# 490d C312 — Same-Gear transport and cumulative field boundary

24 September 2026. One bounded continuation after C311 and the supplemental-studies intake.

**Result:** all six admitted regular comparisons follow one fixed rule with the existing SP rail label: the SP Shem-life pair lies five years later than the same-Gear MT Noah→Flood pair on the primary rail, and six years later on the companion rail. Four comparisons also have admitted antecedents and targets in the two cumulative fields. The fifth admitted SP target has no admitted MT cumulative antecedent. No endpoint is added to repair that boundary.

This gives a source-derived explanation of the regular displacement and its cumulative index relation. It does not establish the source-prescribed annual context selector left open by C311.

## Fixed domain and meaning of the rule

The MT pair is Noah birth→Flood start, Cainan OFF, Terah0, full430. The SP pair is Shem birth→Shem death, Cainan OFF, Terah+60, minimum215. Only regular Gears1–3 are used. Write `r=0` for SP primary and `r=1` for its local companion. All numerical pairs below are positive BC paired bases; a smaller BC label is later in time.

At the same Gear, both endpoints obey:

\[
B_{SP}=B_{MT}+650-500+60-215-r=B_{MT}-5-r.
\]

Each term has a declared role: +650 changes the MT/SP tradition frame; −500 changes Noah/Flood endpoints to Shem birth/death; +60 changes Terah; −215 changes the sojourn frame. The final −r applies only to the resulting SP person stations. It creates no MT companion and does not alter Flood start/close or a cumulative source row.

The source basis is File_18 §§1.1, 3.1.4 and 3.2.1–3.2.2. Its SP table directly supplies the six Shem pairs before +60; its MT normal Noah row and admitted Gear spacing supply the three MT pairs. The retained C286 register authenticates every selected endpoint. These are source-controlled derived states, not six independent historical witnesses.

**One rail-blind translation fails:** the two rails require different shifts. The successful object is one rule using an already declared rail label, independent of Gear. Both600-year widths remain intact; the distinct event roles remain explicit.

## Coupling to the six-year cumulative weave

The already admitted cumulative fields are:

\[
M_i=(5896-i,5296-i),\qquad L_j=(5890-j,5290-j),\qquad i,j\in\{0,\ldots,7\}.
\]

Here M is MT ON Shem/Flood and L is LXX OFF Shem/Flood. MT's fields are printed in the axis report §9.2; supplemental Engine Report III §5 explicitly documents the LXX OFF derivation. At equal indices the fields differ by6. These annual indices are not cumulative Gears.

The held bridge/anchor comparison is `C−B=2832`. Requiring the cumulative pair to have the same displacement as its regular pair gives:

\[
6-i+j=5+r,\qquad \boxed{j=i-1+r}.
\]

The primary comparison therefore moves one annual index; the companion comparison preserves the index. This index relation follows from the fixed source differences, with no fitted anchor or additional shift.

| Regular Gear | SP rail | MT Noah→Flood | SP Shem life | Required MT i | Required LXX j | Cumulative-field result |
| --- | --- | --- | --- | ---: | ---: | --- |
| G3 | Primary | 3060→2460 | 3055→2455 | 4 | 3 | Both admitted |
| G3 | Companion | 3060→2460 | 3054→2454 | 4 | 4 | Both admitted |
| G2 | Primary | 3058→2458 | 3053→2453 | 6 | 5 | Both admitted |
| G2 | Companion | 3058→2458 | 3052→2452 | 6 | 6 | Both admitted |
| G1 | Primary | 3056→2456 | 3051→2451 | 8 | 7 | Target admitted; antecedent outside |
| G1 | Companion | 3056→2456 | 3050→2450 | 8 | 8 | Both cumulative bindings outside |

Every regular pair in this table is admitted. Index8 is only a diagnostic of failed cumulative membership. In the G1 cases, the required MT cumulative pair would be5888/5288, beyond the retained terminal5889/5289. G1 primary still has the admitted LXX target5883/5283. G1 companion would require5882/5282, beyond the LXX terminal5883/5283. Those diagnostic values are not promoted to source nodes.

Thus the regular rule accounts for all five existing SP targets, but only four have complete transports from admitted MT ON antecedents. The rule cannot extend the finite cumulative correspondence to the fifth merely by extrapolating its formula.

## What this changes in the interpretation

C310's exact cover is retained: MT contributes LXX OFF indices0/2; SP contributes3…7. Index1 remains absent. C312 compresses the SP family's displacement into named source terms and explains precisely where a stronger cumulative correspondence stops. It supplies no new529 identity, statistical evidence count or historical selection rule.

The existing quarter-phase components are preserved by translation: for signed BC center `x=−B`, the operation is `x→x+5+r`, leaving each ±1/4 component and pair width unchanged. No25/23 dilation, new seasonal attribution or Mirror identity is inferred. The global Mirror convention remains January AD1, with no civil year zero.

The supplemental Gear register §5 expressly distinguishes comparison permission from source-state identity or a universal generator. That guard remains applicable: this successful labelled relation does not turn the two regular biographies into one event or explain why C310 should choose one family at each index.

## Verification and next decision

The portable verifier passed33 arithmetic, source-binding, phase and finite-domain assertions, plus15 source-authentication/predecessor checks. Its two outputs replay byte identically. A separate mathematical script independently obtained the six/four/one/one result; a separate source review checked the650−500+60−215 decomposition. All frozen inputs and standing controls remain unchanged. These checks establish the conditional calculation, not historical intentionality.

**Best next bounded step — C313, proposed only:** put the five existing SP Shem-life targets into the already admitted full430/Terah0 frame, using +215−60=+155, then test their500-year Noah→Shem birth binding against the SP Noah endpoint selected by the retained17K relation. Require matching Gear and rail and preserve the complete Shem-life pair. This directly connects C312's result to529 through a different event-role relation from the earlier Noah/Flood600 audit. Use only the existing five target cases and source register; record failures. High is sufficient. **C313 has not been executed.**