# Independent review of C632–C641

**PASS.** No numerical correction is required. The conditional minimum in C640 is correct, and the source/derived distinctions are appropriate.

## Evidence and replay

Read File70 §§6.1–6.7 directly, the ten completed journal records, and all ten scripts. Replayed C632–C641 using the read-only verifier:10 steps and27 recorded checks matched,25 bundled source snapshots and25 current originals matched, and the journal bytes were unchanged. The artifact helper computed the C632 packet's exact hash without writing it. The detailed result is `REVIEW_C632_C641_REPLAY.json`.

`SOURCE_BINDINGS.json` was created from root's `SOURCE_ORIGINALS.json`, converting workspace paths to portable relative paths. No journal, research.py, numbered script, or canonical source was changed.

Separately recomputed the nine-entry matrix, exact Key normalizations, calendar LCMs, count increments, and a full positive residue period of the three-congruence system.

## Findings by step

| Step | Result |
|---|---|
|C632|The frozen node packet and all three primary registers match File70. Restored Cainan remains a comparison state.|
|C633|All corresponding spans equal2366, and both7+7 rails are retained. The actual BC-label map is Joseph_i=Creation_i−2366.|
|C634|The complete cross-rail matrix and multiplicities1/2/3/2/1 are correct. Generated intermediate comparisons are distinguished from source-selected alignments.|
|C635|13+2340+13=2366, Joseph's ages17/30, and the numerical6.5×(364−360)=26 relation are exact and source-controlled.|
|C636|The three normalized counts are6.5/7/6.5. P/E/J produce54600/23,58800/23,54600/23, preserving13/14/13 half-unit coefficients.|
|C637|The two generated endpoints differ13 because their held heads differ13. A frame translation aligns them. Fractional outputs remain diagnostics.|
|C638|13×182−14=14(169−1)=14×168 is exact. The rail-width14 explains this source member's change in half-unit count.|
|C639|182m−14=168n reduces to13m−12n=1, with all integer solutionsm=1+12q,n=1+13q. Positivity requiresq≥0; the source member hasq=1.|
|C640|The complete system has exactlyT=2366+32760q, withm=13+180q,n=14+195q,p=13+182q. Its least positive member is2366 and13/14/13.|
|C641|lcm(168,180,182)=32760; full-calendar LCM65520 is twice that value; the prior common-volume coefficient327600 is numerically ten times it. The comparison must retain its declared units.|

## Independent proof of C640

With the source offsets14 and26 held fixed, writeT=182m. The other equations yield:

$$
182m-14=168n\quad\Longrightarrow\quad13m-1\equiv0\pmod{12}
\quad\Longrightarrow\quad m\equiv1\pmod{12},
$$

$$
182m-26=180p\quad\Longrightarrow\quad2m-26\equiv0\pmod{180}
\quad\Longrightarrow\quad m\equiv13\pmod{90}.
$$

The residues are compatible modulo gcd(12,90)=6. Their joint period is lcm(12,90)=180, and the unique residue is13. Hence:

$$
m=13+180q,\qquad T=182m=2366+32760q.
$$

The other two counts follow by substitution:

$$
n=14+195q,\qquad p=13+182q.
$$

For positive integer counts, q≥0. At q=−1, T and all counts are negative. Thus q=0 is the unique least positive member. An independent scan over the entire period1≤T≤32760 found onlyT=2366, agreeing with the algebraic proof.

This theorem is conditional on the supplied offsets and three half-calendar divisibilities. It neither proves that the offsets were historically selected by this rule nor establishes global minimality, rarity, or a statistical holdout prediction. C640 already makes the essential limitation explicit.

## Synthesis guidance

- Present the rail map with its sign: subtract2366 from positive BC labels to reach Joseph's rail.
- Call C640 a **least positive solution under the fixed14/26 geometry**. The stronger result is that adding the exposure condition excludes the smaller two-calendar solutionm=n=1; it is not that all chronology has been uniquely derived.
- Preserve C641's numerical distinction.32760 is the compatibility period for the scalar congruences;327600 is an earlier common-volume coefficient. Their factor10 relationship is exact, but it does not by itself identify their physical or chronological units or authorize a new endpoint translation.
- Calendar-half counts in this file are the source's schematic modules of180/168/182. A multi-millennial BC span is not being redescribed as literally6.5 elapsed civil years.

No research-record correction is needed.