### An integral completion can retain fractional internal positions

File_52c’s four admitted completed paths all have one-pass total 14720. The stated Priestly conversion gives 16000, and holding 6 BC places the completed endpoint at 16006 BC. Examining the entire paths reveals which further information that clean endpoint preserves.

The regular Flood path has transformed components 4100,5010,5610. Its successive prefix sums have residues 6,2,0 modulo 23. The regular Noah path gives 6,4,0; the finer Flood–Noah path gives 6,2,4,0; and the cumulative primary path gives 6,9,0. Consequently every proper prefix remains fractional after multiplication by 25/23, while the final sum becomes integral.

Uniform conversion is still additive over exact rational values. Transforming each component and then summing agrees with transforming the whole duration. The extra requirement that every intermediate position remain on an integer grid is a separate condition. A consecutive block has integral transformed length precisely when its original sum is divisible by 23; these four paths permit only the final cut.

This gives a coherent mathematical comparison without changing any source boundary. A coarse numerical total can display integer closure while the retained internal path carries rational coordinates and its original event roles. The componentwise rational display is a conditional extension of the source’s whole-span conversion, rather than an independently adopted historical chronology.

*Sources:* File52c §3.4, especially the four-path table; Interface packet IQ01–IQ03 and calculations.json:completed_path_prefix_field; Key-constraints packet §C.

*Research record:* C932–C952, C1058–C1068.