490d C411 — Source formula and record-path lift of the four-step 500 chain

24 September 2026. Step 8 of 10 in the C404–C413 authorization.

The four-step500 chain has two continuous source-record lifts. Its single extra bridge is forced by an extreme but already permitted combination of context, Gear and SP companion choices.

For SP Shem birth compared with the corresponding SP/LXX Flood-start field, the gap formula is

100 +130ΔCainan +60ΔTerah +215Δfull430 +2ΔGear −companion.

To equal500, the variant contribution must lie between396 and405. The only allowed value in that interval is405=130+60+215. The remaining correction must therefore be−5, forcing the Shem endpoint to be SP companionG1 and the Flood endpointG3. Thus

100+405−4−1=500, giving3395→2895.

The two endpoint labels at2895 are SP and theoretical Cainan-OFF LXX; neither receives an SP companion rail. This is a derived cross-context comparison, not an additional biblical age.

Before admitting this comparison into the derived analysis graph, the240 role-defined record arrows have192 maximal record paths, all at most two arrows long. Adding its two source labels yields191 maximal paths:143 of length1,46 of length2 and two of length4. The two long paths retain the same record at each join and differ only in the SP versus LXX label for the Flood/death pair. They represent one coordinate chain, not two independent witnesses. This analysis graph is not a canonical repository edit.

All32 assignments of existing±¼ components along the five coordinates were checked. Exactly the two uniformly matched assignments pass all four translations. Since translation has slope1, one physical translation preserves both matched components; this differs from the E dilations' seasonal-width distortion.

The opening Noah's own death is2945, after950; the opening Shem's own death is2795, after600. Neither is1895. The full2000-year chain therefore compares multiple source contexts and traditions rather than extending either lifespan. The source warrant for these records, and the distinction between attested inputs and derived combinations, should now be checked directly.

Verification and limits

10 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 bridge and path are derived comparisons, not a newly established source prescription. Coordinate aliases and coherent record lifts are not independent witnesses. No canonical graph edit or new endpoint was made.

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

C412, proposed and unexecuted: Reconstruct the regular register and the newly used loop/chain records directly from the current source tables and stated variant rules. Audit the warrant for the500 role correspondence, SP companion/Flood-close handling, LXX removal and the new cross-context bridge; keep explicit source data distinct from derived path constructions.

The positive record lift and the boundary-forced bridge now need source-level verification before the final synthesis. This closes the evidential scope of the batch without expanding its source menu or adopting a new protocol.