490d C399 — Complete inverse-E closure test of common P/J source edges
24 September 2026. Step 6 of 10 in the C394–C403 authorization.
Exactly two of the ten admitted common P/J coordinate edges have a sourced E return. They are the two lower-full430 regular MT edges already found by the independent family enumeration. None of the six cumulative edges or the two original full430 regular edges supplies the required pivot.
Let a common P/J edge be u→v with signed gain h=v−u. Its existing J and P pivots are A=u−299h and B=u−69h. The unique E pivot returning v to u is
V=u+25h/2.
The resulting signed geometry is
| Relation | Exact value |
|---|---|
| V−A | 623h/2 |
| V−B | 163h/2 |
| Original529 outer point minus V | 435h/2 |
| Original outer point minus A | 529h |
These formulas apply to h=+2 in the regular field and h=−2 in the cumulative field. They explain the algebra without supplying missing sources.
| Existing common edges | Required E return pivots | Source result |
|---|---|---|
| Six cumulative edges | −5913 through−5908 | All absent from the48-record cumulative menu |
| Two lower-full430 regular edges | 3056,3058 | Existing MT Noah births |
| Two original full430 regular edges | 3271,3273 | Absent from the528-record regular menu |
Membership was checked across every role in the respective menus, not merely Noah. The two positive return pivots have the exact MT Noah Cainan-OFF/Terah0/full4301 records. Their correspondence with C396 is exact.
The new source circuits therefore have a complete, finite domain within the existing common-P/J graph. They have not become a global extension of the original17K pairing or a source-supported cumulative circuit. The earlier C386 E/J square remains a separate valid construction with its different held pivots and source-role change.
Verification and limits
44 main checks and 17 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 signed formulas are exact but do not admit their missing pivots. Source closure is established only for the two existing regular coordinate graphs; source aliases remain separate provenance.
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
C400, proposed and unexecuted: Classify every admissible path in the two sourced E/P/J graphs by its endpoints and operator counts. Derive an exact normal form for open and closed paths and check when distinct words represent the same affine map, without expanding the source graph.
The source domain is now complete. Extending the circuit result to the full admitted path grammar can identify the actual reduced algebra rather than relying on isolated commuting examples.