Emergence Capacity Criterion

• The Emergence Capacity Criterion is a framework that quantifies when interactions at the micro level give rise to robust, novel macroscopic behaviors.
• It integrates deformed Floer theory, stochastic models, and information-theoretic approaches to reveal algebraic, topological, and dynamical conditions of emergence.
• Applications span symplectic topology, evolutionary biology, glass transitions, and neural networks, offering precise tests for emergent phenomena.

The Emergence Capacity Criterion is a formal framework—present across several domains of mathematical physics, dynamical systems, complex systems science, symplectic topology, population genetics, information theory, collective behavior, and machine learning—for diagnosing and quantifying the ability of a system to exhibit robust macroscopic properties, structures, or behaviors that cannot be straightforwardly deduced from the system’s microscopic description. It encodes the precise conditions under which a system’s micro-level components and their interactions yield novel, efficient, or predictive macro-level theories through a process of coarse-graining, information aggregation, or structural phase transition.

1. Formalization Through Deformed Floer Theory and Spectral Invariants

In symplectic topology, the Emergence Capacity Criterion emerges in the context of deformed Hamiltonian Floer theory and quantum homology (Usher, 2010). Here, the Floer complex is deformed by a parameter $$\eta\in \bigoplus_{i=0}^{n-1}H_{2i}(M; \Lambda_{\omega}^{0})$$, altering the differential and the pair-of-pants product. The primary features are:

• The deformed differential counts Floer trajectories weighted by incidence data derived from $\eta$, leading to a deformed homology $HF(H)_\eta$.
• Via the deformed PSS (Piunikhin–Salamon–Schwarz) isomorphism, homology with the deformed quantum product $*_\eta$ is isomorphic to $HF(H)_\eta$, realizing the so-called big quantum homology at the chain level.
• Deformed Oh–Schwarz spectral invariants $\rho(a;H)_\eta$ encode deep structural constraints, admitting triangle and continuity properties and relating the algebraic structures to topological invariants and capacity bounds.
• These frameworks yield sharp bounds on the Hofer–Zehnder capacity, e.g., when certain genus-0 GW invariants are nonzero:

$$c_{HZ}^{\circ}(M,\omega) \leq \langle[\omega],A\rangle$$

whenever $$\langle [pt],a_0,[pt],a_1,\dots,a_k\rangle_{0,k+3,A} \neq 0$$.

• The criterion for the existence of Calabi quasimorphisms and symplectic quasi-states is encoded in the algebraic property that the deformed quantum homology ring $QH(M,\omega)_\eta$ is semisimple or field-split for some (generic) deformation $\eta$.
• Generically semisimple quantum homology is characterized by an open dense set of deformation parameters for which $QH(M,\omega)_\eta$ is semisimple; this ensures the existence of spectral invariants and quasimorphisms with strong rigidity properties.

This formalization links the emergence of rigid, global, macroscopic invariants (capacity bounds, quasi-states) to algebraic and topological features of the deformed microscopic Floer-theoretic structure.

2. Stochasticity, Demographic Structure, and Evolutionary Emergence

In evolutionary biology, the Emergence Capacity Criterion describes how, in a stochastic finite-population regime, the evolutionary success ("emergence") of behaviors is contingent not only on deterministic fitness but also on a demographic parameter: the carrying capacity (Houchmandzadeh, 2014). Main features include:

• Under deterministic selection, deleterious types (cooperators) inevitably decline.
• In the Wright–Fisher model with variable carrying capacity $N(x)$ that increases with the fraction of cooperators, the fixation probability for cooperators can exceed that of defectors if the relative increase in carrying capacity outweighs the selection cost:

$$\delta > \bar{N} s\,, \quad \text{with} \quad 2\delta = \frac{N_f-N_i}{\bar{N}}$$

where $s$ is the selection coefficient.

• This is a strictly stochastic effect, absent in deterministic theory, and sharply defines the emergence capacity of cooperation as a balance between demographic benefit and cost.

3. Information-Theoretic Criteria for Causal and Downward Emergence

In complex systems and multivariate data settings, the Emergence Capacity Criterion is grounded in information theory, particularly Partial Information Decomposition (PID) and related indices (Rosas et al., 2020, Jansma et al., 3 Oct 2025). Key points:

• Emergence is formalized by the presence of high-order synergy in information transfer from the microscopic parts $X_t$ to macroscopic features $V_t=F(X_t)$, with

$\mathrm{Un}^{(k)}(V_t; X_t' | X_t) > 0$

• Downward causation is quantified by the flow of unique information from collective variables to individual components.
• Causal decoupling is identified when macroscopic features exhibit self-predictability not already present at the micro-scale.
• Scalable practical criteria (e.g., $\Psi, \Delta, \Gamma$) are provided to operationalize these concepts for real-world data, enabling detection of emergent regimes in neural activity, flocking, and cellular automata.

This establishes a rigorous criterion: emergence capacity is achieved when macroscopic observables realize predictive or causal power irreducible to combinations of a fixed number of microscopic elements.

4. Emergent Capacity in Physical and Model Systems

Across nonlinear dynamical lattices and glassy systems, emergence capacity is defined in terms of critical transitions or thresholds in collective behavior:

• For discrete breathers in nonlinear lattices (Kevrekidis et al., 2016), the criterion for the emergence of instability (the loss of localized modes) is given by the vanishing of the derivative of the energy with respect to frequency:

$H'(\omega_c) = 0$

This analogizes the Vakhitov–Kolokolov criterion and links the system's capacity to support robust structures to a precise monotonicity change in a macroscopic observable.

• In colloidal glass transitions (Yang et al., 2017), the percolation of rigid domains is tracked by the emergence of a system-spanning cluster at a critical packing fraction (φ ≈ 0.69), detectable in local Debye–Waller fluctuations (α_i) and validated by the Lindemann-type criterion (L ≈ 0.2 d). This empirical threshold demarcates the system's emergence capacity for rigidity and glassy dynamics.

5. Singular Value and Matrix Analysis as an Emergence Test

A recent approach operationalizes the Emergence Capacity Criterion by singular value decomposition (SVD) of system trajectory matrices (Faruque et al., 20 Jun 2024). Essential elements:

• The "singular value decay curve" exhibits a "knee" or elbow indicating the transition from coherent, structured dynamics to noise.
• The "knee angle" metric, derived from the triangle method, quantifies the sharpness of this transition and is validated against noise bounds from random matrix theory (Marcenko–Pastur law).
• The emergence criterion is satisfied when a substantial fraction of singular values and a sharp knee angle signal the dominance of structure over noise, independent of embodiment or measurement specifics.

This technique provides a robust, data-driven emergence test applicable to swarms, cellular automata, and biological systems.

6. Phase Transitions, Criticality, and Structure Acquisition in Neural and AI Systems

In neural network and machine learning contexts, the Emergence Capacity Criterion manifests as a phase transition in learning dynamics (Lubana et al., 22 Aug 2024, Krakauer et al., 10 Jun 2025):

• In formal language learning by Transformers (Lubana et al., 22 Aug 2024), emergence is identified by a nonlinear jump in downstream task performance coinciding with the acquisition of general structures (such as grammar and type constraints), modeled as a percolation transition on a bipartite graph. Criticality is thus determined by a macroscopic order parameter reflecting network connectivity in representational space.
• In LLMs, emergence is more rigorously linked to the development of internal compressed representations or new coarse-grained variables, detectable via scaling laws, critical points, and changes in spectral statistics (Krakauer et al., 10 Jun 2025).

The criterion therefore ties emergence capacity to the ability to reorganize or compress internal structure under increasing system scale or complexity, such that new predictive macroscopic behaviors arise.

7. Multiscale, Hierarchical, and Algorithmic Structure in Emergent Descriptions

Modern theoretical treatments recognize that emergence capacity is not binary but reflects the system’s ability to support a hierarchy of scales, each associated with distinct causal or informational contributions (Carroll et al., 20 Oct 2024, Rizi, 7 Jul 2025, Jansma et al., 3 Oct 2025):

• Emergence is characterized as a commutative relationship between micro- and macro-dynamics via an emergence map $\Phi$ satisfying

$\Phi(E_A[a(t)]) = E_B(\Phi(a(t)))$

ensuring compatibility of time evolution under coarse-graining (Carroll et al., 20 Oct 2024).

• The “emergence capacity” is maximized when the coarse-graining map is algorithmically simple, and macro-level prediction is robust to loss of micro-level detail (Rizi, 7 Jul 2025).
• Hierarchies of emergent descriptions can be quantified by tracking the information-theoretic gain in causal primitives (determinism/degeneracy) across all micro-to-macro partitions (Jansma et al., 3 Oct 2025). Notions such as path entropy and row entropy then describe whether a system is “top-heavy,” “bottom-heavy,” or “scale-free” in the causal distribution across levels.

These perspectives formalize the criterion in terms of efficient, predictive, and observable-reducing maps from micro to macro, with emergence capacity maximized for systems where such multiscale, non-redundant coarse-grainings exist and are robust to noise or perturbations.

---

In sum, the Emergence Capacity Criterion delineates the precise algebraic, topological, stochastic, informational, or dynamical conditions under which macroscopic phenomena arise, persist, and can be quantitatively predicted from—or are irreducible to—a system’s microscopic or component-level dynamics. It provides a unifying framework across fields, linking notions of capacity, rigidity, information synergy, phase transition, and hierarchy in the emergence of complex collective behavior.