C21 — The remaining 10-/13-fold agreement has a named source decomposition

22 September 2026 · First of the two authorized steps after C20.

The remaining cumulative/regular agreement reduces to a natural division at Abraham:

The 1910 is the sum of eight named lifespan-minus-begetting differences. The 269 is the difference between the cumulative Abraham-to-Moses lifespan total and the regular Abraham-to-Conquest span. The final six comprises the retained cumulative opening offset of five and the Flood closing offset of one.

This is an exact reorganization of the remaining source constraint. It does not add an independent witness to the 10-/13-fold result. Its equivalence depends on keeping C20's separate unit condition explicit.

The source cut at Abraham is fixed by the genealogy. The LXX OFF Arphaxad-to-Moses subtotal T=3879 separates into:

The corresponding SP regular Arphaxad-to-Abraham begetting total is s=940. The ordinary retained Abraham-to-Conquest suffix is A=760. Thus

This split follows the existing shared Abraham suffix. No selection among arbitrary subsets of the fifteen lifespans was performed.

The first block consists of named longevity remainders. File_18 §4.2 supplies the LXX lifespans and selected begetting ages. The eight selected ages agree numerically with the SP values in §3.2.1:

Each difference can be read as the remainder of the corresponding LXX lifespan after its retained begetting age. The comparison uses matching fathers and numeric ages; it does not identify different children. Arphaxad begets Cainan in native LXX and Shelah in SP. The derived OFF cumulative comparison removes Cainan's complete 460-year portion, as already verified in C17. It does not create an independently transmitted LXX OFF regular biography.

Terah's 205 here is the retained LXX lifespan. It is not SP's 145 lifespan. Shelah's 460 also remains distinct from the omitted second-Cainan lifespan of 460.

The cumulative and regular coordinate differences telescope through these eight named rows. They begin at 5285−3106=2179 at the selected Arphaxad comparison and end at 2435−2166=269 at Abraham. Each successive difference decreases by the corresponding remainder in the last column. The evidence records every intervening pair; no new dated event is assigned to a subtotal.

The second block is the named Abraham displacement. The cumulative suffix is

using Abraham, Isaac, Jacob, Levi, Kohath, Amram and Moses. The regular suffix remains

Consequently,

Both terminate at the retained Conquest/Moses boundary, so the common anchor cancels. With the source anchor restored for comparison, the cumulative Abraham boundary is 2435 BC and regular Abraham's birth is 2166 BC, reproducing the same 269. The regular total includes the retained full-430 sojourn and forty years to Conquest; this calculation does not reopen either primer hypothesis or supply missing Kohath/Amram begetting ages.

The last six years have two distinct boundary origins. The named LXX OFF cumulative comparison seam is 5285 BC. Its retained opening is five years earlier, at 5290 BC. Fixed SP Arphaxad co-registers with G1 Flood start at 3106 BC; the original handoff uses the Flood close one year later, at 3105 BC. Therefore

or directly

X is a same-side comparison of coordinates from the two chronology constructions. It is not the Mirror width or a new historical elapsed interval within a single chronology. In the retained paired display, both components give the same difference:

The corresponding paired Mirror width remains 8395, with the normal no-year-zero civil checks: 5291+3105−1=8395 and 5290+3106−1=8395. The difference 2185 and the width 8395 retain their separate operations.

The exact dependency on C20 can now be stated. Retain C19/C20's notation:

Here Φ is the remaining 10-/13-fold compatibility and Ψ is the doubled regular/local unit difference audited in C20. Define the new balance residual, distinct from earlier Creation endpoint labels, as

Substitution gives the coefficient identity

It holds before any source numbers are inserted. Therefore, with Ψ=0 retained, Φ=0 if and only if C=0. The old constraint pair (Φ,Ψ) and the new pair (C,Ψ) have the same rank two. This is a change of basis for the source conditions, not an additional condition or witness.

At the source,

which produces the compact form

The coefficients five and three come from eliminating the old calibration equations. They are not counts of five identical genealogical intervals followed by three identical intervals. The named source decomposition on the other side is 1910+269+6; these two decompositions should not be confused.

The cancellation of MT begetting inputs and the SP Creation head from X is conditional in its explanatory use: their constraint has been retained in Ψ. They have not disappeared from the joint 10/13/17 construction. C21 also verifies C=−2J17, where J17 is C19's anchor-eliminated 10-/17-fold compatibility. The compact gap balance therefore fits the same closed system already established.

The named decomposition does not uniquely determine the individual lifespans. All fifteen non-Cainan lifespans enter X with coefficient +1; the eight SP begetting ages and the five regular suffix portions enter with coefficient −1. A formal +1 change to one lifespan, compensated by −1 in another, preserves X and the calibration totals. Source provenance remains necessary to identify the actual named values.

Nine bounded control rows, including the source, check the dependency:

Changed source durations are formal diagnostics, not adopted chronology variants. The two boundary controls use available source positions without replacing the retained opening or original closing calibration. Every row satisfies Φ+Ψ+2C=0.

Validation and checkpoint. The checker passes 75 exact checks. It re-extracts the relevant current source lifespans and begetting ages, verifies the natural Abraham cut and the full named difference ladder, proves the coefficient identity and rank statement, and records the nine controls. An isolated replay reproduces all three JSON outputs byte for byte. All nine original dependencies remain unchanged.

The report and evidence continue C20. No earlier research campaign was executed and no canonical file was edited. Regular Gears remain confined to Noah/Shem/Flood; SP person companions remain confined to Noah/Shem; native LXX ON and derived OFF are explicitly distinguished. This continuation C21 does not establish the interrupted historical Reports 11–25 as recovered or completed.

Source locators. File_18 §§3.2.1, 3.3 and 3.4 control the SP begetting and shared regular suffix; §§4.2, 6A and 6D control the lifespan inputs and cumulative seams; §3.1.4 supplies G1 Flood start/close and fixed Arphaxad. File_09 §3.1 controls paired suffix inversion. C17 supplies the completed named stack; C19 the relative calibration equations; C20 the primitive ledger and unit condition. This is an internal audit of the supplied project model, not a new external textual or historical authentication.

Best next bounded step — High sufficient. Determine the separate roles of the two factors of 23 in 529: distinguish the ratio that supplies a factor of 23 from the source value that fixes the remaining scale. Use the completed source balances to test the resulting 10/13/17 and 8395 coefficients without rerunning endpoint campaigns.