{
  "step": "C413",
  "status": "completed",
  "date": "2026-09-24",
  "title": "Source Shem return graphs, the finite500 domain and Mirror boundaries",
  "previous_certificate_sha256": "c4260f014f05ae100e9bff4259e4aa8eabf14af155bb8f2041eec0e2c8ed343c",
  "bindings": {
    "step": "STEP.json",
    "source_manifest": "SOURCE_MANIFEST.json",
    "report": "490d_C413_Source_Shem_Return_Graphs_500_Domain_and_Mirror_Boundaries_Synthesis_20260924.md",
    "checkpoint": "490d_Mirror_Checkpoint_After_C413_20260924.md",
    "data": "evidence/DATA.json",
    "results": "evidence/RESULTS.json",
    "calculation": "evidence/CALCULATION.py",
    "verifier": "evidence/check.py",
    "next_step_decision": "NEXT_STEP_DECISION.json",
    "replay": "REPLAY.json",
    "control_integrity": "CONTROL_INTEGRITY.json",
    "readme": "README.md",
    "review_0": "evidence/ROOT_SECOND_METHOD.py",
    "review_1": "evidence/ROOT_SECOND_METHOD_RESULTS.json"
  },
  "step_sha256": "bdb9c03d71d2198279fbc3a33fa4914d8c2ffdac615dbd9a0ddb47adc6921562",
  "source_manifest_sha256": "9060ba01e3126107b6d29c0d40f6166917da82995905c712cefb05df37be7525",
  "report_sha256": "2e4bfd38d68b2b0d65da49c8372e04c165d7b7fe577884ed287f74354a99dc9d",
  "checkpoint_sha256": "d7e3e0389bd31a14505f8e55b5630c0217601ab16a80d1c18d5e5f45643c0f1b",
  "data_sha256": "1c6377a173a375e5457c93005217407af7192e0826e2b1ce56057efa4019f41c",
  "results_sha256": "b0138ddd16ebde8fc12bc115ec4627bbb315ab6ca39a841a9c6c5a2f59dee362",
  "calculation_sha256": "29c2d4667f11c931a2e9abe33f44c8fe61a3b88ad275865fd59300b248f25fff",
  "verifier_sha256": "9f175670abb70f7819dd7f6d997ff0bef2e58df8dffd763b433f6c6cd62e332d",
  "next_step_decision_sha256": "348674b580293a782da505af005024a67df9ad34db449414065eb6de950ba5df",
  "replay_sha256": "d03e0e9b7cd4f3f3dbc1c47b563b3410fcaa95c0110c216a4cfe3fb3e8e833de",
  "control_integrity_sha256": "9810807cb07b02f43f568f68bb82cb41a6eeff6799902254b4932e5c350ac838",
  "readme_sha256": "9fd7c211aa3765a6fabdf82d5846aaf1238dd253b358545f933d209ab775e9cb",
  "review_0_sha256": "3f79abfac1307b2cc97ab845f2f1e9a6dbbd89ec644606f58e0e8733acf2dcb8",
  "review_1_sha256": "3b3ecac5feca528dec7ad20bd99756c5180b5124cb85e52b4a5271d294eb21cf",
  "assertions_passed": 75,
  "source_checks_passed": 68,
  "main_result": "C404–C413 complete. The existing Noah-to-Shem E charts retain only the two sourced J arrows; required Shem E pivots3016/3018 and P pivots2853/2855 are absent. The fixed source500 correspondence succeeds: all12 labelled/six coordinate Noah E/J loops translate to Shem births/deaths, including both complete E/P/J graphs. The complete target graph has MT Shem2531,2533,2535, E pivots2556/2558, J death pivots1933/1935 and P death pivots2393/2395. Every path is conjugated byT(t)=t−500 with unchanged slope and circuit multipliers1750/(3K),7500/(13K). A direct2048-case Shem-family census recovers exactly those loops. The valid502/98 binding maps only originalG2–G3 to targetG1–G2; the other six labelled cases would require unavailableGear0. The older translated squared-E endpoints were already established inC361/C363; the new return paths cannot replace their missing2056/2058 intermediate. Three slope-compatible EE paths fail all six whole-map comparisons and48 phase comparisons; the closest discrepancy is−4/529. The existing Mirror corner exchanges admit none of104 reflected loop-record slots or208 inherited components. Translating the original regular2K side−500 preserves1058 but changes both17K cross-spans to8493. The full fixed−500 source-domain census has169 coordinate arrows,392 labelled pairings and141 maximal coordinate chains. A single extra coordinate bridge3395→2895 produces3895→3395→2895→2395→1895. Its formula100+405−4−1=500 forces all three variant differences, SP companionG1 and FloodG3. It has two continuous record lifts, choosing SP or LXX CainanOFF for the Flood/death pair; the sequence is not one biography. All528 regular records reconstruct from twelve current source excerpts. The new graphs and bridge remain derived comparisons; no new source endpoint, protocol prescription, canonical edit or global529/Mirror action is established.",
  "limits": "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.",
  "canonical_or_graph_edits": false,
  "new_independent_agent_review": false,
  "review_provenance": "Root-agent analysis and deterministic replay; no new independent-agent review",
  "next_proposed_step": "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.",
  "next_step_executed": false
}