C288 — Finite path closure and the formal K module
24 September2026. This checks the operational limits of C287's normal form.
The complete four-node rows support four increasing paths to the same19K endpoint. They supply no K or4K edge. The arithmetic gcd of their widths is K, but a gcd is not automatically an available chronological translation.
At each complete row, normalize the first source to0. The four positions are0,2,17,19 in K units. The admitted endpoint comparisons have widths2,15,17,19. All increasing paths from the first to the last are:
| Path widths in K units | Total |
|---|---|
| 19 | 19 |
| 2+17 | 19 |
| 17+2 | 19 |
| 2+15+2 | 19 |
These are four decompositions of one source-to-target comparison, not four independent endpoints or witnesses. At the two lower rows, only positions0,2,17 have supplied source identities, giving17 and2+15. The required fourth Noah node remains absent.
The formal identity17K−8(2K)=K shows that the additive group generated by the numerical widths contains K. The finite source structure does not supply eight repeatable2K moves. Reverse traversal must start at the actual endpoint of the preceding edge; arbitrary subtraction of widths is not composition of tagged paths. Every legitimate closed traversal has zero net coordinate displacement.
Likewise19K−15K=4K is a difference between two interval widths, not a difference between a pair of the four nodes. It therefore does not supply an event endpoint at2116 by itself.
The shared offsets i0–5 are distinct residues modulo529. K comparisons remain within each residue class. Annual adjacency between cumulative field members has width1 year, so the complete annual source domain is not a529-spaced lattice. The K structure is a qualified family of comparisons selected within it.
This distinction retains the useful reduced arithmetic without claiming an unrestricted group action on chronology states. The evidence lists every positive edge and increasing path at all six rows. Twenty-one mathematical/domain checks and six authentication checks pass; exact replay agrees. Author review only; no canonical graph amendment.
Best next bounded step — C289: express the retained local branch maps in the new normal form, including their common-translation freedom. Determine which map constants become pure K multiples and which assumptions still fix their absolute placement. High is sufficient.