# C326 independent four-endpoint geometry review

24 September 2026. The six C325 triples are completed only with the already admitted MT counterpart of their SP terminal, read directly from frozen C322 records. File_09 is compared using its own named cumulative boundaries. No endpoint, expansion target, phase, anchor or global map is generated; C327 is not executed.

**Result:** the six completed SP/MT configurations have ordered gaps `460 | 40 | 460`, overlapping500 intervals with overlap40, and full outer width960. File_09's prescribed field has gaps `40 | 460 | 40`, overlap460, and outer width540. They share exact three-point500 patterns, but their complete four-point metric configurations are different.

## The admitted fourth corner

A and C remain SP OFF/Terah0/full430; B and D are MT restoredON/Terah60/full430. A/B share the anchor role, C/D share the terminal role, and all four retain the same regular Gear. The source modal binding remains Noah/Flood-bound500/100. No textual502 substitution or imported companion is made.

| Gear / i | Anchor role A/B | Terminal role C/D | A_SP | B_MT | C_SP | D_MT |
|---|---|---|---:|---:|---:|---:|
|G1 /3|Noah birth|Shem birth|3706|3246|3206|2746|
|G1 /3|Flood start|Shem death|3106|2646|2606|2146|
|G2 /5|Noah birth|Shem birth|3708|3248|3208|2748|
|G2 /5|Flood start|Shem death|3108|2648|2608|2148|
|G3 /7|Noah birth|Shem birth|3710|3250|3210|2750|
|G3 /7|Flood start|Shem death|3110|2650|2610|2150|

Every row is strictly ordered `A>B>C>D` in the held BC numerical/reflected-coordinate convention. The two500 intervals are A→C within SP and B→D within MT. Their intersection is B→C, width40; their union reaches A→D, width960. The source-preserving context legs A→B and C→D are each460.

## Comparison with the positively prescribed File_09 field

File_09 §3A.2.3 names `14511→14471→14011→13971 BC`. It is retained exactly in its declared cumulative context.

| Invariant or interval | Admitted SP/MT quadruples | File_09 named completion field |
|---|---|---|
|Ordered gaps|460,40,460|40,460,40|
|Two overlapping intervals|500,500|500,500|
|Overlap|40|460|
|Full outer width|960|540|
|Six pairwise distances, sorted|40,460,460,500,500,960|40,40,460,500,500,540|
|Outer-gap / middle-gap ratio|23/2|2/23|

The unequal pairwise distances exclude full metric congruence. The unequal gap ratios also exclude uniform similarity up to order reversal, without fitting any map or scale. Reflection cannot repair the difference because both ordered gap profiles are already palindromic. This is an exact result about these fixed four-point configurations, not an assertion that all possible source constructions are exhausted.

The SP/MT A→B→C triple has gaps460+40, matching the second through fourth File_09 boundaries. Its B→C→D triple has40+460, matching the first through third File_09 boundaries. Thus both500 three-point forms are present. Their assembly into four corners differs: the same two500 widths overlap by40 in one field and by460 in the other. Equal sums of interval lengths do not make their outer widths equal: `500+500−40=960`, whereas `500+500−460=540`.

The source meanings also differ. The SP/MT460 legs compare same-role endpoints across two declared chronology contexts; its500 intervals are same-tradition biography relations. File_09's460 core is the cumulative-Cainan interval and its40 shoulders are generated local completions. The comparison does not transfer that cumulative endpoint prescription to the regular cross-tradition configuration.

## Exact phase census and dependency

For each of the six quadruples, all16 independent choices of the held `±1/4` component at A/B/C/D were tested. There are96 trials,12 successes and84 failures. Each of the four separate conditions passes48 trials; both460 conditions together pass24, and both500 conditions together pass24. Requiring all four leaves only the all-minus and all-plus component in each quadruple.

The residuals for `A−B=460`, `C−D=460`, `A−C=500`, `B−D=500` are respectively `pA−pB`, `pC−pD`, `pA−pC`, `pB−pD`. They vanish together exactly when all four physical output offsets agree. The four length equations have one dependence; three connected differences fix the shape, with common translation still free. These checks are not four independent historical confirmations.

The successful components retain C325's already verified ancestry bindings: the SP Noah anchor comes from cumulative OFF Flood_i; the SP Flood anchor from cumulative OFF Shem_i; the Shem terminal retains reversed index `j=10−i` and opposite inherited cumulative phase. The fourth MT endpoint receives no newly invented cumulative ancestry. Equal output phases do not erase any of these distinctions. No quarter-year components are assigned to the File_09 comparison field in this audit.

The next useful bounded question is whether the two successful event-role families are linked by one source-qualified600 transport across all four corners, and whether that transport preserves or changes their recorded cumulative role/index/phase ancestry. This would assess dependence between the two observed families using the same held endpoints. It remains an unexecuted C327 proposal.

## Reproduction

`C326_Independent_Audit.py --sources DIR --out DIR` uses frozen `PREVIOUS_DATA.json` (C325), `C322_DATA.json` and `File_09.md`, with Python's standard library only. Its JSON retains every source corner, all96 phase trials, inherited ancestry labels and the separately typed File_09 comparison. **313 checks passed.** Global paired Mirror January AD1, local junction x=1/January AD2, no civil year zero, and the existing Gear/companion boundaries remain unchanged.