490d C317 — Source-qualified600 rectangles at17K

24 September 2026. Step 4 of the ten-step authorization C314–C323.

The five selected biographies support a coherent 600-year rectangle with two17K translations, using the existing C271 source identities:

For i3…7, q−p=n−f=600 and f−p=n−q=8993=17K. Source pairs, regular roles, Gear/person rail and both exact phases are admitted. In ordinary BC chronology Shem precedes cumulative Flood; the reflected output is a later-numbered Noah compared with the Flood boundary. Coordinate order is not a substitution of event identity.

The cross-width matrix, rows p/q and columns f/n, is [[8993,9593],[8393,8993]]. This is not an all-K rectangle:600 is not an integer multiple of529. The six pairwise duration equations have rank3 and leave one common translation free. In particular, once the source and output600 sides and one8993 path are fixed, the second8993 path is forced. Five annual stations and ten phase checks are not independent historical measurements of17K.

Comparison with the stronger C287 form retains q and n but replaces the other corners by S=p−458 and R=f−458. Its side becomes600+458=1058=2K and its cross-widths become[[17K,19K],[15K,17K]]. The new corners change source type/context: cumulativeON Shem at i+2 and regular MT restored-context Flood. This is not a free movement of an unchanged source event.

Both corners are admitted only at i3,4,5 within the current five-case domain. For i6/7 the ON-Shem indices8/9 and regular required Flood2651/2652 remain unavailable. The coherent direct600 rectangles at those rows do not supply the missing all-K corners.

Verification and limits

46 main assertions and 14 frozen-input/predecessor checks passed. A second main replay produced byte-identical DATA and RESULTS. Separate review scope and agreement are recorded in the evidence. All frozen originals and standing controls remain unchanged. These are verification counts, not independent-witness counts.

The two rectangle paths are algebraically dependent given equal600 sides. The458 replacement changes source labels; it is not an unrestricted translation rule.

Mirror/529 remains the focus: K=529, E=25/23, P=70/69. Global paired Mirror January AD1; local junction x=1 January AD2; no civil year zero. Regular Gears remain Noah/Shem/Flood only. SP minus-one companion is local to SP Noah/Shem. No cumulative/upstream Gear, new endpoint or phase, fitted anchor, global axis, z-law, canonical or graph edit. Capture now; canonize later. Source state, event identity, generated coordinate and phase ancestry remain distinct. Exact arithmetic and source membership do not establish intention or independent historical witnesses.

Decision after this result

C318, proposed and unexecuted: Audit the paired458 replacement as explicit source-state operations: cumulative CainanON plus annual-index change, and regular SP→MT Flood comparison plus declared restoration and Gear change. Check every intermediate source state and boundary.

This is the substantive link between a valid600 biography and the2K+15K framework. Its source operations and domain must be established before treating the numeric458 coincidence as a permitted composition.