third review round: mathematical corrections + operator separation + headline calibration

The five priority fixes, in the PNAS draft and propagated to the
long-form document and results documentation:

1. The averaging proposition now proves what it claims: a FIRST-ORDER
   cancellation of the multi-parent retention gain under output-mean
   inheritance in the rare-item regime (n·p/K << 1), with the convexity
   boundary stated (averaging's variance reduction can reduce extinction
   outside that regime — the reviewer's argument) and the union
   operator's renormalisation + oracle requirement explicit. "Adding
   parents cannot help" deleted everywhere.
2. Grounding: g*~=0.05 restated as an operational threshold (equilibrium
   smooth in g — no phase transition); m·p floor restated as
   1−exp(−m·p) per-batch observation probability with
   retention/occupancy/reintroduction distinguished; the deep-tail rule
   de-categoricalised (stratified sampling; recombination recovers only
   what parents retain).
3. Grounded INHERITANCE (data channel) separated from grounded
   EVALUATION (fitness channel) in the society section; retitled to
   "complementary contributions"; general joint necessity disclaimed.
   Table 1 + v6 ledger updated.
4. Alignment contradiction removed everywhere ("cannot be an alignment
   failure" -> the reviewer's formulation); abstract says "remaining
   after permutation-and-rescaling alignment"; group = search space,
   control recovery != global optimality; "specialisation is merge-safe"
   -> "do not treat divergence/specialisation alone as evidence of
   incompatibility".
5. Significance headline matched to the bounded evidence; seed-
   dependence sensitivity added (per-seed rho stable +0.37..+0.53 for
   functional measures, ~0 for geometry, gradient alignment
   seed-UNSTABLE −0.11..−0.55 — reported as its own caveat; LOSO ranges
   in stats script).

Presentation: review-process meta-language stripped; "exact" reserved
for closed forms ("analytic model" labels); headroom rule qualitative;
directed-sex phrasing per review; ratchet = consequence-level
correspondence; compact results table (Table 2) added. Response letter:
paper/response-to-review-3.md. Both PDFs rebuilt; 151 tests green.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01BkRLcc18rwT2Lysu6PbG7v

figures/stats_llm_epistasis.py

@@ -96,6 +96,18 @@ def main() -> None:
     for p, (rho, pv, rmse) in loco(df).items():
         print(f"{p:>11}: LOCO rho={rho:+.3f} (p={pv:.3g})  rmse={rmse:.3f}")
 
+    # Shared task-data seeds create dependence ACROSS conditions within a seed, which
+    # condition-clustering does not capture; per-seed and leave-one-seed-out correlations are the
+    # sensitivity check (3 seeds -> a range, not an estimate).
+    print("\n== seed sensitivity (per-seed rho; leave-one-seed-out range) ==")
+    for p in PREDICTORS:
+        per = [spearmanr(df[df.seed == s][p], df[df.seed == s]["pen_oracle"])[0]
+               for s in sorted(df.seed.unique())]
+        loso = [spearmanr(df[df.seed != s][p], df[df.seed != s]["pen_oracle"])[0]
+                for s in sorted(df.seed.unique())]
+        print(f"{p:>11}: per-seed " + " ".join(f"{v:+.2f}" for v in per)
+              + f" | LOSO [{min(loso):+.3f}, {max(loso):+.3f}]")
+
     print("\n== between- vs within-axis ==")
     print("mean penalty by axis:", df.groupby("mode")["pen_oracle"].mean().round(3).to_dict())
     c_df = df[df["mode"] == "conflict"]

paper/arxiv/body.tex

@@ -23,7 +23,7 @@ AI is turning from single frozen models to \textbf{populations of agents} that p
 
 We take one diagnosis as settled and cite it as such: training each generation on the last is \textbf{genetic drift}, and the resulting \textbf{model collapse} is the loss of rare variants a finite population always suffers (the Wright--Fisher process; formalised for language models by Shumailov et al., 2024, and Riis, 2026). We claim none of that. Our contribution is on the remedy side. Single- teacher copying is \textbf{asexual} reproduction, and the irreversible arm of its decay corresponds to \textbf{Muller's ratchet} (a correspondence we state with its scope, not as identity); the remedy biology found for the ratchet is \textbf{sex}. A society of models should reproduce sexually --- each new model \textbf{recombined from several complementary parents} (which the field already does, as \emph{model merging}), selection \textbf{anchored to a reality that can say no} (not to the consensus of other models), and diversity actively \textbf{preserved}. In our models --- from closed-form to trained networks to a language-model prototype --- those three ingredients together let a lineage not merely avoid collapse but \textbf{climb}, producing models fitter than any ancestor (the \textbf{Fisher--Muller effect}) while each specialty is re-earned and exceeded; whether the full recipe holds at frontier scale is the open question the framework is built to test.
 
-From the geneticist's apparatus we extract falsifiable, load-bearing claims (each stated with its operator and scope in the text): (i) \textbf{``merge, don't average''} --- a conservation result: refitting a child to the \emph{mean of its parents' output distributions} conserves rare-capability mass at the single-parent level, so adding parents cannot help, while union-preserving operators realise the gain --- exact in the minimal model, with its weight-space image verified as the headroom rule below; (ii) \textbf{offspring can exceed every parent} (Fisher--Muller), the real argument for sex in model societies; (iii) on \textbf{rugged, epistatic} task landscapes, blind recombination causes \textbf{outbreeding depression}, yielding a design rule --- \emph{merge freely when skills are additive, sparingly and with selection when entangled, and route rather than blend under overlap}; (iv) \textbf{grounding is immigration} from a non-drifting reality, giving a critical real-data fraction far below one; and (v) --- the sharpest new prediction --- sex has a \textbf{limit}: as two models diverge they undergo \textbf{speciation}, a merge-compatibility cliff (compatible \(\rightarrow\) outbreeding depression \(\rightarrow\) hybrid inviability) whose onset is set by divergence \emph{and} epistasis via \textbf{Bateson--Dobzhansky--Muller incompatibilities}, and whose damage grows \emph{super-linearly} (the Orr--Turelli snowball). We introduce and model this ``model speciation'' directly, and confirm it in real trained weights: a merge barrier that survives alignment under the \emph{full} function-preserving symmetry group of the network (not just Git Re-Basin permutations), rising with functional conflict while hybrid fitness falls to inviability --- with an honest converse we pre-registered and found: absent conflicting training signals, divergently-specialised lineages of shared ancestry developed \emph{no} isolation at any divergence tested, the merge instead \emph{rescuing} the forgetting specialists. Isolation must be provoked by conflict; specialisation alone did not speciate. AI also has an advantage biology lacks: \textbf{directed sex} --- unbounded parents, chosen mates, and offspring screened before they are kept --- which converts recombination from a gamble into a reliable engine and has no biological analogue.
+From the geneticist's apparatus we extract falsifiable, load-bearing claims (each stated with its operator and scope in the text): (i) \textbf{``merge, don't average''} --- a conservation result: refitting a child to the \emph{mean of its parents' output distributions} conserves expected rare-capability mass at the single-parent level, cancelling the multi-parent gain \emph{to first order in the rare-item regime} (outside it, variance reduction from averaging can help --- the result is a first-order cancellation, not a universal impossibility), while union-preserving operators realise the gain in all regimes --- derived in the minimal model, with its weight-space image the headroom rule below; (ii) \textbf{offspring can exceed every parent} (Fisher--Muller), the real argument for sex in model societies; (iii) on \textbf{rugged, epistatic} task landscapes, blind recombination causes \textbf{outbreeding depression}, yielding a design rule --- \emph{merge freely when skills are additive, sparingly and with selection when entangled, and route rather than blend under overlap}; (iv) \textbf{grounding is immigration} from a non-drifting reality, giving a critical real-data fraction far below one; and (v) --- the sharpest new prediction --- sex has a \textbf{limit}: as two models diverge they undergo \textbf{speciation}, a merge-compatibility cliff (compatible \(\rightarrow\) outbreeding depression \(\rightarrow\) hybrid inviability) whose onset is set by divergence \emph{and} epistasis via \textbf{Bateson--Dobzhansky--Muller incompatibilities}, and whose damage grows \emph{super-linearly} (the Orr--Turelli snowball). We introduce and model this ``model speciation'' directly, and confirm it in real trained weights: a merge barrier that survives alignment under the \emph{full} function-preserving symmetry group of the network (not just Git Re-Basin permutations), rising with functional conflict while hybrid fitness falls to inviability --- with an honest converse we pre-registered and found: absent conflicting training signals, divergently-specialised lineages of shared ancestry developed \emph{no} isolation at any divergence tested, the merge instead \emph{rescuing} the forgetting specialists. Isolation must be provoked by conflict; specialisation alone did not speciate. AI also has an advantage biology lacks: \textbf{directed sex} --- unbounded parents, chosen mates, and offspring screened before they are kept --- engineered recombination with a flexibility of parent choice and pre-deployment screening that natural mating systems do not approach.
 
 We support the argument with \textbf{minimal, reproducible models} --- a closed-form-exact account of drift and grounding, the same effects in small trained networks and an MNIST image generator, a real-weight demonstration of the speciation cliff (a Git Re-Basin residual that survives neuron alignment), and evolutionary simulations of the whole society --- and a first \textbf{language-model prototype}: merging LoRA-specialised Qwen models (to 7B on a GPU cluster) yields a generalist that beats every specialist parent, with the sharp headroom condition under which ``merge, don't average'' bites. The scope is honest: these are existence proofs and design rules; the \emph{whole grounded society} on a large language model is the open step. We position the work carefully against the crowded 2025--2026 landscape of evolutionary-AI and merging methods --- conceding what they own and marking, precisely, what a genuine population-genetics of sex adds.
 
@@ -39,7 +39,7 @@ The unit that matters is therefore the \textbf{generation}, and the event that m
 
 This axis is suddenly crowded. By 2026 several groups build \textbf{populations of models or agents that improve across generations}: societies of independently-specialised models that self-improve for more rounds than a single agent (Multiagent Finetuning --- Subramaniam et al., 2025); open-ended archives of self-rewriting coding agents (the Darwin--Gödel Machine --- Zhang et al., 2025); groups that evolve by sharing experience across branches (Weng et al., 2026); persistent agent \emph{ecologies} with reproduction and cumulative culture (TerraLingua --- 2026). In parallel, \textbf{model merging} has become a small industry with an overtly evolutionary vocabulary: crossover-mutation-selection over LLM populations (GENOME --- 2025), niching and ``mate choice'' (Sakana's M2N2 --- 2025), and evolutionary search over merge recipes (Akiba et al., \emph{Nature Mach. Intell.} 2024/25).
 
-We are candid about the consequence. Three things we do \textbf{not} claim. First, that collapse is Wright--Fisher drift: formalised independently (Riis, 2026; Shumailov et al., 2024), sharpened to a closed-form first-extinction law whose onset coincides with collapse (Benati et al., 2025) and to a quantitative-trait-genetics account for diffusion models (Yoon et al., ICLR 2025), and conceded here. Second, the bare empirical facts that a merged model can beat its parents, that decorrelated parents merge better, and that naive averaging is inferior to sign- or routing-based merges (TIES, DARE, mixture-of-experts routing): all established. Third, that merge success can be \emph{predicted at all}: machine-learning-native predictors exist, from interpretable pairwise metrics (gradient alignment --- Zhou et al., 2026) to capacity/rate-distortion accounts of ``merging collapse'' (2026); what they lack, and we supply, is the \emph{mechanism} --- when and why the failure is a coordinate artefact versus genuine functional incompatibility, and what moves the cliff. What a geneticist is placed to supply is a \textbf{framework} rather than a search heuristic. The nearest precursor is a theory-of-computation tradition reading sex as an algorithm for \emph{mixability} (Livnat \& Papadimitriou, 2016), pre-dating model merging; the works above use evolution chiefly as vocabulary over an optimiser, and --- to our knowledge --- the quantitative apparatus of the evolution of sex (Fisher--Muller, outbreeding depression, migration--drift balance, reproductive isolation) has not previously been carried over as more than metaphor. We are also candid about what \emph{kind} of contribution each of our claims is, because three different things are easily conflated: \textbf{interpretation} (an existing result is usefully understood in these terms --- e.g., merged offspring beating their parents as Fisher--Muller), \textbf{explanation} (the transferred mechanism accounts for observations existing accounts leave open --- e.g., which merge failures are coordinate artefacts and which are functional), and \textbf{prediction} (the framework forecasts an unmeasured outcome and improves a design decision --- e.g., an epistasis measure taken \emph{before} merging that beats geometry-based predictors of merge success). This paper is strongest on the first, makes concrete progress on the second, and states the third as its open, decisive test --- proposed here with pre-registered falsifiers, not claimed as done. The organising shift we argue for is prior to any single mechanism: \textbf{treat multigenerational model populations as systems whose inheritance, diversity, and compatibility must be managed --- not merely as collections of models to optimise.}
+We are candid about the consequence. Three things we do \textbf{not} claim. First, that collapse is Wright--Fisher drift: formalised independently (Riis, 2026; Shumailov et al., 2024), sharpened to a closed-form first-extinction law whose onset coincides with collapse (Benati et al., 2025) and to a quantitative-trait-genetics account for diffusion models (Yoon et al., ICLR 2025), and conceded here. Second, the bare empirical facts that a merged model can beat its parents, that decorrelated parents merge better, and that naive averaging is inferior to sign- or routing-based merges (TIES, DARE, mixture-of-experts routing): all established. Third, that merge success can be \emph{predicted at all}: machine-learning-native predictors exist, from interpretable pairwise metrics (gradient alignment --- Zhou et al., 2026) to capacity/rate-distortion accounts of ``merging collapse'' (2026); what they lack, and we supply, is the \emph{mechanism} --- when and why the failure is a coordinate artefact versus genuine functional incompatibility, and what moves the cliff. What a geneticist is placed to supply is a \textbf{framework} rather than a search heuristic. The nearest precursor is a theory-of-computation tradition reading sex as an algorithm for \emph{mixability} (Livnat \& Papadimitriou, 2016), pre-dating model merging; the works above use evolution chiefly as vocabulary over an optimiser, and --- to our knowledge --- the quantitative apparatus of the evolution of sex (Fisher--Muller, outbreeding depression, migration--drift balance, reproductive isolation) has not previously been carried over as more than metaphor. We are also candid about what \emph{kind} of contribution each of our claims is, because three different things are easily conflated: \textbf{interpretation} (an existing result is usefully understood in these terms --- e.g., merged offspring beating their parents as Fisher--Muller), \textbf{explanation} (the transferred mechanism accounts for observations existing accounts leave open --- e.g., which merge failures are coordinate artefacts and which are functional), and \textbf{prediction} (the framework forecasts an unmeasured outcome and improves a design decision). This paper is strongest on the first, makes concrete progress on the second, and reports a first, bounded step on the third: a \textbf{controlled predictive test} at small scale in which pre-merge \emph{functional-disagreement} measures --- chosen by the framework --- showed a detectable, held-out-robust association with merge damage on a constructed task grid, while the selected weight-geometry baselines did not. We are precise about that result's boundary where it is reported: it is a small-model demonstration on a constructed grid; the proposed epistasis-specific refinement did not outperform plain disagreement; predictor differences are not individually significant head-to-head; and whether the prediction improves a budget-matched operator choice remains open. The organising shift we argue for is prior to any single mechanism: \textbf{treat multigenerational model populations as systems whose inheritance, diversity, and compatibility must be managed --- not merely as collections of models to optimise.}
 
 \section*{2. Why today's models cannot do this}
 
@@ -87,7 +87,7 @@ Geneticists call this the \textbf{Fisher--Muller effect} (Fisher, 1930; Muller,
 
 This is no longer only a simulation. In a first language-model prototype --- LoRA specialists on disjoint task families, recombined and judged by an exact verifier --- a merge of three specialist Qwen models (7B, on a GPU cluster) \textbf{beats every single specialist}, overall and on every family: the Fisher--Muller effect, in real weights. The same prototype pins down \emph{when} the finer ``inherit the union, don't average'' rule actually bites. Keeping each parent whole and \textbf{routing} each input to the right one beats the tail-thinning average --- but only when the task is hard enough to leave room to lose: on easy tasks a strong model's plain average is already at the ceiling, so the crude soup is fine, whereas on hard tasks the average dilutes a hard-won specialist so badly it falls below even the best single parent, and routing wins by a wide margin. The rule is therefore precise: \textbf{the union beats the average in exact proportion to how far the average is from the best attainable} --- a caveat that sharpens rather than weakens the claim, and that a practitioner needs before spending compute on the fancier operator.
 
-\textbf{The operator boundaries (stated, because ``merge, don't average'' is not one claim but a family).} Four different operators travel under these words, and the conservation result belongs to exactly one of them. What is \emph{derived} is this: when a pupil's knowledge is refit to the \textbf{mean of the parents' output distributions}, the expected mass on any rare item is conserved at the single-parent level --- in the rare-item regime the 1/K dilution of averaging exactly cancels the union gain of having K parents --- so adding parents cannot help; whereas an operator that keeps, per item, its \textbf{strongest source} realises the union. That statement is exact in the minimal model, and it presupposes an oracle (or verifier) able to say which source is strongest. The two operators the LLM prototype tests --- \textbf{weight averaging} (a nonlinear network's weight-mean does not compute the mean of its parents' outputs) and \textbf{routing among intact specialists} (which keeps K models' storage and an input classifier, a different parameter and inference budget from one fixed-size child) --- are \emph{empirical cousins} of the two sides of that law, not instances of it. The headroom rule above is precisely the empirical bridge: it says when the weight-average behaves like the diluting mean (hard tasks, weak base) and when a capable base absorbs the dilution (easy tasks). And all of it operates within a capacity boundary: when parental capabilities genuinely cannot coexist in the child's capacity, no operator preserves the union --- that regime is the subject of the speciation section below.
+\textbf{The operator boundaries (stated, because ``merge, don't average'' is not one claim but a family).} Four different operators travel under these words, and the conservation result belongs to exactly one of them. What is \emph{derived} is this: when a pupil's knowledge is refit to the \textbf{mean of the parents' output distributions}, the expected mass on any rare item is conserved at the single-parent level --- in the rare-item regime (\texttt{n\(\cdot\)p/K ≪ 1}) the 1/K dilution of averaging cancels the union gain of having K parents to first order --- outside that regime, survival is convex in mixed mass and averaging's variance reduction can help, so this is a first-order cancellation, not a universal impossibility; whereas an operator that keeps, per item, its \textbf{strongest source} (and renormalises, which itself redistributes mass) realises the union in all regimes. That statement is exact in the minimal model, and it presupposes an oracle (or verifier) able to say which source is strongest. The two operators the LLM prototype tests --- \textbf{weight averaging} (a nonlinear network's weight-mean does not compute the mean of its parents' outputs) and \textbf{routing among intact specialists} (which keeps K models' storage and an input classifier, a different parameter and inference budget from one fixed-size child) --- are \emph{empirical cousins} of the two sides of that law, not instances of it. The headroom rule above is precisely the empirical bridge: it says when the weight-average behaves like the diluting mean (hard tasks, weak base) and when a capable base absorbs the dilution (easy tasks). And all of it operates within a capacity boundary: when parental capabilities genuinely cannot coexist in the child's capacity, no operator preserves the union --- that regime is the subject of the speciation section below.
 
 Three results keep this honest, and all are results, not hand-waving.
 
@@ -125,7 +125,7 @@ This is where a geneticist's lens earns its keep. The machine-learning literatur
 \end{figure*}
 
 
-\textbf{The real-weight confirmation.} The obvious objection to the analytic model is that its ``incompatibility'' is a re-labelled loss barrier, and loss barriers between independently trained networks are famously a \emph{coordinate} artefact --- two nets that learned the same function in a permuted basis look incompatible until their neurons are aligned (Git Re-Basin), and recent work shows that symmetry groups \emph{richer} than permutations remove still more of the barrier (functionality-preserving rescalings and rotations --- Scaling LMC, 2026; neuron-identifiability approaches). We therefore ran the experiment the objection demands, in real trained weights, aligning modulo the \textbf{full} function-preserving unit symmetry group of the architecture (per-unit positive rescaling composed with permutation --- for a plain ReLU network, all of it). Two small MLPs are forked from a shared MNIST base, trained, weight-averaged, and their linear-mode-connectivity error barrier is measured \emph{before and after} alignment; the after-alignment \textbf{residual} is the part of the incompatibility that no re-coordination can explain away. The decomposition is clean : two nets trained \emph{from different random initialisations on the same task} have a real naive barrier that alignment removes almost entirely (residual \(\approx\) 0.001, and the aligned merge performs at parent level) --- same species, different basis, the canonical Re-Basin result, which also proves the aligner works. Two nets that learned \emph{conflicting} label maps have a large barrier of which the full symmetry group removes \textbf{essentially nothing} (0.502 \(\rightarrow\) 0.497) --- genuine reproductive isolation, not a missed symmetry, and it cannot be dismissed as a failure to align because the very same aligner erased the same-task barrier. It also carries a floor no future alignment method can breach: models loyal to label maps that conflict on a fraction \emph{\(\mu\)} of inputs cannot both be served by \emph{any} single merged model, which must err at rate \(\geq\) \emph{\(\mu\)}/2 against at least one parent (SI proposition). Sweeping the fraction of conflicting classes traces the \textbf{isolation cliff in real weights}, now readable directly as \emph{hybrid fitness}: the residual barrier climbs monotonically while the merged model's accuracy falls from 0.97 to 0.03 --- E12's compatible \(\rightarrow\) depression \(\rightarrow\) inviability trajectory, measured.
+\textbf{The real-weight confirmation.} The obvious objection to the analytic model is that its ``incompatibility'' is a re-labelled loss barrier, and loss barriers between independently trained networks are famously a \emph{coordinate} artefact --- two nets that learned the same function in a permuted basis look incompatible until their neurons are aligned (Git Re-Basin), and recent work shows that symmetry groups \emph{richer} than permutations remove still more of the barrier (functionality-preserving rescalings and rotations --- Scaling LMC, 2026; neuron-identifiability approaches). We therefore ran the experiment the objection demands, in real trained weights, aligning modulo the \textbf{full} function-preserving unit symmetry group of the architecture (per-unit positive rescaling composed with permutation --- for a plain ReLU network, all of it). Two small MLPs are forked from a shared MNIST base, trained, weight-averaged, and their linear-mode-connectivity error barrier is measured \emph{before and after} alignment; the after-alignment \textbf{residual} is the part of the incompatibility that no re-coordination can explain away. The decomposition is clean : two nets trained \emph{from different random initialisations on the same task} have a real naive barrier that alignment removes almost entirely (residual \(\approx\) 0.001, and the aligned merge performs at parent level) --- same species, different basis, the canonical Re-Basin result, which also proves the aligner works. Two nets that learned \emph{conflicting} label maps have a large barrier of which the full symmetry group removes \textbf{essentially nothing} (0.502 \(\rightarrow\) 0.497) --- a conflict-associated barrier the tested alignment leaves largely unchanged --- supporting a functional-conflict interpretation without proving optimal alignment (control recovery validates a special case; the removable share is a lower bound, the residual an upper bound). It also carries a floor no future alignment method can breach: models loyal to label maps that conflict on a fraction \emph{\(\mu\)} of inputs cannot both be served by \emph{any} single merged model, which must err at rate \(\geq\) \emph{\(\mu\)}/2 against at least one parent (SI proposition). Sweeping the fraction of conflicting classes traces the \textbf{isolation cliff in real weights}, now readable directly as \emph{hybrid fitness}: the residual barrier climbs monotonically while the merged model's accuracy falls from 0.97 to 0.03 --- E12's compatible \(\rightarrow\) depression \(\rightarrow\) inviability trajectory, measured.
 
 \begin{figure*}[t]\centering
 \includegraphics[width=\textwidth]{figs/speciation_real.pdf}
@@ -135,7 +135,7 @@ This is where a geneticist's lens earns its keep. The machine-learning literatur
 
 \textbf{And its honest converse: speciation must be provoked; it did not emerge.} A true Dobzhansky--Muller incompatibility is \emph{emergent} --- each lineage's changes harmless alone, incompatible only in combination --- whereas the conflict condition above \emph{imposes} contradiction. So we pre-registered the emergent test: fork two children from a shared base and let them diverge with \textbf{no conflicting training signal anywhere} --- one pair as complementary class specialists (one child trains only on digits 0--4, the other only on 5--9), one pair with divergent input conventions (views shifted in opposite directions) --- out to divergences 6.4\(\times\) the base training. The result is the second pre-registered reading, and it sharpens the theory's scope rather than confirming its most dramatic form: the residual barrier is \textbf{0.000 at every divergence in both conditions}, and far from failing, the merge \emph{rescues} the two specialists --- each parent decays toward \textasciitilde{}0.50 on the full task (catastrophically forgetting the classes it no longer sees) while the merged model holds \textasciitilde{}0.95 throughout, a sustained Fisher--Muller rescue at zero barrier. In real weights, at least in this regime of shared ancestry and compatible tasks, \textbf{reproductive isolation requires functional conflict; it does not arise spontaneously from divergent specialisation.} The design rule sharpens accordingly: \emph{merge freely across divergently-specialised lineages of shared ancestry --- what speciates model populations is conflicting conventions, not specialisation per se.} Whether long-horizon over-specialisation erodes mergeability at language-model scale --- as the empirical merging literature hints (experts trained longer merge worse under averaging) --- is exactly the next tier's question, and the theory now makes the prediction crisp: it should depend on whether extended training induces \emph{conflicting conventions on shared circuitry}, not on divergence time itself.
 
-\textbf{What these experiments do and do not establish.} Stated at exactly the strength of the evidence: they establish that \emph{some merge failures reflect incompatible functional requirements rather than a mismatch of coordinates} --- a residual that survives the full unit-symmetry group of the architecture tested, rises with functional conflict, and is absent under compatible specialisation. Three qualifiers. First, the impossibility at the heart of the conflict condition --- one deterministic model cannot satisfy two contradictory answer conventions --- is information-theoretic and needs no population genetics; what the genetic frame adds is \emph{structure around it}: which divergences generate such conflicts, the prediction that epistasis rather than distance sets the cliff's position, and the snowball's super-linear onset --- the latter two verified so far only in the analytic model, and therefore carried as \textbf{hypotheses at the neural tier, not results}. Second, our alignment removes the symmetries we enumerate for this architecture class; richer transformation families for other architectures could reapportion removable vs residual, though not below the conflict floor. Third, ``unmergeable'' here means by aligned linear interpolation of weights --- a barrier to that operator does not preclude every conceivable recombination method (routing, for one, sidesteps it by not blending). Emergent Dobzhansky--Muller incompatibilities in real weights remain the flagship \emph{hypothesis} of this programme: our tested regimes found none, which bounds where they can live --- longer horizons, shifted data distributions, capacity pressure --- and the decisive experiment (predicting merge success \emph{before} merging from an operational epistasis measure, against geometry- and gradient-based predictors) is posed in the closing section.
+\textbf{What these experiments do and do not establish.} Stated at exactly the strength of the evidence: they establish that \emph{some merge failures reflect incompatible functional requirements rather than a mismatch of coordinates} --- a residual that survives the full unit-symmetry group of the architecture tested, rises with functional conflict, and is absent under compatible specialisation. Three qualifiers. First, the impossibility at the heart of the conflict condition --- one deterministic model cannot satisfy two contradictory answer conventions --- is information-theoretic and needs no population genetics; what the genetic frame adds is \emph{structure around it}: which divergences generate such conflicts, the prediction that epistasis rather than distance sets the cliff's position, and the snowball's super-linear onset --- the latter two verified so far only in the analytic model, and therefore carried as \textbf{hypotheses at the neural tier, not results}. (On the snowball, one more distinction: super-linear growth in the \emph{number} of incompatibilities does not by itself entail a sharp \emph{performance} cliff --- that needs the link from incompatibility count through effect sizes to measured performance, which the analytic model supplies under its assumptions and any neural test must establish separately.) Second, our alignment removes the symmetries we enumerate for this architecture class, and exactly recovering a permuted-and-rescaled copy validates a special case rather than proving global optimality for independently trained networks --- so the removable share is a lower bound and the residual an upper bound; richer transformation families for other architectures could reapportion the split, though not below the conflict floor. Third, ``unmergeable'' here means by aligned linear interpolation of weights --- a barrier to that operator does not preclude every conceivable recombination method (routing, for one, sidesteps it by not blending). Emergent Dobzhansky--Muller incompatibilities in real weights remain the flagship \emph{hypothesis} of this programme: our tested regimes found none, which bounds where they can live --- longer horizons, shifted data distributions, capacity pressure --- and the decisive experiment (predicting merge success \emph{before} merging from an operational epistasis measure, against geometry- and gradient-based predictors) is posed in the closing section.
 
 One question remains, and the rest of the paper is largely about it: recombination combines what the parents kept --- but \emph{who decides what each parent keeps, and which offspring are worth keeping?}
 
@@ -206,7 +206,7 @@ Three honest riders, because re-minting is the most consequential step in the sc
 \item \textbf{Speciation.} A re-minting is a founder event. Different laboratories, re-basing on different criteria, will mint divergent bases; the lineage branches. This is not a defect but \emph{adaptive radiation}, and it is exactly what open weights make possible. The society grows not as one heavy trunk but as a branching tree of bases.
 \end{itemize}
 
-So the answer to ``can it grow forever?'' is \textbf{yes --- but only because it forgets and consolidates at every level, including the base.} Nothing is retained without bound anywhere; unbounded growth of \emph{capability} is bought by \emph{bounded} storage plus periodic consolidation.
+So the honest answer to ``can it grow forever?'' is: *\emph{the architecture removes the }storage\emph{ obstacle to indefinite accumulation}* --- nothing is retained without bound anywhere, and consolidation resets the soft budget each epoch --- but that is a statement about bookkeeping, not a demonstration of unbounded capability growth, which no fixed-capacity system can promise and our finite models (deliberately scoped as ``effectively open-ended relative to the sample size, not astronomically open-ended'') do not test. What the design claims is the weaker, defensible thing: at no level does a full store force the lineage to stop learning.
 
 \section*{12. One process, four timescales}
 
@@ -255,14 +255,17 @@ Offspring exceed every parent (Fisher--Muller) & Interpretation + empirical & Co
 Outbreeding depression on rugged landscapes; operator design rule & Exact-model result; hypothesis at LLM scale & NK epistasis stands in for skill entanglement & E9--E10; directed selection rescues & Not yet mapped onto a real task-entanglement measure \\[3pt]
 Optimal mate-pool breadth shrinks with ruggedness & Exact-model result; hypothesis for merging populations & Ring population, local selection & E14 & Phenomenon known to island-model evolutionary computation; our contribution is the mapping and the diversity/mean decomposition \\[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]
-Epistasis (not divergence) sets the cliff; snowball onset & Exact-model result; \textbf{hypothesis} at the neural tier & BDM incompatibility structure & E12 & The decisive pre-merge prediction test is proposed, not run \\[3pt]
+Epistasis (not divergence) sets the cliff; snowball onset & Exact-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]
+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 ρ\(\approx\)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{} & ρ\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]
-Grounding + sex + diversity jointly necessary & Exact-model result; hypothesis at LLM scale & Conformity stands in for self-consumption & E11 four-arm ablation, each arm failing distinctly & The full grounded LLM society is unbuilt \\[3pt]
+Grounding + sex + diversity complementary (each ablation fails distinctly) & Analytic-model result; hypothesis at LLM scale & Conformity stands in for self-consumption; general joint necessity not established & E11 four-arm ablation & The full grounded LLM society is unbuilt; alternative schemes untested \\[3pt]
 \hline\end{tabular}\end{center}\medskip
 
 \textbf{What is borrowed, and what is ours.} We are deliberate about the ledger, because the surrounding literature is crowded and a reader deserves to know exactly where the line falls. \textbf{Conceded as prior art:} (a) \emph{model collapse is genetic drift} --- derived independently and cleanly (Riis, 2026; the Wright--Fisher collapse literature following Shumailov et al., 2024; the closed-form first-extinction law of Benati et al., 2025; the quantitative-trait account of Yoon et al., 2025); (b) the empirical facts that a merged model can \emph{beat its parents}, that \emph{decorrelated} parents merge better, and that \emph{naive averaging is inferior} to sign-reconciled or routed merges (model soups, TIES, DARE, mixture-of-experts routing); (c) that a \emph{population} of merging or self-improving models can climb (GENOME, M2N2, Multiagent Finetuning, the Darwin--Gödel Machine); (d) that merge success has machine-learning-native \emph{predictors} --- interpretable pairwise metrics (Zhou et al., 2026), capacity/rate-distortion accounts of merging collapse (Cao et al., 2026), and stability/scaling analyses of multi-task degradation; and (e) that verifier-screened synthetic data can avert collapse (Yi et al., 2025) --- the statistical cousin of our grounding operator. We claim none of these.
 
-\textbf{Ours} is the framework those results invite: a \textbf{population-genetics of sex} applied to model societies, generative where the incumbents are empirical. Concretely --- the \textbf{``merge, don't average'' conservation law} (recombination preserves the union; blending inheritance cancels it), derived not observed; \textbf{Fisher--Muller} named and used to explain \emph{why} offspring exceed parents; \textbf{outbreeding depression on rugged/epistatic landscapes}, which turns ``when does merging help vs hurt'' from a thing you must run a search to discover into a thing the landscape's ruggedness \emph{predicts}, with the operator-choice design rule that follows (average / union-route / directed-select); \textbf{grounding as migration--drift balance}, giving a critical real-data fraction and a phase boundary a closed self-consuming loop cannot have; \textbf{directed sex} as the distinctly-AI advantage (unbounded parents, offspring preview, mate choice); and the \textbf{integrated society} whose four operators are shown \emph{jointly necessary}. The value-add over the machine-learning-native merge theory is that ours predicts \emph{which operator to use and when it will backfire}, not merely how fast quality decays. And it opens --- and begins to occupy --- a question nobody has framed: \textbf{model speciation}, the population-genetics of \emph{reproductive isolation} (Bateson--Dobzhansky--Muller incompatibilities) as the account of \emph{when two models are too diverged to be merged at all}. We model it explicitly (§5), predicting the compatible \(\rightarrow\) outbreeding-depression \(\rightarrow\) inviability curve, its super-linear (snowball) onset, and its control by epistasis rather than divergence alone --- the one place the merge literature has phenomena (Pari et al., 2024; Zhou et al., 2026) but no theory --- and we confirm it in real trained weights, where a merge barrier survives alignment under the \emph{full} function-preserving symmetry group (not only Re-Basin permutations) as a residual, functional reproductive isolation with an information-theoretic floor --- together with the pre-registered emergent converse: absent conflicting training signals, divergently-specialised lineages of shared ancestry showed \emph{no} isolation at any divergence tested, the merge instead rescuing the forgetting specialists (isolation must be provoked; specialisation alone did not speciate). In one sentence: the field agrees on the disease and tinkers at the cure with evolutionary metaphors; we bring the evolutionary \emph{theory}, and it makes falsifiable predictions --- a merge-compatibility cliff among them --- that the metaphors do not.
+\textbf{Ours} is the framework those results invite: a \textbf{population-genetics of sex} applied to model societies, generative where the incumbents are empirical. Concretely --- the \textbf{``merge, don't average'' conservation law} (recombination preserves the union; blending inheritance cancels it), derived not observed; \textbf{Fisher--Muller} named and used to explain \emph{why} offspring exceed parents; \textbf{outbreeding depression on rugged/epistatic landscapes}, which turns ``when does merging help vs hurt'' from a thing you must run a search to discover into a thing the landscape's ruggedness \emph{predicts}, with the operator-choice design rule that follows (average / union-route / directed-select); \textbf{grounding as migration--drift balance}, giving a critical real-data fraction and a phase boundary a closed self-consuming loop cannot have; \textbf{directed sex} as the distinctly-AI advantage (unbounded parents, offspring preview, mate choice); and the \textbf{integrated society} whose operators make \emph{complementary, distinctly-failing contributions} in the tested model (general joint necessity is not established). The value-add over the machine-learning-native merge theory is that ours predicts \emph{which operator to use and when it will backfire}, not merely how fast quality decays. And it opens --- and begins to occupy --- a question nobody has framed: \textbf{model speciation}, the population-genetics of \emph{reproductive isolation} (Bateson--Dobzhansky--Muller incompatibilities) as the account of \emph{when two models are too diverged to be merged at all}. We model it explicitly (§5), predicting the compatible \(\rightarrow\) outbreeding-depression \(\rightarrow\) inviability curve, its super-linear (snowball) onset, and its control by epistasis rather than divergence alone --- the one place the merge literature has phenomena (Pari et al., 2024; Zhou et al., 2026) but no theory --- and we confirm it in real trained weights, where a merge barrier survives alignment under the \emph{full} function-preserving symmetry group (not only Re-Basin permutations) as a residual, functional reproductive isolation with an information-theoretic floor --- together with the pre-registered emergent converse: absent conflicting training signals, divergently-specialised lineages of shared ancestry showed \emph{no} isolation at any divergence tested, the merge instead rescuing the forgetting specialists (isolation must be provoked; specialisation alone did not speciate). In one sentence: the field agrees on the disease and tinkers at the cure with evolutionary metaphors; we bring the evolutionary \emph{theory}, and it makes falsifiable predictions --- a merge-compatibility cliff among them --- that the metaphors do not.
 
 \textbf{What is still open --- honestly.} The old hole (what to select) we fill in kind: don't design the selector, evolve it. But the hole has \emph{moved}, not closed, and the new one is harder: \textbf{the fitness function} --- what reality-anchored measure selects for \emph{truth} without also selecting for \emph{persuasion}, given that in our own species the two have been at war for the whole history of ideas. Alongside it: the \textbf{institutions} that let contemporaries correct one another before error is inherited (§8), which we do not solve; and the \textbf{calibration} of everything the results left as knobs --- how many parents, how complementary, at what ratio of inherited-to-real data, and how healthy a lineage must be before its knowledge is safe to make irreversibly innate. These are, at least, \emph{measurable} --- which is the difference between an open problem and a hole. And the largest gap of all: the \emph{recombination} claims now hold in real language models, but the \emph{society} --- the grounded, diversity-preserving, continually reproducing loop --- does not yet. The real test is to build that whole system out of actual open-weight language models, and see whether all the signs survive contact with a system too big to write down. The operators, checked; the living society, next.
 

paper/pnas/body.tex

@@ -1,10 +1,10 @@
 \section*{Significance statement}
 
-Artificial intelligence is shifting from single, frozen models to populations of models that specialise, are retrained on each other's output, and are combined (``merged'') into new models. Trained on their own output, model lineages degenerate --- a process already recognised as the mathematics of genetic drift. This paper imports the other half of population genetics: the biology of sexual reproduction. It treats model merging as recombination, real data as immigration, and merge failure as reproductive isolation, and tests each correspondence in simulations, small neural networks, and language models. The framework yields design rules --- when to average models, when to keep them separate, how much real data suffices --- and a first controlled test showing that measured functional conflict, not weight distance, predicts when merging fails.
+Artificial intelligence is shifting from single, frozen models to populations of models that specialise, are retrained on each other's output, and are combined (``merged'') into new models. Trained on their own output, model lineages degenerate --- a process already recognised as the mathematics of genetic drift. This paper imports the other half of population genetics: the biology of sexual reproduction. It treats model merging as recombination, real data as immigration, and merge failure as reproductive isolation, and tests each correspondence in simulations, small neural networks, and language models. The framework yields design rules --- when to average models, when to keep them separate, how much real data suffices --- and a controlled small-model test in which pre-merge functional disagreement predicted merge damage, motivating further comparison with weight-space measures.
 
 \section*{Abstract}
 
-AI development increasingly resembles a population process: models are specialised, retrained on model output, and recombined by weight merging, with an openly evolutionary vocabulary but little use of evolutionary theory. Here we treat multigenerational model populations as systems whose inheritance, diversity, and compatibility must be managed, and transfer the quantitative apparatus of the evolution of sex. We take as settled that training on model output is genetic drift (model collapse). In a minimal inheritance model that is exactly Wright--Fisher --- and measurably Wright--Fisher-plus-bias in trained networks --- we derive and test the remedies: grounding as immigration, with a critical real-data fraction far below one but a per-capability floor that leaves the rarest knowledge unrescuable; recombination, where averaging parents' output distributions exactly cancels the benefit of multiple parents while union-preserving operators realise it; the Fisher--Muller effect, with merged language-model specialists exceeding every parent in replicated experiments; outbreeding depression on rugged task landscapes, converted into reliable gains by directed, offspring-screened recombination; and population structure, where the optimal mating breadth shrinks as skills entangle. Sex has a limit: we introduce model speciation --- merge failure as reproductive isolation --- and show in trained networks that a merge barrier surviving the full function-preserving symmetry group tracks functional conflict, that isolation did not emerge from compatible specialisation, and, in a controlled predictive test, that pre-merge functional disagreement predicts merge damage where weight-geometry baselines do not. We state precisely what is exact, what is measured, and what remains hypothesis.
+AI development increasingly resembles a population process: models are specialised, retrained on model output, and recombined by weight merging, with an openly evolutionary vocabulary but little use of evolutionary theory. Here we treat multigenerational model populations as systems whose inheritance, diversity, and compatibility must be managed, and transfer the quantitative apparatus of the evolution of sex. We take as settled that training on model output is genetic drift (model collapse). In a minimal inheritance model that is exactly Wright--Fisher --- and measurably Wright--Fisher-plus-bias in trained networks --- we derive and test the remedies: grounding as immigration, where a real-data fraction far below one retained most equilibrium diversity in the tested settings, with a per-capability observation floor that makes the rarest knowledge expensive under unstratified sampling; recombination, where refitting to the mean of parents' output distributions cancels the multi-parent gain to first order in the rare-item regime while union-preserving operators realise it; the Fisher--Muller effect, with merged language-model specialists exceeding every parent in replicated experiments; outbreeding depression on rugged task landscapes, converted into reliable gains by directed, offspring-screened recombination; and population structure, where the optimal mating breadth shrinks as skills entangle. Sex has a limit: we introduce model speciation --- merge failure as reproductive isolation --- and show in trained networks that a merge barrier remaining after permutation-and-rescaling alignment tracks functional conflict, that isolation did not emerge from compatible specialisation, and, in a controlled predictive test, that pre-merge functional disagreement predicted merge damage while the tested weight-geometry baselines showed no detectable association. We state precisely what is exact, what is measured, and what remains hypothesis.
 
 \medskip\hrule\medskip
 
@@ -14,17 +14,17 @@ The unit of AI progress is quietly changing. Multi-agent systems arrange many mo
 
 One half of the transfer is settled and is not our contribution. Training each generation of a model on the previous generation's output degrades it --- \emph{model collapse}: rare capabilities vanish first and the lineage drifts toward its own most common behaviour (6). That this is the mathematics of \textbf{genetic drift} in a finite population is now established from several directions (7--9); a closed-form first-extinction law even places collapse onset at the Wright--Fisher first-extinction time (8). We cite this literature as the diagnosis and build on it.
 
-Our contribution is on the remedy side, and we are explicit about what kind of contribution each claim is, distinguishing \textbf{interpretation} (an existing result understood in population-genetic terms), \textbf{explanation} (the transferred mechanism accounts for observations existing accounts leave open), and \textbf{prediction} (the framework forecasts an unmeasured outcome). The paper is strongest on the first; makes concrete progress on the second --- separating merge failures that are coordinate artefacts from those that are functional; and reports a first, bounded step on the third --- a controlled predictive test in which pre-merge functional-disagreement measures, chosen by the framework, predicted merge damage on a constructed task grid while weight-geometry baselines did not.
+Our contribution is on the remedy side, and we are explicit about what kind of contribution each claim is, distinguishing \textbf{interpretation} (an existing result understood in population-genetic terms), \textbf{explanation} (the transferred mechanism accounts for observations existing accounts leave open), and \textbf{prediction} (the framework forecasts an unmeasured outcome). The paper is strongest on the first; makes concrete progress on the second --- separating merge failures that are coordinate artefacts from those that are functional; and reports a first, bounded step on the third --- a controlled predictive test in which pre-merge functional-disagreement measures, chosen by the framework, predicted merge damage on a constructed task grid while the tested weight-geometry baselines showed no detectable association.
 
-The correspondences we develop, summarised in Table 1: single-teacher retraining is \textbf{asexual reproduction}, and the irreversible arm of its decay corresponds to \textbf{Muller's ratchet} (10) --- once every copy of a rare capability is gone from all parents and sources, no recombination can rebuild it, which is precisely why remedies must act before fixation-by-loss. Injecting verified real data is \textbf{immigration} from a non-drifting source (11--13). Model merging is \textbf{recombination}, and its celebrated payoff --- a merged model exceeding every parent --- is the \textbf{Fisher--Muller effect} (14, 15). Merging entangled skills courts \textbf{outbreeding depression}; screening many candidate merges is a form of \textbf{directed sex} with no biological analogue; restricting who merges with whom is \textbf{population structure}. And merging's hard limit --- models too diverged in function to combine --- is \textbf{reproductive isolation}, for which the Bateson--Dobzhansky--Muller theory of incompatibilities (16, 17) supplies the structure. The nearest precursor to this programme reads sex as an algorithm for mixability in the theory of computation (18), pre-dating model merging; the model-merging literature itself has strong empirical operators (1, 19, 20) and emerging merge-success predictors (21, 22), to which our delta is mechanism: \emph{when and why} failure is coordinate versus functional, and what moves the boundary.
+The correspondences we develop, summarised in Table 1: single-teacher retraining is \textbf{asexual reproduction}, and the irreversible arm of its decay shares the defining consequence of \textbf{Muller's ratchet} (10) --- once every copy of a rare capability is gone from all parents and sources, no recombination can rebuild it, which is why remedies must act before fixation-by-loss (a consequence- level correspondence: the minimal model lacks the ratchet's recurrent deleterious-mutation mechanism, so irreversible loss alone does not identify that specific mechanism). Injecting verified real data is \textbf{immigration} from a non-drifting source (11--13). Model merging is \textbf{recombination}, and its celebrated payoff --- a merged model exceeding every parent --- is the \textbf{Fisher--Muller effect} (14, 15). Merging entangled skills courts \textbf{outbreeding depression}; screening many candidate merges is engineered recombination with unusually flexible parent choice and pre-deployment screening (we use the shorthand \textbf{directed sex}); restricting who merges with whom is \textbf{population structure}. And merging's hard limit --- models too diverged in function to combine --- is \textbf{reproductive isolation}, for which the Bateson--Dobzhansky--Muller theory of incompatibilities (16, 17) supplies the structure. The nearest precursor to this programme reads sex as an algorithm for mixability in the theory of computation (18), pre-dating model merging; the model-merging literature itself has strong empirical operators (1, 19, 20) and emerging merge-success predictors (21, 22), to which our delta is mechanism: \emph{when and why} failure is coordinate versus functional, and what moves the boundary.
 
-We support the framework at three tiers of evidence, in ascending realism and descending exactness: a \textbf{minimal analytic model} validated against closed forms to a fraction of a percent; \textbf{small trained networks} (MLPs, recurrent networks, an MNIST image generator) where the operators are measured in real weights; and \textbf{language models} (LoRA-specialised Qwen models, 0.5B locally and 7B on a compute cluster) where the claims are tested as signs under seed replication. Throughout, we report negative and tempering results with the same prominence as confirmations: they include the failure of an internal pre-registered prediction, a null on emergent speciation that bounds the analogy, and the sensitivity analyses that temper the predictive test.
+We support the framework at three tiers of evidence, in ascending realism and descending exactness: a \textbf{minimal analytic model} validated against closed forms to a fraction of a percent; \textbf{small trained networks} (MLPs, recurrent networks, an MNIST image generator) where the operators are measured in real weights; and \textbf{language models} (LoRA-specialised Qwen models, 0.5B locally and 7B on a compute cluster) where the claims are tested as signs under seed replication. Negative results are reported with the same prominence as confirmations; they include the failure of an internal pre-registered prediction, a null on emergent speciation that bounds the analogy, and the sensitivity analyses on the predictive test.
 
 \section*{The minimal model, and where its exactness ends}
 
 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*. In this minimal inheritance model the resampling step \textbf{is} the Wright--Fisher process --- the same equations, which we exploit as an engineering gate: our 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.
 
-The boundary of the exactness matters, and we measured it 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 \textbf{learning kernel} (a smoothing knob and a sharpening knob on the refit) reproduces both. The honest statement, used throughout: a real learner is 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 we 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. 1).
+The boundary of the exactness matters, and we measured it 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 \textbf{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 we 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. 1).
 
 \textbf{Table 1.} The dictionary. Each correspondence is stated with the level of support it currently has (exact = closed form in the minimal model; empirical = measured in trained systems; hypothesis = stated with a falsifier, untested or unconfirmed). The full claim-by-claim ledger with assumptions and known limits is SI Appendix, Table S1.
 
@@ -36,18 +36,18 @@ Genetic drift in a finite population & Training on finite samples of model outpu
 Immigration from a fixed source & Grounding with verified real data & Exact equilibrium; signs in RNN/MLP/VAE/MNIST \\[3pt]
 Muller's ratchet (asexual decay) & Irreversible arm of model collapse & Correspondence, scoped: applies to unrecoverable loss \\[3pt]
 Recombination / sexual reproduction & Model merging & Empirical at 0.5B--7B \\[3pt]
-Fisher--Muller effect & Merged specialists exceed every parent & Exact-model result; replicated in LLMs \\[3pt]
+Fisher--Muller effect & Merged specialists exceed every parent & Analytic model; replicated in LLMs \\[3pt]
-Outbreeding depression under epistasis & Merging entangled skills harms offspring & Exact-model (NK landscapes); hypothesis at LLM scale \\[3pt]
+Outbreeding depression under epistasis & Merging entangled skills harms offspring & Analytic model (NK landscapes); hypothesis at LLM scale \\[3pt]
-Mating systems / population structure & Who merges with whom (breadth of the parent pool) & Exact-model result; hypothesis for real populations \\[3pt]
+Mating systems / population structure & Who merges with whom (breadth of the parent pool) & Analytic model; hypothesis for real populations \\[3pt]
 Reproductive isolation (BDM incompatibilities) & Merge failure from functional conflict & Empirical (MLP + LLM tiers, conflict-associated); emergent form not observed \\[3pt]
-Selection on a fitness function & Verifier-anchored selection (``reality that can say no'') & Exact-model result (jointly necessary with sex and diversity) \\[3pt]
+Selection on a fitness function & Verifier-anchored selection (``reality that can say no'') & Analytic model (complementary with recombination and diversity in the tested society) \\[3pt]
 \hline\end{tabular}\end{center}\medskip
 
 \section*{Results}
 
 \subsection*{Grounding is immigration: cheap, with a floor}
 
-In the minimal model, grounding from a fixed real source is immigration into a drifting population, and the equilibrium diversity has a closed form our simulator matches exactly. The engineering headline is the \emph{magnitude}: a critical grounding fraction \texttt{g* \(\approx\) 0.05} retains most diversity indefinitely --- real data is cheap insurance. But the same analysis yields a floor the field's average-loss framing misses: an individual capability of rarity \texttt{p} survives only if the \emph{absolute} real-data budget satisfies \texttt{m\(\cdot\)p \(\gtrsim\) 1}. Protecting the rarest knowledge is priced per item, at cost \texttt{\(\propto\) 1/p}, and no affordable grounding fraction rescues the deepest tail --- that requires recombination (next section). In trained networks the \emph{sign} of the grounding response transfers everywhere we looked, with two honest 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. 1B), dry self-training collapses a convolutional VAE to one mode while \textasciitilde{}10\% grounding holds all thirty (the trained model needs roughly twice the exact-operator fraction --- the measured price of the estimator bias).
+In the minimal model, grounding from a fixed real source is immigration into a drifting population, and the equilibrium diversity has a closed form our 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 we 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). The engineering point survives the definition: verified real data is 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 we 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. 1B), dry self-training collapses a convolutional VAE to one mode while \textasciitilde{}10\% grounding holds all thirty (the trained model needs roughly twice the exact-operator fraction --- the measured price of the estimator bias).
 
 \begin{figure*}[p]\centering  % fig1
 \includegraphics[width=\textwidth,height=0.63\textheight,keepaspectratio]{figs/fig1_E2.pdf}\par\smallskip
@@ -57,7 +57,7 @@ In the minimal model, grounding from a fixed real source is immigration into a d
 
 \subsection*{Recombination: a conservation law, its operators, and offspring that exceed every parent}
 
-Merging is where the evolution-of-sex apparatus pays for itself, beginning with a result about the obvious operator. \textbf{Averaging is blending inheritance, and it cancels the benefit of multiple parents:} when a child is refit to the \emph{mean of its parents' output distributions}, the expected mass on any rare item is conserved at the single-parent level --- in the rare-item regime the 1/K dilution of averaging exactly cancels the union gain of K parents, so adding parents cannot help. An operator that keeps, per item, its strongest source (which presupposes a verifier or oracle to say which) realises the union. That statement is exact for those operators in the minimal model. The practically important operators --- \textbf{weight averaging} (a nonlinear network's weight-mean does not compute its parents' output-mean) and \textbf{routing among intact specialists} (different storage and inference budgets from a single child) --- are its empirical cousins, and the measured bridge is a \textbf{headroom rule}: in language models, union-preserving operators beat the weight-average in proportion to how far that average is from the best attainable. On easy tasks a capable base's average is already at ceiling and refinements add nothing; on hard tasks the average dilutes a fragile specialist below even the best single parent and routing wins by a wide margin (Fig. 6A--B).
+Merging is where the evolution-of-sex apparatus pays for itself, beginning with a result about the obvious operator, stated with its assumptions. \textbf{Proposition (blending inheritance, rare-item regime).} Let K parents independently retain a rare item (mass \texttt{p} when retained), and let the child draw \texttt{n} samples either from one parent chosen at random or from the \emph{mean of the parents' output distributions}. Expected item mass is identical under the two schemes; and in the rare-item regime \texttt{n\(\cdot\)p/K \(\ll\) 1}, where per-item survival is first-order in sampled mass, expected \emph{survival} is also identical --- the 1/K dilution of averaging cancels the K-parent union gain to first order, so in this regime adding parents through the output-mean does not increase expected tail retention. Two boundaries: outside that regime, survival is a convex function of mixed mass, so the variance reduction from averaging can \emph{reduce} extinction relative to a randomly chosen single parent --- the cancellation is a first-order result about rare items, not a universal impossibility; and the contrasting union operator (keep each item's strongest source, then renormalise --- which itself redistributes mass, and presupposes a verifier or oracle to identify the strongest source) increases expected retention with K in all regimes in the minimal model. The practically important operators --- \textbf{weight averaging} (a nonlinear network's weight-mean does not compute its parents' output-mean) and \textbf{routing among intact specialists} (different storage and inference budgets from a single child) --- are its empirical cousins, and the measured bridge is a \textbf{headroom rule}, stated qualitatively: in language models, union-preserving operators beat the weight-average where that average falls short of attainable performance, and add nothing where it does not (easy-versus-hard contrasts at two scales; a quantitative form of the relationship is untested). On easy tasks a capable base's average is already at ceiling and refinements add nothing; on hard tasks the average dilutes a fragile specialist below even the best single parent and routing wins by a wide margin (Fig. 6A--B).
 
 The generative payoff is the \textbf{Fisher--Muller effect}: recombination assembles, in one offspring, complementary variants that arose in different lineages, producing a genotype fitter than any parent. In the multi-locus model, sexual merging of decorrelated specialists climbs to the global optimum --- a genotype no parent held --- while the best single parent and the blended average both plateau below (Fig. 2). In real language models the signature replicates under seed replication: merges of three LoRA specialists beat every parent overall (decisively at 7B: 0.87 vs 0.77), and on the sharper worst-family metric the merged models are the only ones competent everywhere, in every seed (Fig. 6A).
 
@@ -76,9 +76,9 @@ Sex has risks and, for AI, an unfair advantage --- both quantified on rugged (ep
 \caption{Rugged (epistatic) landscapes: risk, remedy, and structure. (A, top) Outbreeding depression: blind recombination of specialists drops offspring below their parents, worsening with ruggedness; the optimal recombination rate shrinks as skills entangle. (B, middle) Directed sex --- unbounded parents, chosen mates, verifier-screened offspring --- converts the catastrophe into a reliable gain at every ruggedness. (C, bottom) Mating structure: wide (promiscuous) mixing maximises the population mean but monotonically destroys diversity; the champion-optimal mate-pool breadth narrows as the landscape roughens.}\label{fig3}
 \end{figure*}
 
-\subsection*{The society: grounding, sex, and diversity are jointly necessary}
+\subsection*{The society: grounding, recombination, and diversity make complementary contributions}
 
-Composing the operators closes the loop (Fig. 4). A finite population of agents evolves on a rugged NK landscape, with selection acting on a grounded score --- \texttt{g}\(\cdot\)true-fitness + (1−g)\(\cdot\)conformity to the population's own consensus, the analogue of training on the crowd's output. A four-arm ablation separates the failure modes: the \textbf{full} system (grounding + directed recombination + diversity-preserving selection) climbs to near the global optimum while keeping its specialists; remove \emph{grounding} and the population converges confidently on an unfit consensus (self-consumption); remove \emph{sex} and it strands on local optima; remove \emph{diversity} and it converges prematurely to a worse answer. Each removal fails \emph{differently} --- the operators are jointly necessary, which is the system-level claim the single-operator results build toward. At language-model scale this composed loop remains unbuilt; it is the paper's largest stated gap.
+Composing the operators (Fig. 4) requires one definitional distinction first. In the inheritance model, grounding is \textbf{grounded inheritance}: external samples added to the reproduction process (the data channel). In the society model, grounding is \textbf{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 --- both couple the lineage to a non-drifting external signal --- but they are different operators, and we 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) 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  % fig4
 \includegraphics[width=\textwidth,height=0.98\textheight,keepaspectratio]{figs/fig4_E11.pdf}\par\smallskip
@@ -89,9 +89,9 @@ Composing the operators closes the loop (Fig. 4). A finite population of agents
 
 Recombination presupposes compatible parents. In biology, lineages pushed far enough apart become separate species --- \textbf{reproductive isolation} --- through Bateson--Dobzhansky--Muller incompatibilities: changes harmless on their own background but deleterious in combination. A merged model is exactly the exposed hybrid. We built the analytic model (Fig. 5A): hybrid fitness tracks the parents while compatible, then peels off and crashes below the ancestor; the isolation cliff arrives earlier the denser the incompatibilities; and the incompatibility \emph{count} snowballs quadratically with divergence (17) --- noting that a super-linear count does not by itself entail a sharp performance cliff without the count-to-effect-size link, which the analytic model supplies under its assumptions and any neural test must establish separately.
 
-In trained networks, the claim must survive a known alternative: merge barriers between independently trained networks are famously \emph{coordinate artefacts}, removable by re-aligning hidden units (23), and richer symmetry groups remove more (24). We therefore aligned modulo the \textbf{complete} function-preserving unit symmetry group of the architecture tested (permutation composed with per-unit positive rescaling, for plain ReLU MLPs) and decomposed the barrier (Fig. 5B): two networks trained from different initialisations on the \emph{same} task have a barrier that alignment removes essentially entirely (residual \(\approx\) 0.001, the aligned merge performing at parent level) --- coordinate, not functional; two networks trained on \emph{conflicting} label maps have a barrier the full group leaves intact (0.502 \(\rightarrow\) 0.497), with the merged model functionally dead --- and this cannot be an alignment failure, because the same aligner succeeded on the control. Sweeping conflict traces the cliff as hybrid fitness, 0.97 \(\rightarrow\) 0.03. Two scope notes: exact recovery of a permuted-and-rescaled copy validates a special case rather than global optimality, so the removable share is a lower bound and the residual an upper bound; and the conflict floor itself is information-theoretic --- no single model can satisfy contradictory conventions (SI Appendix, Proposition S2) --- with the framework's role being the \emph{structure around it}: which divergences generate conflict, and what moves the cliff.
+In trained networks, the claim must survive a known alternative: merge barriers between independently trained networks are famously \emph{coordinate artefacts}, removable by re-aligning hidden units (23), and richer symmetry groups remove more (24). We therefore aligned under the composition of permutation matching and exact per-unit rescaling (the unit symmetry group of plain ReLU MLPs, as the search space) and decomposed the barrier (Fig. 5B): two networks trained from different initialisations on the \emph{same} task have a barrier that this alignment removes essentially entirely (residual \(\approx\) 0.001, the aligned merge performing at parent level) --- coordinate, not functional; two networks trained on \emph{conflicting} label maps have a barrier the same alignment leaves largely unchanged (0.502 \(\rightarrow\) 0.497), with the merged model functionally dead. The tested alignment removes the same-task barrier but leaves the conflict-associated barrier intact --- supporting a functional-conflict interpretation without proving optimal alignment: exact recovery of a permuted-and-rescaled copy validates a special case, so the removable share is a lower bound and the residual an upper bound. Sweeping conflict traces the cliff as hybrid fitness, 0.97 \(\rightarrow\) 0.03. The conflict floor itself is information-theoretic --- no single model can satisfy contradictory conventions (SI Appendix, Proposition S2) --- with the framework's role being the \emph{structure around it}: which divergences generate conflict, and what moves the cliff.
 
-The sharpest honesty comes from the pre-registered \textbf{emergent test}: true BDM incompatibilities are emergent (each lineage's changes harmless alone), so we let children diverge with \emph{no conflicting signal anywhere} --- complementary class specialists, and divergent input conventions --- to 6.4\(\times\) the base training. \textbf{No isolation emerged} (residual 0.000 throughout); instead the merge \emph{rescued} the two catastrophically-forgetting specialists (parents \(\approx\) 0.50, merge \(\approx\) 0.955 --- a sustained Fisher--Muller rescue). The same double result appears at the language-model tier (Fig. 5C): conflicting conventions produce \textbf{function-specific} hybrid breakdown (the merge scores below both parents on the conflicted function, while a budget-controlled design shows the disjoint skills merge unharmed), and over-training disjoint specialists 1\(\rightarrow\)12 epochs produces no isolation at all --- the merge improves. Across every tier tested, \textbf{isolation had to be provoked by functional conflict; specialisation alone did not speciate} --- a bound on the analogy that sharpens the design rule: what breaks merging is conflicting conventions on shared circuitry, not divergence per se.
+The strongest constraint comes from the pre-registered \textbf{emergent test}: true BDM incompatibilities are emergent (each lineage's changes harmless alone), so we let children diverge with \emph{no conflicting signal anywhere} --- complementary class specialists, and divergent input conventions --- to 6.4\(\times\) the base training. \textbf{No isolation emerged} (residual 0.000 throughout); instead the merge \emph{rescued} the two catastrophically-forgetting specialists (parents \(\approx\) 0.50, merge \(\approx\) 0.955 --- a sustained Fisher--Muller rescue). The same double result appears at the language-model tier (Fig. 5C): conflicting conventions produce \textbf{function-specific} hybrid breakdown (the merge scores below both parents on the conflicted function, while a budget-controlled design shows the disjoint skills merge unharmed), and over-training disjoint specialists 1\(\rightarrow\)12 epochs produces no isolation at all --- the merge improves. Across every tier tested, \textbf{isolation had to be provoked by functional conflict; specialisation alone did not speciate} --- a bound on the analogy that sharpens the design rule: what breaks merging is conflicting conventions on shared circuitry, not divergence per se.
 
 \begin{figure*}[p]\centering  % fig5
 \includegraphics[width=\textwidth,height=0.42\textheight,keepaspectratio]{figs/fig5_E12.pdf}\par\smallskip
@@ -102,9 +102,9 @@ The sharpest honesty comes from the pre-registered \textbf{emergent test}: true
 
 \subsection*{A controlled predictive test: functional conflict, measured pre-merge, predicts merge damage}
 
-The framework's prediction-level claim was put to a designed test (Fig. 6C). Thirty-nine parent pairs (13 conditions \(\times\) 3 seeds; rows are not independent --- parents share task-data seeds across conditions --- so all inference is condition-clustered) 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: \textbf{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 (21), LoRA-delta cosine and distance, and a cross-task performance baseline. The pre-registered outcome is the merge penalty against oracle parent potential (the hybrid-load analogue), also reported against best- and mean-parent references because the predictor ordering is sensitive to that choice.
+The framework's prediction-level claim was put to a designed test (Fig. 6C). 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 we 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: \textbf{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 (21), LoRA-delta cosine and distance, and a cross-task performance baseline. The pre-registered outcome is the merge penalty against oracle parent potential (the hybrid-load analogue), also reported against best- and mean-parent references because the predictor ordering is sensitive to that choice.
 
-The supported conclusion, stated conditionally: \textbf{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, and only these baselines were tested. Two further results earn their place by tempering: 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 \textbf{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.
+The supported conclusion, stated conditionally: \textbf{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 \textbf{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.
 
 \begin{figure*}[p]\centering  % fig6
 \includegraphics[width=\textwidth,height=0.42\textheight,keepaspectratio]{figs/fig6_llm_seeds.pdf}\par\smallskip
@@ -113,11 +113,27 @@ The supported conclusion, stated conditionally: \textbf{across this controlled g
 \caption{The language-model tier. (A, top) Seed-replicated recombination claims (fixed test sets, training seed varied, 95\% CI): merges beat every specialist; union-preserving routing and directed offspring selection beat the blend in every seed on headroom tasks, including one catastrophic blend failure they avoided. (B, middle) The headroom rule at 7B on hard (unsaturated) tasks: the weight-average dilutes a fragile specialist below the best single parent; routing preserves it. (C, bottom) The controlled predictive test: across a task grid with conflict, compatible-overlap, and duration axes decorrelated by construction, pre-merge functional disagreement predicts merge penalty (held-out $\rho \approx 0.4$) while weight-geometry baselines show no detectable association; paired predictor differences are not individually significant.}\label{fig6}
 \end{figure*}
 
+\textbf{Table 2.} Headline quantitative results with sample sizes, uncertainty, and outcome definitions (full per-experiment tables and falsifier status in SI Appendix and per-experiment documentation).
+
+\medskip\noindent\begin{center}\footnotesize
+\begin{tabular}{p{0.230\textwidth} p{0.230\textwidth} p{0.230\textwidth} p{0.230\textwidth}}
+\hline
+Result & Setting / n & Outcome definition & Headline \\ \hline
+Closed-form validation & Analytic tier; 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]
+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]
+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]
+\hline\end{tabular}\end{center}\medskip
+
 \section*{Discussion}
 
-\textbf{Design rules.} Read as engineering, the results compress into rules an operator of a model population can apply. \emph{Ground every generation} in verified reality --- a few percent retains most diversity --- but price the rarest capabilities individually (\texttt{m\(\cdot\)p \(\gtrsim\) 1}) and use recombination, not grounding, to reach the deep tail. \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 the society result shows grounding, recombination, and diversity are jointly necessary. \emph{Before merging, measure functional conflict} --- cheap, pre-merge, and in our controlled setting predictive where weight distance was not; and expect specialisation alone to be merge-safe, with conflicting conventions on shared circuitry as the thing to detect and avoid.
+\textbf{Design rules.} Read as engineering, the results compress into rules an operator of a model population can apply. \emph{Ground every generation} in verified reality --- a few percent retained most diversity in our 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 our 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 we tested, what broke merging was conflicting conventions on shared circuitry, which is the thing to detect.
 
-\textbf{What is borrowed and what is ours.} The diagnosis --- collapse as drift --- is prior art (6--9), as 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 (1, 19, 20), that model populations can climb (2--5), and that merge success admits ML-native predictors (21, 22). Ours 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 jointly-necessary society; model speciation as a named, tested question, with the coordinate-versus-functional decomposition under a complete symmetry group and the emergent null that bounds it; and the controlled predictive test with its controls. We claim the framework generated these measurements and experiments; we do not claim their outcomes validate a uniquely population-genetic mechanism, and one refinement it proposed was not supported.
+\textbf{What is borrowed and what is ours.} The diagnosis --- collapse as drift --- is prior art (6--9), as 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 (1, 19, 20), that model populations can climb (2--5), and that merge success admits ML-native predictors (21, 22). Ours 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. We claim the framework generated these measurements and experiments; we do not claim their outcomes validate a uniquely population-genetic mechanism, and one refinement it proposed was not supported.
 
 \textbf{Limits and open problems.} The demonstrations are deliberately small: exact where small is a virtue, sign-level and seed-replicated at the language-model tier, on constructed task families with a trivially separable router and one model lineage (Qwen, 0.5B--7B). The composed society has not been built at language-model scale. The predictive test's next bars, in order of value: generalisation to \emph{unfamiliar} conflict structures and real task pairs; a demonstrably better \emph{budget-matched} merging decision; then scale replication. Beyond engineering, the framework's hardest open problem is the fitness function itself: selection optimises what is measured, and for knowledge systems the persuasive and the true compete --- grounding against a reality that can refuse is the only anchor we trust, and institutionalising that anchor (verification, replication, and challenge among models) is the society-level problem we pose but do not solve. What biology receives in return is a new model system: populations of learners where every genotype, environment, and mating decision is observable and manipulable --- where the evolution of sex can be studied with interventions (unbounded parents, offspring preview, directed mating) that no living system permits.
 
@@ -125,7 +141,7 @@ The supported conclusion, stated conditionally: \textbf{across this controlled g
 
 \textbf{Analytic tier.} Pure NumPy/SciPy Wright--Fisher simulator over \texttt{K}-item distributions (knowledge as \texttt{p\_t}; Zipf-tailed truth \texttt{p*}; drift--grounding--refit generations), extended with a learning kernel (smoothing/sharpening refit), multi-locus genotypes on additive and Kauffman NK landscapes, n-parent crossover, and finite-population society loops. All parameters live in per-experiment YAML configs; every run derives all randomness from one master seed (\texttt{SeedSequence.spawn}) and is bitwise reproducible; scientific-validation tests assert the closed forms (heterozygosity decay, immigration equilibrium, closed-form union) to <0.5\% and run in CI with 151 further correctness tests.
 
-\textbf{Neural tier.} Trained-network experiments realise the same abstractions with an exact oracle: histogram/RNN/MLP/VAE generators on a synthetic mode universe (the histogram model reduces the harness exactly to the analytic tier --- the bridge gate), and a convolutional VAE on MNIST with a frozen CNN oracle (98.5\% mode accuracy; confusion matrix recorded as the measurement floor). Speciation experiments fork no-BatchNorm MLPs (784--512--512--10) from a shared base, weight-average, and measure linear-mode-connectivity error barriers before and after alignment; alignment composes deterministic Git Re-Basin permutation matching with exact per-unit scale canonicalisation (the complete unit symmetry group for this class), gated by exact recovery of a permuted-and-rescaled copy.
+\textbf{Neural tier.} Trained-network experiments realise the same abstractions with an exact oracle: histogram/RNN/MLP/VAE generators on a synthetic mode universe (the histogram model reduces the harness exactly to the analytic tier --- the bridge gate), and a convolutional VAE on MNIST with a frozen CNN oracle (98.5\% mode accuracy; confusion matrix recorded as the measurement floor). Speciation experiments fork no-BatchNorm MLPs (784--512--512--10) from a shared base, weight-average, and measure linear-mode-connectivity error barriers before and after alignment; alignment composes deterministic Git Re-Basin permutation matching with exact per-unit scale canonicalisation (the unit symmetry group of this class, as the alignment search space; control recovery does not establish global optimality), gated by exact recovery of a permuted-and-rescaled copy.
 
 \textbf{Language-model tier.} LoRA specialists (rank 16) on procedurally generated task families with an exact-match verifier, on frozen Qwen2.5-Instruct bases (0.5B on one 16 GB GPU; 7B on one L40S). Operators: weight-space merges (soup/TIES via adapter arithmetic), per-input routing, and Dirichlet-sampled offspring populations screened on held-out validation splits. Multi-seed protocols fix the test sets and vary the training seed. The predictive test computes all predictors pre-merge (generation confidence from token log-probabilities; base-model gradient cosines; exact r-space LoRA-delta geometry) and evaluates merges on held-out tests; robust statistics (condition-clustered bootstrap, paired predictor contrasts, leave-one-condition-out prediction, multi-reference outcomes) are produced by a committed script. Statistical, per-seed reproducibility is documented for GPU tiers.
 

paper/pnas/main.md

@@ -13,8 +13,9 @@ genetic drift. This paper imports the other half of population genetics: the bio
 reproduction. It treats model merging as recombination, real data as immigration, and merge failure as
 reproductive isolation, and tests each correspondence in simulations, small neural networks, and
 language models. The framework yields design rules — when to average models, when to keep them
-separate, how much real data suffices — and a first controlled test showing that measured functional
+separate, how much real data suffices — and a controlled small-model test in which pre-merge
-conflict, not weight distance, predicts when merging fails.
+functional disagreement predicted merge damage, motivating further comparison with weight-space
+measures.
 
 ## Abstract
 
@@ -24,18 +25,20 @@ evolutionary theory. Here we treat multigenerational model populations as system
 diversity, and compatibility must be managed, and transfer the quantitative apparatus of the evolution
 of sex. We take as settled that training on model output is genetic drift (model collapse). In a
 minimal inheritance model that is exactly Wright–Fisher — and measurably Wright–Fisher-plus-bias in
-trained networks — we derive and test the remedies: grounding as immigration, with a critical
+trained networks — we derive and test the remedies: grounding as immigration, where a real-data
-real-data fraction far below one but a per-capability floor that leaves the rarest knowledge
+fraction far below one retained most equilibrium diversity in the tested settings, with a
-unrescuable; recombination, where averaging parents' output distributions exactly cancels the benefit
+per-capability observation floor that makes the rarest knowledge expensive under unstratified
-of multiple parents while union-preserving operators realise it; the Fisher–Muller effect, with merged
+sampling; recombination, where refitting to the mean of parents' output distributions cancels the
-language-model specialists exceeding every parent in replicated experiments; outbreeding depression on
+multi-parent gain to first order in the rare-item regime while union-preserving operators realise it;
-rugged task landscapes, converted into reliable gains by directed, offspring-screened recombination;
+the Fisher–Muller effect, with merged language-model specialists exceeding every parent in replicated
-and population structure, where the optimal mating breadth shrinks as skills entangle. Sex has a
+experiments; outbreeding depression on rugged task landscapes, converted into reliable gains by
-limit: we introduce model speciation — merge failure as reproductive isolation — and show in trained
+directed, offspring-screened recombination; and population structure, where the optimal mating breadth
-networks that a merge barrier surviving the full function-preserving symmetry group tracks functional
+shrinks as skills entangle. Sex has a limit: we introduce model speciation — merge failure as
-conflict, that isolation did not emerge from compatible specialisation, and, in a controlled
+reproductive isolation — and show in trained networks that a merge barrier remaining after
-predictive test, that pre-merge functional disagreement predicts merge damage where weight-geometry
+permutation-and-rescaling alignment tracks functional conflict, that isolation did not emerge from
-baselines do not. We state precisely what is exact, what is measured, and what remains hypothesis.
+compatible specialisation, and, in a controlled predictive test, that pre-merge functional
+disagreement predicted merge damage while the tested weight-geometry baselines showed no detectable
+association. We state precisely what is exact, what is measured, and what remains hypothesis.
 
 ---
 
@@ -67,17 +70,20 @@ and **prediction** (the framework forecasts an unmeasured outcome). The paper is
 first; makes concrete progress on the second — separating merge failures that are coordinate artefacts
 from those that are functional; and reports a first, bounded step on the third — a controlled
 predictive test in which pre-merge functional-disagreement measures, chosen by the framework,
-predicted merge damage on a constructed task grid while weight-geometry baselines did not.
+predicted merge damage on a constructed task grid while the tested weight-geometry baselines showed
+no detectable association.
 
 The correspondences we develop, summarised in Table 1: single-teacher retraining is **asexual
-reproduction**, and the irreversible arm of its decay corresponds to **Muller's ratchet** (10) — once
+reproduction**, and the irreversible arm of its decay shares the defining consequence of **Muller's
-every copy of a rare capability is gone from all parents and sources, no recombination can rebuild it,
+ratchet** (10) — once every copy of a rare capability is gone from all parents and sources, no
-which is precisely why remedies must act before fixation-by-loss. Injecting verified real data is
+recombination can rebuild it, which is why remedies must act before fixation-by-loss (a consequence-
+level correspondence: the minimal model lacks the ratchet's recurrent deleterious-mutation mechanism,
+so irreversible loss alone does not identify that specific mechanism). Injecting verified real data is
 **immigration** from a non-drifting source (11–13). Model merging is **recombination**, and its
 celebrated payoff — a merged model exceeding every parent — is the **Fisher–Muller effect** (14, 15).
-Merging entangled skills courts **outbreeding depression**; screening many candidate merges is a form
+Merging entangled skills courts **outbreeding depression**; screening many candidate merges is
-of **directed sex** with no biological analogue; restricting who merges with whom is **population
+engineered recombination with unusually flexible parent choice and pre-deployment screening (we use
-structure**. And merging's hard limit — models too diverged in function to combine — is **reproductive
+the shorthand **directed sex**); restricting who merges with whom is **population structure**. And merging's hard limit — models too diverged in function to combine — is **reproductive
 isolation**, for which the Bateson–Dobzhansky–Muller theory of incompatibilities (16, 17) supplies the
 structure. The nearest precursor to this programme reads sex as an algorithm for mixability in the
 theory of computation (18), pre-dating model merging; the model-merging literature itself has strong
@@ -88,10 +94,9 @@ We support the framework at three tiers of evidence, in ascending realism and de
 **minimal analytic model** validated against closed forms to a fraction of a percent; **small trained
 networks** (MLPs, recurrent networks, an MNIST image generator) where the operators are measured in
 real weights; and **language models** (LoRA-specialised Qwen models, 0.5B locally and 7B on a compute
-cluster) where the claims are tested as signs under seed replication. Throughout, we report negative
+cluster) where the claims are tested as signs under seed replication. Negative results are reported with the same prominence as confirmations; they include the failure of
-and tempering results with the same prominence as confirmations: they include the failure of an
+an internal pre-registered prediction, a null on emergent speciation that bounds the analogy, and the
-internal pre-registered prediction, a null on emergent speciation that bounds the analogy, and the
+sensitivity analyses on the predictive test.
-sensitivity analyses that temper the predictive test.
 
 ## The minimal model, and where its exactness ends
 
@@ -109,8 +114,8 @@ approximation, optimisation noise, and inductive bias, and when trained networks
 exact drift null they deviate in *opposite, architecture-specific* directions: a smoothing recurrent
 network resists collapse (keeping spurious variants alive), while a sharpening image generator
 accelerates it. A one-parameter **learning kernel** (a smoothing knob and a sharpening knob on the
-refit) reproduces both. The honest statement, used throughout: a real learner is Wright–Fisher *plus a
+refit) reproduces both. Throughout, a real learner is therefore treated as Wright–Fisher *plus a signed, measurable
-signed, measurable estimator bias* — and the drift signs (rare-first loss; the grounding response)
+estimator bias* — and the drift signs (rare-first loss; the grounding response)
 survived that bias in every architecture we 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. 1).
@@ -126,25 +131,33 @@ known limits is SI Appendix, Table S1.
 | Immigration from a fixed source | Grounding with verified real data | Exact equilibrium; signs in RNN/MLP/VAE/MNIST |
 | Muller's ratchet (asexual decay) | Irreversible arm of model collapse | Correspondence, scoped: applies to unrecoverable loss |
 | Recombination / sexual reproduction | Model merging | Empirical at 0.5B–7B |
-| Fisher–Muller effect | Merged specialists exceed every parent | Exact-model result; replicated in LLMs |
+| Fisher–Muller effect | Merged specialists exceed every parent | Analytic model; replicated in LLMs |
-| Outbreeding depression under epistasis | Merging entangled skills harms offspring | Exact-model (NK landscapes); hypothesis at LLM scale |
+| Outbreeding depression under epistasis | Merging entangled skills harms offspring | Analytic model (NK landscapes); hypothesis at LLM scale |
-| Mating systems / population structure | Who merges with whom (breadth of the parent pool) | Exact-model result; hypothesis for real populations |
+| Mating systems / population structure | Who merges with whom (breadth of the parent pool) | Analytic model; hypothesis for real populations |
 | Reproductive isolation (BDM incompatibilities) | Merge failure from functional conflict | Empirical (MLP + LLM tiers, conflict-associated); emergent form not observed |
-| Selection on a fitness function | Verifier-anchored selection ("reality that can say no") | Exact-model result (jointly necessary with sex and diversity) |
+| Selection on a fitness function | Verifier-anchored selection ("reality that can say no") | Analytic model (complementary with recombination and diversity in the tested society) |
 
 ## Results
 
 ### Grounding is immigration: cheap, with a floor
 
 In the minimal model, grounding from a fixed real source is immigration into a drifting population,
-and the equilibrium diversity has a closed form our simulator matches exactly. The engineering
+and the equilibrium diversity has a closed form our simulator matches exactly. That equilibrium is
-headline is the *magnitude*: a critical grounding fraction `g* ≈ 0.05` retains most diversity
+*smooth* in the grounding fraction — there is no phase transition in aggregate diversity — so the
-indefinitely — real data is cheap insurance. But the same analysis yields a floor the field's
+practical number is an operational threshold, and we define it as such: under the tested population
-average-loss framing misses: an individual capability of rarity `p` survives only if the *absolute*
+size and Zipf source distribution, `g ≈ 0.05` retained most (≥95%) of equilibrium diversity
-real-data budget satisfies `m·p ≳ 1`. Protecting the rarest knowledge is priced per item, at cost
+indefinitely, with the required fraction depending on sample size, source distribution, and the
-`∝ 1/p`, and no affordable grounding fraction rescues the deepest tail — that requires recombination
+chosen retention target (dependencies in SI). The engineering point survives the definition: verified
-(next section). In trained networks the *sign* of the grounding response transfers everywhere we
+real data is cheap insurance at fractions far below one. But the same analysis yields a floor the field's
-looked, with two honest deviations, both traced to the estimator bias above: sharp thresholds soften,
+average-loss framing misses: under unstratified sampling from the source, a capability of rarity `p`
+appears in a real-data batch of size `m` with probability `1 − e^{−m·p}`, so `m·p ≈ 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 `∝ 1/p`; targeted
+or stratified sampling changes that cost, and recombination can recover rare capabilities *that are
+still retained across complementary parents* (next section). In trained networks the *sign* of the grounding response transfers everywhere we
+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. 1B), dry self-training collapses a convolutional VAE to one
 mode while ~10% grounding holds all thirty (the trained model needs roughly twice the exact-operator
@@ -155,17 +168,25 @@ fraction — the measured price of the estimator bias).
 ### Recombination: a conservation law, its operators, and offspring that exceed every parent
 
 Merging is where the evolution-of-sex apparatus pays for itself, beginning with a result about the
-obvious operator. **Averaging is blending inheritance, and it cancels the benefit of multiple
+obvious operator, stated with its assumptions. **Proposition (blending inheritance, rare-item
-parents:** when a child is refit to the *mean of its parents' output distributions*, the expected mass
+regime).** Let K parents independently retain a rare item (mass `p` when retained), and let the child
-on any rare item is conserved at the single-parent level — in the rare-item regime the 1/K dilution of
+draw `n` samples either from one parent chosen at random or from the *mean of the parents' output
-averaging exactly cancels the union gain of K parents, so adding parents cannot help. An operator that
+distributions*. Expected item mass is identical under the two schemes; and in the rare-item regime
-keeps, per item, its strongest source (which presupposes a verifier or oracle to say which) realises
+`n·p/K ≪ 1`, where per-item survival is first-order in sampled mass, expected *survival* is also
-the union. That statement is exact for those operators in the minimal model. The practically important
+identical — the 1/K dilution of averaging cancels the K-parent union gain to first order, so in this
+regime adding parents through the output-mean does not increase expected tail retention. Two
+boundaries: outside that regime, survival is a convex function of mixed mass, so the variance
+reduction from averaging can *reduce* extinction relative to a randomly chosen single parent — the
+cancellation is a first-order result about rare items, not a universal impossibility; and the
+contrasting union operator (keep each item's strongest source, then renormalise — which itself
+redistributes mass, and presupposes a verifier or oracle to identify the strongest source) increases
+expected retention with K in all regimes in the minimal model. The practically important
 operators — **weight averaging** (a nonlinear network's weight-mean does not compute its parents'
 output-mean) and **routing among intact specialists** (different storage and inference budgets from a
-single child) — are its empirical cousins, and the measured bridge is a **headroom rule**: in language
+single child) — are its empirical cousins, and the measured bridge is a **headroom rule**, stated qualitatively: in language models,
-models, union-preserving operators beat the weight-average in proportion to how far that average is
+union-preserving operators beat the weight-average where that average falls short of attainable
-from the best attainable. On easy tasks a capable base's average is already at ceiling and refinements
+performance, and add nothing where it does not (easy-versus-hard contrasts at two scales; a
+quantitative form of the relationship is untested). On easy tasks a capable base's average is already at ceiling and refinements
 add nothing; on hard tasks the average dilutes a fragile specialist below even the best single parent
 and routing wins by a wide margin (Fig. 6A–B).
 
@@ -196,18 +217,23 @@ structured-population search, mapped onto merging populations.
 
 *(FIG:fig3)*
 
-### The society: grounding, sex, and diversity are jointly necessary
+### The society: grounding, recombination, and diversity make complementary contributions
 
-Composing the operators closes the loop (Fig. 4). A finite population of agents evolves on a rugged NK
+Composing the operators (Fig. 4) requires one definitional distinction first. In the inheritance
-landscape, with selection acting on a grounded score — `g`·true-fitness + (1−g)·conformity to the
+model, grounding is **grounded inheritance**: external samples added to the reproduction process (the
-population's own consensus, the analogue of training on the crowd's output. A four-arm ablation
+data channel). In the society model, grounding is **grounded evaluation**: selection weights true
-separates the failure modes: the **full** system (grounding + directed recombination +
+fitness against conformity to the population's own consensus — `g`·true-fitness + (1−g)·conformity —
-diversity-preserving selection) climbs to near the global optimum while keeping its specialists;
+the analogue of scoring models by the crowd's approval (the fitness channel). These are related design
-remove *grounding* and the population converges confidently on an unfit consensus (self-consumption);
+ideas — both couple the lineage to a non-drifting external signal — but they are different operators,
-remove *sex* and it strands on local optima; remove *diversity* and it converges prematurely to a
+and we name them separately. In the tested society (a finite agent population on a rugged NK
-worse answer. Each removal fails *differently* — the operators are jointly necessary, which is the
+landscape), a four-arm ablation separates the failure modes: the full system (grounded evaluation +
-system-level claim the single-operator results build toward. At language-model scale this composed
+directed recombination + diversity-preserving selection) climbs to near the global optimum while
-loop remains unbuilt; it is the paper's largest stated gap.
+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 *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.
 
 *(FIG:fig4)*
 
@@ -225,21 +251,22 @@ test must establish separately.
 
 In trained networks, the claim must survive a known alternative: merge barriers between independently
 trained networks are famously *coordinate artefacts*, removable by re-aligning hidden units (23), and
-richer symmetry groups remove more (24). We therefore aligned modulo the **complete**
+richer symmetry groups remove more (24). We therefore aligned under the composition of
-function-preserving unit symmetry group of the architecture tested (permutation composed with per-unit
+permutation matching and exact per-unit rescaling (the unit symmetry group of plain ReLU MLPs, as the
-positive rescaling, for plain ReLU MLPs) and decomposed the barrier (Fig. 5B): two networks trained
+search space) and decomposed the barrier (Fig. 5B): two networks trained from different
-from different initialisations on the *same* task have a barrier that alignment removes essentially
+initialisations on the *same* task have a barrier that this alignment removes essentially entirely
-entirely (residual ≈ 0.001, the aligned merge performing at parent level) — coordinate, not
+(residual ≈ 0.001, the aligned merge performing at parent level) — coordinate, not functional; two
-functional; two networks trained on *conflicting* label maps have a barrier the full group leaves
+networks trained on *conflicting* label maps have a barrier the same alignment leaves largely
-intact (0.502 → 0.497), with the merged model functionally dead — and this cannot be an alignment
+unchanged (0.502 → 0.497), with the merged model functionally dead. The tested alignment removes the
-failure, because the same aligner succeeded on the control. Sweeping conflict traces the cliff as
+same-task barrier but leaves the conflict-associated barrier intact — supporting a functional-conflict
-hybrid fitness, 0.97 → 0.03. Two scope notes: exact recovery of a permuted-and-rescaled copy validates
+interpretation without proving optimal alignment: exact recovery of a permuted-and-rescaled copy
-a special case rather than global optimality, so the removable share is a lower bound and the residual
+validates a special case, so the removable share is a lower bound and the residual an upper bound.
-an upper bound; and the conflict floor itself is information-theoretic — no single model can satisfy
+Sweeping conflict traces the cliff as hybrid fitness, 0.97 → 0.03. The conflict floor itself is
-contradictory conventions (SI Appendix, Proposition S2) — with the framework's role being the
+information-theoretic — no single model can satisfy contradictory conventions (SI Appendix,
-*structure around it*: which divergences generate conflict, and what moves the cliff.
+Proposition S2) — with the framework's role being the *structure around it*: which divergences
+generate conflict, and what moves the cliff.
 
-The sharpest honesty comes from the pre-registered **emergent test**: true BDM incompatibilities are
+The strongest constraint comes from the pre-registered **emergent test**: true BDM incompatibilities are
 emergent (each lineage's changes harmless alone), so we let children diverge with *no conflicting
 signal anywhere* — complementary class specialists, and divergent input conventions — to 6.4× the base
 training. **No isolation emerged** (residual 0.000 throughout); instead the merge *rescued* the two
@@ -257,8 +284,9 @@ on shared circuitry, not divergence per se.
 ### A controlled predictive test: functional conflict, measured pre-merge, predicts merge damage
 
 The framework's prediction-level claim was put to a designed test (Fig. 6C). Thirty-nine parent pairs
-(13 conditions × 3 seeds; rows are not independent — parents share task-data seeds across conditions —
+(13 conditions × 3 seeds; rows are not independent — parents share task-data seeds across conditions,
-so all inference is condition-clustered) span three axes decorrelated by construction: *conflict*
+so inference is condition-clustered, and because shared seeds also couple rows *across* conditions we
+report per-seed and leave-one-seed-out sensitivity alongside) span three axes decorrelated by construction: *conflict*
 (contradictory conventions on shared prompts, private budgets fixed), *compatible overlap* (the same
 shared prompts under the same convention — overlap and volume without conflict), and *duration* (weight
 divergence with zero conflict). Before merging, six predictors are computed: **confidence-weighted
@@ -274,8 +302,11 @@ The supported conclusion, stated conditionally: **across this controlled grid, p
 disagreement predicted merge penalties (clustered bootstrap CIs excluding zero; held-out
 leave-one-condition-out ρ ≈ 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, and only these baselines were
+differences are not individually significant at this sample size; only these baselines were tested;
-tested. Two further results earn their place by tempering: the initial two-axis grid's best predictor
+and with three seeds, uncertainty about seed generalisation remains substantial — though the seed
+sensitivity favours the functional measures (per-seed ρ stable at +0.37 to +0.53 in each seed alone,
+geometry ≈ 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 (ρ = +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 **failed** (they are statistically indistinguishable as rank
@@ -287,21 +318,37 @@ are the experiment's open front.
 
 *(FIG:fig6)*
 
+**Table 2.** Headline quantitative results with sample sizes, uncertainty, and outcome definitions
+(full per-experiment tables and falsifier status in SI Appendix and per-experiment documentation).
+
+| Result | Setting / n | Outcome definition | Headline |
+|---|---|---|---|
+| Closed-form validation | Analytic tier; standing tests | Simulated vs closed-form H-decay, immigration equilibrium, multi-teacher union | Agreement < 0.5% |
+| Grounding retention | Minimal model; 18+ replicates per point | Fraction of equilibrium diversity retained at grounding g (operational threshold) | g ≈ 0.05 retained ≥95% (tested setting); smooth in g |
+| MNIST collapse & rescue | Conv-VAE, 4 replicates; frozen oracle (98.5% mode acc.) | Mode support / forward-KL over generations | Dry: 30→1 modes; 10% grounding: 30/30 held |
+| Fisher–Muller in LLMs | 5 seeds (0.5B), fixed tests; single 7B run | Merged vs best-specialist accuracy (overall; worst family) | Ties 0.647±0.027 vs 0.592±0.009; 7B 0.87 vs 0.77 |
+| Union vs blend (headroom) | 3 seeds (0.5B hard); single 7B-hard run | Paired per-seed ordering, routing vs weight-average | Routing > blend in 3/3 seeds; one catastrophic blend failure avoided |
+| Speciation decomposition | MLPs, 3 replicates | LMC error barrier residual after permutation+rescaling alignment | Same-task 0.001; conflict 0.497 (naive 0.502) |
+| Emergent isolation | MLPs 4 reps to 6.4× base training; LLM 1→12 epochs | Residual barrier; merged vs parent accuracy | 0.000 everywhere; merge rescues parents (≈0.955 vs ≈0.50) |
+| Predictive test | 13 conditions × 3 seeds (0.5B) | Merge penalty vs oracle parent potential (pre-registered; ±: clustered 95% CI) | Functional ρ +0.45/+0.46, CI excl. 0; LOCO ρ ≈ 0.4; geometry n.s.; paired differences n.s. |
+
 ## Discussion
 
 **Design rules.** Read as engineering, the results compress into rules an operator of a model
-population can apply. *Ground every generation* in verified reality — a few percent retains most
+population can apply. *Ground every generation* in verified reality — a few percent retained most diversity in our tested
-diversity — but price the rarest capabilities individually (`m·p ≳ 1`) and use recombination, not
+settings — but price the rarest capabilities individually (observation probability `1 − e^{−m·p}` per
-grounding, to reach the deep tail. *Merge, don't blend, when there is headroom*: keep specialists
+batch under unstratified sampling), consider targeted sampling for the deep tail, and use
+recombination to recover rare capabilities still retained across complementary parents. *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. *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. *Preserve diversity as a first-class objective*, because selection can only preserve variety
+roughen. *Preserve diversity as a first-class objective*, because selection can only preserve variety that
-that exists, and the society result shows grounding, recombination, and diversity are jointly
+exists, and in the tested society its removal produced a distinct failure mode. *Before merging,
-necessary. *Before merging, measure functional conflict* — cheap, pre-merge, and in our controlled
+measure functional conflict* — cheap, pre-merge, and in our controlled setting predictive where the
-setting predictive where weight distance was not; and expect specialisation alone to be merge-safe,
+tested weight-distance baselines were not; and *do not treat divergence or specialisation alone as
-with conflicting conventions on shared circuitry as the thing to detect and avoid.
+evidence of incompatibility* — in every regime we tested, what broke merging was conflicting
+conventions on shared circuitry, which is the thing to detect.
 
 **What is borrowed and what is ours.** The diagnosis — collapse as drift — is prior art (6–9), as are
 the empirical facts that merges can beat parents, that decorrelated parents merge better, and that
@@ -309,9 +356,9 @@ naive averaging loses to interference-aware or routed merges (1, 19, 20), that m
 climb (2–5), and that merge success admits ML-native predictors (21, 22). Ours 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 jointly-necessary society; model speciation as a named, tested question, with the
+the society ablation with its complementary failure modes; model speciation as a named, tested question, with the
-coordinate-versus-functional decomposition under a complete symmetry group and the emergent null that
+coordinate-versus-functional decomposition under permutation-and-rescaling alignment and the emergent
-bounds it; and the controlled predictive test with its controls. We claim the framework generated
+null that bounds it; and the controlled predictive test with its controls. We claim the framework generated
 these measurements and experiments; we do not claim their outcomes validate a uniquely
 population-genetic mechanism, and one refinement it proposed was not supported.
 
@@ -345,8 +392,9 @@ exactly to the analytic tier — the bridge gate), and a convolutional VAE on MN
 oracle (98.5% mode accuracy; confusion matrix recorded as the measurement floor). Speciation
 experiments fork no-BatchNorm MLPs (784–512–512–10) from a shared base, weight-average, and measure
 linear-mode-connectivity error barriers before and after alignment; alignment composes deterministic
-Git Re-Basin permutation matching with exact per-unit scale canonicalisation (the complete unit
+Git Re-Basin permutation matching with exact per-unit scale canonicalisation (the unit symmetry
-symmetry group for this class), gated by exact recovery of a permuted-and-rescaled copy.
+group of this class, as the alignment search space; control recovery does not establish global
+optimality), gated by exact recovery of a permuted-and-rescaled copy.
 
 **Language-model tier.** LoRA specialists (rank 16) on procedurally generated task families with an
 exact-match verifier, on frozen Qwen2.5-Instruct bases (0.5B on one 16 GB GPU; 7B on one L40S).

paper/response-to-review-3.md

@@ -0,0 +1,96 @@
+# Response to the third review (of the PNAS-format draft)
+
+*All five priority fixes are made, plus the presentation items. The revised draft is
+`paper/pnas/main.md` (rebuilt PDF alongside); the long-form document and the results documentation
+were corrected wherever they carried the same overstatements. Point-by-point:*
+
+## 1. The averaging proposition (your §2) — you are right, and the text now proves what it claims
+
+Your convexity argument is correct: conservation of expected mass does not establish that averaging
+cannot help, because extinction is convex in mixed mass and averaging reduces its variance. Our result
+is, exactly as you diagnosed, a **first-order cancellation in the rare-item regime**, and the main
+text now states the actual proposition with its quantities and assumptions: K parents with independent
+retention; child draws `n` samples from one random parent vs the parents' output-mean; expected mass
+identical; and in the regime `n·p/K ≪ 1`, where per-item survival is first-order in sampled mass,
+expected survival is identical too. Two boundaries follow in the same paragraph: outside that regime
+averaging's variance reduction can *reduce* extinction relative to a random single parent (your
+argument, credited to the review process); and the union operator's renormalisation (which itself
+redistributes mass) and oracle requirement are stated. "Adding parents cannot help" is deleted here
+and in every other document that carried it. We agree the interesting content is the consequence for
+retention, not the elementary conservation of a mean — which is how the proposition is now framed.
+
+## 2. Grounding (your §3) — threshold made operational, floor made probabilistic, rule de-categoricalised
+
+- `g* ≈ 0.05` is now explicitly an **operational threshold**, with the text stating what our own
+  analysis always showed: the immigration–drift equilibrium is *smooth* in the grounding fraction (no
+  phase transition in aggregate diversity). New wording: under the tested population size and Zipf
+  source, `g ≈ 0.05` retained ≥95% of equilibrium diversity, with dependence on sample size, source,
+  and retention target (SI).
+- `m·p ≳ 1` is restated as what it is: `1 − e^{−m·p}` observation probability per batch (~63% at
+  `m·p = 1`), confidence-dependent, with retention vs stationary occupancy vs reintroduction
+  distinguished (immigration can restore an absent item).
+- The design rule now reads in your form: under unstratified grounding rare capabilities are expensive
+  (targeted sampling changes the cost); recombination recovers rare capabilities *still retained
+  across complementary parents*.
+
+## 3. Grounded inheritance vs grounded evaluation (your §4) — separated and named
+
+The society section now opens with the definitional distinction: **grounded inheritance** (external
+samples in the reproduction process — the data channel) vs **grounded evaluation** (true fitness vs
+conformity in selection — the fitness channel), related but different operators, connected only in
+that both couple the lineage to a non-drifting external signal. The section is retitled to your
+formulation ("…make complementary contributions"), the ablation is described as separating failure
+modes *under the tested conditions*, and general joint necessity is explicitly disclaimed (alternative
+mutation/restart/archive/selection schemes noted). Table 1's corresponding row now says
+"complementary… in the tested society"; the same fix is propagated to the long-form document.
+
+## 4. The alignment contradiction (your §5) — deleted, both statements reconciled
+
+"This cannot be an alignment failure, because the same aligner succeeded on the control" is removed
+everywhere (manuscript, long-form document, results documentation), replaced by your formulation: the
+tested alignment removes the same-task barrier but leaves the conflict-associated barrier largely
+unchanged — supporting a functional-conflict interpretation without proving optimal alignment. The
+abstract now says "remaining after permutation-and-rescaling alignment" (not "surviving the full
+symmetry group"), and the Methods note that the group is the alignment's *search space*, with control
+recovery not establishing global optimality. The discussion's "expect specialisation alone to be
+merge-safe" is replaced by the supported lesson: **do not treat divergence or specialisation alone as
+evidence of incompatibility.**
+
+## 5. Headline vs detail (your §6) — matched, and the seed-dependence analysed
+
+The significance statement now ends with your suggested sentence (a controlled small-model test…
+motivating further comparison). On the clustering point: you are right that condition-clustering does
+not capture cross-condition dependence through shared task-data seeds. We added the sensitivity you
+asked for (committed to the statistics script): **per-seed correlations** — each seed alone, n = 13
+conditions — are stable for the functional measures (+0.37 to +0.53 in every individual seed) and ≈0
+for geometry in every seed; leave-one-seed-out ranges are [+0.38, +0.56] (functional) vs
+[−0.04, +0.28] (geometry). One informative surprise: gradient alignment is *seed-unstable*
+(−0.11 to −0.55), which the manuscript now reports as its own caveat. The text also states plainly
+that with three seeds, uncertainty about seed generalisation remains substantial.
+
+## 6. Presentation (your §7) — done
+
+Meta-language removed ("the honest statement", "sharpest honesty", "earn their place by tempering",
+"honest deviations" — all gone; results are stated, not described as disclosures). "Exact" is now
+reserved for closed-form mathematics — NK/simulation results are labelled "analytic model" in Table 1
+and the text. The headroom relationship is stated qualitatively with "a quantitative form is
+untested". "Directed sex with no biological analogue" is replaced by your phrasing (the shorthand kept,
+defined as engineered recombination with flexible parent choice and pre-deployment screening).
+Muller's ratchet is now a *consequence-level* correspondence, with the text stating that irreversible
+loss alone does not identify the ratchet's mechanism. A compact results table (Table 2: setting/n,
+outcome definition, headline with uncertainty, for the eight headline results) is added before the
+Discussion. Reference numbering and the figure files accompany the rebuilt PDF; the bespoke unified
+figures and journal-format reflow remain flagged as submission-time work.
+
+## One point of information, not disagreement
+
+On §2's closing remark — that conservation of an arithmetic mean's expectation is elementary and the
+contribution must lie in its consequences — we agree, and would only note that the consequence now
+stated (first-order cancellation of the multi-parent retention gain under output-mean inheritance,
+against union-operator retention growth, in the regime where the deep tail actually lives) is the
+claim we intended all along; the earlier wording claimed more than this and is gone.
+
+Your bottom-line formulation — minimal models establish conditional results; neural experiments reveal
+where the correspondences hold and break; a controlled predictive test motivates measuring functional
+conflict before merging — is now, near-verbatim, how the paper describes itself. Thank you for three
+rounds of genuinely improving review.

paper/the-evolution-of-sex-for-ai.md

@@ -81,9 +81,11 @@ question the framework is built to test.
 
 From the geneticist's apparatus we extract falsifiable, load-bearing claims (each stated with its
 operator and scope in the text): (i) **"merge, don't average"** — a conservation result: refitting a
-child to the *mean of its parents' output distributions* conserves rare-capability mass at the
+child to the *mean of its parents' output distributions* conserves expected rare-capability mass at
-single-parent level, so adding parents cannot help, while union-preserving operators realise the gain
+the single-parent level, cancelling the multi-parent gain *to first order in the rare-item regime*
-— exact in the minimal model, with its weight-space image verified as the headroom rule below; (ii)
+(outside it, variance reduction from averaging can help — the result is a first-order cancellation,
+not a universal impossibility), while union-preserving operators realise the gain in all regimes —
+derived in the minimal model, with its weight-space image the headroom rule below; (ii)
 **offspring can exceed every parent** (Fisher–Muller), the real argument for sex in model societies;
 (iii) on **rugged, epistatic** task landscapes, blind recombination causes **outbreeding depression**,
 yielding a design rule — *merge freely when skills are additive, sparingly and with selection when
@@ -100,8 +102,8 @@ pre-registered and found: absent conflicting training signals, divergently-speci
 shared ancestry developed *no* isolation at any divergence tested, the merge instead *rescuing* the
 forgetting specialists. Isolation must be provoked by conflict; specialisation alone did not speciate.
 AI also has an advantage biology lacks: **directed sex** — unbounded parents, chosen mates,
-and offspring screened before they are kept — which converts recombination from a gamble into a
+and offspring screened before they are kept — engineered recombination with a flexibility of parent
-reliable engine and has no biological analogue.
+choice and pre-deployment screening that natural mating systems do not approach.
 
 We support the argument with **minimal, reproducible models** — a closed-form-exact account of drift
 and grounding, the same effects in small trained networks and an MNIST image generator, a real-weight
@@ -322,9 +324,11 @@ the fancier operator.
 Four different operators travel under these words, and the conservation result belongs to exactly one
 of them. What is *derived* is this: when a pupil's knowledge is refit to the **mean of the parents'
 output distributions**, the expected mass on any rare item is conserved at the single-parent level —
-in the rare-item regime the 1/K dilution of averaging exactly cancels the union gain of having K
+in the rare-item regime (`n·p/K ≪ 1`) the 1/K dilution of averaging cancels the union gain of having
-parents — so adding parents cannot help; whereas an operator that keeps, per item, its **strongest
+K parents to first order — outside that regime, survival is convex in mixed mass and averaging's
-source** realises the union. That statement is exact in the minimal model, and it presupposes an
+variance reduction can help, so this is a first-order cancellation, not a universal impossibility;
+whereas an operator that keeps, per item, its **strongest source** (and renormalises, which itself
+redistributes mass) realises the union in all regimes. That statement is exact in the minimal model, and it presupposes an
 oracle (or verifier) able to say which source is strongest. The two operators the LLM prototype
 tests — **weight averaging** (a nonlinear network's weight-mean does not compute the mean of its
 parents' outputs) and **routing among intact specialists** (which keeps K models' storage and an input
@@ -436,8 +440,9 @@ on the same task* have a real naive barrier that alignment removes almost entire
 and the aligned merge performs at parent level) — same species, different basis, the canonical Re-Basin
 result, which also proves the aligner works. Two nets that learned *conflicting* label maps have a
 large barrier of which the full symmetry group removes **essentially nothing** (0.502 → 0.497) —
-genuine reproductive isolation, not a missed symmetry, and it cannot be dismissed as a failure to align
+a conflict-associated barrier the tested alignment leaves largely unchanged — supporting a
-because the very same aligner erased the same-task barrier. It also carries a floor no future alignment
+functional-conflict interpretation without proving optimal alignment (control recovery validates a
+special case; the removable share is a lower bound, the residual an upper bound). It also carries a floor no future alignment
 method can breach: models loyal to label maps that conflict on a fraction *μ* of inputs cannot both be
 served by *any* single merged model, which must err at rate ≥ *μ*/2 against at least one parent
 (SI proposition). Sweeping the fraction of conflicting classes traces the **isolation cliff in real
@@ -754,8 +759,9 @@ shape.)
   offspring, beats the naive average — but *only when the task leaves headroom*. On easy tasks a strong
   model's plain average is already at the ceiling and the refinements add nothing; on hard tasks the
   average dilutes a specialist below even the best single parent, and the union-preserving operators win
-  clearly. The practical rule is exact: these tricks pay off in proportion to how far the naive average
+  clearly. The practical rule, stated qualitatively: these tricks pay off where the naive average falls
-  is from the best attainable. This is a prototype (three task families, one seed), so we read it as
+  short of attainable performance, and add nothing where it does not (a quantitative form is untested).
+  This is a prototype (three task families, one seed), so we read it as
   signs, not magnitudes; the *whole grounded society* on a language model remains the open step.
 - *The whole society, and why every part is needed.* In a population evolving on a "reality" landscape,
   the full system — grounding + sexual recombination + preserved diversity — climbs to the top while
@@ -786,7 +792,7 @@ falsifier, not yet established):
 | Confidence weighting improves rank prediction over raw disagreement | **Not supported** (pre-registered internal prediction) | — | Paired Δ\|ρ\| ≈ −0.02, CI [−0.13, +0.06] | Weighting does double the conflict-vs-compat level contrast |
 | The predictor improves budget-matched operator choice | **Open** | — | Soup-vs-route gap readout noise-dominated at 0.5B | The practical payoff; untested |
 | Emergent speciation without conflict | **Not observed** (pre-registered) | Shared ancestry, compatible tasks, tested divergences | E13b: residual 0.000; merge rescues specialists | Bounds the hypothesis; longer horizons/distribution shift/capacity pressure untested |
-| Grounding + sex + diversity jointly necessary | Exact-model result; hypothesis at LLM scale | Conformity stands in for self-consumption | E11 four-arm ablation, each arm failing distinctly | The full grounded LLM society is unbuilt |
+| Grounding + sex + diversity complementary (each ablation fails distinctly) | Analytic-model result; hypothesis at LLM scale | Conformity stands in for self-consumption; general joint necessity not established | E11 four-arm ablation | The full grounded LLM society is unbuilt; alternative schemes untested |
 
 **What is borrowed, and what is ours.** We are deliberate about the ledger, because the surrounding
 literature is crowded and a reader deserves to know exactly where the line falls. **Conceded as prior
@@ -811,8 +817,9 @@ from a thing you must run a search to discover into a thing the landscape's rugg
 the operator-choice design rule that follows (average / union-route / directed-select); **grounding as
 migration–drift balance**, giving a critical real-data fraction and a phase boundary a closed
 self-consuming loop cannot have; **directed sex** as the distinctly-AI advantage (unbounded parents,
-offspring preview, mate choice); and the **integrated society** whose four operators are shown *jointly
+offspring preview, mate choice); and the **integrated society** whose operators make
-necessary*. The value-add over the machine-learning-native merge theory is that ours predicts *which
+*complementary, distinctly-failing contributions* in the tested model (general joint necessity is not
+established). The value-add over the machine-learning-native merge theory is that ours predicts *which
 operator to use and when it will backfire*, not merely how fast quality decays. And it opens — and
 begins to occupy — a question nobody has framed: **model speciation**, the population-genetics of
 *reproductive isolation* (Bateson–Dobzhansky–Muller incompatibilities) as the account of *when two

results/speciation_real/README.md

@@ -29,9 +29,11 @@ Statistically reproducible (seeded); the alignment itself is deterministic NumPy
 - **`independent`**: the barrier is a coordinate artefact — permutations already remove ~98%, and the
   full symmetry group confirms (residual 0.001). The aligned merge performs **at parent level**
   (0.960): same species, different basis.
-- **`conflict`**: the full symmetry group removes essentially nothing (0.502 → 0.497). The residual is
+- **`conflict`**: permutation+rescaling alignment removes essentially nothing (0.502 → 0.497), and the
-  **functional**, not a missed symmetry — and the hybrid is functionally dead (accuracy 0.037).
+  hybrid is functionally dead (accuracy 0.037). The tested alignment removes the same-task barrier but
-  Because the same aligner erased the independent-init barrier, this cannot be a failure to align.
+  leaves the conflict-associated barrier largely unchanged — supporting a functional-conflict
+  interpretation without proving optimal alignment (control recovery validates a special case;
+  removable = lower bound, residual = upper bound).
 - Formal floor (SI note S1, `paper/si-notes.md`): for label maps conflicting on mass `μ(S)`, *any*
   single merged model errs at rate ≥ `μ(S)/2` against at least one parent, under *any* alignment
   group and merge operator — hybrid disadvantage is information-theoretic, and endpoints/chord are