# C313 independent arithmetic review

The fixed five-case test independently confirms one typed center match, at cumulative LXX OFF annual index **j5**, and four failures. The calculation uses completed C312 data, the frozen C286 source-qualified target register, C287's four-node data, and C307's phase-orientation record. It does not rerun an endpoint census or add source states.

## Center and complete-pair results

Normalize the existing SP Shem-life birth/death pair from OFF/Terah+60/minimum215 to OFF/Terah0/full430 by adding **215−60=155** to both BC labels. Every normalized endpoint is already present in C286 with the stated SP Gear and rail, and every pair retains600 years. The same cumulative LXX OFF Flood source q supplies a17K Noah requirement n=q+8993; each required Noah is also present in the same frozen register.

|j|Shem Gear/rail|Normalized Shem life, BC|17K Noah, BC label of mirrored source|Noah Gear/rail|Noah−Shem birth|Typed500 result|
|---|---|---|---|---|---|---|
|3|G3 primary|3210→2610|3706|G1 primary|496|Fail|
|4|G3 companion|3209→2609|3707|G2 companion|498|Fail|
|5|G2 primary|3208→2608|3708|G2 primary|500|Pass at centers|
|6|G2 companion|3207→2607|3709|G3 companion|502|Fail|
|7|G1 primary|3206→2606|3710|G3 primary|504|Fail|

Every source-qualified Shem pair has its own same-Gear, same-rail Noah500 years earlier in the source register. Only j5's own Noah is also the same-q17K-selected Noah. Thus no genealogical age was altered: the failed rows compare distinct, explicitly labelled states.

The direct17K legs for j6 and j7 are supported by the admitted cumulative Flood and regular SP Noah endpoints. They do **not** supply complete C287 four-node instances: the additionally required cumulative LXX ON Shem indices j+2 would be8 and9, outside the inherited0…7 field.

## Phase restriction

Write reflected coordinates in the existing Mirror convention. With q=−5290+j, the normalized reflected Shem birth and death are s=−q−2077 and d=s−600; the17K Noah is n=q+8993. Consequently n−s=490+2j.

For a common cumulative component q+ε, C307's recorded action gives n+ε but s−ε and d−ε. Normalization adds a fixed state shift and cannot reverse that orientation. The resulting Noah→Shem gap is490+2j+2ε. At the unique j5 center match, ε=−1/4 and+1/4 yield **499.5 and500.5**, respectively. The full Shem-life component width remains600 throughout.

The source biography at G2 primary supports Noah3708→Shem3208 with500 years, including equal output phase offsets. The unordered phase-pair sets also translate by500. Joining those equal output phases to the two different transports, however, assigns them opposite cumulative ancestry. Therefore C313 establishes a valid source biography and center coincidence; it does not establish a500-year binding for both components transported from the **same** cumulative component.

## Scope and reproducibility

`C313_Independent_Math.py` was written independently of the author's main C313 calculation. It supports `--sources DIR --out DIR`; the source directory must contain PREVIOUS_DATA.json (C312 data; C312_DATA.json is also accepted), C286_DATA.json, C287_DATA.json and C307_DATA.json. Its results include source hashes and the exact Fraction-based phase ledger. All finite-domain, state-membership, center-gap and phase assertions passed. From the evidence packet root, run `python3 evidence/C313_Independent_Math.py --sources sources --out independent_replay`.

No source-prescribed selector, historical intent, new axis, new source node, interpolated center or broader Gear scope follows. No C314 pairing or further scan was executed.