Mathematical Duality Navigator
Ki · Single Ability
The Problem
The algorithm is stuck because the problem is too hard to solve from this single perspective, and no alternative view seems viable.
The Operation
This ability makes the model identify the current problem representation and scan for dual forms -- concrete vs. Map the problem into the candidate dual and compare tractability. Solve in whichever representation reduces complexity, then invert the solution back. Verify that the solution holds in both representations. If no simpler dual exists, halt the search and document why. It will not force a duality that distorts the problem.
The Structure
Structurally, this is a disassemble-transform-reassemble pattern that breaks the problem apart, modifies each piece, and reconstructs.
If a problem proves intractable in one representation without testing whether an alternative formulation would simplify the solution, perspective switching was skipped.
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.