# C289 — Pure-K maps and the remaining placement freedom

24 September2026. This lifts the retained local maps into C287's source normal form.

**At the retained placement, four map constants become10K,12K,7K and5K. A common translation u still remains free.** The simplification makes the source correspondence clearer without independently determining the absolute placement.

|Branch|Map|
|---|---|
|Cumulative LXX OFF Flood|q+10K+u|
|Cumulative LXX ON Shem, before Mirror|q+12K+u|
|The same Shem branch, after Mirror|−q−12K−u|
|Reflected regular SP Noah|y−7K+u|
|Reflected restored MT Flood|5K−u−y|

With the source coordinates S=−12K+i, F=−10K+i, R=5K+i and N=7K+i, both forward cumulative images are i+u. The regular Noah image is also i+u wherever its source record exists; the reflected Shem and regular Flood images are −i−u. The routes commute for every u, including both physical seasonal components with the correct reversal.

These constants come from the retained state lift, not a new numerical fit:5290=10K;5888+460=6348=12K;3703=7K;3105−460=2645=5K. C270's actual coefficients are checked against those identities. The mixed-state forward translation difference is2K=1058; removing the cumulative460 lift restores598.

The familiar placement is u=0. Source correspondence, coincidence of the selected forward images and their global reflection all persist as u varies. Requiring every intercept to be a K multiple restricts u to an integer multiple of K; it still does not uniquely select zero. This follows immediately from the coefficient10K+u, with the other coefficients giving the same congruence.

No new historical state is selected from that formal family. Integer u preserves annual-center labeling, while the existing placement remains the reference. The missing Noah records at i0/1 stay absent for every u: moving a plotted image cannot create its source ancestry.

The pure-K form is therefore a useful state-qualified coordinate description. It supports a simpler algebraic account of the observed maps, while retaining C276's distinction between relative alignment and absolute placement. Thirty-three mathematical/domain checks and eight authentication checks pass; exact replay agrees. Author review only.

**Best next bounded step — C290:** apply the fixed25/23 dilation to the four-node source pattern, both globally and about its stated annual offset. Check actual source-center and seasonal compatibility, especially the apparent5290→5750 Cainan correspondence. High is sufficient.