# 490d C479 — Complete Interval Families, Sourced Factorizations and Mirror Limits

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

**All ten authorized steps, C470–C479, are complete. C480 remains unexecuted.**

The principal advance is a complete source comparison around the inherited $2K=1058\to1150\to1250$ ladder. The new censuses show how broad the1150 and1250 interval families are, while preserving the narrow conditions under which a complete sourced factorization exists. The fixed Mirror comparison retains the known17K corners but does not extend them to a sourced cumulative E ladder in the declared menu.

## 1. Complete1150→1250 census

There are **51 sourced coordinate intervals of1150**, with101 labelled endpoint pairs. Applying each of the eight fixed E pivots gives408 trials. Sixteen yield two integer outputs; exactly **three** have both outputs in the frozen register:

| Input pair, BC | E pivot | Output pair, BC | Sourced2K predecessor under the same fixed menu? |
|---|---:|---|---|
|1955 / 3105|3680|1805 / 3055|No; the sole integer candidate2093/3151 is absent|
|2020 / 3170|2756|1956 / 3206|**Yes:2148/3206, first pivot3620**|
|2148 / 3298|3620|2020 / 3270|No; none of the eight inverse pivots gives integer endpoints|

Thus the earlier complete interval ladder remains the unique one in this fixed menu. All three1250 outputs are matched MT Shem-death / SP-primary-or-LXX Shem-birth intervals. Their source gap is600+650; the coordinate-equal LXX labels remain qualified by their native or theoretical Cainan state.

## 2. Complete1250 source grammar

The entire register contains **74 coordinate pairs and130 labelled pairs** at1250. Seven disjoint label classes explain them:

| Class | Source binding | Coordinate pairs | Label pairs |
|---|---|---:|---:|
|A|Matched MT Shem death → SP-primary/LXX Shem birth|24|48|
|B|Matched MT Flood close → SP-companion Noah birth|24|24|
|C|Matched MT Flood start → SP-primary/LXX Noah birth|24|48|
|D|MT Noah-death alias → SP-companion Noah birth|2|2|
|E|Extra MT Noah death → SP-companion Shem birth|2|2|
|F|SP-companion Shem-death alias → SP-primary/LXX Noah birth|1|2|
|G|SP-primary/LXX Shem-death alias → SP-companion Noah birth|2|4|

The coordinate counts in the last four rows are not additive: D/F/G are aliases at existing B/C coordinates. A/B/C occupy72 distinct coordinate pairs, and E supplies the remaining two.

At every one of the24 contexts, A→C adds500 to both endpoints, and SP C→B subtracts1. Consequently B=A+499. The lower−1 is a Flood boundary change; the upper−1 is an SP person-companion change. They are distinct source mechanisms. These correspondences do not create translation operators or transport a companion rail into MT/LXX.

All six +499/+500 translations of the three admitted E comparisons retain sourced output intervals, but each needs an absent upper input and an absent pivot. Endpoint-family correspondence alone therefore fails to supply the translated path.

## 3. Equal maps, different source support

The +500 translation of the current split word has exactly the same overall affine map as the inherited C429 Noah-held word:

\[
E_{3256}\circ E_{4120}=E_{3706}^{\,2}
=\frac{625}{529}x-\frac{96\cdot3706}{529}.
\]

The translated path2648/3706→2520/3670→2456/3706 lacks pivots4120/3256 and middle3670. The inherited held path2648/3706→2556/3706→2456/3706 is fully sourced. The3706-held diagnostic remains outside the active eight-pivot menu.

The complete four-ladder comparison gives16 profiles. Five are fully sourced: the four original Noah-held words and the established Shem split partner of3706. Only the latter uses the active menu. This reproduces the earlier C434 split boundary in a wider factorization comparison; it is not another independent discovery of that boundary.

The full24-context held-family derivation shows why the rails differ. Every primary middle is matched MT Shem birth and every primary final is matched MT Flood start. Exactly four primary inputs exist, forcing Cainan0, Terah-base and G1/G2, with either sojourn setting. All four paths were already recorded in C429.

Every companion final is matched MT Flood close, but **none of the24 required companion middles occurs in the declared register**. Five companion inputs are sourced: four matched Flood-close inputs and the inherited3899/2841 exception, whose lower input is MT Noah birth in a different context. Retaining this alias avoids incorrectly reducing the companion endpoint count to four. No MT person-companion middle is licensed.

## 4. Fixed Mirror boundary

The nine source-endpoint Noah-held cases give18 trials under the two existing exchanges−2639−x and−2635−x. Their five complete initial images are exactly those already recorded in C461. Only the original C287 i3/i5 assignments give8993=17×529.

At the i3 corner, the held regular path2648→2556→2456 about3706 reflects formally to−5287→−5195→−5095 about−6345. The initial cumulative points are sourced; the two expanded stations are absent. The same failure occurs at every complete input image. Across all18 trials, no cumulative middle exists in the frozen48-record menu. Two isolated final hits from the3899 alias case lack their pivot, input and middle and do not supply a trajectory.

The conclusions concern exactly the declared regular and cumulative domains. They establish arithmetic and source-admission boundaries, not historical design intent, a new global Mirror axis, or a combined chronological prescription.

## 5. Ten-step result locator

| Step | Completed bounded result |
|---|---|
|C470|All51 source1150 intervals; three fixed-E expansions; one complete2K ancestry|
|C471|Source roles and inverse ancestry of the three1250 outputs|
|C472|Complete74-coordinate/130-label1250 grammar|
|C473|24 typed +500/+499 family triplets and six failed translated E comparisons|
|C474|Equal whole maps for translated split and inherited Noah-held words|
|C475|Four inherited ladders compared across16 factorizations|
|C476|Complete primary/companion context conditions, including the fifth companion input alias|
|C477|18 fixed-Mirror held-family trials and their source boundary|
|C478|Different implementation:98 agreeing checks and sealed predecessor audit|
|C479|This synthesis, final checkpoint, report locator and recovery packages|

The individual reports preserve every next-step decision made after its predecessor result. C478 regenerated all528 regular labels,373 coordinates and48 cumulative records; reverse pivot recovery and direct integer coefficient evaluation agreed with the main calculations. Its98 checks are included in the aggregate main-check total. The final completion/integrity record supplies the all-ten-packet totals and archive hashes after publication.

File_18, File_09 and the C265 checkpoint/certificate are accessible and match the inherited controlling bytes. The canonical graph remains unchanged. C465’s mixed-cycle12→12 label count still refers only to vertices; the full anchored count is48→12. C432’s E excerpt locator remains corrected to File_46 lines117–133. The corrected LXX8464 and supplemental SKL observations remain awareness items.

Read this synthesis and the C479 checkpoint first, then the C470–C479 reading index. Use the C479 compact restart kit as the active packet and the full continuation archive for older calculation files. Prior archives remain selective reference libraries, with current corrections and scope overriding stale next-step paragraphs.

## Verification and scope

51 calculation checks and 65 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 ten steps use the inherited source domains. No new date, pivot, phase, global axis, cumulative/upstream Gear or canonical edit has been admitted. The held-Noah diagnostics and formal translated factors remain distinct from the active eight-pivot menu. Original unprefixed reports, Engine reports and C-series reports remain distinct.

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

**C480, proposed and unexecuted:** Reduce the complete51-coordinate/101-label1150 interval inventory already frozen in C470 to disjoint source-role and context templates, retaining aliases separately. Use that grammar to explain the three admitted fixed-E expansions and the failed cases without adding a date, pivot, phase or operator.

The1250 output grammar is complete, while the complete1150 inventory has only been classified for its three admitted E expansions. Its finite source grammar is the remaining direct dependency for explaining the full selection pattern. C480 needs a new instruction and no such classification has been executed.