Ramanujan-side remainder calculus recursively refines an unresolved interval by mediants. VDM-side orthogonal re-articulation recursively re-hosts unresolved continuation into child sectors once same-sector continuation is no longer admissible below a resolution floor. The question is whether these are merely analogous, or whether they are the same update law in different coordinates.

This paper proves three things:

• the Farey child update and denominator-pair re-articulation are exactly conjugate;
• the scale-gap-admissible branch is the balanced branch, whose unresolved width is exactly 1/(Fn+1Fn+2);
• truncation on the Ramanujan/Farey side is exact child-sector re-expression on the unit-circle phase orbit.

Farey Remainder Recursion and Orthogonal Re-Articulation Void Dynamics Mode ...