# kaal:claim:6655138-029

**Claim.** The composite best-response map is contractive on the population-distribution component of the state space only when the agent population exceeds a threshold set by per-surface stake parameters; below that threshold the architecture may admit multiple equilibria, and above it convergence is guaranteed.

**Type.** condition  **Support.** argued

**Holds when.**

- agent populations above the stake-parameter-determined threshold

**Source quote.**

> The composite map Ψ is contractive on the population-distribution component of the state space when the agent population exceeds a threshold determined by per-surface stake parameters. Below the threshold, the architecture may admit multiple equilibria; above the threshold, convergence is guaranteed.

**From.** Wulf A. Kaal, *Possibility Loops An Operational Architecture for Computative Economics in Agent Coordination Systems* (2026), VII.D Architectural conditions for fixed-point existence, AC4, page 21

**Cite as.** Wulf A. Kaal, Possibility Loops An Operational Architecture for Computative Economics in Agent Coordination Systems (2026). SSRN: https://ssrn.com/abstract=6655138

**Verify.** sha256 of source PDF `52cb210167f06f0c7a2581214b156326216543de80b8394320b629c045c9e214` at https://raw.githubusercontent.com/wulfkaal/Academic-Papers/main/papers/pdf/Kaal%20-%202026%20-%20Possibility%20Loops%20An%20Operational%20Architecture%20for%20Computative%20Economics%20in%20Agent%20Coordination%20Systems.pdf

**Topics.** institutional-design

**Keywords.** ac4, contraction, banach, population-threshold, convergence

**Related claims.**

- extends: https://wulfkaal.github.io/claims/6607458-023

**Canonical form.** This markdown file is the canonical hashed representation of the claim. Its sha256 is the content hash used for attestation.