Brouwer Fixed-Point Applier
Ki · Single Ability
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 · Multi Ability
Synergy Topology
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, these four abilities don't run in sequence. They merge into a single injection where the dependency grounds the reasoning context, the amplifier sharpens the primary's output, and the alternative provides a fallback path if the primary's topology cannot converge. The result is a multi-angle reasoning scaffold that covers failure modes no single ability can reach alone.