# C12 — A minimal endpoint basis for Conquest and the 3+2+5 partition

22 September 2026 · Bounded continuation of C06 and C10–C11.

**C06's eleven-endpoint reconstruction extends to eighteen named endpoints with seventeen independent linear relations and one fixed source anchor.** Seven relations place the seven added endpoints. In a compact choice of basis, only one adds a new nonzero numerical span: **460**. The other six specify interval correspondence, paired reflection and the supplied partition ratios. Those endpoint assignments and ratios remain inputs; the reduction does not establish them from the number 460 alone.

The resulting **1058, 190, 5819 and 5+5 equality are dependent consequences**. A further source-qualified consequence is

\[
\boxed{20=3+17=10+10\quad\text{in units of }529.}
\]

Across the five supplied 3+7 chains, the Christ-period landing is the midpoint of a **mixed cumulative 20×529 Mirror interval**. This remains valid for the principal 6 BC landing without requiring the unsupported 2651n regular Flood midpoint. It is another endpoint realization of the 20-fold width, not an independent calibration of that width.

**What is held.** The principal reconstruction is C06's N02 packet, with its explicitly retained apparent-Creation comparison. Regular Gears remain confined to Noah/Shem/Flood. Creation stays Normal/G2; its ordinary and apparent fields remain distinguished. SP's person companion is confined to Noah/Shem, while Flood close remains an independent event boundary. The main regular profile is full-430/Terah-0/MT-SP-Cainan-OFF; the alternate MT Flood midpoint retains its separate full-430/Cainan-ON/Terah+60 profile. Native LXX Cainan ON and derived OFF remain explicitly identified.

All coordinates below are **signed paired centers**, with the fixed Mirror axis at January AD1. A BC coordinate C is −C, displayed as (C+1)t/Cn BC; its paired reflection has center +C, displayed as AD Ct/(C+1)n. There is no civil year zero.

**The retained C06 variables and equations.**

| Symbol | Signed center | Source role |
|---|---:|---|
| a | −4444 | Apparent SP Creation closing |
| o | −4421 | Ordinary SP Creation opening |
| b | −4419 | Ordinary SP Creation offset 2 |
| f | 2456 | Reflected MT G1 Flood start |
| g | 2458 | Reflected MT G2 Flood start |
| m | 2757 | Reflected SP G2 companion Noah death |
| n | 3056 | Reflected MT G1 Noah birth |
| h | 3108 | Reflected SP G2 Flood start |
| s | 3706 | Reflected SP G1 primary Noah birth |
| c | −5287 | Cumulative LXX OFF later member |
| c₋ | −5885 | Cumulative LXX OFF earlier member |

The original ten relations are retained as written:

| ID | Equation |
|---|---|
| R01 | o−a=23 |
| R02 | b−o=2 |
| R03 | g−f=2 |
| R04 | n−f=600 |
| R05 | 25f−23n−2a=0 |
| R06 | 25n−23s−2b=0 |
| R07 | h−g=s−n |
| R08 | 2m=g+n |
| R09 | b−c=868 |
| R10 | c−c₋=n−g |

R05 and R06 retain their source-qualified anchored dilation meanings from C05–C06. The measured **868 attachment remains an input**. These relations give L=n−g=598, s−n=650, g−b=6877 and s−c=8993; C12 does not rerun the earlier search that identified their endpoints.

**Seven added endpoints require seven added relations.**

| Symbol | Signed center | Source role |
|---|---:|---|
| u | −6345 | Cumulative LXX native-ON earlier member |
| q | 1400 | Reflected Conquest member, AD1400t/1401n |
| v | 2648 | Reflected MT G2 Flood, retained ON/+60 profile |
| A | −5293 | C10 cumulative MT ON Flood member |
| X | −3706 | Actual BC SP G1 primary Noah birth |
| Y | −2648 | Actual BC MT ON/+60 Flood midpoint |
| Z | −3 | C10 Christ-period landing, 4t/3n BC |

Write **E=c−u**. The compact extension is:

| ID | Equation | What the relation supplies |
|---|---|---|
| R11 | c₋−u=460 | Cumulative Cainan insertion |
| R12 | s−v=E | The retained cumulative/regular equal-interval correspondence |
| R13 | g−q=E | The Conquest/Flood correspondence |
| R14 | X+s=0 | Paired reflection of SP Noah |
| R15 | Y+v=0 | Paired reflection of alternate MT Flood |
| R16 | 2(X−A)=3E | The supplied 3-fold first leg, measured against the 2-fold interval |
| R17 | Z−A=5E | The supplied 10-fold outer interval, measured against the 2-fold interval |

R12 and R13 are statements about the **identified endpoints**, not permission to assign a source label to any point produced by the equations. R16 and R17 retain the dimensionless allocation choices **3/2 and 5** from the supplied construction. In particular, the presence of just one new nonzero right-hand-side constant does not reduce the seven source and incidence choices to one historical assumption.

The checker uses the expanded forms, for example R12 as v−s−u+c=0 and R16 as 2X−2A−3c+3u=0. It solves the complete system exactly over rational numbers.

**The square-of-23 unit is recovered from the retained spans.**

\[
L=600-2=598,
\qquad E=L+460=1058,
\qquad\boxed{K=E/2=529=23^2.}
\]

No separate 529-length datum is added to this basis. The commensuration

\[
10\times598=13\times460=5980
\]

connects C06's 13-fold Creation width to E. It follows that

\[
g-b=13K,\quad g-q=2K,\quad\boxed{q-b=11K=5819}.
\]

The Conquest **attachment** R13 is an independent endpoint relation; once it is retained, the numerical 5819 does not require another calibration. Its membership in the Conquest field is verified separately in C09–C11 and preserved here.

Likewise,

\[
v-g=(s-g)-E=(650+598)-(598+460)=\boxed{190}.
\]

File_18 and the retained profile independently identify this as **130+60**. The number 190 follows in the compact reconstruction, but the regular Cainan and Terah operations do not become the cumulative 460 insertion.

There is an equivalent source-first basis: replace R12 by the directly documented **v−g=190**. The two sets have identical affine row spaces. This makes the distinction precise: one may use the regular profile displacement or the equal-interval correspondence as the reconstruction relation; neither constitutes extra independent evidence once the other relations are held.

**The retained 5+5 equality adds no further constraint.** Reflection and R12 give

\[
Y-X=E=2K.
\]

Together with R16–R17 this yields

\[
X-A=3K,\quad Y-X=2K,\quad Z-Y=5K,
\]
\[
\boxed{Y-A=Z-Y=5K,\qquad Z-X=7K.}
\]

The midpoint equation **A+Z−2Y=0** has the exact certificate

\[
\boxed{2R12+2R14-2R15-R16+R17.}
\]

Here each R denotes its zero-form linear equation. Thus the old 5+5 observation is compatible with, and algebraically implied by, the retained equal interval, reflection and 3+7 allocation. The source qualification of Y is still a separate requirement; this implication does not supply a 2651n Flood event in the principal opening row.

**The independence claim is narrow and reproducible.**

| Exact linear-algebra result | Value |
|---|---:|
| Named endpoint variables | 18 |
| Rank of the seventeen relations | 17 |
| Rank after deleting any one relation | 16 |
| Rank after adding a=−4444 | 18 |
| Rank after appending all 21 derived relations | 17 |

Without the final source anchor, the remaining free direction moves the original eleven points and u/q/v by +δ, while A/X/Y/Z move by −δ. This is the reflection behavior already found in C11. It is not a common translation of all eighteen coordinates and is not claimed as a source-preserving chronological action.

For every new relation the packet supplies a formal deletion witness. For example, deleting R13 permits q to move while the entire 3+2+5 partition stays fixed. Deleting R16 permits A and Z to move together while leaving the Conquest comparison and the 10-fold outer length fixed. These demonstrate the independent endpoint-placement constraints; the perturbed values are not admitted chronologies.

Deleting R11 exposes the remaining numerical calibration: with the old anchored core held and u increased by λ, q and v increase by λ, Y decreases by λ, A increases by 3λ/2 and Z decreases by 7λ/2. The supplied 460 insertion fixes that freedom. Once the stipulated correspondences and ratios are held, an equivalent numerical calibration could use the regular 190 offset or the Conquest 11-fold relation; their source meanings remain distinct.

This rank calculation reconstructs one selected packet. It does not reduce the original three-coordinate configuration space to one dimension, count independent textual witnesses, or prove a minimal axiom system for the whole repository.

**The 20-fold interval has a sourced Christ-period midpoint.** This consequence has a particularly simple proof in positive BC Nisan coordinates. Let N be one of the supplied Noah births, A its cumulative MT ON predecessor, Z its Christ-period landing, and D the admitted cumulative LXX OFF member in the 8993 comparison. Then

\[
A-N=3K,\qquad N-Z=7K,\qquad N+D=17K.
\]

Reflect the cumulative D boundary to its AD paired position. The source path through actual BC Noah gives

\[
\boxed{A\text{ BC}\ \longrightarrow\ N\text{ BC}\ \longrightarrow\ D\text{ reflected}
\quad=3K+17K=20K.}
\]

The path through the landing gives

\[
A-Z=10K,\qquad Z+D=17K-7K=10K.
\]

Therefore **Z is the midpoint of the same 20K interval**. This also gives the common refinement **3+7+10**, without requiring an intermediate regular Flood node.

| MT cumulative ON A, Nisan BC | SP Noah N, Nisan BC | Landing Z, Nisan BC | Reflected LXX cumulative OFF D | Two equal halves |
|---:|---:|---:|---|---|
| **5296** | **3709**, G3 companion | **6** | AD5284t/5285n | 5290+5290 |
| 5295 | 3708, G2 primary | 5 | AD5285t/5286n | 5290+5290 |
| 5294 | 3707, G2 companion | 4 | AD5286t/5287n | 5290+5290 |
| **5293** | **3706**, G1 primary | **3** | AD5287t/5288n | 5290+5290 |
| 5292 | 3705, G1 companion | 2 | AD5288t/5289n | 5290+5290 |

All five cumulative endpoints were checked against the held source fields. The principal paired chain is

\[
\boxed{5297t/5296n\text{ BC}
\ \xrightarrow{5290}\ 7t/6n\text{ BC}
\ \xrightarrow{5290}\ \text{AD }5284t/5285n.}
\]

For its second half, the civil component checks are **7+5284−1=6+5285−1=5290**. Its full width is **5297+5284−1=5296+5285−1=10580**. Thus the no-year-zero and paired-season conventions agree.

This is a **mixed MT-ON / LXX-OFF cumulative comparison**, distinct from the previously retained 5290-opening self-Mirror construction of the same width. The interval's midpoint is the selected Christ-period landing, while the fixed reflection axis remains January AD1. The five rows retain the supplied landing field; they are not five independent historical nativity claims.

The extension does not add a new 20-fold calibration: **20=17+3** and **10=17−7** supply both halves. In the N02 linear register these appear as **A+c=−10580**, **Z+c=−5290**, and **A−c=2Z**. Their certificates retain the old R09 **868 attachment**; that attachment has not been derived from the 598/460 core. The principal row's 2651n midpoint failure is unaffected, because the new 20-fold path uses the Noah and Christ-period nodes directly.

**Source limits remain active.** The eighteen-node packet is fully sourced at the principal N02 attachment. C11's N04 core follows the corresponding +2/−2 displacement, but translating the entire C06 apparent packet would require **4442 BC**, outside the held apparent Creation field. N01/N03 keep their valid partial Conquest/Flood-close paths; their full Creation expansions still require unsupported MT Noah births **3055/3057**. No missing Shem join is supplied. The proposed full-week 300→299 compression and its extra 4413 boundary remain conditional and are not used in this basis.

The five mixed cumulative 20-fold comparisons were checked separately against the actual fields, so their support does not depend on extending the full eighteen-node packet to every row. The previously recorded seasonal-envelope interpretation and the author's confirmation of the 5292 typo correction remain unresolved.

**Evidence and checkpoint.** `evidence/check_c12.py` passes **156 exact checks**. `C12_BASIS_CERTIFICATE.json` contains the endpoint register, seventeen equations, oriented free direction, twenty-one exact affine consequence certificates and seven deletion witnesses. `C12_RESULTS.json` contains source checks, ranks, the alternative-basis equivalence, all five mixed cumulative paths and the retained exclusions. Ten frozen inputs include File_18/File_09 and the relevant C06/C10/C11 reports and evidence. Both generated result files reproduce byte for byte in an isolated replay; input and artifact SHA256 manifests accompany the packet.

No canonical source or historical graph was changed, and no completed campaign was rerun. The completed continuation is now **C01–C12**, alongside the recovered original **Reports 01–10**. Interrupted original **Reports 11–25** remain unestablished as completed; C12 is a new continuation supplement.

**Best next bounded step:** leave the **868 attachment and 460 insertion symbolic** in this reduced system, then distinguish identities that survive from 529-family alignments that require their exact calibrations. Use the retained equations and source limits without rerunning the older endpoint campaigns. **High is sufficient.**