490d C337 — Source factorization of the two 2K sides
24 September2026. Step4 of10 in the C334–C343 authorization.
The all-K rectangle requires three distinct source selections: one for each1058 side, then equality of their annual comparison indices. The result factors C287's four complete SP paths without introducing any endpoint.
For cumulative Cainan states s,f∈{0,1}, Shem index j and Flood index i in0…7:
q−S = 600 + 460(s−f) + i−j.
Therefore1058 requires ON-Shem, OFF-Flood, and j=i+2. This leaves six left pairs, i0…5.
For SP Noah in Gear g, MT Flood in Gear g+h, restored-Cainan c and Terah t flags, SP companion flag e and Flood-close flag k:
N−R = 1250 − 130c − 60t − 2h − e + k.
Within the existing three-Gear register,1058 requires c=t=1, h=1 and e=k. Thus a primary Noah pairs with Flood start, and a companion Noah with Flood close. Four right pairs remain, i2…5. The flags represent distinct person and event classifications; their equality does not turn Flood close into a person companion.
Joining the six left pairs and four right pairs gives24 admissible endpoint combinations. Each has two1058 sides, but its cross-spans are 8993+i_right−i_left. Exactly four joins have both17K spans: the equal-index joins i2,i3,i4,i5. Their full coordinates agree exactly with C287. The off-diagonal joins are controls; they do not create new source states or undermine the matched examples.
| Requirement | What it selects | What remains free |
|---|---|---|
| Left2K | ON/OFF profiles and relative annual offset2 | Six annual positions |
| Right2K | Restored MT context, next Gear, matching primary/start or companion/close | Four source paths |
| Cross17K | Equal left/right comparison index | Four complete C287 paths |
The regular formula also exposes a useful next question: its unadjusted source comparison is1250, while the selected1058 satisfies 1058×(25/23)²=1250. The intervening value1150 may correspond to an already admitted MT Shem endpoint. This deserves an endpoint-qualified test before being treated as an expansion chain.
Verification and limits
427 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.
Necessary and sufficient only within the frozen domains and stated SP/MT contexts. The24 pairings are mathematical controls, not independent observations. No source prescription for selecting the criteria has been established.
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
C338, proposed and unexecuted: Test the duration ladder1058→1150→1250 under E=25/23 against already admitted same-Gear MT Shem and Flood endpoints, with SP Noah held as the existing comparison endpoint. Check primary and companion cases and make no new endpoint or anchor assignment.
The factorization produces1250 as an actual unadjusted source interval. Its exact double-E relation to1058 offers a constructive endpoint test;1150 must be checked against source records, not presumed.