extinction seen one model at a time. The mechanisms differ (forgetting is largely deterministic interference, collapse is sampling drift) but the victims and the remedies coincide, and to my knowledge no prior work carries population-genetic formalism into continual learning. Read into that field, the results offer: (i) an equilibrium theory for the replay ratio, with the sharper prediction that the required fraction is set by the rarest capability one refuses to lose (the \texttt{1 \(-\) e\textasciicircum{}{\(-\)m\(\cdot\)p}} law) rather than by average loss, testable against published replay sweeps; (ii) a \emph{failure theory for generative replay}: self-generated rehearsal is safe for short horizons and compounds into collapse across generations unless verifier-filtered back into grounding (29, 50--52); (iii) \emph{pre-merge interference prediction with a mechanism}: where the current state of the art fits regressions over candidate metrics (47), the functional-conflict measure arrives at a convergent signal from principle and comes with an operator prescription --- when conflict is high, do not average; route or breed-and-screen; (iv) a candidate \emph{decision rule for the consolidate-versus-stay-modular question} that currently splits the field's practice (keep adapters separate vs merge them; 57--61): union-preserving operators where headroom exists, fusion where the base composes, consolidation as the slow-store step; and (v) \emph{tail monitoring as the leading indicator}: continual-learning evaluation that averages over capabilities hides exactly the losses that drift theory says come first and, past a threshold, become irreversible. On that last point I note the standing objection that apparent forgetting can be skewed task-inference over latent capability rather than erasure (65); the irreversibility results here concern oracle-measured behavioural distributions, and distinguishing latent from extinct capability at language-model scale is an open experiment whose outcome would be decisive for both readings.
 
 \textbf{What is borrowed and what is new.} The collapse-as-drift diagnosis is established prior work (21--25); so are the empirical facts that merges can beat parents, that decorrelated parents merge better, and that naive averaging loses to interference-aware or routed merges (4, 66, 67), that model populations can climb (5, 8--10), and that merge success admits ML-native predictors (47, 68), correlational where this framework supplies mechanism; the reading of sex as an algorithm for mixability in the theory of computation (69) anticipated the transfer before model merging existed. New here is the framework-level synthesis --- inheritance, diversity, and compatibility as managed quantities --- together with: the conservation law for blending inheritance and its operator boundaries; the per-item grounding floor; the society ablation with its complementary failure modes; model speciation as a named, tested question, with the coordinate-versus-functional decomposition under permutation-and-rescaling alignment and the emergent null that bounds it; and the controlled predictive test with its controls. I claim the framework generated these measurements and experiments; I do not claim that their outcomes validate a uniquely population-genetic mechanism, and one refinement it proposed was not supported.
 

paper/pnas/build.py

@@ -106,9 +106,17 @@ FIGURES: dict[str, tuple[list[str], str]] = {
         "between predictors are not individually significant."),
 }
 
-UNICODE = {"—": "---", "–": "--", "→": r"\(\rightarrow\)", "≈": r"\(\approx\)", "≥": r"\(\geq\)",
+UNICODE = {
+    # Multi-character sequences first: esc() applies these in insertion order.
+    "⁽ᵏ⁾": r"\(^{(k)}\)", "⁻³": r"\(^{-3}\)", "⁻⁴": r"\(^{-4}\)",
+    "—": "---", "–": "--", "→": r"\(\rightarrow\)", "≈": r"\(\approx\)", "≥": r"\(\geq\)",
     "≳": r"\(\gtrsim\)", "×": r"\(\times\)", "·": r"\(\cdot\)", "μ": r"\(\mu\)",
-           "ρ": r"\(\rho\)", "≤": r"\(\leq\)", "≪": r"\(\ll\)", "∝": r"\(\propto\)"}
+    "ρ": r"\(\rho\)", "≤": r"\(\leq\)", "≪": r"\(\ll\)", "∝": r"\(\propto\)",
+    # Greek and math symbols: the typewriter font has no Greek, so these must become math.
+    "α": r"\(\alpha\)", "β": r"\(\beta\)", "ε": r"\(\varepsilon\)", "Δ": r"\(\Delta\)",
+    "Σ": r"\(\Sigma\)", "∈": r"\(\in\)", "≠": r"\(\neq\)", "±": r"\(\pm\)",
+    "∼": r"\(\sim\)", "−": r"\(-\)", "²": r"\(^{2}\)", "³": r"\(^{3}\)", "⁴": r"\(^{4}\)",
+}
 SPECIALS = {"&": r"\&", "%": r"\%", "#": r"\#", "_": r"\_", "$": r"\$",
             "~": r"\textasciitilde{}", "^": r"\textasciicircum{}"}
 
@@ -123,19 +131,21 @@ def esc(s: str) -> str:
 
 
 def inline(s: str) -> str:
-    parts = re.split(r"(`[^`]*`)", s)
+    # Code spans are stashed behind sentinels *before* emphasis is applied, so that an italic or
-    out = []
+    # bold span containing `code` is still matched as one span (splitting first would break it).
-    for p in parts:
+    codes: list[str] = []
-        if p.startswith("`") and p.endswith("`") and len(p) >= 2:
+
-            out.append(r"\texttt{" + esc(p[1:-1]) + "}")
+    def stash(m: re.Match) -> str:
-        else:
+        codes.append(r"\texttt{" + esc(m.group(0)[1:-1]) + "}")
-            p = esc(p)
+        return f"\x00{len(codes) - 1}\x00"
-            p = re.sub(r"\[([^\]]+)\]\((https?://[^)]+)\)", r"\\href{\2}{\1}", p)
+
-            p = re.sub(r"\*\*([^*]+)\*\*", r"\\textbf{\1}", p)
+    s = re.sub(r"`[^`]*`", stash, s)
-            p = re.sub(r"\*([^*]+)\*", r"\\emph{\1}", p)
+    s = esc(s)
-            p = re.sub(r'"([^"]+)"', r"``\1''", p)
+    s = re.sub(r"\[([^\]]+)\]\((https?://[^)]+)\)", r"\\href{\2}{\1}", s)
-            out.append(p)
+    s = re.sub(r"\*\*([^*]+)\*\*", r"\\textbf{\1}", s)
-    return "".join(out)
+    s = re.sub(r"\*([^*]+)\*", r"\\emph{\1}", s)
+    s = re.sub(r'"([^"]+)"', r"``\1''", s)
+    return re.sub(r"\x00(\d+)\x00", lambda m: codes[int(m.group(1))], s)
 
 
 def figure_env(name: str) -> str:
@@ -192,6 +202,8 @@ def convert(text: str) -> str:
             emit_table(block, out)
         elif first == "---" and len(block) == 1:
             out.append("\\medskip\\hrule\\medskip"); out.append("")
+        elif first.startswith("# ") and not first.startswith("## "):
+            continue                                       # document title: the wrapper supplies it
         elif first.startswith("## "):
             out.append(f"\\section*{{{inline(first[3:])}}}"); out.append("")
         elif first.startswith("### "):

paper/pnas/si.md

@@ -1,101 +1,124 @@
 # SI Appendix — The evolution of sex for artificial intelligence
 
-*Every experiment has a committed config (`configs/`), an artifact triple
+## Reproducibility
-(`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.*
 
-## 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
+**Setting.** Two models, A and B, are trained on the same input distribution. Their label functions
-`f_A` and `f_B` agree except on a conflict set `S` of probability mass `μ(S)` (in E13's conflict
+`f_A` and `f_B` agree everywhere except on a *conflict set* `S`, whose size is its probability mass
-condition, the cyclically-relabelled classes; `μ(S) ≈ conflict_frac` up to class balance). A
+`μ(S)`. In the conflict condition of the trained-network speciation experiment, `S` consists of the
-*function-preserving transformation* `T` (any composition of hidden-unit permutations and, for ReLU
+cyclically relabelled classes, so `μ(S)` is approximately the configured conflict fraction, up to
-networks, positive per-unit rescalings — the full unit symmetry group of a plain ReLU MLP) satisfies
+class-balance corrections.
-`T(B)(x) = B(x)` for all `x` by construction.
 
-**Proposition 1 (endpoint invariance — with the term "chord" defined precisely).** Here "chord"
+A *function-preserving transformation* `T` is any change to a network's weights that leaves its
-means the α-linear interpolation **of the endpoint loss values**, `(1−α)·L(A) + α·L(B)` — the
+outputs untouched. For a plain ReLU multilayer perceptron these transformations are exactly the
-baseline in the barrier definition, a function of the endpoints only — NOT the weight-space
+permutations of hidden units and the positive rescalings of individual units: scaling a unit's
-interpolation path. For every function-preserving `T`, the endpoint functions, hence the endpoint
+incoming weights up and its outgoing weights down by the same factor does not change what the network
-losses and this chord, are identical for `(A, T(B))` and `(A, B)`. The **interpolation path itself is
+computes. Together they form the *unit symmetry group* of the architecture. By construction `T(B)`
-generally NOT invariant** — losses along `(1−α)·A + α·T(B)` change with `T`, which is precisely why
+computes the same function as B, that is `T(B)(x) = B(x)` for every input `x`.
-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.
 
-**Proposition 2 (no merged model can serve both parents).** Let `h` be *any* single classifier (in
+**Proposition 1 (endpoint invariance).** Define the *chord* as the straight line connecting the two
-particular, any interpolated/merged model, under any alignment). On every `x ∈ S`, `f_A(x) ≠ f_B(x)`,
+endpoint loss values, `(1−α)·L(A) + α·L(B)`. It depends only on the endpoints and is the baseline used
-so `h(x)` disagrees with at least one of them. Hence
+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
+**Proposition 2 (no merged model can serve both parents).** Let `h` be any single classifier; in
-conflict on mass `μ(S)` errs at rate at least `μ(S)/2` against at least one parent — **hybrid
+particular, any interpolated or merged model, under any alignment. On every input `x ∈ S` the two
-disadvantage with an information-theoretic floor, independent of the alignment group, the
+parents disagree, `f_A(x) ≠ f_B(x)`, so `h` must disagree with at least one of them. Writing `ε_P(h)`
-architecture, and the merging operator.** This is reproductive isolation in the fitness sense: past a
+for `h`'s error rate against parent `P`'s labels,
-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*
+  `ε_A(h) + ε_B(h) ≥ μ(S)`,  hence  `max(ε_A(h), ε_B(h)) ≥ μ(S)/2`.
-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
+When two models' conventions conflict on a set of mass `μ(S)`, any hybrid of the two is wrong on at
-incompatibility; for ReLU MLPs I align modulo the full unit symmetry group, so the estimate is not
+least one parent's task at least `μ(S)/2` of the time. This floor is information-theoretic, holding
-confounded by missed symmetries of that architecture class.
+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
+**Terminology used in the paper.** "Residual (after alignment)" denotes the estimated functional
-`μ(S) > 0` and activates Proposition 2. A true Bateson–Dobzhansky–Muller incompatibility is
+incompatibility: the part of the merge barrier that remains after the architecture's unit symmetries
-*emergent*: each lineage's substitutions are harmless on their own background (`μ(S) = 0` — the
+have been divided out. For ReLU MLPs I align modulo the full unit symmetry group, so the estimate is
-training signals never contradict), and incompatibility, if any, arises only in the *combination*.
+not confounded by symmetries of that architecture class that the aligner might have missed.
-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
+## SI Text S2. Emergent versus imposed incompatibility
-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).
 
+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) |
+| 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 = 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 |
+| 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 |
+| 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 + 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 |
+| 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 |

paper/pnas/si_body.tex

@@ -1,30 +1,36 @@
-\# SI Appendix --- The evolution of sex for artificial intelligence
+\section*{Reproducibility}
 
-*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.*
+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.
 
-\section*{SI Text S1--S2: formal statements}
+\section*{SI Text S1. The incompatibility floor: what no alignment can remove}
 
-\section*{S1. The incompatibility floor: what no alignment can remove (E13c)}
+\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.
 
-\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.
+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}.
 
-\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.
+\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.
 
-\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
+\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.
 
-\texttt{ε\_A(h) + ε\_B(h) \(\geq\) \(\mu\)(S)}, and therefore \texttt{max(ε\_A(h), ε\_B(h)) \(\geq\) \(\mu\)(S)/2},
+\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,
 
-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.
+\texttt{\(\varepsilon\)\_A(h) + \(\varepsilon\)\_B(h) \(\geq\) \(\mu\)(S)}, hence \texttt{max(\(\varepsilon\)\_A(h), \(\varepsilon\)\_B(h)) \(\geq\) \(\mu\)(S)/2}.
 
-\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.
+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.
 
-\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.
+\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.
 
-\section*{S2. Emergent vs imposed incompatibility (E13b framing)}
+\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.
 
-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).
+\section*{SI Text S2. Emergent versus imposed incompatibility}
 
-\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).
+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.
+
+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.
+
+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.
+
+\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.
 
 \section*{SI Table S1: the claims ledger (status / assumptions / evidence / limits)}
 
@@ -32,19 +38,19 @@ The conflict condition \emph{imposes} contradiction (the two label maps disagree
 \begin{tabular}{p{0.184\textwidth} p{0.184\textwidth} p{0.184\textwidth} p{0.184\textwidth} p{0.184\textwidth}}
 \hline
 Claim & Status & Key assumptions & Evidence & Known limits \\ \hline
-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]
+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]
-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]
+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]
 ``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]
 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]
 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]
 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]
-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]
+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]
-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]
+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]
 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]
-Confidence weighting improves rank prediction over raw disagreement & \textbf{Not supported} (pre-registered internal prediction) & --- & Paired Δ\textbackslash{} & \(\rho\)\textbackslash{} \\[3pt]
+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]
 The predictor improves budget-matched operator choice & \textbf{Open} & --- & Soup-vs-route gap readout noise-dominated at 0.5B & The practical payoff; untested \\[3pt]
-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]
+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]
-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]
+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]
 \hline\end{tabular}\end{center}\medskip
 
 \section*{SI Table S2: headline quantitative results}
@@ -56,13 +62,13 @@ Headline quantitative results with sample sizes, uncertainty, and outcome defini
 \hline
 Result & Setting / n & Outcome definition & Headline \\ \hline
 Closed-form validation & Biological model; standing tests & Simulated vs closed-form H-decay, immigration equilibrium, multi-teacher union & Agreement < 0.5\% \\[3pt]
-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]
+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]
 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]
-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]
+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]
 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]
 Speciation decomposition & MLPs, 3 replicates & LMC error barrier residual after permutation+rescaling alignment & Same-task 0.001; conflict 0.497 (naive 0.502) \\[3pt]
 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]
-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]
+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]
 \hline\end{tabular}\end{center}\medskip
 
 \section*{SI Methods: experimental procedures}
@@ -126,15 +132,15 @@ Knowledge is a distribution over \texttt{K} discrete items; reality is a fixed Z
 
 \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.
 
-\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.
+\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.
 
-\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.
+\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.
 
 \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.
 
-\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).
+\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).
 
-\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.
+\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.
 
 \subsection*{M4. The trained-network tier}
 
@@ -142,9 +148,9 @@ Knowledge is a distribution over \texttt{K} discrete items; reality is a fixed Z
 
 \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.
 
-\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.
+\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.
 
-\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.
+\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.
 
 \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.
 
@@ -156,7 +162,7 @@ Knowledge is a distribution over \texttt{K} discrete items; reality is a fixed Z
 
 \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.
 
-\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.
+\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.
 
 \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.
 
@@ -164,7 +170,7 @@ Knowledge is a distribution over \texttt{K} discrete items; reality is a fixed Z
 
 \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.
 
-\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.
+\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.
 
 \subsection*{M6. Negative controls}