490d C405 — Source-qualified 500-year translation of Noah return loops to Shem

24 September 2026. Step 2 of 10 in the C404–C413 authorization.

The fixed500-year source relation carries all six Noah E/J loops to sourced Shem loops, including both complete E/P/J graphs. Every one of the52 labelled endpoint/pivot slots has the required source record.

All displayed pivots in the translated graphs are existing Shem records. E holds a Shem birth; J and P hold Shem deaths. Noah births map to Shem births in the same tradition, chronology context, Gear and person rail. The corresponding Flood boundary maps to a Shem death. This is a role correspondence, not a claim that Flood and Shem death are the same event.

For the SP companion loops, the original J boundary is Flood close. Its minus500 image is the companion Shem death. This preserves the independent Flood-close event; it does not assign a person-companion rail to the Flood. LXX Cainan-ON/OFF status is unchanged for each record.

Writing T(t)=t−500 and D_(k,a)(t)=a+k(t−a), the exact identity is T∘D_(k,a)∘T⁻¹=D_(k,a−500). It holds for E,P,J, for each uniformly matched±¼ source component, and therefore for every finite admitted path. All402 recorded path maps were checked. The three Shem vertices2531,2533,2535 have the same operator-count normal form and the same circuit multipliers1750/(3K),7500/(13K) as their Noah counterparts.

These are different Shem points from the failed C404 E transport, whose image vertices were2991,2993,2995. The source500 shift is fixed by the retained500/100 mode; it was not fitted to close a loop. The result proves closure of this selected source correspondence, not that every Shem loop is its image. That completeness question is now worth checking directly.

Verification and limits

545 main checks and 18 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.

Translation is qualified by source roles, context and the500/100 mode. It is not an added global date operator, a claim of historical prescription or a new independent witness.

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

C406, proposed and unexecuted: Enumerate adjacent-Gear Shem birth pairs independently across the frozen coherent-family register. Require an existing held Shem birth for E and the moving family's next-Gear Shem death for J, then test the unique P pivot anywhere in the fixed register. Compare the complete results with C405, without prefiltering by500 or529.

The positive500 correspondence warrants an independent source-side completeness test; this will distinguish a full structural match within the specified family roles from a selected example.