# 490d C118 — Minimum relations behind the Peleg–Joseph complement

2026-09-22. One bounded step after C117. **Completed.**

**The strongest simplification is 191−190=1.** Within the held 8×529 Abram–Isaac Mirror alignment, Peleg’s generated birth complement necessarily falls one year after the retained Entry boundary. This numerical result requires no assignment of Joseph’s land purchase to that year.

The 241 offset also has a genealogical derivation:

\[
\boxed{241=50+191,\qquad 2=241-239.}
\]

Thus Peleg’s birth and its complement can be derived before using his lifespan. His 239-year life then locates his death two years before the 2116 BC fulcrum. This refines C117’s equivalent decomposition, **241=239+2**, by identifying which source inputs are needed for each conclusion.

## Source relations and selected state

Actual File_18 §§2.2–2.3 supplies the minimum regular MT rows. Applying the already-selected +215/full430 state gives Peleg2357–2118, Abram2166, Isaac2066, Jacob2006 and Entry1876 BC. This comparison uses the **no-+60 Terah path** for Peleg’s ancestry. The separate qualified Terah-firstborn2226 record in C117 is not used in this derivation.

| Source interval | Components | Total |
|---|---|---:|
| Peleg birth → Abram birth | Peleg30 + Reu32 + Serug30 + Nahor29 + Terah70 | **191** |
| Abram birth → Isaac birth | Abram’s age at Isaac’s birth | **100** |
| Isaac birth → Entry | Isaac→Jacob60 + Jacob→Entry130 | **190** |
| Peleg birth → death | Peleg’s lifespan | **239** |

The source decomposition uses Nahor as a genealogical link; it adds no new claim about the importance of his death. No Gear is applied. Regular Gears remain limited to Noah/Shem/Flood.

## Derivation within 8×529

Let **K=529**, **C=4K=2116**, **W=8K=4232**. Let A,I,P,D,E,J denote the BC pair-base labels for Abram birth, Isaac birth, Peleg birth, Peleg death, Entry, and the generated Peleg-birth complement.

The admitted Abram–Isaac relation and their 100-year separation give:

\[
A+I=W,\quad A-I=100
\quad\Longrightarrow\quad A=C+50,\ I=C-50.
\]

The genealogical birth interval gives:

\[
P=A+191=C+241=2357,
\]
\[
J=W-P=C-241=1875.
\]

Peleg’s lifespan supplies the death relation:

\[
D=P-239=C+2=2118.
\]

For Entry:

\[
E=I-190=C-240=1876.
\]

Subtracting the complement from Entry then yields:

\[
\boxed{E-J=191-190=1.}
\]

Because BC labels decrease forward in time, J is one year **after** E. Both the 529 width and Abram’s 100-year begetting interval cancel from this relative difference once the held Mirror registration is in force. K still fixes the absolute placement. **The 191−190 relation is a consequence within the 529 construction; it is not an independent derivation of 529.**

## Which conditions are necessary?

The finite audit used these six relations:

| ID | Relation | Role |
|---|---|---|
| R1 | A+I=W | Held 8K Mirror registration |
| R2 | A−I=100 | Abram–Isaac birth separation |
| R3 | P−A=191 | Peleg–Abram birth chain |
| R4 | P−D=239 | Peleg lifespan |
| R5 | I−E=190 | Isaac–Entry interval |
| R6 | P+J=W | Definition of the generated complement |

| Quantity determined | Unique smallest support in this six-relation model |
|---|---|
| Peleg birth2357 | R1, R2, R3 |
| Peleg death2118 | R1, R2, R3, R4 |
| Generated complement1875 | R1, R2, R3, R6 |
| Entry−complement =1 | R1, R3, R5, R6 |

“Smallest” refers to these explicitly named aggregate relations. R3 expands to five genealogical intervals and R5 to two; this is not a claim that the Bible contains only six relevant data points.

The last result has a direct proof: **R1+R3−R5−R6 gives E−J=191−190**. It uses neither R2 nor R4. All64 subsets were checked for each of four targets. Fifteen omission witnesses show that deleting any member of a listed minimum set allows its target to change while the remaining members hold. These are logical diagnostics, not alternative historical chronologies.

## Famine boundaries and the narrative assignment

File_16 retains the primary famine envelope **1878n→1871n BC** and Entry at1876n, two elapsed years after its opening. With the derived one-year offset:

\[
\text{elapsed famine at }J=2+(191-190)=3.
\]

The next Nisan boundary lies four years after the opening.

| Boundary | After Entry | Elapsed / remaining famine | Span to reflected Peleg birth |
|---|---:|---:|---:|
| 1875n BC | 1 year | 3 / 4 years | Paired span4232=8K |
| 1874n BC | 2 years | 4 / 3 years | Paired span4231=8K−1 |

“Paired span” uses File_09’s exact component convention: **1876t/1875n BC → AD2357t/2358n** is4232 on both components; **1875t/1874n BC → that same reflected Peleg pair** is4231. No civil year zero is inserted.

The primary famine midpoint is **1875t BC**, halfway between the Nisan boundaries1875n and1874n. Accordingly, the central year surrounds the exact midpoint. Under translation by8K, its bounding pairs map to **AD2357t/2358n** and **AD2358t/2359n**; the first is Peleg’s ordinary reflected birth pair. This is a complete coordinate correspondence even before assigning the transaction to it.

Dean’s latest reading retains **1875 or perhaps1874 BC** for the land-purchase notation. Genesis47:17–18 supplies the livestock year followed by another year; equating the livestock year with Entry, or choosing a precise transaction season, is additional narrative placement. C118 does not resolve that placement by selecting whichever date gives an exact multiple.

The inherited Genesis41 universal famine/gathering and Peleg–nations interpretation remains recorded. The Genesis47 distinctions between Egypt/Canaan’s famine market, Egyptian land purchase, and Israel’s provision/possession remain unchanged. These theological and narrative correspondences are not counted as additional linear constraints. See the frozen `USER_AND_SCRIPTURE_UPDATE.json` for the primary-text review and URLs carried forward from C117; no new textual research was undertaken in this step.

## Checkpoint and next step

**56 checks passed**, including source extraction, exact equation solutions, all64 subsets for each target,15 omission witnesses, agreement with the six relevant C117 records, and componentwise famine/translation checks. One isolated replay reproduced all three generated evidence outputs byte-for-byte. Seven inputs are frozen with hashes.

The C117 working graph remains unchanged at **178 nodes,310 durations,72 reflections,8 coordinate equalities,20 component expressions,27 component equations; rank175**. No prior campaign was rerun, no source/canonical file was edited, and no quarantined primer was reopened.

**Best next bounded step:** compare the admitted **8×529** construction with the retained **11×529 Conquest relation**, using their source-qualified Moses/Conquest anchors, and determine whether the bridge adds a placement condition. **High sufficient.**