# 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.

|Noah graph|Translated Shem graph|E pivot|J pivot|P pivot|
|---|---|---:|---:|---:|
|3031↔3033|2531↔2533|2556|1933|2393|
|3033↔3035|2533↔2535|2558|1935|2395|

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.