490d C375 — Paired P/J source-corner conjugacy
24 September 2026. Step 2 of 10 in the C374–C383 authorization.
The two source fields carry exact conjugate P/J comparisons at the paired Flood pivots. Let r_C(x)=C−x be the endpoint exchange already determined by the source corners, and let F_a^λ(x)=a+λ(x−a). Direct comparison of affine coefficients gives
r_C ∘ F_a^λ = F_(C−a)^λ ∘ r_C.
For every complete C287 row, the cumulative OFF Flood exchanges with the restored-MT Flood, and the cumulative native Flood exchanges with the SP Flood. Therefore the identity applies to both J=300/299 and P=70/69 at their actual paired pivots. This is a whole-affine identity between these specified comparisons, rather than a coincidence at the single meeting input.
Actual input/output membership imposes an additional restriction:
| i | Rail | Cumulative input/output under the exchange | Actual MT Noah input/output | Source lift |
|---|---|---|---|---|
| 2 | Companion | 3245 → 3247 | 3246 → 3248 | No |
| 3 | Primary | 3246 → 3248 | 3246 → 3248 | Yes |
| 4 | Companion | 3247 → 3249 | 3248 → 3250 | No |
| 5 | Primary | 3248 → 3250 | 3248 → 3250 | Yes |
The companion rows still pair the Flood boundaries correctly, but their reflected cumulative Shem members fall one year below the actual MT Noah records. No MT person companion is created to fill those gaps. This identifies the source-membership obstruction behind the primary selection.
The correspondence also reverses phase: r_C(x+u)=r_C(x)−u. Eight opposite-offset pairs pass; none of the eight same-offset pairs does. Moving the diagnostic sum to C+2u would restore a same-offset equation, but would change the reflection with the component; no such physical-time axis is adopted. The four source rows also have four different center sums.
The cumulative and regular P/J results consequently share an exact affine structure and are not independent confirmations. The source identities, role exchange and phase ancestry remain necessary qualifications.
Verification and limits
47 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.
Conjugacy is between specified affine comparisons. It does not supply a common global axis, a textual prescription for joint use, or missing MT companion records.
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
C376, proposed and unexecuted: Enumerate all nontrivial P/J endpoint comparisons at these same frozen Flood pivots within the existing cumulative48-member and regular528-record Noah/Shem/Flood menus. Preserve source identities, separate fixed points, and check whether the common repair extends to further source-supported edges.
Exact conjugacy explains the matching local comparisons but does not establish closure on the source register. A finite source-edge census can show whether the repair is isolated or belongs to a larger existing chain.