490d C390 — Unconditioned finite regular E/J family census

24 September 2026. Step 7 of 10 in the C384–C393 authorization.

A fresh census of the fixed regular family motif recovers exactly the same four coordinate E/J squares, with eight SP/LXX context lifts. No529 test, prior common-P/J membership, or17K requirement was used to select the candidates.

The domain was fixed before calculation:32 source Noah families and32 target Noah/Shem families, each labelled by tradition, rail, Cainan, Terah60 and full430; the same adjacent Gear transition1→2 or2→3; and source-family next-Gear Flood start or close as the lower J pivot. The target-family next-Gear Shem death is the upper J pivot. The target Noah births hold E at each slice. These choices give4096 labelled trials.

Sixteen trials retain both vertical E edges. Choosing the correct existing Flood boundary leaves eight complete squares. They are precisely MT restored-Cainan/Terah60 Noah inputs, with SP primary or LXX Cainan-OFF/Terah0 target contexts, in the two full430 branches and two Gear transitions already identified. The four coordinate squares are not multiplied into eight independent witnesses by their tradition labels.

Every upper Shem J edge works because its source lifespan is600 and the adjacent Gear difference is2. The selective part is the conjunction with the two vertical E edges and the original Flood-pivot J edge. The successful rows have N−x460 and N−A1058 as observed outputs. These numbers were not filters.

This is a complete census within the named coherent same-Gear family motif, not over arbitrary choices of endpoints or pivots. Its denominator4096 is a verification inventory, not a probability model: source families, translations, aliases and operator equations are strongly related. The result supplies an exact finite comparison and motivates deriving why1058 follows from the admitted source geometry.

Verification and limits

67 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 motif fixes coherent same-Gear family labels and stated pivot roles. No significance probability, independent witness claim, arbitrary-source exhaustiveness or source-prescribed pairing is inferred.

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

C391, proposed and unexecuted: Derive the source equations behind the census and test whether the admitted E/J motif is equivalent, within this fixed domain, to the original common P/J529-side construction. State which relations are forced and which still require source membership.

The fresh census found no extra contexts. A derivation can separate a structural consequence of500,600 and the operators from apparent numerical corroboration and connect the two finite constructions directly.