# C292 — All equal-spacing translations of the two source pairs

24 September2026. Specificity test for C291's16K interlacing.

**Equal spacing admits four exact translations:13K,16K,18K and21K.** Only16K and18K interlace the two pairs with spacing K. Thus the supplied16K bridge has a precise role, but equal spacing alone does not uniquely require it.

Write W=mK. After translating the cumulative pair, the four coefficients are m−12,m−10,5,7. The common annual offset cancels from every difference. Both within-pair orders are fixed, leaving six possible orderings. Solving equality of the three successive gaps in each ordering gives:

|W|Ordered coefficients|Spacing|Order|
|---:|---|---:|---|
|13K|1,3,5,7|2K|S,F,R,N|
|16K|4,5,6,7|K|S,R,F,N|
|18K|5,6,7,8|K|R,S,N,F|
|21K|5,7,9,11|2K|R,N,S,F|

The remaining two orderings have inconsistent gap equations. This is an exact affine solution across all real W, not a search over conveniently chosen integer bridges. Each solution is positive and preserves the inherited seasonal offsets.

The13K and16K carriers already occur in the retained research. Their new applications here remain separately source-tagged. In particular, translating this cumulative pair by13K does not identify it with the older SP Creation-to-MT Flood13K path merely because the widths agree. The18K and21K solutions are diagnostic alternatives; this step does not adopt them as new chronology operations.

The result identifies what each geometry actually says. Interlacing by K leaves a two-way orientation choice about the coincident17K correspondence. Separated pairs can also form an arithmetic progression, but with spacing2K. Equal spacing is therefore weaker than a complete source and operation specification.

All four-node statements apply where N is admitted: four SP rows and two additional LXX labels. The missing Noah records at i0/1 remain missing under every W. Changing the translation cannot repair source coverage.

Five aggregate exact checks cover all six ordering domains and the complete solution set; seven authentication checks pass. Exact replay agrees. Author review only; no new bridge or axis is adopted.

**Best next bounded step — C293:** use the retained13K carrier's1,3,5,7 arrangement and test it against the fixed4232=8K reflection composition. Does the row midpoint coincide with that fixed comparison center, with complete source support? High is sufficient.