490d C473 — Typed Translations Between the Three 1250 Families

25 September 2026 UTC. Step 4 of 10, C470–C479.

The complete matched families form 24 source-context triplets. At each Gear/Cainan/Terah/full430 setting:

• A→C adds500 to both coordinates: MT Shem death→MT Flood start, and SP-primary/LXX Shem birth→Noah birth.
• SP C→B subtracts1 from both coordinates: MT Flood start→Flood close, and SP Noah primary→companion.
• Therefore B=A+499 at the coordinate level.

The+500 correspondence has48 typed label maps, including SP and LXX alternatives. The−1 correspondence has24 typed SP maps. LXX Noah records acquire no companion; mapping a LXX primary coordinate to the B-family SP companion would change source tradition and is not counted as a same-tradition map.

All three admitted E outputs translate to sourced B and C output intervals. But their six translated single-step comparisons fail complete source support. Each requires an absent upper input and an absent translated pivot. Affine covariance alone therefore does not supply an admissible translated E path.

A selective reference search located the related Noah-held2K ladder in C429, where2648→2556→2456 about3706 was explicitly retained as a prior result. That packet was authenticated (24 payloads and12 certificate bindings). The+500 translation of the current split-pivot ladder has those same input and final interval endpoints, but a different proposed intermediate and different pivots. Comparing these factorizations is the next precise question; the old Noah-held ladder is not being claimed as new.

Verification and scope

72 calculation checks and 22 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.

The role correspondences are comparison maps, not additions to the active E/reflection operator menu. Flood close and SP companion remain distinct mechanisms. Prior C429 results retain their historical status.

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

C474, proposed and unexecuted: Compare the+500 translation of the established split-pivot2K ladder with the already recorded C429 Noah-held ladder at3706 and the inherited C28717K corner. Verify equality of the whole affine maps while checking each factorization’s anchors and intermediate records separately.

The source-family translations preserve endpoint intervals but fail for the translated stepwise E paths. The authenticated older Noah-held ladder supplies a contrasting, already sourced factorization of the same overall expansion.