QUESTION-132: Under what conditions do initially equivalent adaptive operators differentiate into functionally complementary specialists that each have positive recovery growth when rare, rather than into roles kept in place by external load balancing or by persistent authority asymmetries?

Why it is a question

Differentiation that is merely present does not show coexistence. Three outcomes look alike from usage statistics and differ in kind:

1. Stabilizing coexistence. Each specialist, made rare, recovers because the resident organization leaves its niche open. This is Chesson’s stabilizing mechanism (LIT-1219, unread).
2. Imposed diversity. Specialists persist because a programmed load balancer or a floor on influence keeps them from vanishing.
3. Entrenched asymmetry. One or a few processes hold authority through early advantage, the CLAIM-277 loop, and the rest persist as residue.

A recovery exponent. Suppose operator i has influence w_i(t) and contribution F_i(c_t, w_t), with ẇ_i = w_i F_i(c_t, w_t). Here w_i is an unnormalized participation intensity. If influence is a normalized share, the dynamics are of replicator type and F_i is measured against the population mean (CLAIM-369, “Two boundaries of application”). While the others follow a resident regime and i is very rare, its long-run recovery exponent is

λ_i = lim_{T→∞} (1/T) ∫₀ᵀ F_i(c_t, w_{−i}(t)) dt,

when the limit exists along the resident dynamics. With discrete multiplicative updates it is the average log growth factor. λ_i > 0 means the rare specialist recovers from small positive influence. λ_i < 0 means the resident organization drives it toward exclusion. It is a statement about recovery, not about equal weights: a specialist can hold little influence most of the time and still have λ_i > 0.

A wrinkle. A specialist useful only during rare events can have negative average growth over the full context distribution and still be critical when the event comes. Two quantities are worth separating: λ_i^resident, the long-run recovery exponent, and λ_i^conditional(C), the recovery exponent during a specified context class C. The second is not a substitute for the ecological invasion rate. It may better capture what CLAIM-362 cares about, a process keeping access to consequential roles when its contexts recur. A process may not merit continuous presence in the active coalition and still merit preservation as a dormant capability.

An ablation the answer needs. Because zero is absorbing under multiplicative dynamics, recovery should first be tested from small positive abundance or influence, n_i(0) = ε > 0. The specialist then recovers or declines under the resident dynamics with no continuing intervention. That is how an invasion exponent is normally evaluated. Externally supplied reintroduction and internally maintained exploration are distinct interventions. The first cannot establish endogenous maintenance. The second may be an organizational mechanism of persistence, and so may be one of the achievements under study, but its origin, resource cost and causal role must be established independently. The first filing said testing recovery “needs some reintroduction” and put the supplying mechanism outside the organization. The owner’s review corrected both: the endogenous protection of rare specialists should not be defined away before it is studied. A model does not exhibit coexistence merely because a programmed lower bound on influence keeps every operator above zero. Uniformity-promoting objectives (THEORY-306, the equal-cluster prior of joint-embedding objectives) are the representation-learning version of imposed diversity. A specialist useful 1% of the time need not be active 20% of the time because there are five specialists.

Three ways to keep a specialist. Recovering influence when the context returns is one achievement. The organization might instead:

1. maintain the specialist through active participation;
2. maintain it as a dormant but recoverable capability, kept without being continuously activated;
3. reconstruct a lost specialist through later learning.

These can differ greatly in cost and resilience. Dormant preservation is the analogue of memory, organizational archives and protected minority expertise. It is also why positive long-run invasion growth need not be the only criterion of valuable coexistence. Whether retention without activation is possible when training follows influence is QUESTION-134’s question.

Recovery of a retained specialist against adaptation during recovery. A returning specialist may regain influence because its old capability is still useful, or because new training during the recovery makes it useful again. Both are interesting, and they are different achievements. The first is retention, which is the dormant case above. The second is re-adaptation, which lies between dormancy and reconstruction. A recovery measurement must say whether the specialist’s parameters are frozen or trainable while recovery is measured. With them frozen, a positive recovery exponent shows that the retained capability is still valued. With them trainable, it shows only that the specialist can become valued again.

Nonlinear transitions. Coexistence and exclusion can change abruptly with parameters, which bears on the open nonlinear-transition questions (QUESTION-125; CLAIM-330 on the coupled loop; THEORY-136 on what evidence shows alternative stable states). A model with positive feedback and exclusion is not thereby bistable, and it need not have a cusp. That has to be shown in the model, not inferred from the ingredients.

What would answer it

A population model in which:

• the interaction structure is derived from the operators’ learned contributions, not assumed;
• the recovery criteria λ^resident and λ^conditional are stated;
• the parameter regimes of each of the three outcomes are found.

In the potential case CLAIM-369 gives the recovery test as a marginal value, s_i = −∂_i L. A restricted theorem would be a natural first target. For symmetric competitive interactions and bounded niche adaptation, find sufficient conditions under which a differentiated equilibrium is locally asymptotically stable and every active specialist has positive growth when reintroduced into the resident equilibrium without it. Then ask which conditions fail when symmetry is relaxed, adaptation is delayed (THEORY-548), or training opportunities depend on current influence (QUESTION-134). With more than two operators, coexistence should be judged by invasion graphs (Hofbauer and Schreiber 2022), not pairwise tests.

What would not answer it

Heterogeneous embeddings, unequal routing weights or persistent expert identities. Mixture-of-experts routing already produces specialization, and an answer must show more than it does (QUESTION-083’s note of 2026-10-18).

Note of 2026-10-21: asked of institutions

A19 §5 of the exchange of 2026-10-20: “Symmetry breaking is not inherently a failure … We can distinguish desirable specialization from problematic entrenchment only after specifying criteria beyond the symmetry-breaking dynamics themselves”. This is this question’s alternative, asked of institutions (CLAIM-408). A20’s close (“may also reflect genuine learning, merited expertise, and beneficial specialization”) is QUESTION-100’s and QUESTION-087’s.

Note of 2026-10-29: invasion graphs, the mean-field flow, and three more readings

Hofbauer and Schreiber’s invasion graphs, which “What would answer it” names, are LIT-1233, now read (NOTE-1379). This question’s λ_i is exactly their r_i(µ), and the reading adds two conditions the question should carry.

• Positive recovery at one resident regime is not enough. It must be positive at every −i community, which may be a much smaller community than all the others.
• Signs decide only if the invasion graph is acyclic. With cyclic replacement among operators, rock–paper–scissors among roles, every recovery exponent can be positive and coexistence still fail.

Chizat and Bach, LIT-1232 (NOTE-1367), give the question a hard precondition in their model. Gradient flow transports mass and never creates it, so recovery from rarity is available only where some mass already sits. Units that start identical, at a single point rather than spread over a separating set, cannot differentiate under gradient flow alone. Their experiments show the cost: initialized on {1} × Θ, the flow fails with 15 particles and succeeds with 30.

Three more readings bear.

• Marsland, Cui and Mehta, LIT-1230 (NOTE-1352): under symmetric resource competition, differentiated specialists that each recover from rarity are exactly the binding constraints, and assembly cannot undo an invasion except by killing species. Revision of authority in an analogue built on that class needs an external removal or an asymmetry.
• Power et al., LIT-1235 (NOTE-1368): the long-run breakdown, one attractor outcompeting the other, is an instance of the third outcome, entrenched asymmetry, emerging from initially balanced memory.
• Letten, Ke and Fukami, LIT-1237 (NOTE-1377), locate Chesson’s stabilizing mechanism in impacts: an operator that limits itself more than others through what it does to a shared resource. Divergent requirements alone mostly equalize.

Chesson, LIT-1219, which the summary and the first outcome cite as unread, is now Unavailable. The question is unchanged and stays Open.

Note of 2026-10-29: acyclic in the potential case, a candidate case, and Chesson registered

• The potential case. THEORY-1168 (Active), from Serván and Allesina, LIT-1824 (NOTE-1422): with a symmetric interaction matrix, rᵀx strictly increases along every edge of Hofbauer and Schreiber’s invasion graph, so the graph has no cycles. In that class the second condition the earlier note of this date adds, that signs decide only if the invasion graph is acyclic, is met automatically. Cycles among roles need asymmetry, or a resource that can be excluded (the chapter’s §3.5.4).
• Ito and Kaneko, LIT-1842 (NOTE-1414). Identical chaotic units with similarity-reinforced, normalized attention split into a core and a periphery. That looks like the third outcome, entrenched asymmetry, more than stabilizing coexistence: membership in the core is absorbing, and the core does not depend on the periphery (2 of about 1,344 possible links run from periphery to core). But the paper measures no recovery when rare, no perturbation and no removal of a core unit. So it is a candidate case for this question’s classification, not evidence for one outcome. Removing core units and seeing whether the periphery refills the core would decide it cheaply.
• Chesson. His 1994 paper, LIT-1827, whose “−i community” Hofbauer and Schreiber make rigorous, is registered and Unavailable. His 1990 paper on MacArthur’s consumer–resource model, LIT-1826, is registered and Unfetched: the author’s reprint is located and could not be fetched. The first outcome’s name still rests on no reading of Chesson.

The question is unchanged and stays Open.