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
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.
No separate 529-length datum is added to this basis. The commensuration
connects C06's 13-fold Creation width to E. It follows that
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,
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
Together with R16–R17 this yields
The midpoint equation A+Z−2Y=0 has the exact certificate
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
Reflect the cumulative D boundary to its AD paired position. The source path through actual BC Noah gives
The path through the landing gives
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
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.