{
  "step": "C469",
  "completed_steps": [
    445,
    446,
    447,
    448,
    449,
    450,
    451,
    452,
    453,
    454,
    455,
    456,
    457,
    458,
    459,
    460,
    461,
    462,
    463,
    464,
    465,
    466,
    467,
    468,
    469
  ],
  "predecessor_summaries": [
    {
      "step": "C445",
      "title": "Ordered Compositions of the Four Fixed Local Reflections",
      "main_result": "All 5,968 ordered two-reflection trials give 330 complete paths: 208 zero-shift returns and 122 nonzero paths. Sourced shifts are ±4 (48 each), ±120 (5 each) and ±124 (8 each); both ±116 have zero complete paths."
    },
    {
      "step": "C446",
      "title": "Finite Reachability Under the Four Fixed Reflections",
      "main_result": "The four-reflection graph has 373 vertices, 106 edges including four loops, and 272 components. Its 152 active coordinates form 49 nontrivial components plus two loop-only singletons; 221 coordinates have no admitted reflection. The largest components have six coordinates and the maximum shortest-path distance is four."
    },
    {
      "step": "C447",
      "title": "All-Word 529 Test and Established-Path Connectivity",
      "main_result": "No nonzero multiple of 529 occurs among all 191 reflection-reachable coordinate pairs. The nonzero E-path coordinates 2148, 2020 and 1956 are reflection-isolated. The source pivots 3620/2756 share a four-point component; 3206/3170 share a different four-point component; midpoint3188 is a loop-only singleton."
    },
    {
      "step": "C448",
      "title": "Cycle Words and Fixed-Point Returns",
      "main_result": "The reflection graph has one ordinary simple cycle, 2610→3766→2730→3770→2610, through centers3188,3248,3250,3190. Its four-reflection affine word is the identity because3188+3250=3248+3190. The other four cycles are single-center fixed-point loops, not identity maps."
    },
    {
      "step": "C449",
      "title": "Source Domain of the Four-Center Identity",
      "main_result": "All 8,952 four-distinct-center trials produce exactly eight complete words, all returns: the two directions and four starts of one cycle. Each of the eight identity orders has only one sourced input. The cycle has eight source-label assignments and none keeps one unchanged source context."
    },
    {
      "step": "C450",
      "title": "Primitive E Arrows at the Eight Existing Pivots",
      "main_result": "The eight existing pivots yield37 sourced E arrows in2,984 trials: eight pivot self-arrows and29 distinct nonzero arrows. Another2,854 outputs are fractional and93 are integer coordinates absent from the frozen register."
    },
    {
      "step": "C451",
      "title": "Complete Directed E and Reflection Reachability",
      "main_result": "The combined fixed E/reflection graph has245 operator-labelled arrows and522 reachable directed distinct pairs,140 beyond reflection-only reachability. None has a nonzero529-multiple gap. One merged strong component has eight coordinates; all other strong components retain the reflection partition."
    },
    {
      "step": "C452",
      "title": "Source and Affine Structure of the Unique Mixed Feedback Component",
      "main_result": "The only enlarged strong component joins two former reflection components. Its unique simple E-bearing directed cycle is2731→3765→2735→2733→2731. The two E legs each expand a signed−23 pivot gap to−25, moving the Noah-death coordinate down one Gear. The full word fixes2731 with slope625/529 and is not an identity."
    },
    {
      "step": "C453",
      "title": "The 23-Year Source-State Residual and One-Gear E Arrows",
      "main_result": "The residual215−130−60−2=23 generates88 same-role source-label E templates, representing64 distinct coordinate arrows. Five label templates realize three of the four one-Gear arrows at the fixed pivots. The remaining arrow is MTNoah2841→2843 about2818, explained by950−650−215−60−2=23."
    },
    {
      "step": "C454",
      "title": "Direct 529 Pairs Versus Operator Reachability",
      "main_result": "The frozen register contains60 direct nonzero529-multiple coordinate pairs:34 at529 and26 at1058, with119 labelled source assignments. No pair is connected in either direction by the fixed mixed E/reflection menu. Thus the graph exclusion does not mean the source register lacks529 spans."
    },
    {
      "step": "C455",
      "title": "Two-E Effective Pivots and Sourced 529 Input Gaps",
      "main_result": "Across23,872 input/word trials,64 ordered E pairs yield15 complete paths. Thirteen words have a sourced effective fixed point;11 complete paths use one. The only nonzero529-multiple input gap is the established1058 gap atH3206, with2148→2020→1956. Four additional paths have rational, unsourced effective pivots."
    },
    {
      "step": "C456",
      "title": "E-Square Integrality and Source Admission Gates",
      "main_result": "For integer input and effective pivot, a two-E output is integral exactly when the input gap is divisible by529. The13 sourced-pivot words have17 integer-output cases,15 sourced outputs and11 complete paths. Among nonzero gaps the gates narrow4→2→1; only2148→2020→1956 survives."
    },
    {
      "step": "C457",
      "title": "Shared-Word Transport of Actual 529 Source Intervals",
      "main_result": "Complete shared-word closure of all60 sourced529/1058 pairs reaches63 pairs in756 trials. Only two seeds move:1895/2953→1745/2895 and2148/3206→2020/3170, both1058→1150 underE3620. Only the latter continues, to1956/3206 with span1250 underE2756. No reflection or further E step is admitted."
    },
    {
      "step": "C458",
      "title": "Source Equations for the Two 2K Interval Ladders",
      "main_result": "Both admitted2K ladders resolve exactly into source terms. Their input gaps are350+650+60−2 and600+650−130−60−2, each1058. Both first output gaps are500+650=1150; the complete second output is600+650=1250. Eleven required coordinates have16 retained source records."
    },
    {
      "step": "C459",
      "title": "The Shared 23-Unit Configuration of the Two 2K Seeds",
      "main_result": "The two transported2K seeds are translated rows of one sourced configuration underE3620: input separation253=11×23 expands to275=11×25. Both input spans are46×23=1058 and both output spans46×25=1150. No other fixed pivot supports a529-multiple input pair."
    },
    {
      "step": "C460",
      "title": "Modulo-23 Selection of the Unique Second Expansion",
      "main_result": "Of16 possible second E applications to the two1150 intervals, exactly one has integer outputs and it is fully sourced:2020/3170→1956/3206 about2756. The rows have residues20 and19 modulo23; the eight fixed pivots include residue19 only at2756 and no residue20."
    },
    {
      "step": "C461",
      "title": "Fixed Mirror Images of the Complete 529 Pair Inventory",
      "main_result": "The120 fixed-Mirror pair trials yield five complete images, all2K intervals: four atA−2639 and one atA−2635. Exactly two have17K cross-spans, reproducing2648/3706↔−6345/−5287 and2650/3708↔−6343/−5285. The other three have8991,8995 and8997 cross-spans."
    },
    {
      "step": "C462",
      "title": "Inherited Annual Rows and the 17K Cross-Span Condition",
      "main_result": "C287 already records four complete17K annual rows, with exchange constants−2641,−2639,−2637 and−2635. The active two-constant test preserves exactly thei3 andi5 assignments. Its three other sourced images reassign annual slots and change the cross-span by−2,+2 or+4; none is a new17K corner."
    },
    {
      "step": "C463",
      "title": "Source-Qualified Mirror Conjugacy of the Mixed Operator Menu",
      "main_result": "Across490 fixed-Mirror conjugacy trials, seven arrows have sourced cumulative endpoints but only six also have sourced anchors. All six are reflections: four fixed loops and the two directions of one four-year comparison. The sole E endpoint image,−6344→−6346, requires absent pivot−6321, so no E arrow is fully sourced in this cumulative menu."
    },
    {
      "step": "C464",
      "title": "SP-Companion Dependence of the Positive Witnesses",
      "main_result": "Removing SP-companion labels leaves432 records at282 coordinates and34 direct529/1058 pairs, but no shared-word transport. Both2K ladders lose only pivot3620 while retaining every endpoint. The ordinary reflection cycle loses2730; the mixed E cycle loses3765. The two inherited primary Mirror corner controls retain all regular endpoints."
    },
    {
      "step": "C465",
      "title": "LXX-OFF Alias Dependence Versus Coordinate Dependence",
      "main_result": "Removing72 theoretical LXX-OFF labels leaves456 labels at all373 coordinates. All245 operator arrows and all60 direct529 pairs survive; labelled529 assignments fall119→88. The full two-E ladder retains one complete MT/SP source assignment instead of eight alias combinations. Its source support does not depend solely on LXX-OFF."
    },
    {
      "step": "C466",
      "title": "Seven-Coordinate Source Certificate for the Complete 2K Ladder",
      "main_result": "The complete1058→1150→1250 ladder has a seven-coordinate MT/SP provenance certificate. All seven typed source records are regenerated from six controlling source excerpts. Endpoint equations determine the pivots uniquely as3620 and2756; the SP-companion and primary-death qualifications are explicit."
    },
    {
      "step": "C467",
      "title": "Operator Dependence of the Interval Ladder and Local Cycles",
      "main_result": "The complete interval ladder depends only on E3620 andE2756 among the12 tested operators. Deleting3620 removes all529-pair transport; deleting2756 retains the two first expansions but removes continuation. Each reflection can be removed without changing interval transport. The ordinary and mixed cycles have separate, explicitly verified operator dependencies."
    },
    {
      "step": "C468",
      "title": "Alternate-Method Verification of the Finite 529 Findings",
      "main_result": "A second implementation passes51 checks and reproduces528 source labels,373 coordinates,330 complete reflection compositions,15 complete two-E paths,60 direct529 pairs,522 mixed reachable pairs with no529 gap, the63-pair shared-word closure with three edges, and all fixed-Mirror image counts. The23 predecessor packets and69 publications pass the sealed integrity audit."
    }
  ],
  "predecessor_main_checks": 45547,
  "predecessor_authentication_checks": 518,
  "predecessor_sealed_audit": {
    "payloads": 710,
    "certificate_bindings": 288,
    "source_entries": 398,
    "main_checks": 45547,
    "authentication_checks": 518
  },
  "alternate_method": {
    "source_labels": 528,
    "source_coordinates": 373,
    "R_directed_arrows": 208,
    "E_directed_arrows": 37,
    "complete_R_compositions": 330,
    "complete_E_compositions": 15,
    "direct_K_pairs": 60,
    "reachable_directed_pairs": 522,
    "strong_components": 271,
    "pair_relation_edges_in_full_register": 6122,
    "K_seed_pair_closure": 63,
    "admitted_K_pair_edges": 3,
    "fixed_Mirror_pair_images": 5,
    "qualified_Mirror_operator_images": 6
  },
  "label_count_clarification": {
    "C465_vertex_only_count": [
      12,
      12
    ],
    "complete_anchored_mixed_cycle_count": [
      48,
      12
    ],
    "meaning": "C46512-to12 counts its four listed cycle vertices only. Including the four required operator anchors gives48-to12. No coordinate, arrow, cycle or interval conclusion changes."
  },
  "synthesis": {
    "source_supported_interval_ladder": {
      "pairs": [
        [
          2148,
          3206
        ],
        [
          2020,
          3170
        ],
        [
          1956,
          3206
        ]
      ],
      "spans": [
        1058,
        1150,
        1250
      ],
      "pivots": [
        3620,
        2756
      ]
    },
    "related_shorter_ladder": {
      "pairs": [
        [
          1895,
          2953
        ],
        [
          1745,
          2895
        ]
      ],
      "spans": [
        1058,
        1150
      ],
      "pivot": 3620
    },
    "input_row_translation": 253,
    "output_row_translation": 275,
    "mixed_operator_reachability_K_hits": 0,
    "fixed_Mirror_complete_pair_images": 5,
    "fixed_Mirror_inherited_17K_assignments": 2,
    "C287_archived_complete_17K_rows": 4,
    "essential_companion_pivot": 3620,
    "LXX_OFF_coordinate_dependency": false
  },
  "standing_scope": "Mirror/529; regular Gears Noah/Shem/Flood only; no new source, phase, global axis or canonical graph edit.",
  "canonical_or_graph_edits": false,
  "new_independent_agent_review": false,
  "C470_executed": false,
  "next_proposed_question": "Census all existing1150-year source pairs, test common E transport to1250 at the same eight fixed pivots, and record which have a complete sourced2K predecessor. Do not add pivots, phases or dates."
}