490d C439 — Source Complement Rule for Noah Birth–Death Reflections
25 September 2026 UTC. Step 5 of 10, C435–C444.
The entire Noah birth/death reflection class reduces to one complementary source rule. All36,864 role-pair comparisons at the four fixed centers agree with it; no hit falls outside the rule.
Let the center be c=3188+2(g−1)+60T, where g is1 or2 and T is0 or1. For a Noah birth and Noah death to reflect about c:
- Both belong to SP or the explicitly indexed LXX comparison field.
- Their Cainan flags sum to1, and their sojourn flags sum to1.
- Both retain the center's Terah setting T.
- Their Gear numbers sum to2g.
- Exactly one endpoint is the local SP companion.
The SP-normalized source base sum is3491+2541=6032. The complementary variants contribute130+215−1=344, producing6376=2×3188. Gear and common Terah shifts then supply the other three centers.
Explicitly, the reflection condition is
130(Cb+Cd)+60(Tb+Td)+215(Fb+Fd)+2(Gb+Gd−2)−(εb+εd)=344+4(g−1)+120T.
The finite flag ranges yield only the stated sum solution. The equation is not inferred merely from selected positive rows; all1,620 abstract bounded context-sum candidates were checked as well.
| Center | Allowed endpoint Gear pairs | Coordinate pairs | Labelled pairs |
|---|---|---|---|
| 3188 | G1/G1 | 8 | 16 |
| 3190 | G1/G3, G2/G2, G3/G1 | 24 | 48 |
| 3248 | G1/G1 | 8 | 16 |
| 3250 | G1/G3, G2/G2, G3/G1 | 24 | 48 |
Exactly one endpoint must be SP companion. Its primary partner may carry SP or the appropriate LXX label; LXX does not receive a companion rail. Native LXX Cainan remains ON, while OFF cases retain their theoretical status.
MT cannot participate in this role class at these centers: any pair containing an MT Noah endpoint has maximum coordinate sum6200, below the smallest required6376. The MT Noah record serves as the center, not one of these reflected endpoints.
These64 coordinate-center pairs and128 labelled pairs account for the largest part of C438's inventory. They exchange complementary source contexts and do not constitute128 single-biography lifespan comparisons. Matched biographies continue to preserve950.
Verification and scope
11 calculation checks and 18 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.
Finite source-context complement rule. No companion is copied to MT/LXX, no new biography is assembled, and no global reflection operation is inferred.
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
C440, proposed and unexecuted: Test whether the Noah-birth/Shem-death reflection class and the Shem-birth/Flood-boundary class are exactly related by opposed500-year source shifts. Preserve primary/companion and Flood-start/close distinctions, all aliases, and any change in pair ordering.
The dominant class now has one source rule. The next two role classes appear related by existing500-year source correspondences; testing that relation could reduce the remaining reflection grammar.