Brouwer Fixed-Point Applier
Ki · Reasoning
The Problem
The algorithm will reach a stable state after enough iterations, so we can assume convergence without further proof.
The Operation
This ability makes the model identify any iterative or self-referencing process and ask whether it converges to a stable state. Compute candidate fixed points where further iteration produces no change. Enumerate whether fixed points are unique or multiple, then rank them by stability. Test each equilibrium: simulate a perturbation and verify the system returns to the fixed point. If no equilibrium exists, flag the process as oscillatory or divergent. The reasoning applies a formal computation: fixed points = set(x where f(x) = x).
The Structure
Structurally, this is an iterative convergence loop that cycles until the reasoning stabilizes on a consistent answer. The procedure repeats until diminishing returns trigger an exit.
If an iterative process is left running without identifying whether it converges to a fixed point, equilibrium finding was not performed.
Haki · Reasoning-Multi
Cross-Domain Suppression
In Haki mode, the API retrieves the primary ability first, then fans out to three synergy roles that compound its reasoning.
When retrieved in Haki mode, the primary ability is augmented with failure guards extracted from 3 abilities in different cognitive domains. Each guard blocks a specific reasoning failure the primary alone wouldn't catch. A self-check forces verification before output. The result is cross-domain coverage that no single ability can reach alone.