# 490d C413 — Source Shem return graphs, the finite500 domain and Mirror boundaries

24 September 2026. Step 10 of 10 in the C404–C413 authorization.

**C404–C413 is complete. The main advance is a complete source correspondence between the Noah return graphs and Shem return graphs under the existing500-year relation. A direct target-side census confirms it. The wider fixed500 comparison also has a precisely bounded source domain and one additional four-step chain. Neither result closes the original17K Mirror rectangle. C414 remains unexecuted.**

## The complete return-graph correspondence

The earlier E charts send MT Noah3031/3033/3035 to SP/LXX Shem2991/2993/2995, but they preserve only the J arrows as source-qualified arrows. Required E return pivots3016/3018 and P pivots2853/2855 are absent.

The existing500-year relation gives a different result. With T(t)=t−500, all12 labelled Noah E/J loops translate to sourced Shem birth/death records. The six coordinate configurations include two complete E/P/J graphs:

|Translated segment|MT Shem births|Held E Shem birth|J Shem death|P Shem death|
|---|---|---:|---:|---:|
|G1↔G2|2531↔2533|2556|1933|2393|
|G2↔G3|2533↔2535|2558|1935|2395|

The P pivots retain SP-primary/LXX-OFF source labels. The J pivots and moving births retain their MT context. The held E births belong to the separate MT context specified in the source loop census. They are not combined into one biography.

For a held-pivot comparison D_(k,a)(t)=a+k(t−a),

\[
T\circ D_{k,a}\circ T^{-1}=D_{k,a-500}.
\]

Every admitted path therefore carries over with unchanged slope. The three Shem vertices2531,2533,2535 inherit the Noah graph's operator-count normal form and the circuit multipliers

\[
\lambda_P=\frac{1750}{3\cdot529},\qquad
\lambda_J=\frac{7500}{13\cdot529}.
\]

The complete circuits commute about a shared source point; individual E/P/J operators still do not. This is an isomorphism of the specified finite comparison graphs with source labels retained, not a global action on chronology.

The direct Shem census tests32 moving families×32 held families×2 Gear transitions=2048 cases without prefiltering by500 or529. It returns exactly the same12 labelled/six coordinate loops and two complete E/P/J graphs. Source600 and the Gear increment2 give598→600 under J; the variant difference215−130−60=25 gives the E return.

The valid502/98 mode covers a smaller source domain. OriginalG2–G3 maps to targetG1–G2, admitting six labelled/three coordinate loops and one complete E/P/J graph. The other six cases would require Gear0. This boundary does not invalidate502/98.

## The older E ladder and Mirror limits

C361/C363 already established the−500 translated squared-E endpoint comparison3206→2148→1956, with3206 held, and its+2 counterpart. Its intermediate2056/2058 is absent. This batch uses that result rather than presenting the translation as a new discovery.

The new source graphs do not provide a replacement factorization. Any positive E/P/J word with slope625/529 must have exactly two E arrows and no P or J arrows, by prime valuations at7,13 and5. Only three such sourced paths exist. All six whole-map comparisons fail, as do48 combinations of existing pivot phases. The closest companion path has effective fixed point3206+1/24 and differs from the target held at3206 by−4/529. That discrepancy remains unadopted.

Under the two existing corner exchanges C−x, C=−2639 or−2635, none of104 labelled reflected Shem-loop slots belongs to the frozen cumulative register. All208 inherited component checks also fail. The identity R_C T_−500=T_+500 R_C shows why a regular500-year shift requires new cumulative members500 higher; those members are absent.

For the original rectangle, both short sides remain1058=2K after shifting the regular endpoints. Both former8993=17K cross-spans become8493. Thus the source-qualified regular correspondence preserves local widths without preserving the original cumulative/regular connection. No replacement axis is fitted.

## The finite500 source domain and its extra bridge

Across all373 regular coordinates, the fixed−500 comparison gives169 coordinate arrows and141 maximal nontrivial coordinate chains:115 of length1,25 of length2 and one of length4. All528 source records remain distinguishable; the392 labelled pairings are an inventory, not392 independent findings.

Same-family Noah-to-Shem births, Flood-to-Shem-death relations and the qualified SP/LXX-Shem-death to MT-Noah-death relation account for168 arrows. The sole additional coordinate bridge is3395→2895. It joins

**3895→3395→2895→2395→1895**.

Its source formula is

\[
100+130\Delta C+60\Delta T+215\Delta F
 +2\Delta G-r=500.
\]

The permitted variant sums force405=130+60+215, and the remaining correction forces−4−1. Thus the bridge runs from SP companion ShemG1, with CainanON/Terah60/full430, to SP or LXX-OFF Flood startG3, with Terah0/half430. It is a uniquely determined source-context combination within that field:100+405−4−1=500.

The full chain has two continuous record lifts, choosing SP or LXX-OFF for the G3 Flood/death pair. It does not depend on silently exchanging different identities at a shared coordinate. Before this extra comparison, the240 role-defined record arrows have192 maximal paths, all of length≤2. With its two labels included in the derived analysis, there are191 paths, including two length4 lifts of the same coordinate chain. No canonical graph was edited.

One slope-one translation preserves both matched±¼ components. Of32 assignments along the five chain points, exactly the two uniform assignments pass. The opening Noah's own death remains2945, after950 years; the opening Shem's own death remains2795, after600. The terminal1895 is a different MT Noah death, so the chain is not a2000-year biography.

## Step record, source warrant and recovery

|Step|Result that determined the next question|
|---|---|
|C404|E-chart transport lacks E/P source pivots; test the fixed500 relation|
|C405|All selected return loops translate to Shem; test target-side completeness|
|C406|Direct Shem census matches exactly; check the other retained binding|
|C407|502 works on a shorter Gear domain; compare the older E ladder|
|C408|The older translation is prior work; new return paths cannot replace its missing factorization|
|C409|Fixed Mirror source closure fails; delimit the500 operation's regular domain|
|C410|Complete coordinate census finds one extra bridge; audit its source identities|
|C411|The bridge formula and two continuous record lifts pass; verify source warrant|
|C412|All528 regular records reconstruct from current source tables; consolidate|
|C413|Binds the results, alternate verification and recovery; leaves C414 unexecuted|

Twelve exact source excerpts support the independent reconstruction of all528 records, including52 translated loop slots and ten long-chain slots. File18 preserves both500/502 modes, the SP-local companion, independent Flood boundaries, native LXX Cainan and the qualified950/600 biographies. File02's J remains a gap conversion, not a general date or Residue Protocol operation. The newly assembled graphs and bridge are derived comparisons; a source prescription for their combined use is not established in the reviewed passages.

The alternate root implementation passes1041 checks using integer affine cross-products, forced target pivots, weighted fixed points, output-first membership, residue-class chains, backward record-path enumeration and separate table-cell parsing. Its portable replay is byte-identical. These are alternate methods by the same agent, not independent-agent or historical-witness claims.

The nine predecessor packets pass their sealed-packet audit with27 confirmed individual deliverables and18 identical primary replay outputs. C413's two primary replays and alternate portable replay also agree. The final ten-packet integrity record, reading index and recovery archives accompany this synthesis separately.

Actual File18/File09 bytes match their retained hashes; File02/File46/File11 match the C403 frozen sources. The C265 checkpoint is readable and its certificate matches its retained hash. Current scope and corrections override older archive instructions, including the awareness-only corrected LXX8464 observation.


## Verification and limits

75 main checks and 68 frozen-input/predecessor authentication checks passed. A second main replay produced byte-identical DATA/RESULTS. These checks and the source interpretation were performed by the root agent; no new independent-agent review is claimed. All frozen originals and standing controls remain unchanged. Verification counts are not independent-witness counts.

Exhaustive claims apply only to the stated frozen menus, role/context rules and positive source paths. Whole-path exclusions have explicit prime-valuation proofs. Source reconstruction, repeated coordinates, aliases and verification counts are not independent witnesses or proof of historical intention. No global529/Mirror-preserving action or combined source protocol is established. C414 remains unexecuted.

Mirror/529 focus: K=529, E=25/23, P=70/69. Global paired Mirror January AD1; local junction January AD2; no civil year zero. Regular Gears are Noah/Shem/Flood only. SP minus-one person companions remain local to SP Noah/Shem; Flood close is a separate boundary. No upstream/cumulative Gear, new endpoint or phase, fitted anchor, global axis, primer reopening, canonical or graph edit. Capture now; canonize later. Source identity, context, modal binding, generated coordinates and phase ancestry stay distinct.

## Decision after this result

**C414, proposed and unexecuted:** Within the fixed C410 regular source menu, enumerate existing endpoint pairs exactly2K=1058 apart and test whether both ends remain sourced after one and two applications of the fixed−500 comparison. Preserve source roles and contexts; compare surviving pairs with the established529 rectangle while keeping its cumulative corners fixed. Search no new intervals, shifts, phases or axes.

The new500 graph has a complete finite source domain, but its precise support for paired529 endpoints has not been classified. This brings the new regular structure back to the Mirror/529 question without treating the2000-year chain as a new research direction. C414 is proposed only and requires a new instruction.