Executive Summary
This paper studies how the distribution of acceptable outputs evolves in a system where agents produce outputs, constraint functions score them, and lifecycle operations (fork, decay, merge) inject and remove variance. Three formal questions organize it: when constraints can be composed without circular dependencies, when composition preserves well-formedness, and when the acceptability distribution converges to a unique fixed point, and how fast.
The central result is for the Gaussian case. Under Gaussian scoring with lifecycle variance injection, the Q-dynamics operator is proved to be a global contraction with a unique, globally attracting fixed point Q* and contraction parameter λ* = σκ² / (σ*² + σκ²), where σκ² is the scoring-function variance and σ*² is the fixed-point variance. The parameter is derived from first principles, not assumed; sharper scoring gives faster contraction.
Key Contributions and Methodology
First, a constraint type system with four primitives (schema, threshold, scoring, topological), a dependency graph, and sequential, parallel, and conditional composition. Type compatibility is decidable in polynomial time, and composition preserves types.
Second, compositional well-formedness: composing two well-formed constraint architectures yields a well-formed architecture under type compatibility and energy separability.
Third, a multi-dimensional extension. Mean contraction holds in d dimensions; covariance contraction is proved when the covariance and precision matrices commute, and is locally stable in general.
Fourth, a cage characterization. The cage, the attractor under convergent selection, is the limit Q* → pF (the system's own prior) as scoring sharpens. A generative-preservation theorem gives conditions on selection balance under which the fixed point keeps generative capacity: generative capacity requires nonzero selection balance.
Key Findings
- Gaussian contraction theorem: unique fixed point Q* with explicit contraction parameter λ* = σκ² / (σ*² + σκ²)
- Constraint type system: four primitive types; type compatibility decidable in polynomial time; type preservation under composition
- Compositional well-formedness: well-formed architectures compose into well-formed architectures
- Cage as attractor: convergent selection drives the acceptability distribution toward the system's prior
- Generative preservation: nonzero selection balance is required to retain generative capacity
- Open problems: log-concave and sub-Gaussian extensions, non-commuting covariance contraction for d > 1, general scoring functions, and mixture contraction, each stated with a conjecture and bridging strategy
Key References
Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae, 3.
Diffusions hypercontractives. Séminaire de Probabilités XIX, Lecture Notes in Mathematics 1123. Springer.
Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. Journal of Functional Analysis, 173(2), 361-400.
Nonlinear Perron-Frobenius Theory. Cambridge University Press.