490d C458 — Source Equations for the Two 2K Interval Ladders
25 September 2026 UTC. Step 14 of 25, C445–C469.
Both admitted2K ladders resolve exactly into source terms. Their input gaps are350+650+60−2 and600+650−130−60−2, each1058. Both first output gaps are500+650=1150; the complete second output is600+650=1250. Eleven required coordinates have16 retained source records.
| Pair | Source binding | Gap |
|---|---|---|
| 1895,2953 | MTNoah death G3/C0/T0/F0 to SP or LXX-OFF Flood start G2/C0/T1/F0 | 1058 |
| 1745,2895 | MTShem death to SP or LXX-OFF Flood start, bothG3/C0/T0/F0 | 1150 |
| 2148,3206 | MTShem death G2/C1/T1/F1 to SP-primary or LXX-OFF Shem birth G1/C0/T0/F1 | 1058 |
| 2020,3170 | MTShem death to SP or LXX-OFF Flood start, bothG3/C0/T1/F1 | 1150 |
| 1956,3206 | MTShem death to SP-primary or LXX-OFF Shem birth, bothG1/C0/T0/F1 | 1250 |
The shared first pivot3620 is SP companion Noah birth G1/C1/T0/F0. The second pivot2756 is SP-primary or LXX-OFF Noah death G1/C0/T0/F1. The SP primary death qualification remains inclusive/harmonic.
The signed23→25 units are−75 and−29 for the first seed,−64 and−18 for the second seed, then−32 and+18 for its continuation. Each endpoint equation is checked independently against the source-state formula. The final1250 interval combines the600-year Shem-birth/death separation with the650-year MT-to-SP/LXX-OFF offset; it is a direct sourced span.
Both1150 output pairs differ by the same275-year translation. The relationship between their1058 inputs is therefore worth resolving before the two examples are counted as separate structural findings.
Verification and scope
26 calculation checks and 19 source/predecessor authentication checks passed. Main result files replay byte for byte. These are root-agent checks, not an independent-agent review or independent historical witnesses.
All source labels remain distinct. The650 is the normalized MT-to-SP/LXX-OFF offset in this register, not a blanket native-LXX offset for every state.
Mirror/529 focus: K=529, E=25/23, P=70/69; J=300/299 remains File02 gap conversion, not a calendar date operator or Residue Protocol. Regular Gears are Noah/Shem/Flood only. SP person companions are local to Noah/Shem; Flood close is an independent event boundary. Both500/100 and502/98 bindings remain valid; SP primary Noah death retains its inclusive/harmonic qualification. LXX native Cainan is ON; OFF is theoretical removal. Global paired Mirror January AD1; local junction January AD2; no civil year zero. No new source point, phase, fitted axis, upstream/cumulative Gear, primer reopening or canonical/graph edit. Capture now; canonize later. Coordinates, source labels, context aliases and derived comparisons remain distinct. Counts are not independent historical witnesses.
Best next bounded step
C459, proposed and unexecuted: Place every admitted fixed-pivot E input in its signed23-unit lattice and determine whether the two transported2K seeds are one translated four-point configuration, including the corresponding source-state translation.
The complete source equations suggest that the two first-stage ladders are translated rows of one configuration. The signed23-unit census can verify that relation and its uniqueness in the fixed menu.