490d C475 — Four Inherited Noah Ladders and Their Split Factorizations
25 September 2026 UTC. Step 6 of 10, C470–C479.
For each of the four previously known Noah-held ladders, let the upper coordinate be , its Shem partner , and the formal split pivots , . The identity
holds in all four cases. The Shem split word equals , and its +500 translation equals as a full affine map. Source admission remains a separate requirement.
| Noah upper N | Inherited Noah-held lower path | Shem split lower path | Shem upper path | Split pivots | Split failure |
|---|---|---|---|---|---|
| 3491 | 2433→2341→2241 | 1933→1805→1741 | 2991→2955→2991 | 3405 / 2541 | First pivot3405 absent |
| 3493 | 2435→2343→2243 | 1935→1807→1743 | 2993→2957→2993 | 3407 / 2543 | First pivot and both middle endpoints absent |
| 3706 | 2648→2556→2456 | 2148→2020→1956 | 3206→3170→3206 | 3620 / 2756 | None; both pivots in active menu |
| 3708 | 2650→2558→2458 | 2150→2022→1958 | 3208→3172→3208 | 3622 / 2758 | Both middle endpoints absent |
All four original Noah-held words are fully sourced; these are C429 results. All four Shem-held words have sourced first and last endpoints but an absent lower intermediate (1841,1843,2056,2058). All four +500-translated split words fail source support for pivots and intermediate endpoints. Of the16 profiles, five are fully sourced; only one also uses the active eight-pivot menu.
This comparison recovers and contextualizes the established C434 G1/Terah-base boundary rather than claiming another discovery of it. It adds a common factorization comparison across the four inherited Noah-held ladders and distinguishes missing anchors from missing path stations. In particular, the3491 case shows that every interval endpoint can be sourced while the factorization still lacks an admissible first pivot.
The next unresolved explanatory point is the source grammar of the primary/companion difference in these held ladders. The complete1250 families C and B from C472 already give their matched final intervals. The full24-context test will determine why the intermediate behaves differently.
Verification and scope
49 calculation checks and 25 source/predecessor authentication checks passed. Main result files replay byte for byte. These are root-agent checks, not an independent-agent review or independent historical witnesses.
The held-Noah calculations remain inherited diagnostic comparisons outside the active eight-pivot menu. Formal translated pivots remain generated and are not admitted as sources. C434’s prior six-case result is retained as prior work.
Mirror/529 focus: K=529, E=25/23, P=70/69; J=300/299 remains File02 gap conversion, not a calendar date operator or Residue Protocol. Regular Gears are Noah/Shem/Flood only. SP person companions are local to Noah/Shem; Flood close is an independent event boundary. Both500/100 and502/98 bindings remain valid; SP primary Noah death retains its inclusive/harmonic qualification. LXX native Cainan is ON; OFF is theoretical removal. Global paired Mirror January AD1; local junction January AD2; no civil year zero. No new source point, phase, fitted axis, upstream/cumulative Gear, primer reopening or canonical/graph edit. Capture now; canonize later. Coordinates, source labels, context aliases and derived comparisons remain distinct. Counts are not independent historical witnesses.
Best next bounded step
C476, proposed and unexecuted: Derive and check the complete Noah-held1058→1150→1250 family over all24 existing Gear/Cainan/Terah/full430 contexts, separately for SP primary and companion records. Explain the four complete primary ladders and the companion endpoint-only cases through typed source relations, checking all coordinate aliases.
The factorization comparison is now finite and complete for the four inherited ladders. A source-state derivation of the held primary/companion family will link those older results to the newly classified1250 interval families without admitting another pivot.