WP / 2026 / 04 · 22 jan 2026 · 34 pp

completing the square — a method note.

the canonical statement of the firm’s method. three months to complete, three weeks to hand over, then we leave. with the eight failure modes we have actually observed.

§1 · the figure.

al-khwārizmī's proof of al-jabr is a geometric one. you take a quadratic, you draw it as a square missing a corner, and you complete the corner. the missing piece is named, drawn, and added. the work is the drawing — not the algebra.

every engagement we take is shaped like that figure. a square missing a corner. our work is to name the corner, draw it, and add it. when it is added, we step out.

§2 · three months. three weeks. then we leave.

the canonical cadence is three months to complete the corner, three weeks of paired hand-over, then we leave. quarterly office hours for twelve months — they call us; we do not call them. that is the entire engagement.

this paper is a placeholder for the canonical method note. the public version is in draft.

§3 · the eight failure modes.

~ named in section §3 of the working paper. published when the draft is signed. ~

— founder · khwarizmi · johannesburg · 22 jan 2026

a paper we should write?
commission us.