# C312 — source derivation of the MT/SP 600-year pair comparison

Date: 2026-09-24. Scope: one source-focused review; no new endpoint census or C313 work. The arithmetic below concerns positive BC year labels, so a negative label change moves forward in time. It does not use a year zero.

## Finding

The admitted MT Noah-birth→Flood-start pair and SP Shem-birth→Shem-death pair have the same 600-year width. At each held Gear, their regular BC labels obey

`SP pair = MT pair +650 −500 +60 −215 −r = MT pair −(5+r)`,

where `r=0` is SP primary and `r=1` is the SP-local lower companion. This is a fixed, source-labelled comparison of different event roles and named source settings. It is not evidence that the source prescribes C310's annual-index-dependent choice of context.

## Source route and exact roles

1. **650 tradition offset:** File_18 §3.1.4, lines1542–1552, gives MT/SP full430, Terah0, CainanOFF Flood starts G3 `2460/3110`, G2 `2458/3108`, G1 `2456/3106`. Each difference is650. The same-Gear SP primary Noah births, formed by adding215 to §3.2.1 native rows, are3710/3708/3706. These are650 above the already-admitted MT births3060/3058/3056. The MT G2 base is explicitly2843 in §2.1, line1237, and full430 adds215. G1 is explicitly2841 in §2.2.1, line1263. G3 is coherent Gear transport, not a separately printed native birth row here.
2. **−500 event-role change:** File_18 §3.2.1, lines1577–1581, preserves same-rail Noah-to-Shem birth separation500. Its primary Shem birth is100 before the same-Gear Flood start; a600-year Shem life therefore ends500 after that start. Thus both endpoints move forward500 when the comparison changes SP Noah→Flood roles to SP Shem-life roles. A regular Shem death remains a regular Shem death, not a regular Flood.
3. **+60 Terah, −215 sojourn:** File_18 §1.1, lines539–548, opens the shared regular fork. §3.2.2, line1637, explicitly retains SP+60 as state-valid where opened. This converts the SP full430/Terah0 comparison to the already-opened SP minimum215/Terah+60 context. No cumulative Terah145 is changed.
4. **−r companion:** File_18 §3.2.1, lines1573–1581, supplies the primary/lower companion pairs and says their matched Shem endpoints preserve600. §0.1d, line324, limits this companion to SP Noah/Shem. This last−1 operates on the resulting SP Shem person stations only; it is not applied to MT endpoints, Flood event boundaries, cumulative rows, or upstream people.

## Six admitted regular comparisons

All values are BC year labels. MT context is OFF/Terah0/full430. SP context is OFF/Terah+60/minimum215.

| Gear | MT Noah→Flood | SP primary Shem life | Difference | SP companion Shem life | Difference |
|---|---|---|---:|---|---:|
| G3 |3060→2460|3055→2455|−5|3054→2454|−6|
| G2 |3058→2458|3053→2453|−5|3052→2452|−6|
| G1 |3056→2456|3051→2451|−5|3050→2450|−6|

SP rows reconstruct directly from File_18 lines1577–1579 by adding60. File_18 explicitly calls these derived source-controlled stations, not six independently attested source states. Their permissions survive as qualified comparisons; no historical independence is claimed.

## Cumulative six-year relation

File_18 §6A, lines1945/1947, has MT OFF Shem5436–5429 and Flood4836–4829. §6B, lines1971–1974, inserts460 upstream, giving MT ON Shem5896–5889 and Flood5296–5289; its Flood table at line1997 prints the latter directly. AXIS_REPORT §9.2, lines538–539, prints both MT ON fields.

Supplemental Engine Report III §5, lines296–324, explicitly gives LXX OFF Shem5890–5883 and Flood5290–5283. It derives LXX OFF Flood from native LXX5750–5743 by subtracting cumulative Cainan460, then adds Shem600. Native Flood5750–5743 is also File_18 §6D line2063. Thus the LXX OFF Shem row remains a documented derived field, not a new raw File_18 row.

For already-admitted integer members `i,j ∈ {0,…,7}`:

`M_i=(5896−i,5296−i)`, `L_j=(5890−j,5290−j)`.

At the same index, `M_i−L_i=(6,6)`. Matching the regular pair difference5+r requires `6−i+j=5+r`, hence **`j=i−1+r`**. The one-index primary displacement is forced by these fixed inputs; the companion preserves the index. It must not be described as a Gear applied to cumulative chronology.

## Bounds and concerns for the integrated C312 result

- The finite eight-station field is a seven-year envelope. An algebraically required index8 is outside it. Do not extend that field to rescue a comparison or call the extension an established cumulative source node.
- A target SP primary pair may be valid even if its same-Gear MT_ON antecedent would require index8. Distinguish an antecedent outside the admitted field from an invalid regular target.
- This review establishes the source ingredients and their algebraic composition. It supplies no independent source mandate selecting this combined context or this event-role correspondence in advance. The C311 distinction between ingredient validity, comparison permission, and a prescriptive selector remains operative.
- Report III §53, lines2514–2529, independently guards regular/cumulative separation,130 versus460, envelope membership, and Shem/Flood node identity. Current File_18 corrections govern any older Report III assumptions elsewhere.
- No numerical fit statistic, historical intentionality, unrestricted transport law, canonical revision, extra Gear scope, or new anchor follows from this bounded result.

## Files read and hashes

- File_18: `project_sources/40-File_18.Chronological_Data_Tables-20260919-141048-.md`; SHA256 `68301ab760bfbe2e874aedc5c8d943ced7a6219d34bc2859332a70713f9630d5`.
- Supplemental Report III: `supplement_intake_20260924/01_Cumulative_MT-SP-LXX_Creation-Shem_Comparative_Engine_Report_III_20260902(2).md`; SHA256 `25623b1d84ab1131a1a66c075cf1ff2c19db2738042afd390a58b181b129a428`.
- AXIS_REPORT: `axis_ledger_work/C266/sources/AXIS_REPORT.md`; SHA256 `ed20797a44cc5b4c201012017f66afb5c6a3a3f8951c1d63f21997b0cb2bfdbc`.
- Existing C302 `evidence/DATA.json` was read to confirm the named contexts and already-admitted pairs. No new census was run.