C278 — The finite 598/600/602 correspondence
24 September 2026. The next step was selected after C277's four complete cross-index alignments.
The register supports four598 comparisons, six600 comparisons and four602 comparisons. The two-year shifts are mutual inverses on their permitted domains, but they do not close on the whole finite register.
For Noah index i and regular Flood index j, both in2–7, the reflected source difference is600+i−j. The exact conditions are:
| Width | Index relation | Allowed Noah indices | Count |
|---|---|---|---|
| 598 | i=j−2 | 2–5 | 4 |
| 600 | i=j | 2–7 | 6 |
| 602 | i=j+2 | 4–7 | 4 |
Thus the reverse602 relation is as well bounded as the598 relation. The central600 relation retains the same Gear and matched selector. The outer comparisons change the Gear by one while retaining primary/start or companion/close status. They are cross-event comparisons, not revised Noah lifespans.
The annual indices split into two chains:2→4→6 for companion/close and3→5→7 for primary/start. Upward shift is defined at2–5; downward shift at4–7. Each returns the other on its domain. The terminal rows cannot be extended without generated records outside the held register.
The map algebra explains why these shifts occur. For the existing global Mirror M(x)=−x and local reflection J₁(x)=2−x, J₁M(x)=x+2 and MJ₁(x)=x−2. This holds for centers and physical ±1/4 components. On the ambient coordinate line the two reflections generate translations by even integers. On source-tagged records, however, J₁ supplies six role-swapping pairs while M supplies four. The ambient algebra therefore restricts to partial correspondences, not a global group action on the twelve records.
The role qualification matters. The Flood image at0 is a numerical fixed point of global Mirror, but it does not supply a missing Noah source at0. Counting untyped plotted positions would lose precisely the ancestry restriction tested here. No new axis, calendar phase or regular Gear is added.
The useful advance is a finite operational grammar: same-index600, downward-index598, upward-index602, with explicit inverse domains and boundaries. The arithmetic alone does not show that the two reflection requirements are historically necessary or that every chronology shares this domain.
Thirty-two mathematical/domain checks and seven source-authentication checks pass. Both generated readouts replay identically. The full fourteen-row triad comparison is in DATA.json. Author review only; standing controls remain unchanged.
Best next bounded step — C279: connect the six cumulative598 alignments to the already documented8395 correspondence. Test whether the8395 and8993 routes terminate at the same regular Flood source with their difference supplied by the tagged598 edge. This could join two established width families without introducing a new calibration. High is sufficient.