# C129 — Are the two 3×529 legs the same source construction?

22 September 2026 · One bounded comparison of C10/C12 and C119–C128. Set **K=529**, so **3K=1587**.

**Finding.** Four N01–N04 rows share the *width* 1587, the same branch labels, and the established 1058-year continuations. They do **not** share a 3K source segment: the 3+7 leg joins an actual **MT cumulative Cainan-ON Flood member** to an actual **SP regular Noah birth**; the 3+8+2 leg joins a native **SP regular Creation** member to a **generated BC preimage** of reflected Conquest. The measured offset between the two legs changes by two years from row to row. Their equality establishes a finite, phase-consistent numerical correspondence within the retained packet, not a new source-preserving transfer of events.

## The matched field

The numerical entries below are Nisan **BC pair bases**, except the reflected Conquest column (AD pair base). Thus 5293 BC represents `5294t/5293n BC`, and AD1400 represents `AD1400t/1401n`. N01/N03 have retained *partial* Creation–Flood expansions; N02/N04 have the full prior expansions. All four displayed prefix paths are admitted.

| Arc | 3+8+2: SP Creation → generated preimage | Reflected Conquest AD → ordinary MT Flood AD | 3+7: cumulative MT ON Flood → regular SP Noah | Offset of both 3K endpoints (years) | Prior expansion |
|---|---|---|---|---:|---|
| N01 | 4420 → 2833 | 1399 → 2457 (G2 close) | 5292 → 3705 (G1 companion) | **872** | Partial |
| N02 | 4419 → 2832 | 1400 → 2458 (G2 start) | 5293 → 3706 (G1 primary) | **874** | Full |
| N03 | 4418 → 2831 | 1401 → 2459 (G3 close) | 5294 → 3707 (G2 companion) | **876** | Partial |
| N04 | 4417 → 2830 | 1402 → 2460 (G3 start) | 5295 → 3708 (G2 primary) | **878** | Full |

For the principal N02 row, the actual paired first legs are **5294t/5293n → 3707t/3706n BC** and **4420t/4419n → 2833t/2832n BC**. Both t-to-t and n-to-n differences equal 1587. The generated endpoint has no assigned historical event. Its further paired connection to the reflected Conquest boundary is **2833t BC → AD1400t** and **2832n BC → AD1401n**; both cross-axis counts are 4232 after subtracting the absent civil year zero. Reflecting a 3K leg reverses its order; it does not turn Creation into Noah or a generated preimage into cumulative Flood.

## What the equality depends on

In N01–N04 order, put \(i=0,1,2,3\). For the *BC bases*, the two constructions give

\[
\begin{aligned}
&c_i=4420-i,\quad z_i=2833-i,\quad q_i=1399+i,\quad f_i=2457+i,\\
&A_i=5292+i,\quad X_i=3705+i,\quad Z_i=2+i.
\end{aligned}
\]

The Creation construction holds **\(c_i-z_i=3K\)** because \(c_i+q_i=11K\) and the generated preimage satisfies \(z_i+q_i=8K\); it continues by \(f_i-q_i=2K\). The Noah construction holds **\(A_i-X_i=3K\)** between two *different, actual* source rows; its supplied 3+7 allocation is recorded in C12's R16 and continues by \(X_i-Z_i=7K\). The C12 allocation is an endpoint choice in that reconstruction, whereas the generated boundary \(z_i\) is a definition following the Conquest attachment. Neither construction supplies the other's first-leg source relationship.

There is nevertheless a checkable relationship between the selected rows:

\[
\boxed{A_i-c_i=X_i-z_i=872+2i.}
\]

This is a *measured affine displacement*, not a constant transport rule. It also separates into the existing C12 input \(\ell_i-c_i=868\), where \(\ell_i=5288-i\) is the **LXX cumulative OFF** Flood comparison member, and the cross-cumulative remainder \(A_i-\ell_i=4+2i\). Accordingly **872, 874, 876, 878 = 868 + (4, 6, 8, 10)**. The 868 attachment retains its prior input status; the second term changes with row selection. No extra regular Gear at Creation or cumulative station is introduced to explain it.

The branches have a *shared surrounding scaffold*. C10's N02 and N04 primary records explicitly compare the sourced regular SP Noah → alternate-profile MT Flood **1058 BC** intervals (3706→2648 and 3708→2650) with the distinct reflected Conquest → ordinary MT Flood **1058 AD** intervals (1400→2458 and 1402→2460). The alternate MT Flood profile is Cainan ON/Terah +60, 190 years from the ordinary MT Flood coordinate. These are genuine retained 2K correspondences with separately labeled events; they do not identify the preceding 3K legs. N01/N03 retain their partial/close qualifications, including N03's missing full Shem join.

The fifth C10 row, **5296→3709→6 BC**, sharpens the boundary. One can pair its 3K arithmetic with native Creation **4421→2834 BC**, but the required reflected Conquest member would be **AD1398**, outside the held Conquest field. C122 therefore rejects its 3+8+2 **13K source path** even though both isolated 3K subtractions still equal 1587. Numeric equality alone cannot enlarge the admitted four-row domain.

## Evidence and decision

The C129 checker made **83 exact checks** on the four row matches and the excluded fifth row. Its frozen inputs are controlling File_18/File_09, C10 results/report, C12's basis certificate, C122's certificate/report, the unchanged C127 graph and C128's checkpoint. The input and artifact SHA256 manifests, one isolated byte-identical replay, and all selected dependencies accompany the evidence ZIP. C127 already contains both classes of 3K edges; C129 does not alter that graph or its **208-node, rank-208** state. No earlier campaign was executed, no new regular Gear was used, and original interrupted Reports 11–25 remain unestablished.

**Best next bounded step:** on the fully supported N02/N04 rows, test whether the retained **868 attachment**, the varying cross-cumulative remainder, and the paired **2K continuations** yield a *dependent* closed graph relation, without identifying unlike endpoint roles or adding a source operation. **High sufficient.**