J loop to the17K branch

24 September 2026. Step 5 of 10 in the C394–C403 authorization.

The existing+215 translation carries the common P/J repair to the original17K branch, but it does not carry the complete E/P/J loop within the source menu. Its uniquely required E pivots3271 and3273 are absent.

Two Noah roles must remain separate. In the first lower-branch loop, the original529 geometry uses SP/LXX outer Noah3491, while the E return is held at MT Noah3056. The moving MT Noah edge is3031→3033. Translating all coordinates by215 carries the original P/J comparison to3246→3248, with outer SP/LXX Noah3706 and Flood pivots2648/3108. But E would have to be held at3056+215=3271.

Moving MT P/J edge	E return pivot before translation	Translated P/J edge	Unique translated E pivot
3031→3033	3056, sourced	3246→3248	3271, absent
3033→3035	3058, sourced	3248→3250	3273, absent

Uniqueness follows directly from E_V(x)=y: V=(25x−23y)/2. No alternative existing anchor can give the same return between the fixed endpoints. Retaining the old E pivot instead leaves an exact output error430/23.

The source reason is explicit: the moving Noah family starts in full4300, while its held E Noah is already in full4301. The+215 shift is available to the moving family and original P/J pivots, but would carry the held E Noah beyond the two supplied branch states. The numerical absence was checked against all528 regular records, so no coordinate alias was overlooked.

Against the original cumulative counterparts, the lower branch has cross-spans17K−215=8778; the translated branch has17K=8993. The complete sourced loop therefore does not inherit the original17K pairing merely because its P/J portion does. All eight endpoint-phase combinations were checked; none supplies a required held-Noah component.

Verification and limits

18 main checks and 15 frozen-input/predecessor authentication checks passed. A second main replay produced byte-identical DATA/RESULTS. These checks and the source interpretation were performed by the root agent; no new independent-agent review is claimed. All frozen originals and standing controls remain unchanged. Verification counts are not independent-witness counts.

The positive loop and original17K rectangle use distinct held/outer Noah roles. Diagnostic translated coordinates are not source additions or new epoch anchors.

Mirror/529 focus: K=529, E=25/23, P=70/69. Global paired Mirror January AD1; local junction January AD2; no civil year zero. Regular Gears are Noah/Shem/Flood only. SP minus-one person companions remain local to SP Noah/Shem; Flood close is a separate boundary. No upstream/cumulative Gear, new endpoint or phase, fitted anchor, global axis, primer reopening, canonical or graph edit. Capture now; canonize later. Source identity, context, modal binding, generated coordinates and phase ancestry stay distinct.

Decision after this result

C399, proposed and unexecuted: For all ten admitted common P/J coordinate edges—six cumulative and four regular—solve the unique E return pivot and check the appropriate frozen source menu. Derive one signed formula for the return-loop geometry and state its complete source-supported domain.

The branch test gives a specific closure obstruction. A complete ten-edge check can determine whether any other existing common P/J edge supplies the missing E return, including the cumulative side.