# Independent review — C872–C889

**Result: PASS.** Eighteen completed research steps reviewed;68 independent checks passed. No blocking arithmetic, source-role, or scope issue found. No numbered step or canonical source was edited.

The independent calculations are recorded in `INDEPENDENT_C872_C889.json`. They were computed without importing or replaying the root research scripts. The reviewer also reread current source excerpts and checked all seven packet source hashes against the available originals.

## Mathematical review

- **One-Key grid theorem:** Correct for a nonzero grid spacingh. If input ishm andp/q is reduced, its image is again inhZ precisely whenq dividesm. Integer input domains23Z/69Z/299Z and Rounded same-grid domains115Z/345Z/1495Z are correct; intersections897Z and4485Z refer to separate images, as stated.
- **Five quarter-carrier rows:** Every exact output and membership flag was independently recomputed. The formulas175m/4,245m/6,525m/13 give quarter-grid divisibility conditions1,3,13. The full literalm={3,10,12,30,300} inventory is retained.
- **File46 complete head image:** All five heads, radii, and transformed coordinates agree. The430/30/430 image gaps are fractional independently of the anchor. The stronger C879 statement—no common anchor can make all the heads integral—is justified even if the common anchor itself were allowed to be nonintegral.
- **Nine two-Key words:** All final coefficients, final-only domains, stagewise domains, and5Z domains agree. P→J alone has the larger20631Z stagewise domain against6877Z final-only domain. J→P uses6877Z at both levels.
- **Six distinct-Key orders:** All prefix coefficients and domains agree. J-before-P gives158171Z; P-before-J gives474513Z. The fixed finite inventory remains separate from any claim of route authorization.
- **File51a chains and civil/Rounded distinction:** Both authorized E² chains and55016→59800→60000 agree exactly. The inputs are civil countsC; their neighboring Rounded widthsW_R are not substituted. The one-unit difference propagates as25/23 and625/529, matching the recorded diagnostic responses.
- **Held-anchor order:** Direct expansion gives(k−1)(l−1)(a−b). The complete five-head field has difference18340/1587 at every head and equal scaled adjacent gaps in both orders.
- **NT transfer:** Independently enumeratingr=0,…,76, including the requirement that the transformed index stays in that finite range, gives E={0,23,46,69}, P={0,69}, J={0}. File43's selected Melki/Cosam, Salmon/Hezron, Enoch/Jared/Adam rows were checked directly against the current literal table.

## Claim and source controls

The records consistently distinguish exact arithmetic from a licensed chronological route. File51a's civil-count chains carry source authorization; the complete image matrices and mixed-anchor compositions remain formal diagnostics. Fractional outputs are not rounded away or treated as invalid merely for leaving their starting grid. The familiar inherited completions are used to explain a general criterion, without being recounted as newly discovered scalar coincidences.

The NT names retain ledger roles. The finite slot-return rule neither creates historical dates nor supplies a probability claim. C888 appropriately explains that integer output and slot return are conceptually different requirements even though the finite index sets coincide for these particular Keys on70Z.

The general grid theorem presumesh≠0, conventionallyh>0 for chronological units. This is implicit in the word “grid” and causes no error in any executed case. A later synthesis may state the convention explicitly if presenting the theorem formally.

No second decimal inversion, new Gear transport, primer research, remote extension, or canonical amendment appears in this reviewed block.