Why it is a question
The record already treats a representation as fixed by its kernel up to symmetries (THEORY-004), and holds that a linear probe family sees only differences in normalized kernel (CLAIM-193). CLAIM-369 shows that, in symmetric resource competition, the kernel is also the competition matrix. That suggests that how differentiated a set of features is spectrally might predict whether the processes implementing them coexist.
CASE-118 shows where the suggestion holds and where it fails. With equal growth rates the rare component’s invasion growth is exactly λ_min(A) = 1 − ρ. With growth rates 1 ± δ the same A supports coexistence only for |δ| < (1 − ρ)/(1 + ρ). So feature geometry is necessary context and not sufficient. The supply, meaning the tasks, resources or evaluation opportunities the system offers, also decides. That is CLAIM-152’s target-relativity: what a representation supports depends on what it is asked for.
What would answer it
Conditions, in a stated class of learning systems, under which the Gram spectrum of the learned features predicts the recovery rates of the processes carrying them, together with the further quantities that must be fixed for the prediction to hold. The answer should keep three measurements apart:
- structural differentiation, from feature geometry;
- functional differentiation, from conditional performance across contexts;
- dynamical stabilization, from recovery when rare and from persistence.
It bears on QUESTION-125. A change in feature geometry, such as an eigengap opening, and a change in coexistence, such as a component becoming able to invade, need not happen at the same parameter values. A model in which they come apart in either direction would be informative.
What would not answer it
Heterogeneous embeddings, or a well-conditioned Gram matrix, with no account of recovery dynamics.
Note of 2026-10-29: a large-system instance, and a model with no spectral condition
Two works read on this date bear on the question.
- Mehta, Cui, Wang and Marsland, LIT-1231 (NOTE-1395): in random quadratic programming the variance σ_c of the constraint vectors sets how many components survive, and more differentiated constraints coexist more. That is a large-system instance of the suggestion above, with the supply K and the costs m entering through the growth rates, as CASE-118 found for two components.
- Chizat and Bach, LIT-1232 (NOTE-1367), supply no spectral condition. Their coexistence notion is first-order, the sign of F′, not the growth-rate comparison CASE-118 uses.
The question is unchanged and stays Open.