# C14 — The 12-fold family attaches to the 2 BC companion path

22 September 2026 · Bounded continuation of C05 and C09–C13.

**The retained 6348-year interval now has a contiguous, source-qualified 2+10 decomposition.** Its intermediate boundary is the **MT cumulative ON member 5292 BC**. Refining the existing tenfold interval gives

\[
\boxed{6350\xrightarrow{2\times529}5292
\xrightarrow{3\times529}3705
\xrightarrow{2\times529}2647
\xrightarrow{5\times529}2\quad\text{BC}.}
\]

All displayed numerals in this chain are **Nisan BC**. The opening and landing are exactly the original Mirror handoff’s **6351t/6350n BC → 3t/2n BC**. Among C10’s five retained rows, this selects the **SP G1 companion Noah / MT G2 Flood-close** row. It also extends C12’s mixed twentyfold interval to a dependent **11638=22×529** path. These are additional endpoint realizations; their several partitions are consequences of shared intervals, not independent numerical witnesses.

**Each station retains its source role.**

| Symbol | Paired boundary | Role and profile |
|---|---|---|
| H | **6351t/6350n BC** | Opening of the cumulative native-LXX, Cainan-ON Shem-birth / Noah-death comparison field, 6350–6343 |
| A | **5293t/5292n BC** | Member of the cumulative MT restored-Cainan-ON Flood field, 5296–5289; opening offset 4 |
| N | **3706t/3705n BC** | Regular SP G1 **companion Noah birth**; full-430, Terah 0, Cainan OFF |
| M | **2648t/2647n BC** | Regular MT G2 **Flood close**, in the explicitly retained full-430 / Cainan ON / Terah +60 comparison |
| Z | **3t/2n BC** | The original held Christ-period comparison landing |

The mixed cumulative and regular profiles are deliberate. In particular, M is the close of the alternate-frame **2648→2647** Flood year; it is not an ordinary-profile MT Flood. The SP companion identifies N and does not generate the Flood close. LXX derived-OFF Noah births do not include 3705. The terminal pair remains a retained comparison landing, without a new assertion of historical Christ-birth dating.

File_18 §6D supplies the native LXX Flood field 5750–5743 and cumulative Cainan lifespan 460; the retained 600-year Shem stack gives 6350–6343. File_18 §6B independently supplies MT ON 5296–5289. File_18 §§3.1.4 and 3.2.1 supply the regular event and person stations, with the existing frame conversions. The Mirror handoff §§3.1 and 10 supplies the cumulative role labels and exact original opening/landing pair. C10 provides the completed five-row comparison.

The 3+7 branch handoff’s **5295/3705/2 → 5292/3705/2** correction remains arithmetic; explicit author confirmation of that typo is not claimed. Here 5292 is independently checked inside the controlling MT field. The proposed interpretation of the author’s seasonal shorthand also remains unconfirmed; the present calculation explicitly uses the paired dates above.

**The source path supports three exact views of the twelvefold interval.** Write K=529. Direct subtraction gives

\[
H-A=1058=2K,\quad A-N=1587=3K,\quad
N-M=1058=2K,\quad M-Z=2645=5K.
\]

| Partition | Actual Nisan-BC path | Widths |
|---|---|---|
| **2+10** | 6350 → 5292 → 2 | 1058 + 5290 |
| **5+7** | 6350 → 3705 → 2 | 2645 + 3703 |
| **7+5** | 6350 → 2647 → 2 | 3703 + 2645 |
| **2+3+2+5** | 6350 → 5292 → 3705 → 2647 → 2 | 1058 + 1587 + 1058 + 2645 |

The two 1058 intervals are separated by a **2645=5K translation** of corresponding signed endpoints: **[−6350,−5292]+2645=[−3705,−2647]**. This is an exact finite correspondence between the specified cumulative and regular endpoints. It does not identify their event roles or introduce a universal chronological translation operator.

The new first leg can also be read directly from the cumulative source openings:

\[
6350-5292=600+(5750-5296)+4
=600+454+4=1058.
\]

Thus both endpoints of this first leg are Cainan ON, in different cumulative traditions. Its width equals the previously retained **598+460**, but that equality does not turn it into the older ON-to-OFF interval.

**This extends C09’s endpoint qualification.** C09 correctly observed that splitting 6350→2 at the **LXX OFF opening 5290** gives **1060+5288**. Those endpoints and lengths remain unchanged. C14 uses the distinct **MT ON member 5292**, giving **1058+5290**. The relation 12=2+10 therefore gains a contiguous source realization after the C10 attachment. It does not relabel the old LXX opening or treat the numerical coordinate 5290 as though it were already this tenfold interval.

**The finite boundary test selects one of the five C10 rows.** Prepending 2K to a C10 opening A requires a cumulative LXX Shem member H=A+1058.

| C10 opening A | SP Noah N | MT geometric midpoint M | Landing Z | Required H for 2+10 | Inside LXX ON Shem 6350–6343? |
|---:|---:|---:|---:|---:|---|
| 5296 | 3709 | 2651 | 6 | 6354 | No |
| 5295 | 3708 | 2650 | 5 | 6353 | No |
| 5294 | 3707 | 2649 | 4 | 6352 | No |
| 5293 | 3706 | 2648 | 3 | 6351 | No |
| **5292** | **3705** | **2647** | **2** | **6350** | **Yes — opening** |

All numerals in this table are Nisan BC. The first midpoint 2651 remains geometric only in this domain; the next four have the alternate-frame Flood identifications established in C10. The new attachment does not repair that first row’s missing event qualification.

With the actual opening 6350 held, the five prefix widths are **1054,1055,1056,1057,1058**, and the whole intervals are **6344,6345,6346,6347,6348**. Only the last is twelvefold. Conversely, holding the original 2 BC landing and testing the eight admitted LXX ON Shem members gives widths **6348 down to 6341**; only the opening is divisible by 529. These are bounded checks of retained fields, not an expanded selector search.

The numeral **6351t** belongs to the valid paired opening **6351t/6350n**. It does not admit **6351n** for the preceding row. Reading both seasonal components consistently reproduces the same interval lengths without enlarging the source field.

**The mixed twentyfold continuation supplies a dependent twenty-twofold path.** C12’s final mixed row already retains

\[
5292\text{ BC}\longrightarrow2\text{ BC}
\longrightarrow\text{reflected }5288,
\]

with widths **5290+5290=10580=20K**. The reflected endpoint is **AD 5288t/5289n**, from the derived LXX cumulative-OFF field member 5288. Prepending C14’s 1058 interval gives

\[
\boxed{6350\text{ BC}\longrightarrow\text{reflected }5288
\quad=11638=22K.}
\]

The paired civil calculations agree:

\[
6351+5288-1=6350+5289-1=11638.
\]

There is no civil year zero, and the fixed paired Mirror axis remains January AD1.

| Partition of 22K | Signed paired-center path | Widths |
|---|---|---|
| **2+20** | −6350 → −5292 → +5288 | 1058 + 10580 |
| **12+10** | −6350 → −2 → +5288 | 6348 + 5290 |
| **5+17** | −6350 → −3705 → +5288 | 2645 + 8993 |
| **2+3+2+5+10** | −6350 → −5292 → −3705 → −2647 → −2 → +5288 | 1058 + 1587 + 1058 + 2645 + 5290 |

The 17-fold leg uses the same SP companion Noah boundary and reflected cumulative OFF member: **3705+5288=8993**. The 22-fold total follows from **(H−A)+(A+D)=2K+20K**. It requires no additional numerical calibration after the new opening attachment. Its status is a dependent source-qualified comparison, with no claim of a separate statistical witness or a new canonical family rule.

**N01’s older 1058 interval keeps different endpoints.** The retained N01 arc has cumulative OFF earlier/later members **5886→5288**, and SP companion Noah→Flood-close **3705→3107**, both separated by 598. Restoring the cumulative 460 on the earlier side gives **6346→5288**, width 1058. Compare:

| 1058 construction | Earlier BC member | Later BC member |
|---|---:|---:|
| N01 older LXX ON→OFF interval | 6346 | 5288 |
| C14 LXX ON opening→MT ON member | 6350 | 5292 |

The new interval lies **four years earlier at both ends**. In signed coordinates it is the old interval translated by −4. Consequently, **6350−5288=1062**, and **6350+2647=8997**. They must not be substituted for the older **6346−5288=1058** or **6346+2647=8993** relations. The four-year distinction is part of the source attachment.

**The extension fits C11’s existing coordinate language.** On its retained L=598 layer, put t=−D, so N01 has t=−5288. C11 and C12 give

\[
A_s=-t-10580,\quad Z_s=-t-5290,
\quad U_s=t-1058,
\]

where the subscript denotes signed centers, and U is the older cumulative ON member. C14 adds

\[
\boxed{H_s=A_s-1058=-t-11638.}
\]

This introduces no additional free continuous coordinate. It adds a specified endpoint relation and a source-membership requirement. Across the five retained C10 placements the required H members are precisely 6354 through 6350, admitting only **t=−5288**. At that placement,

\[
H_s-U_s=-2t-10580=-4.
\]

The coordinate identity alone does not establish source support at the other placements or make the whole earlier Creation expansion packet valid at N01.

**The connection to 6877 remains a qualified companion prefix.** C11 already retained the N01 path

\[
4420\text{ BC}\longrightarrow\text{reflected Conquest }1399
\longrightarrow\text{reflected MT G2 Flood close }2457,
\]

with **5819+1058=6877=(11+2)K**. Its ordinary regular MT profile is full-430/Terah-0/Cainan-OFF. This differs explicitly from C14’s BC MT midpoint profile.

The older N01 interval **[−6346,−5288]** reaches the reflected Conquest→Flood interval **[1399,2457]** by C11’s translation **+7745**. The new prefix **[−6350,−5292]** reaches that same interval by **+7749**, retaining the four-year difference. These checked correspondences connect the twelvefold attachment to the companion thirteenfold prefix without merging their origins.

The full C03 6877→7475→8125 attachment at N01 would still require an ordinary MT Noah birth at **3055**, absent from the held MT three-Gear field **3056,3058,3060**. The new twelvefold path does not supply that missing source node. N01’s recorded J01 join remains the existing join; no historical join or full Creation-chain witness is added.

**Later 6900/7500 realizations retain C05’s separate qualification.** Holding the original 2 BC landing, the twelvefold expansions remain

| Width | Fixed-landing interval, Nisan BC | Outer boundary status |
|---:|---|---|
| 6348 | 6350 → 2 | Retained opening |
| 6900 | 6902 → 2 | Generated by 25/23 expansion |
| 7500 | 7502 → 2 | Generated by a second 25/23 expansion |

C05 independently realizes widths **6900** and **7500** from the apparent SP Creation closing boundary **4444 BC** to reflected MT G1 Flood **2456** and Noah **3056**. Its formal alignment **x↦−4446−x**, with endpoint-order reversal, maps the generated intervals to those retained upper intervals. Applied to the 6348 base it still gives **[−4444,1904]**; C05’s recorded absence of a 1904 Noah/Shem/Flood target remains unchanged. C14 qualifies the original cumulative base more fully but supplies no sourced transport of every expanded station. The previous 66-record search was not repeated.

**Evidence and checkpoint.** `evidence/check_c14.py` passes **98 exact checks**. Its two JSON outputs reproduce byte for byte in an isolated replay. Ten frozen inputs include the controlling File_18/File_09, both relevant handoffs, the C05/C09–C12 dependencies and the original N01/J01 certificate. `C14_SOURCE_GRAPH.json` records each paired endpoint, profile, contiguous path, affine attachment and obstruction; `C14_RESULTS.json` records the complete five-row boundary test and controls. Manifests record source and artifact hashes. Check counts are verification assertions, not independent witness counts.

Regular Gears remain restricted to **Noah/Shem/Flood**; SP companions to **Noah/Shem**. Creation remains fixed Normal/G2. The principal regular profile, explicitly named midpoint exception, native/derived LXX distinction and phase rules remain active. No canonical source, historical graph, source field or earlier report has been edited. No completed campaign was rerun. Recovery status remains **original Reports 01–10 complete**, continuation supplements **C01–C14 complete**; interrupted original Reports 11–25 remain unestablished as completed.

**Best next bounded step:** consolidate the now-qualified **2, 10, 11, 12, 13, 17, 20 and 22** constructions into one endpoint-relation graph, marking which relations follow from shared paths and which require distinct source attachments. Keep the N01 companion and N02/N04 complete Creation-chain domains explicit. **High is sufficient.**