# 490d C476 — Primary and Companion Source Conditions for Noah-Held Ladders

25 September 2026 UTC. Step 7 of 10, C470–C479.

The complete24-context comparison explains the four inherited primary ladders and the absence of a companion middle. Set

\[
N=3491+2(g-1)+130C+60T+215F,
\quad g\in\{1,2,3\},\ C,T,F\in\{0,1\}.
\]

Here $N$ is the existing SP-primary Noah-birth coordinate; its local companion is $N-1$. For the held-primary expansion, the lower stations are

\[
N-1058\ \longrightarrow\ N-1150\ \longrightarrow\ N-1250.
\]

The middle is matched **MT Shem birth**, and the final is matched **MT Flood start**, at the same context as $N$. Both are sourced in all24 contexts. The input is a matched MT Flood start with Cainan+1, Terah+1 and one Gear higher. Those source settings exist exactly when **C=0, T=0 and g=1 or2**, with either F setting. The full-register lookup confirms no additional primary input alias. This yields precisely the four C429 complete ladders.

| Rail | Contexts | Sourced inputs | Sourced middles | Sourced finals | Complete paths |
|---|---:|---:|---:|---:|---:|
| SP primary |24|4|24|24|4|
| SP companion |24|5|0|24|0|

Subtracting1 gives matched **MT Flood-close** final endpoints, which account for the C472 B family. Four companion inputs also use Flood close at the shifted context. But the proposed middle is one less than the matched MT Shem birth. No such coordinate is supplied by any of the528 labels in the declared register. A local SP person companion does not license an MT person companion.

The all-label check also retains the fifth, already recorded C429 companion endpoint case:

| Function | BC coordinate | Source |
|---|---:|---|
| Held upper |3899|SP companion Noah birth, G3/C1/T1/F1|
| Input |2841|MT Noah birth, G1/C0/T0/F0|
| Required middle |2749|Absent from the declared register|
| Final |2649|MT Flood close, G3/C1/T1/F1|

Its input gap is $650+215+130+60+4-1=1058$. This is a different role/context match, not another instance of the shifted matched-Flood rule. The five companion lower paths are2432→2340†→2240,2434→2342†→2242,2647→2555†→2455,2649→2557†→2457, and2841→2749†→2649, where † denotes an absent coordinate.

The result unifies the existing C429 endpoint inventory with the newly classified C4721250 families. It explains the admission conditions; it does not claim those earlier ladders as new.

## Verification and scope

341 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.

All absence claims are limited to the frozen528-label/373-coordinate Noah/Shem/Flood register. Flood-close boundaries and SP person companions remain separate. These held-pivot diagnostics do not enlarge the active eight-pivot E menu.

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

**C477, proposed and unexecuted:** Test the nine source-endpoint Noah-held diagnostic cases under the two already fixed Mirror exchanges A−2639 and A−2635. Check conjugacy, cumulative anchor/input/middle/final membership, inherited17K phase assignments and complete-path support separately, without enlarging either menu.

The complete held family now distinguishes four genuine two-step primary paths from five companion endpoint-only cases. The fixed Mirror comparison can therefore test the expanded paths without treating either an endpoint match or an inherited17K corner as a sourced E trajectory.