# 490d C406 — Independent coherent-family census of Shem return loops

24 September 2026. Step 3 of 10 in the C404–C413 authorization.

**The independent Shem census matches the complete500-year image exactly within the declared family grammar.** None of its surviving loops lies outside C405.

The enumeration tests32 moving Shem families×32 held Shem families×2 adjacent-Gear transitions=2048 labelled cases. It requires the E pivot to be a same-lower-Gear Shem birth and the J pivot to be the moving family's next-Gear Shem death. No500,529 or prior-Noah filter is applied. The unique P pivot is then checked against the entire regular register.

Twelve labelled E/J comparisons yield six coordinate configurations; two also admit P. The SP/LXX aliases and SP companion records retain separate identities.

|Held Shem E pivot|Higher birth|Lower birth|J death pivot|
|---:|---:|---:|---:|
|2556|2533|2531|1933|
|2558|2535|2533|1935|
|3205|3182|3180|2582|
|3206|3183|3181|2583|
|3207|3184|3182|2584|
|3208|3185|3183|2585|

The source geometry explains the match. Adjacent Shem births differ by2; the next-Gear death lies598 below the lower birth and600 below the higher. Hence J gives the forward arrow. The E return requires a held birth25 above the lower birth, or23 above the higher. Every surviving context obtains25 from215−130−60: held full430/Cainan-OFF/Terah0 versus moving half430/Cainan-ON/Terah60. This is the same permitted variant displacement as in the Noah census.

The two complete graphs use MT Shem2531/2533/2535, MT held Shem2556/2558, MT death1933/1935 and SP/LXX death2393/2395. J's and P's pivots remain460 apart. The full correspondence is therefore supported by both a forward translation and a direct enumeration of the target records.

The result is exhaustive for these specified roles and family rules. It does not assert that no other operator graph can be built from the register. The500/100 mode supports the full three-Gear correspondence; the equally retained502/98 mode still needs its own boundary accounting.

## Verification and limits

28 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.

Completeness is conditional on the specified Shem birth/death family grammar. Aliases and the independent source-side census are not new historical witnesses.

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

**C407, proposed and unexecuted:** Check the500/100 and502/98 Noah-to-Shem bindings separately on all12 source-labelled loops. Test every endpoint and pivot with its required Gear and person rail, identify unavailable boundaries without creating Gear0, and distinguish repeated coordinate images from new source configurations.

The complete correspondence is now established under500/100. Testing the other valid binding is necessary to state its exact scope without treating502/98 as an invalid chronology.