Through four sessions and fifteen papers, the modulus/remainder framework decomposed every physical computation into two pieces. The modulus — R₂ = π/4 = β, the circle-to-square conversion — appeared in 22 engineering domains, in every loop integral, in every gauge coupling running equation, in every angular integration. It was the geometric bridge between circular physics and rectilinear measurement. It was understood. The remainder — everything left after the modulus was removed — was not. The gap ratio cancelled the modulus, leaving pure integers (218/115, 38/27). The Koide formula cancelled the mass dimension, leaving a shape parameter (K = 2/3). The sin²θW correction cancelled the loop factor, leaving group theory fractions (15/104). In every case, the modulus departed and the remainder survived, carrying the physics-specific content. But the remainder itself was opaque. What was 197/144? What was ζ (3)? What was Li₄ (½)? The framework could separate them from the modulus but could not explain what they were geometrically. They were tagged as "number-theoretic β⁰" — terms with no π content, arising from diagram combinatorics and radial integrations. They were parked. The notebook said: "the remainder is where the physics lives, but we don't yet know the geometry of the remainder. " This paper un-parks it.
Geoffrey Howland (Wed,) studied this question.