Documentation

Reasoning Library

Designing how an AI reasons — the shape of the algorithm it thinks through, distinct from the model underneath and the prompts on top — turned out to need a name, and we did not find a good one waiting for us. The term we kept coming back to is cognitive engineering: treating the structure of reasoning as something you can specify, measure, and improve, the way you would any other engineered artifact. We are not precious about the label — it is simply what best fit the work once we were doing it. Read through these ideas and see whether it fits for you too; if you have a better name for it, we would genuinely like to hear it at info@arcusx.ai.

This library is a manual in two parts. Foundations lays out the theory — why output quality decomposes into factors and structure is the leverage point, how that structure decomposes into levels you can design, and how it is measured. The Graph Library catalogs the reasoning strategies as executable graphs, each with a full walkthrough and worked use cases. New strategies are introduced family by family in the Reasoning Strategies series on the blog.

Section 1

Foundations

The theory, formally — why structure matters, how it decomposes, and how it is measured.

The R/K/G Framework

A mathematical framework for cognitive system output quality, extending the Cattell–Horn–Carroll tradition

R = reasoning capability · K = knowledge sufficiency · G = graph design quality

Appendix A

Live

↳

Why the Curve Has This Shape

Companion to R/K/G — an intuitive derivation of the equation from three assumptions, and why it is a candidate to be fit, not a law

diminishing returns → log · compounding → power law · gap-closing → linear amplification

Companion

Live

Strategy, Tactics, and Implementation

The three levels of reasoning design — and the three-layer meta-strategy every effective strategy moves through

strategy = the shape of thought · tactics = the topology · implementation = the prompt set

Foundations

Live

Reasoning Circuit Complexity

A formal cost model and structural analysis of reasoning graphs, in the lineage of circuit complexity theory

D(G) = depth · W(G) = width · ν(G) = verification density · classes G₁–G₅

Appendix B

Live

Section 2

Graph Library

The reasoning strategies as executable graphs — each with a full walkthrough, its gates, and worked use cases.

Strategies grouped by the family they come from. Epistemic is published and expands to its graphs; the rest are named here and arrive one family at a time.

Epistemic The three canonical modes of inference

3 graphs