490d C396 — Complete regular family census of E return loops
24 September 2026. Step 3 of 10 in the C394–C403 authorization.
The full finite family census finds six coordinate E/J loops, including two MT loops that also have a sourced P return. The initial SP/LXX cases were part of a larger, precisely bounded motif.
No tradition,23/25 distance,529 relation or prior P/J hit was used as a filter. The4096 trials pair32 moving-Noah families with32 held-Noah families, use the two existing adjacent Gear transitions, and test Flood start and close separately. E must carry higher-Gear Noah to lower-Gear Noah, and J must return it using the moving family’s next-Gear Flood boundary.
| Held E Noah | E arrow | J return pivot | P return pivot |
|---|---|---|---|
| 3056 | 3033→3031 | 2433, MT Flood start | 2893, sourced SP/LXX Flood start |
| 3058 | 3035→3033 | 2435, MT Flood start | 2895, sourced SP/LXX Flood start |
| 3705 | 3682→3680 | 3082, SP Flood close | 3542, absent |
| 3706 | 3683→3681 | 3083, SP/LXX Flood start | 3543, absent |
| 3707 | 3684→3682 | 3084, SP Flood close | 3544, absent |
| 3708 | 3685→3683 | 3085, SP/LXX Flood start | 3545, absent |
There are12 labelled E/J comparisons at these six coordinate loops. SP primary and LXX source aliases account for several labels; they are not independent witnesses. The companion loops use existing SP companion Noah endpoints and an independently typed Flood close. No companion rail is imported into a Flood record, MT or LXX.
Every successful case has the same recorded context relation: held Noah Cainan-OFF/Terah0/full4301 versus moving Noah Cainan-ON/Terah60/full4300. The net25-year offset and adjacent Gear difference2 force23→25 under E. This emerged from the census.
The two MT return edges3031→3033 and3033→3035 are exactly the earlier lower-full430 common P/J repairs. Their existing P pivots2893/2895 have correctly labelled SP/LXX Flood records. Thus all three operators now occur in a finite two-vertex source graph: P and J move forward, while E returns. This does not say that the affine maps are inverses; C395 already explains why endpoint return and map identity differ.
Verification and limits
94 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.
Exhaustiveness is limited to the named coherent same-Gear family motif. Labels, source aliases and repeated arithmetic do not multiply historical witnesses; no missing pivot or new Gear is introduced.
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
C397, proposed and unexecuted: Determine the affine composition structure of the two fully sourced E/P/J loops. Test whether the closed P–E and J–E circuits commute at each source vertex, classify their multipliers, and separate loop equality from equality of the three individual operators.
The complete census reveals two source graphs containing all three operators. Their closed-loop algebra is the strongest new structural question before testing broader transport or Mirror claims.