C04 — The 868 interface has a formal continuation, with a precise Creation-source limit
21 September 2026 · Bounded continuation of C03.
The established 868-year attachment between 8125 and 8993 survives as an exact algebraic relation under expansion, but neither subsequent stage supplies another sourced ordinary SP Creation attachment. Holding C01’s Creation member fixed changes the gap to 1328 and 1828. Preserving proportional expansion requires a new, fractional Creation coordinate outside the held week.
This leaves C01’s two complete, sourced 6877 → 7475 → 8125 chains intact. C03’s four qualified chart placements still represent those same two chains. The result distinguishes their source realization from a proposed continuation of the entire expanded configuration.
The domain is the original N02/N04 interface, its four retained Shem placements, and the two expansion stages already present in the recovered evidence. Creation remains Normal/G2. Regular Gear variation remains confined to Noah/Shem/Flood; the SP ordinary Creation field remains 4421 through 4414 BC Nisan coordinates, with eight boundaries and seven elapsed years. No additional selector search is undertaken.
The archived expansion law fixes the member index. Let
The original expanded-join certificate gives
For N02 and N04, respectively, j=3 and j=5. Their later cumulative boundary and reflected SP Noah-birth image therefore obey
The certificate explicitly classifies expanded outputs as generated comparisons and admits no new source dates. Its source labels identify the original endpoints from which those outputs were generated. They do not independently date an ordinary SP Noah birth at each expanded coordinate.
Holding ordinary Creation gives an exact, different sequence. C01’s two Creation coordinates are C*=4422−j, hence 4419 and 4417. Keeping each fixed gives
Here δ is the left-side gap, and B is the Creation-to-generated-Noah width. The semicolon-separated entries below identify N02 and N04, respectively; they are numerical coordinates rather than seasonal pair notation.
| Old configuration stage | Cumulative coordinate, N02; N04 | Reflected Noah image, N02; N04 | Gap from fixed Creation, δ | Width from fixed Creation, B | Old 17-fold width |
|---|---|---|---|---|---|
| n=0, u=23 | 5287; 5285 | 3706; 3708 | 868 | 8125 | 8993 |
| n=1, u=25 | 5747; 5745 | 4028; 4030 | 1328 | 8447 | 9775 |
| n=2, u=625/23 | 6247; 6245 | 4378; 4380 | 1828 | 8797 | 10625 |
Every row retains δ+B=391u. The gap increases by 460 and then 500; the other width increases by 322 and then 350. These divide the old total gains of 782 and 850 between the two sides of a held Creation boundary.
For example, holding 4420t/4419n BC while using the first generated reflected endpoint AD 4028t/4029n gives 4420+4028−1=4419+4029−1=8447. Both seasonal civil calculations agree. The generated endpoint is not an ordinary SP primary Noah birth in the retained 3706/3708/3710 field.
There is a genuine partial source coincidence at the first expansion: cumulative 5747 and 5745 are members of File_18 §6D’s native LXX Cainan-ON field, 5750–5743. This agrees with the earlier 5290→5750 opening relation. It does not supply the missing ordinary SP Noah birth or a transported SP Creation boundary. The cumulative OFF source label and the native ON coordinate match remain separately identified.
Keeping the number 868 fixed requires moving Creation. If instead one imposes δ_n=868, then
| Witness | Original Creation | Creation required after one expansion | Creation required after two expansions |
|---|---|---|---|
| N02 | 4419 | 4879 | 5379 |
| N04 | 4417 | 4877 | 5377 |
These are 460 and 960 years earlier than the original members, outside the ordinary SP week. Their corresponding widths are 8125 → 8907 → 9757, which are not proportional expansions of 8125. Keeping the gap fixed therefore preserves neither the ordinary Creation boundary nor the 13-fold continuation. No Creation shift is admitted from this calculation.
Proportional continuation is unique once the old shared endpoint is retained. To require the new width to become 8125λ_n while retaining the archived Noah image, the needed Creation coordinate is forced:
Its gap is then
| Stage | Required Creation coordinate, N02 | Required gap | Proportional new width |
|---|---|---|---|
| n=0 | 4419 | 868 | 8125 |
| n=1 | 110481/23 | 21700/23 | 203125/23 |
| n=2 | 2762163/529 | 542500/529 | 5078125/529 |
N04’s required Creation coordinate is two less at every stage, with the same gaps and widths. The fractional entries are formal coordinate numerals, not newly assigned calendar dates. Both later Creation coordinates lie outside the held SP field.
The obstruction is exact: 4422≡6 mod 23. Consequently 4422(25/23)^n−j has denominator 23^n for positive n and integer j. At the first two stages its fractional parts are 12/23 and 254/529. No integer member selection or ordinary two-year Gear adjustment removes those fractions. This statement concerns the proposed affine continuation, not every possible anchoring construction.
The original shared SP Noah parameter has a different arithmetic property: R₀−j=3703=7×23². Its first two generated images remain integral. Integral generated coordinates still require independent source support; in the present field, 4028/4030 and 4378/4380 are not ordinary SP primary Noah births.
The complete C03 extension also has a consistent formal continuation. Apply
to each of its five signed positions. The four widths become
the midpoint halves become 299λ_n, and the final MT/SP separation becomes 650λ_n. The last endpoint agrees exactly with the archived SP Noah image. This captures a coherent rational extension while retaining its status as a proposed numerical construction. It does not authorize a new source transport on Creation or rescale a historical lifespan.
The two expansion indices must remain distinct. Define the scalar families
The established source attachment is
because the shared endpoint uses 8125, the second expanded width of the 13-fold family. Multiplying the relation gives the formal identity
Thus the first whole-configuration expansion already requires A₃=203125/23, beyond C01’s three sourced square-carrier widths. It must not be confused with C01’s original first or second endpoint realization. Direct comparison of the three archived configuration stages with C01’s three source landings finds only (n,k)=(0,2) at each witness.
The held-Creation discrepancy is compact:
It vanishes at the established attachment and records exactly why a fixed ordinary Creation endpoint does not support the proposed proportional continuation.
Checkpoint: the two source-qualified 868 attachments remain established; zero additional such attachments occur at the two tested expansion stages. An exact formal continuation is now specified, with its endpoint and integrality limits. C01’s 299 midpoint, 598/650 gains and sourced 8125 completion are unchanged. C02’s other midpoint controls remain separate. No primer, Rounded state, localized −33, upstream Gear, canonical amendment or new historical graph count is activated.
Evidence: evidence/check_c04.py passes 301 exact checks, with a byte-identical isolated replay. Eleven frozen inputs retain File_18/File_09, the original expansion law and generated arc records, the existing coordinate chart, Reports 06/09, and the relevant C01/C03 results and reports. The checker reconstructs the necessary ordinary SP fields, compares all three anchoring choices at both witnesses, verifies the four inherited placements, checks the full formal extension, and records the finite endpoint comparison. Check counts are verification assertions, not counts of independent witnesses. Earlier campaigns were not rerun; interrupted Reports 11–25 remain unestablished.
Best next bounded step: compare the 12×529 and 13×529 families at their source-qualified endpoints, focusing on the 552/598 and 600/650 gains and the shared 6900/7500 lengths. This returns to the retained source constructions with the formal-expansion limit now explicit. High is sufficient.