490d C313 — SP 500-year binding to 17K Noah endpoints
24 September 2026. One bounded continuation after C312. K=529; 17K=8993.
Result: one of the five same-index comparisons recovers the exact source500-year Noah→Shem birth relation with matching Gear and rail: j5, SP Gear2 primary, Noah3708 BC → Shem3208 BC → Shem death2608 BC. The source intervals are500+600=1100. All five Shem-life pairs remain valid after normalization.
The inherited paths require a separate qualification. At this successful center match, their components descended from the same cumulative phase give499½ and500½ years. The exact source biography and equality of unordered phase-pair sets survive; the stronger500-year correspondence preserving the same cumulative component does not.
Fixed five-case test
C312 supplied SP Shem-life pairs at cumulative LXX OFF indices j3…7 in Cainan OFF/Terah+60/minimum215. Convert both birth and death to the already admitted OFF/Terah0/full430 state:
File_18 §1.1, lines541–551, documents this variant difference. Its §3.2.1, lines1573–1594, supplies the SP primary/companion rows and explicitly retains matched Noah/Shem birth separation500 and Shem life600. The normalization changes neither Gear nor rail. All normalized endpoints are already in the frozen C286 register.
For the same cumulative Flood index, let q=−5290+j. The retained17K relation selects the reflected Noah base n=q+8993=3703+j. Each selected Noah has an admitted SP source counterpart. The table gives the corresponding regular BC source bases, retaining the states attached to those values.
| j | Normalized Shem life, BC | Shem Gear/rail | 17K Noah BC source / Gear/rail | Birth gap | Same-Gear/rail500 binding |
|---|---|---|---|---|---|
| 3 | 3210→2610 | G3 primary | 3706 / G1 primary | 496 | No |
| 4 | 3209→2609 | G3 companion | 3707 / G2 companion | 498 | No |
| 5 | 3208→2608 | G2 primary | 3708 / G2 primary | 500 | Yes |
| 6 | 3207→2607 | G2 companion | 3709 / G3 companion | 502 | No |
| 7 | 3206→2606 | G1 primary | 3710 / G3 primary | 504 | No |
Every row has its own source-matched Noah500 years earlier than Shem. Only j5's source-matched Noah is also the Noah chosen by17K from the same cumulative index. The other rows compare different declared states; they do not revise a genealogical age. Shem's complete600-year life is retained in all five cases.
Why the center match occurs at j5
The cross-bridge equation fixes the original Shem birth at 3058−j. After +155 its reflected base is:
Subtracting from the17K endpoint gives:
Consequently500 forces j5. The496/498/500/502/504 pattern follows from these opposite slopes; its five entries are not five independent witnesses. Source labels supply the additional fact that the middle case is G2 primary on both sides.
Direct17K endpoint support must also remain separate from the stronger C287 four-node configuration. Within this five-case domain, j3/4/5 belong to that retained configuration. The admitted Noah endpoints at j6/7 do not complete it: its additional cumulative ON Shem indices would be8/9, outside the retained field. This does not invalidate their separate17K legs.
Seasonal components: biography, sets and path ancestry
Use the established signed quarter-year coordinates around each paired center. C271 supplies exact Noah component records; File_09 Appendix A supplies the paired Mirror convention. C307 records the different orientations of the inherited paths.
For one cumulative component q+ε, with ε=±1/4:
Thus Shem's600-year component span is preserved, but:
At j5 this is499½/500½. None of the five rows supplies500 for both components with the same cumulative ancestry.
The ordinary G2 primary source biography still has exactly500 years between matched physical Noah/Shem phases. Likewise, subtracting500 from the unordered reflected Noah pair gives the reflected Shem pair. Connecting those equal output phases to the two inherited paths pairs opposite cumulative phase components. That reversal must be declared; it cannot be reported as preservation of the original component correspondence. No source date or global Mirror axis is changed by this distinction.
Evidential gain and next decision
C313 locates one exact source-labelled intersection of the500-year biography with the17K endpoint rule and identifies the remaining phase condition. It neither supplies C310's source-prescribed context selector nor repairs the earlier C305–C307600-year interface. No new endpoint, anchor, Gear scope, historical intention or global law is inferred.
The main verifier passed44 arithmetic, source-label, domain and exact-phase assertions plus16 source/predecessor authentication checks. Its outputs replay byte identically. A separate mathematical script obtained the same five gaps, unique typed center match and zero same-ancestry500 matches; a separate source/method review confirms the normalization and phase distinctions. Standing inputs and C127 remain unchanged.
Best next bounded step — C314, proposed only: test the reversed annual-index pairing required by the source500 relation, together with explicit phase reversal, within these same five cases. Allowing Noah index i and Shem index j gives the necessary condition i+j=10; the proposed finite test must verify each source Gear/rail, both Shem endpoints, phase ancestry and the existing stronger-chain boundaries. This examines a precisely forced correspondence without fitting another anchor or claiming a new global Mirror. High is sufficient. C314 has not been executed.