# C312 independent mathematical review

24 September 2026. Read C311's checkpoint and next-step decision; independently calculated from the frozen C286 register, C301 cumulative coverage and C302 segment records. File_18 §§3.1.4, 3.2.1 and 3.2.2 were checked for the source-frame and local-companion meanings. No root packet or canonical source was edited.

## Finding

All six regular, same-Gear comparisons have a uniform **rail-labelled** translation. Write `r=0` for SP primary and `r=1` for its local companion. At both endpoints,

`B_SP = B_MT − 5 − r`.

This is not one unconditioned scalar translation: primary requires −5, companion −6. The role map remains MT Noah birth/Flood start → SP Shem birth/Shem death. Its upper-endpoint decomposition is the already supported MT→SP frame difference +650, Noah→Shem −500, Terah +60 and full430→minimum215 −215, totaling −5; the companion subtracts one more. Equal 600-year widths carry the same displacement to the lower endpoints. The derivation neither identifies these roles nor prescribes which comparison to select at each cumulative index.

With `C−B=2832`, MT_ON cumulative Shem/Flood is `(5896−i,5296−i)` and LXX_OFF is `(5890−j,5290−j)`. The required relation is `j=i−1+r`.

| Gear | Rail | MT regular BC pair | SP regular BC pair | Required MT index | Required LXX index | Admitted whole transport |
|---|---|---|---|---:|---:|---|
| 3 | Primary | 3060→2460 | 3055→2455 | 4 | 3 | Yes |
| 3 | Companion | 3060→2460 | 3054→2454 | 4 | 4 | Yes |
| 2 | Primary | 3058→2458 | 3053→2453 | 6 | 5 | Yes |
| 2 | Companion | 3058→2458 | 3052→2452 | 6 | 6 | Yes |
| 1 | Primary | 3056→2456 | 3051→2451 | 8, absent | 7 | No: target exists, antecedent absent |
| 1 | Companion | 3056→2456 | 3050→2450 | 8, absent | 8, absent | No: both cumulative bindings absent |

The finite result is six valid regular transports, four complete cumulative transports, one target-only boundary case and one case with neither cumulative binding admitted. All five existing SP LXX_OFF segment labels in C302 are recovered as the target-admitted cases. Four have admitted MT_ON antecedents; the fifth does not. An algebraically required index8 is a diagnostic, not an added endpoint. Cumulative annual fields remain 0…7 and receive no Gear operation.

## Phase and interpretation limits

Translation preserves paired width and the same selected quarter-year component: `(B+ε)−5−r=(B−5−r)+ε`. This algebra holds for the existing `ε=±1/4` test coordinates. It supplies no new source attribution of seasons and does not test harmonic expansion. When these BC coordinates are passed through a Mirror or expressed as signed ordinary coordinates, the corresponding signs and retained ancestry must be converted explicitly; equality of center sets is not a licence to exchange component ancestries.

This supports a source-derived structural relation. It leaves C311's bounded finding about the unlocated C310 selector intact. C312's successful regular maps do not establish a unique chronology or historical intentionality.

## Reproducibility

From the self-contained packet root, run `python3 evidence/C312_Independent_Math.py --sources sources --out replay/INDEPENDENT_RESULTS.json`. With both options omitted, the script retains its original workspace source routes and writes `C312_Independent_Math_Results.json` beside itself. The script filters source-labelled records independently, verifies each endpoint and existing field binding, and writes its results with input SHA256 hashes. It does not inspect a root C312 calculation or use its output. The cumulative coverage source also contains Noah records, which the script explicitly excludes from this Shem/Flood test.

## Suggested next bounded question; not executed

C313 could move the five existing SP Shem-life targets into the already admitted OFF/Terah0/full430 frame using +215−60=+155, then compare them with SP Noah selected by the retained17K relation. Test the explicit 500-year Noah→Shem birth binding, matching Gear and rail, and preserve the complete Shem-life pair. This is a distinct source-role question from C303–C307's Noah/Flood600 comparison, and it directly connects C312 to529 without enlarging the endpoint census. The proposed algebra and source-label test require C313 authorization and verification; they are not part of this review's completed results.