# C15 — Eight tracked 529 families in one endpoint-relation graph

22 September 2026 · Bounded consolidation of C05 and C09–C14.

**All eight tracked base widths — 2, 10, 11, 12, 13, 17, 20 and 22 times 529 — fit the N01 companion comparison.** Its thirteenfold contribution is the retained **Creation→Conquest→Flood-close prefix**. The complete fixed-Creation **6877→7475→8125** expansions remain confined to **N02 and N04**. The graph therefore unifies the base-width relations while preserving different domains for their later continuations.

The principal reduction is a distinction among **path addition**, **paired reflection**, **calibrated equal widths**, and **source attachments**. In particular, **12=2+10** and **22=12+10** have literal shared-endpoint paths. The relation **17=13+2×2** also uses the measured attachment equation **868+190=1058**. Its outer flanks remain **868 and 1248**, as established in C08.

“Eight” names the families being consolidated, not a census of every multiplier obtainable by adding graph edges. The 3-, 5- and 7-fold auxiliary legs remain explicit. No all-pairs search, new historical join search or completed campaign was rerun.

**The N01 view places all eight base families on checked endpoints.** The diagram uses signed paired centers: a BC Nisan coordinate C is −C, with pair (C+1)t/Cn BC; its paired reflection is +C, with pair AD Ct/(C+1)n. K=529. Solid arrows are elapsed intervals. Dotted arrows are the explicitly checked **coordinate correspondence T(x)=x+7749**, not additional elapsed-span claims or source identities.

```mermaid
flowchart TD
    H["H: LXX ON Shem opening −6350"] -->|"2K"| A["A: MT ON cumulative member −5292"]
    A -->|"3K"| X["X: SP G1 companion Noah −3705"]
    X -->|"7K"| Z["Z: retained landing −2"]
    Z -->|"10K"| D["D: reflected LXX OFF member +5288"]
    H -->|"12K"| Z
    A -->|"20K"| D
    X -->|"17K"| D
    H -->|"22K"| D
    b["b: SP Creation member −4420"] -->|"11K"| q["q: reflected Conquest +1399"]
    q -->|"2K"| f["f: reflected MT G2 Flood close +2457"]
    b -->|"13K"| f
    H -. "T +7749" .-> q
    A -. "T +7749" .-> f
```

Here H and A belong to different cumulative traditions, both Cainan ON. X is regular SP, full-430/Terah-0/Cainan-OFF, with its SP-only companion label. The reflected ordinary MT Flood at f is also full-430/Terah-0/Cainan-OFF. Creation is a member of the fixed Normal/G2 field; Conquest has no Gear. The graph’s separate regular midpoint **Y=−2647** belongs to the explicitly retained MT **full-430/Cainan-ON/Terah+60** profile. It refines X→Z into **2K+5K**, without changing the profile of f.

The exact paired endpoints and profiles for all registered branches are in `C15_ENDPOINT_GRAPH.json`. D is a reflection of a cumulative member, not a historical AD event. Z remains a retained Christ-period comparison landing, without a new historical birth-date claim. The paired Mirror axis remains January AD1; there is no civil year zero.

**The family register keeps the construction attached to the number.**

| Family | Base width | Retained realization in the consolidated graph | Principal dependency |
|---:|---:|---|---|
| 2 | 1058 | Cumulative ON-earlier→OFF-later; Conquest→MT Flood; alternate MT Flood→SP Noah; their BC reflection; N01 H→A | E=598+460, with each endpoint correspondence separately retained |
| 10 | 5290 | A→Z and Z→D in the five C10/C12 rows | Supplied outer partition and the reflected 17-fold comparison |
| 11 | 5819 | Ordinary SP Creation→reflected Conquest | Seven native pairs; four enter the Flood-linked graph |
| 12 | 6348 | N01 H→Z, original 6350→2 BC | New head interval H→A followed by the existing tenfold path |
| 13 | 6877 | Ordinary SP Creation→reflected MT Flood | Five retained base intervals including C05’s opening case; four have the Conquest split; two have the full fixed-origin expansion |
| 17 | 8993 | Cumulative OFF→reflected SP Noah, its paired reflection, and retained opposite Noah/Flood diagonals | Existing source calibration and equal-interval attachment |
| 20 | 10580 | Five mixed A→D intervals; separately, LXX OFF opening to its own reflection | Mixed 3+17 / 10+10 paths; independently typed opening self-Mirror construction |
| 22 | 11638 | N01 H→D | Dependent 2+20, 12+10 or 5+17 path |

The original **5290 opening coordinate** remains distinct from a 5290-year interval. Actual tenfold intervals now have their own source endpoints in C10–C14. Likewise, the new 22-fold representation does not open File_53’s separately controlled 22-fold genealogy subject.

These registrations are selective constructions already supported by the completed reports. The table does not assert that no other endpoint pair in the repository can have one of these widths. Later expansion states retain C09/C14’s qualifications: the base graph does not confer source labels on every 25/23 image. In particular, C05’s separately anchored 6900/7500 realizations remain separate from the generated fixed-2-BC outer boundaries.

**Shared endpoints give exact path certificates.** For any registered interval e from endpoint a to endpoint b, its endpoint expression is b−a. The checker verifies path identities by cancelling endpoint coefficients, not merely by substituting the selected numbers. These identities hold for arbitrary endpoint coordinates as long as the same incidences are retained.

| Relation | Source path | Domain |
|---|---|---|
| 13=11+2 | b→q→f | N01–N04 |
| 10=3+7 | A→X→Z | All five C10 rows |
| 10=3+2+5 | A→X→Y→Z | N01–N04; principal 2651n midpoint excluded |
| 20=3+17 | A→X→D | All five C10 rows |
| 20=10+10 | A→Z→D | All five C10 rows |
| 12=2+10 | H→A→Z | N01 |
| 22=2+20 | H→A→D | N01 |
| 22=12+10 | H→Z→D | N01 |
| 22=5+17 | H→X→D | N01 |

The numerical family labels still depend on the retained widths of the component intervals. Path cancellation explains why, **after those components are established**, each displayed total adds no further constraint. The evidence contains **53 path certificates**, including the refinements and ordinary non-529 flanks.

Paired reflection has a different certificate. Let c be the BC cumulative OFF member and s the reflected SP Noah coordinate. Then D=−c and X=−s. Consequently

\[
(D-X)-(s-c)=(D+c)-(X+s)=0.
\]

This gives the equal 17-fold width on **[c,s]** and **[X,D]** by reflection with reversed endpoint order. It is not an additive path from one interval to the other, and it does not identify the different endpoint roles.

**The 17/13 relation exposes one compact attachment balance.** In each N01–N04 branch, retain

\[
\alpha=b-c=868,\qquad W=f-b=6877,
\qquad p=v-f=190,\qquad s-v=E=1058,
\]

where v is the reflected alternate-profile MT Flood. The actual contiguous decomposition is

\[
s-c=\alpha+W+p+E.
\]

The source values satisfy

\[
\boxed{\alpha+p=868+190=1058=E,}
\]

so

\[
\boxed{s-c=W+2E=8993.}
\]

This identifies precisely how the width equality depends on source attachments. The left flank is α=868, and the right flank is **p+E=1248**. They remain unequal. The 190 is the documented regular **130+60** profile displacement; it is not the cumulative Cainan 460 operation.

The checker records an exact dependency certificate:

\[
\begin{aligned}
F_{17}-F_{13}-2E
&=(\alpha+p-E)+\bigl[(s-v)-E\bigr].
\end{aligned}
\]

Both terms vanish on the admitted packet, using the actual-flank path identity. This expression has nonzero endpoint coefficients before the extra equalities are imposed. For example, a formal one-unit movement of b alone changes the relation while all genuine path-cancellation identities remain valid. That diagnostic is not a proposed chronology.

The relation **17=11+3×2** then follows by combining this result with the b→q→f path. It does not establish three new consecutive 1058 event segments between the original 17-fold endpoints.

**One unit and two branch residuals organize the eight base widths.** This is a projection of the completed reconstruction, with its incidence choices held. Let

\[
K=E/2,\qquad
\gamma=W-13K,\qquad
\delta=\alpha+p-E.
\]

The source calibration is **E=1058**, **W=6877**, **α=868**, **p=190**, so K=529 and γ=δ=0. The square unit is inherited from **(600−2+460)/2**; C15 adds no separate 529 datum.

Hold the existing correspondences and partition assignments:

\[
f-q=E,\quad s-v=E,\quad X=-s,\quad Y=-v,\quad D=-c,
\quad X-A=\tfrac32E,\quad Z-A=5E,
\]

and, where source-admitted, C14’s **A−H=E**. The resulting widths are:

| Registered width | Symbolic result | Departure from its named K multiple |
|---|---|---|
| E | 2K | 0 |
| A→Z | 10K | 0 |
| b→q | 11K+γ | γ |
| H→Z | 12K | 0 |
| b→f | 13K+γ | γ |
| c→s, or X→D | 17K+γ+δ | γ+δ |
| A→D | 20K+γ+δ | γ+δ |
| H→D | 22K+γ+δ | γ+δ |
| Z→D, the second tenfold half | 10K+γ+δ | γ+δ |

Thus the attached branch widths require **γ=0 and δ=0** for simultaneous exact closure. Adding the 12- and 22-fold representations after C14’s endpoint attachment adds no new independent branch-width condition. The 12-fold result follows from the held head and outer-span assignments; the 22-fold departure is the already existing mixed-20/17 departure.

This does not reduce the source choices to two historical assumptions. The interval identifications, reflection assignments, **3/2 and 5 allocation ratios**, and C14 head attachment remain explicit inputs. Source membership is checked separately. Symbolically constructing H does not admit the required 6351 or 6353 Nisan opening in N02 or N04.

The projected description has **12 endpoint variables and 11 independent relative equations**. Adding H gives 13 variables and rank 12. Its remaining displacement parameter moves the original side and reflected side in opposite signed directions. This is a smaller projection focused on base-family relations: it omits the independent regular Shem comparison coordinate and the fuller Creation-expansion apparatus. It neither replaces C12’s **18 endpoints / rank 17** nor revises the original three-coordinate chart.

**The opening self-Mirror comparison retains an absolute registration.** Write O=5290 for the retained LXX OFF opening, and I₀=[−O,+O]. Its self-Mirror width is 2O=10580. In every C12 mixed row,

\[
\boxed{[A,D]=T_Z(I_0),\qquad T_Z(x)=x+Z,}
\]

with signed Z equal to −6, −5, −4, −3 or −2. For N01 this is

\[
[-5292,+5288]=[-5290,+5290]-2.
\]

This exact interval correspondence changes the left endpoint’s source role from **LXX OFF opening** to **MT ON member**. It is not a source-preserving global translation. Z is the midpoint of this mixed interval, while the project’s fixed paired Mirror axis remains January AD1.

The opening also registers the absolute branch position. If j=O−D is the selected OFF-member offset, then **c=j−O**; the five rows have j=6,5,4,3,2. This removes the otherwise free placement once O and the member selection are held.

If O is independently released in a formal diagnostic, it has a separate departure **ω=O−5E**. The opening’s coordinate departure is ω and its self-Mirror width departure is 2ω. It is therefore incorrect to silently infer every opening claim from the two branch residuals alone. Three independently released quantities γ, δ, ω have formal rank three; this is not an independent-witness count.

C13’s actual compact diagnostic did **not** independently release O. With its α=868+x, E=1058+y, W=6877 and p=190−y, these quantities become

\[
\gamma=-\tfrac{13}{2}y,\qquad
\delta=x-2y,\qquad
\gamma+\delta=x-\tfrac{17}{2}y,\qquad
\omega=-5y=\tfrac{10}{13}\gamma.
\]

That is exactly C13’s two-residual result. The graph consolidation identifies the extra source registration; it does not overturn the earlier diagnostic or admit nonzero x/y as source chronologies.

**The source domains remain narrower than the formal graph template.**

| Branch | C10 BC path A→Noah→landing | Sourced alternate MT midpoint | 11+2 Creation prefix | Full fixed-origin 13 expansion | Retained 12/22 attachment | Existing complete join labels |
|---|---|---|---|---|---|---|
| Principal, 6 BC | 5296→3709→6; SP G3 companion | No: 2651n is only geometric | No | No | No | None assigned by this attachment |
| N04 | 5295→3708→5; SP G2 primary | 2650, G3 start | Yes | **Yes** | No: requires H=6353n | J05 |
| N03 | 5294→3707→4; SP G2 companion | 2649, G3 close | Yes | No: requires MT Noah 3057 | No: requires H=6352n | No retained complete Shem join |
| N02 | 5293→3706→3; SP G1 primary | 2648, G2 start | Yes | **Yes** | No: requires H=6351n | J02/J03/J04 |
| N01 | 5292→3705→2; SP G1 companion | 2647, G2 close | Yes | No: requires MT Noah 3055 | **Yes, H=6350n** | J01 |

All five rows have the source-qualified **3+7**, reflected **17-fold**, and mixed **20-fold** paths. The principal row’s standalone 17-fold endpoints do not give it a completed Noah/Flood/Shem join. N01’s J01 remains its previously retained SP native-215 companion Shem comparison; N02’s three join labels remain one Creation-chain witness with distinct Shem comparisons. No new join is inferred from the edge counts.

The additional C05 base interval **4421 BC→reflected MT G1 Flood 2456** supplies the fifth registered 6877 interval. Its implied Conquest point 1398 is outside the held Conquest field. It remains a valid base interval without the 11+2 split in this domain. The seven native 11-fold pairs likewise remain distinct from the four that have the present ordinary MT Flood extension. The proposed full-week 300→299 compression remains conditional and is not needed for this graph.

**The reduced grammar remains conditional on typed attachments.** Elapsed lengths add along source paths. Paired reflection preserves interval length after endpoint reversal. Finite translations relate the explicitly checked interval pairs. Equal lengths alone do not identify typed endpoints, and the retained source domains are not closed under every formal translation or dilation. The appropriate current description is therefore a set of **typed paths and finite affine correspondences**, with the γ/δ width conditions and opening registration written separately.

**Evidence and checkpoint.** `evidence/check_c15.py` passes **149 exact checks**. Its three JSON outputs reproduce byte for byte in an isolated replay. The consolidated record contains **77 typed endpoint nodes, 124 interval edges and 15 paired-reflection relations**, with **53 path identities**, **17 further equality certificates**, and **15 finite interval correspondences**. These describe registered representations and proof steps, not independent observations.

Twelve frozen dependencies contain File_18/File_09, the Mirror handoff, and the completed C05/C09–C14 records needed here. The checker verifies source membership, civil seasonal spans, endpoint-incidence cancellation, calibrated dependency decompositions, branch restrictions, exact symbolic residuals and consistency with C13. It reads prior results without rerunning their searches. File_18/File_09 remain byte-identical to the controlling checkpoint.

Regular Gears remain confined to **Noah/Shem/Flood**; SP companions to **Noah/Shem**. Canonical files, source fields, the original historical graph and all earlier reports remain unchanged. Capture now; canonize later. Completed status is **original Reports 01–10 plus continuation supplements C01–C15**. Interrupted original Reports 11–25 remain unestablished as completed.

**Best next bounded step:** test an alternative basis in which the retained opening self-Mirror correspondence and 8993 Noah comparison generate the **3+7 and 5+5** partitions. Identify exactly which source-matching assumptions replace C12’s explicit 3/2 and 5 allocations, and whether the same finite domains survive. **High is sufficient.**