C115 — Isaac and Peleg attach the local 8×529 construction to the 20×529 Mirror

22 September 2026 · One bounded continuation after C114.

Isaac supplies the named 360+180+720 division, Peleg supplies a sourced ±2 bracket about 2116, and the local 8×529 interval has an exact endpoint attachment to the retained 20×529 opening span. The controlling unit remains K=529. Isaac’s 180-year life measures the subdivision of the derived 1260; it does not replace K.

This step follows Dean’s supplied Moses–Isaac observation and his instruction to give Peleg priority, with Nahor as a secondary comparison. It also completes C114’s proposed bounded comparison of local 4/8-fold widths with the established 10/20-fold opening structure.

1. Source settings and the Moses–Isaac sequence

File_18 §§2.2–2.3 supplies the MT minimum rows. The retained +215/full-430 selection gives:

Abram’s birth is 2166=1951+215. Moses’ birth 1526 and Exodus 1446 remain fixed by §2.4. These Peleg/Nahor records use the ordinary MT profile; no +60 Terah displacement or regular Gear is applied to them.

For W=8K=4232, retain the complementary pair-base rule C_W(b)=W−b. Its exact operation is a forward translation by W followed by reflection about the existing January AD 1 axis, with component order reversed. It generates the requested BC sequence:

Every interval holds on both seasonal components:

This verifies the proposed time + half a time + times ordering within the 529 construction. The BC complement places Isaac’s derived death before his derived birth. Its birth landing exactly reuses the Abram source coordinate admitted in C113. Under the ordinary global reflection, the AD sequence reads forward as Exodus→Abram/derived Isaac birth→derived Isaac death→derived Moses birth, with intervals 720+180+360.

The dependency is compact:

The advance is therefore a named biographical partition of the already verified residual. The translated and reflected copies are dependent representations of those same source relations.

2. Peleg gives the primary near-center source comparison

The local complementary center is C=4K=2116. Peleg’s sourced death lies two years before it in BC chronological order. His complementary boundary lies two years after it:

Nahor supplies the adjacent one-year relation. The exact AD comparison is:

Thus Peleg’s ordinary and translated death pairs are four years apart, bracketing the generated center at ±2. Nahor’s are two years apart, bracketing it at ±1. The BC complements are respectively 2115t/2114n and 2116t/2115n. The AD translations and BC complements retain their different directions and labels.

Peleg also supplies a named division of the Abram-to-Isaac birth interval:

Including his generated complementary boundary exposes its symmetry:

Nahor gives 49+51, or 49+2+49 when his complement is included. These are componentwise pair relations. They do not shift either source death. The center itself remains 50 years after Abram’s birth and 50 years before Isaac’s birth.

This gives Peleg interpretive priority within Dean’s Arphaxad–Abram division/nations reading: his actual death boundary sits next to the center of the sourced Abram/Isaac pairing. The endpoint remains labelled as his death. The ±2 geometry follows from 2118−2116; the Noah/Shem Gear operation is not applied here.

3. The local construction attaches to the retained 10/20-fold opening

The existing C112 graph preserves the LXX cumulative Cainan-OFF opening pair 5291t/5290n BC and its ordinary reflection AD 5290t/5291n, joined by opening_self20. File_18 §6D reproduces its radius from the actual native opening: 5750−460=5290=10K. Its self-Mirror width is 10580=20K.

The generated local fulcrum has radius 2116=4K. The radius difference is therefore 3174=6K, giving the symmetric containment:

The named Abram and Isaac endpoints supply the following source attachment:

Both components verify, including the middle civil-span equations 2167+2066−1=4232 and 2166+2067−1=4232. Reversing the Abram/Isaac selection exchanges the two flanks: 3224+4232+3124.

This path attaches to the existing opening node identities. Its endpoint types remain cumulative LXX OFF at the outside and regular MT at the inside. The 5290 radius is distinguished from the separately registered tenfold elapsed intervals in earlier campaigns.

4. Peleg and Nahor inside the same outer frame

Peleg’s source death pair and its own ordinary reflection delimit 4236=8K+4. Each outer flank is 5290−2118=3172=6K−2. Nahor supplies the neighboring case:

The Peleg route has actual sourced endpoints at both ends of each same-side flank: cumulative LXX opening to regular MT death, and their reflections. Peleg’s inner self-Mirror width is four years larger than 8K; his source-to-complement translation remains exactly 8K. Those two comparisons explain the ±2 bracket without identifying the 2116 generated center as Peleg’s death.

All three outer partitions are dependent on the retained opening and their selected inner coordinates. They attach the new local source comparisons to the known 20K interval; they do not create three independent calibrations of K.

5. Checkpoint and next bounded question

52 exact source, phase and attachment checks passed. A single isolated replay reproduced the certificate, results and report body byte for byte. Frozen inputs and archived file hashes were verified. C114’s complete 3.5-year witness construction remains admitted; C115 adds the Isaac subdivision and Peleg/Nahor source attachments.

The C112 integration graph is retained at 104 nodes, 193 durations, 35 reflections, rank 101. The new finite paths are recorded in the C115 certificate against its existing opening node identities; they have not yet been merged into that graph. Regular Gears remain confined to Noah/Shem/Flood. No completed campaign, quarantined primer, Rounded analysis or localized −33 branch was reopened.

Sources: Dean’s current supplied sequence and Peleg instruction; updated branch handoff §§3–4 and 8; File_18 §§2.2–2.4 and 6D; File_09 Appendix A.3; the original Mirror handoff §7; C114 certificate/checkpoint; and the frozen C112 graph/certificate. These are calculations within the selected project chronology. Dean’s integrated chiastic/fractal interpretive framework is retained.

Best next bounded step — High sufficient: examine the other end of Peleg’s biography, 2357+1875=4232, against the fixed Entry and famine boundaries. Determine the source status of the proposed 1875 Joseph boundary while preserving the 8×529 relation. The present step stops at C115.