# 490d C387 — Companion and502/98 controls for the E/J square

24 September 2026. Step 4 of 10 in the C384–C393 authorization.

**The E/J square realizes the primary500/100 binding, while the companion and502/98 alternatives remain valid source states.** The alternatives fail the E side of this particular diagram; their existing horizontal Shem J edges continue to work.

The finite test uses the four C385 regular repairs, SP primary and companion contexts, and LXX Cainan-OFF contexts. It tests500/100 and the recorded previous-Gear Shem realization of502/98. There are18 available labelled cases. Six additional proposed cases would require Gear0 Shem and were left unavailable rather than generated.

|Available mode|E image minus its selected Shem birth|Result|
|---|---:|---|
|SP primary or LXX,500/100|0|All four source edges pass|
|SP companion,500/100|25/23|E edge fails|
|SP primary or LXX,502/98|2|E edge fails|
|SP companion,502/98|71/23|E edge fails|

Eight labelled cases are the already established positive squares. Ten alternatives fail the vertical E comparison. In **all18 available cases**, the existing Shem-birth edge still has598 and600 as its distances from the next selected Shem death and therefore supports J exactly.

This cannot be repaired by silently changing the MT input. Inverting E at the selected Shem endpoint gives a unique required input. The primary500/100 cases recover the admitted MT Noah input. SP companion500/100 would require the absent corresponding MT minus-one Noah companion. The502/98 alternatives require fractional MT inputs. None is admitted as a new source endpoint.

The500/100 and502/98 source modes are retained, following File18 and C339. The result is conditional on the chosen diagram; it is not evidence for deleting a valid source mode. The SP companion mechanism stays local, and no MT or LXX companion is manufactured.

## Verification and limits

86 main checks and 16 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 test uses only existing regular Gear1–3 records. Conditional success does not select one valid historical binding over another; coordinate aliases and verification counts are not 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

**C388, proposed and unexecuted:** Compose the admitted E/J squares across their complete two-edge regular source paths. Check whole-map equality, distinguish it from endpoint agreement of the other P/J words, and locate the actual Gear and E-iteration boundaries.

The source and mode qualification is now explicit. The next bounded question is whether the surviving squares form a finite compatible strip, rather than isolated endpoint coincidences.