C116 — Peleg’s 8×529 complement and the central famine year

22 September 2026 · One bounded continuation after C115.

Peleg’s birth at 2357 BC has the exact complement 1875 within 4232=8×529. In the retained Nisan famine chronology, 1875n opens the central fourth year; the exact halfway instant is 1875t. Identifying that year with Joseph’s land transaction remains conditional on the assignment of the preceding livestock year to the 1876 Entry year.

The main advance is a precise admission of the coordinate, its place in the famine window, and its dependence on Peleg’s full biography. The controlling unit remains 529, with local radius 2116 and width 4232.

1. The complementary pair verifies; Entry retains its own date

File_18 §2.2 gives MT Peleg 2142–1903 BC, lifespan 239, in the minimum frame. The retained +215/full-430 selection gives 2357–2118 BC. Its §2.3 Jacob row gives Entry 1661+215=1876. C115 already admitted Peleg’s death at 2116+2.

Set K=529, C=4K=2116 and W=8K=4232. Then:

The proposed Joseph pair is also the BC reflection of Peleg’s translated birth pair. Its two cross-axis component checks are:

Entry itself gives 1876+2357=4233 at paired-base level, or 8K+1. Its own +W translation therefore remains one year earlier than the required Peleg mirror. No adjustment to Entry 1876n is needed or made.

2. The birth-side relation follows through Peleg’s 239-year life

Peleg’s birth is 241 years before the local center, and its complement is 241 years after it:

This connects the new birth correspondence directly to C115’s death bracket. The complete translated biography is:

Reflecting the translated biography gives the forward BC sequence 2115t/2114n→1876t/1875n, again 239 years, with the biographical order reversed. Thus the 1875 landing is determined by the same eightfold bridge, the sourced 239-year life and the already verified two-year death offset. It adds a new proposed narrative attachment, while its arithmetic is dependent on those retained inputs.

3. Its exact position in the famine window

File_16 §3.3A.2 explicitly supplies the Nisan Plenty/Famine triad 1885n/1878n/1871n BC. Sections 4.1 and 7.1 preserve the seven-year famine window and the 1878−1876=2 Entry comparison. Only these source-boundary settings are used here; the quarantined 1876 primer is not reopened.

With Nisan boundaries held, the annual intervals are 1878n→1877n, 1877n→1876n, 1876n→1875n, and so forth. The fourth, central interval is therefore:

These are elapsed-boundary calculations within the selected chronology. File_16’s narrative “Year 2” wording and its explicit window qualification are preserved; the table does not recast that wording as an exact biblical event month.

The distinction is useful: 1875 can name the central famine year, with 1875n as its opening boundary and 1875t as its halfway instant. The exact 8K correspondence attaches to the proposed boundary pair 1876t/1875n. On the primary Nisan component, that boundary stands six months before the famine’s exact halfway instant.

Translation of the full famine pair envelope gives AD 2354t/2355n→AD 2361t/2362n. The Peleg birth mirror AD 2357t/2358n divides that translated envelope componentwise into 3+4 years. The primary Nisan halfway instant translates instead to AD 2358t, half a year after its selected AD 2358n component.

4. The paired Tishri diagnostic

File_09 Appendix A.3’s phase coordinates also permit a check of the parallel Tishri companions. These are the completion of the retained paired dates, rather than an independently dated famine season.

The proposed 1875n component is therefore the midpoint of the parallel Tishri span, while it opens the central year in the primary Nisan span. The whole proposed pair does not equal both rail midpoints: those midpoints, in chronological order, are 1875n and 1875t, each half a year later than the corresponding component of 1876t/1875n.

This specifies the phase choices behind the central-year language. The primary Nisan famine boundaries and the global January Mirror axis remain fixed.

5. Source status of the Joseph identification

Genesis 45:6 reports two famine years experienced and five remaining. Genesis 47:17–20 places the land-purchase request in the year following the livestock exchange. It does not synchronize that livestock year with Jacob’s arrival.

The proposed attachment is therefore conditional: assign the livestock interval to 1876n→1875n, and the following land-transaction year becomes 1875n→1874n. That central year contains the exact famine midpoint. Its selection remains an additional narrative assignment; no exact transaction season is established here.

Dean’s Peleg/division-of-nations and Joseph/famine correspondence remains the proposed interpretation. Its exact numerical support can now be preserved without promoting the transaction-year assignment into an independently sourced date.

6. Evidence and continuation

36 exact source, phase and boundary checks passed. One isolated replay reproduced the certificate, results and report body byte for byte. The arithmetic checker verifies the selected source rows and phase equations; the short Scripture review is recorded separately as a textual reading. Frozen source files and archived outputs were hash-verified.

C113–C115 admissions remain unchanged. The C112 graph stays at 104 nodes, 193 durations, 35 reflections, rank 101; this report records the new coordinate classification separately. Regular Gears remain confined to Noah/Shem/Flood. Neither the quarantined primers nor File_16’s separate Rounded AD 1875 target was used.

Sources: File_18 §§2.2–2.3; File_09 Appendix A.3; File_16 §§3.3A.2, 4.1 and 7.1; updated branch handoff §4; C115 certificate/checkpoint; frozen C112 graph; and the two primary Scripture passages linked above.

Best next bounded step — High sufficient: integrate the verified C113–C116 local 4/8-fold records with the retained 10/20-fold Mirror graph. Keep 1875 as a generated coordinate with a provisional Joseph role, and determine which added relations supply new constraints. Stop at C116 pending instruction.