# C314 independent finite arithmetic review

Date: 2026-09-24. Scope: the five C313 SP Shem-life cases only. This review is an independent calculation from frozen C313 data, the C286 source-state register, C287 completeness records, and the C271 ancestry register. It is not an independent historical witness or a source-prescription audit.

**Result: the explicitly reversed index and ancestry phase correspondence succeeds in all five source-labelled cases and all ten seasonal components.**

With the retained formulas

- `N(i,eN) = 3703 + i + eN`,
- `S(j,eS) = 3213 − j − eS`,
- `D(j,eS) = 2613 − j − eS`,

we obtain `N−S = 490 + i + j + eN + eS`. Integer annual indices and quarter-year phases imply that an exact 500 requires both `i+j=10` and `eN=−eS`: the phase sum can be only −1/2, 0, or +1/2, so a nonzero phase sum cannot compensate for an integer index error. All ten matches are unique within the fixed ten-by-ten component comparison.

| Shem index j | Noah index i | SP Gear / rail | Noah → Shem → Shem death, BC | C287 complete at i | C287 complete at j |
|---:|---:|---|---|---|---|
| 3 | 7 | G3 primary | 3710 → 3210 → 2610 | No | Yes |
| 4 | 6 | G3 companion | 3709 → 3209 → 2609 | No | Yes |
| 5 | 5 | G2 primary | 3708 → 3208 → 2608 | Yes | Yes |
| 6 | 4 | G2 companion | 3707 → 3207 → 2607 | Yes | No |
| 7 | 3 | G1 primary | 3706 → 3206 → 2606 | Yes | No |

All these source records use Cainan OFF, Terah0, and full430 after the inherited +155-year BC normalization. Each has 500 years Noah→Shem, 600 years Shem's life, and 1100 years Noah→Shem's death. The Noah endpoint and its exact cumulative ancestry and output phase are checked against C271. Regular source membership retains the same tradition, Gear, rail, and variants at all three endpoints.

The finite label map `J(j,e)=(10−j,−e)` is closed on `{3,4,5,6,7} × {−1/4,+1/4}` and satisfies `J²=id`. At the annual-index level its orbits are `{3,7}`, `{4,6}`, and `{5}`. The ten phase labels instead form five two-cycles: even j5 exchanges the two cumulative phase labels and has no fixed exact component. This is an involution of these comparison labels; it does not identify Noah with Shem or replace one event class with another.

The cumulative coordinate sum `qN+qS=−10570` follows directly from the same equations. It does not establish a new global Mirror axis or an independently source-prescribed choice. The match is a qualified comparison between different cumulative ancestries, with explicit reversal, and is not a same-component extension of C313.

**Boundary:** direct17K Noah legs exist in every row. C287's entire four-node form is a stricter requirement. At paired i7 and i6 it would need cumulative Cainan-ON Shem indices9 and8, outside the retained0…7 field. Accordingly three rows are complete at the paired Noah index, and only the center row is complete at both compared indices. The successes do not repair the failed full-form boundaries.

The reproducible script `C314_Independent_Math.py` accepts `--sources DIR` containing `PREVIOUS_DATA.json` (C313), `C286_DATA.json`, `C287_DATA.json`, and `C271_REGISTER.json`; `--out DIR` writes its results. All checks passed. No expanded census, new source endpoints, fitted anchors, or later step was executed.

Suggested best next bounded question: test whether this finite 500-year correspondence also preserves the admitted regular SP Noah/Flood/Shem event bindings, with primary/companion rails and exact output phases kept distinct. This tests how the successful biography interacts with the actual Flood event rather than accumulating another rearrangement of the same three equations.