490d — C692–C711: sequential research record

Working continuation of C631 · packet date20260928.

Each completed step retains its question, declared inputs, exact results, finding, and reassessment. These are bounded research actions, not a count of independent discoveries. Canonical sources remain unchanged; second decimal inversion remains deferred.

C692 — Recover the dual anchored4346 convergence

Question. Do the two supplied MT states converge under their declared anchors and Keys?

Sources. Supplement A §§9.1–9.7; File22 dual-route ownership

Inputs

{
  "routes": [
    {
      "source": 4114,
      "anchor": 1446,
      "key": "E"
    },
    {
      "source": 4304,
      "anchor": 1406,
      "key": "P"
    }
  ]
}

Results

{
  "routes": [
    {
      "input_radius": 2668,
      "output_radius": 2900,
      "gain": 232,
      "head": 4346
    },
    {
      "input_radius": 2898,
      "output_radius": 2940,
      "gain": 42,
      "head": 4346
    }
  ]
}

Finding. Standard regular MT about Exodus under E and the+60/+130 source state about Conquest under P both generate4346. The agreement is between two explicitly anchored constructions;4346 remains their common generated parent label.

Reassessment. Identify the exact balance of source, anchor and Key-gain differences that forces convergence.

C693 — Explain the dual-route convergence balance

Question. Which differences are premises and which are consequences of the common head?

Sources. C692; Supplement A §9

Inputs

{
  "source_states": [
    4114,
    4304
  ],
  "anchors": [
    1446,
    1406
  ],
  "gains": [
    232,
    42
  ],
  "completed_radii": [
    2900,
    2940
  ]
}

Results

{
  "source_gap": 190,
  "gain_gap": 190,
  "anchor_gap": 40,
  "completed_radius_gap": 40,
  "source_components": [
    60,
    130
  ]
}

Finding. The two source states differ190=60+130, exactly balanced by their Key-gain difference232−42. Their completed radii differ40, matching the anchor separation. These equations explain convergence without turning the generated head into a transmitted Creation date.

Reassessment. Recover the internal source partition under its declared anchoredP route.

C694 — Map the full4304 source path throughP

Question. Does the declared common-parent route preserve the source’s internal ratio?

Sources. Supplement A §9.8

Inputs

{
  "source_path": [
    4304,
    2648,
    1406
  ],
  "held_anchor": 1406
}

Results

{
  "source_parts": [
    1656,
    1242
  ],
  "generated_path": [
    4346,
    2666,
    1406
  ],
  "generated_parts": [
    1680,
    1260
  ],
  "ratio": "4/3"
}

Finding. The full anchoredP map takes1656|1242 to1680|1260 and2648 to2666 while retaining the4:3 partition ratio. Both internal labels are generated seams in this comparison, so shared numbers elsewhere do not transfer event identity.

Reassessment. Reconstruct the4346 carrier ladder across the civil seam.

C695 — Generate the anchored1470 carrier ladder

Question. Does one civil1470 step recover every supplied node and its five-year companion?

Sources. Supplement A §§9.1–9.7

Inputs

{
  "source": [
    "4346BC",
    "2876BC",
    "1406BC",
    "AD65"
  ],
  "translation_forward": 5
}

Results

{
  "native_civil_coordinates": [
    -4345,
    -2875,
    -1405,
    65
  ],
  "translated_civil_coordinates": [
    -4340,
    -2870,
    -1400,
    70
  ],
  "translated_labels": [
    "4341BC",
    "2871BC",
    "1401BC",
    "AD70"
  ],
  "gaps": [
    1470,
    1470,
    1470
  ]
}

Finding. One1470-step civil ladder yields4346→2876→1406→AD65. Uniform translation by five years yields the companion4341→2871→1401→AD70. The Enoch close2876 and Long center2871 occupy different roles despite the whole-path relation.

Reassessment. Test the source’s directional paired-Key convergence as a complete four-row family.

C696 — Reconstruct the fixed directional Key matrix

Question. Do all four source-declared anchored rows converge as stated?

Sources. Supplement A §§8.1–8.6

Inputs

{
  "rows": [
    [
      3626,
      2936,
      "E"
    ],
    [
      3576,
      2886,
      "P"
    ],
    [
      2216,
      2906,
      "P"
    ],
    [
      2166,
      2856,
      "E"
    ]
  ],
  "scope": "fixed admitted comparison; no new SKL state"
}

Results

{
  "declared_outputs": [
    2876,
    2876,
    2916,
    2916
  ],
  "swapped_key_diagnostic": [
    2926,
    2826,
    2966,
    2866
  ],
  "input_magnitudes": [
    690,
    690,
    690,
    690
  ]
}

Finding. All four admitted rows use magnitude690. The declared forward assignments meet at2876 and reverse assignments at2916; swapping Keys breaks each pairing. Direction and held anchor are essential parts of this bounded family, whose existing SKL comparison status is retained.

Reassessment. Explain the50 offset algebraically instead of treating the convergences as separate matches.

C697 — Derive directional convergence from the Key gap

Question. What source offset makes the paired anchored outputs equal?

Sources. C696; Supplement A §8

Inputs

{
  "span": 690,
  "held_forward": [
    3626,
    3576
  ],
  "held_reverse": [
    2216,
    2166
  ]
}

Results

{
  "Key_output_gap": 50,
  "forward_residual": 0,
  "reverse_residual": 0,
  "individual_gains": [
    10,
    60
  ]
}

Finding. The50 source offset equals(E−P)×690. With the declared direction and assignment, that one equality forces both convergences. The four outputs are structured consequences of the offset and operators, not four independent confirmations.

Reassessment. Recover the complete70-wide quartet and its supplied1470 translation.

C698 — Reconstruct the quartet and its translated comparison

Question. Which structural information survives the supplied1470 shift?

Sources. Supplement A §§8.5–8.6

Inputs

{
  "quartet": [
    2946,
    2916,
    2906,
    2876
  ],
  "translated": [
    1476,
    1446,
    1436,
    1406
  ],
  "role_classes": [
    "generated companion",
    "direct Key output",
    "Short physical source anchor",
    "Key output and schematic Enoch close"
  ]
}

Results

{
  "translated": [
    1476,
    1446,
    1436,
    1406
  ],
  "gaps": [
    30,
    10,
    30
  ],
  "center": 2911,
  "translated_center": 1441
}

Finding. The complete quartet has30|10|30 topology and center2911. Its declared1470 translation preserves that shape across1476/1446/1436/1406, while Joshua, Exodus, generated comparison and Conquest roles remain separate. Only2916 and2876 are direct outputs of the paired-Key matrix.

Reassessment. Refine the source’s full Danielic4900 path through the same4346 hinge.

C699 — Recover the complete refined4900 civil path

Question. Does the source’s detailed Danielic path close with every retained junction?

Sources. Supplement A §§11.1–11.6

Inputs

{
  "labels": [
    "4836BC",
    "4346BC",
    "2946BC",
    "2876BC",
    "1406BC",
    "6BC",
    "AD30",
    "AD65"
  ]
}

Results

{
  "civil_coordinates": [
    -4835,
    -4345,
    -2945,
    -2875,
    -1405,
    -5,
    30,
    65
  ],
  "gaps": [
    490,
    1400,
    70,
    1470,
    1400,
    35,
    35
  ],
  "70_unit_counts": [
    7,
    20,
    1,
    21,
    20,
    "1/2",
    "1/2"
  ],
  "total": 4900
}

Finding. The source’s entire refined path closes as490|1400|70|1470|1400|35|35=4900. It joins the4346 hinge, Enoch body, carrier ladder and civil terminal in one labeled path. The source’s schematic Christological roles retain their prior status.

Reassessment. Contract only the admitted interior junctions to recover the coarse Danielic template.

C700 — Explain coarse and refined path agreement

Question. What information is retained when the detailed4900 path becomes the Danielic template?

Sources. Supplement A §§11.1–11.6; C699

Inputs

{
  "refined": [
    490,
    1400,
    70,
    1470,
    1400,
    35,
    35
  ],
  "grouping": [
    [
      0
    ],
    [
      1,
      2,
      3,
      4
    ],
    [
      5
    ],
    [
      6
    ]
  ]
}

Results

{
  "coarse": [
    490,
    4340,
    35,
    35
  ],
  "tenth_scale_template": [
    49,
    434,
    "7/2",
    "7/2"
  ],
  "retained_total": 4900
}

Finding. Grouping the four middle intervals yields490|4340|35|35, whose tenth-scale template is49|434|3.5|3.5. Coarse and fine rows are two resolutions of one source path; the template alone does not establish a separate literal434-year historical interval.

Reassessment. Examine the two69-coefficient arms meeting at4346.

C701 — Recover the hinge’s two coefficient arms

Question. How do the admitted cumulative and projective arms fit around4346?

Sources. Supplement A §§10.1–10.5

Inputs

{
  "upper_BC": 14006,
  "hinge_BC": 4346,
  "projective_AD": 29465,
  "projective_status": "inherited comparison boundary"
}

Results

{
  "arms": [
    9660,
    33810
  ],
  "69_units": [
    140,
    490
  ],
  "partition_4830_units": [
    2,
    7
  ],
  "fixed_projective_generated_heads": [
    4836,
    7286
  ],
  "whole": 43470
}

Finding. The admitted4346 hinge divides the whole43470 civil span as9660|33810=69×(140|490), or2|7 in4830 units. With the supplied projective endpoint held, P/E generate4836/7286; those retain co-registration/generated roles and are not new primary anchors.

Reassessment. Test the supplied cumulative Cainan pair within the same anchoredE family.

C702 — Recover the cumulative insertion pair-difference law

Question. What happens to the supplied460 cumulative Cainan increment under anchoredE?

Sources. Supplement A §10; regular/cumulative role distinction

Inputs

{
  "heads": [
    14006,
    14466
  ],
  "anchor": 4346,
  "source_increment": 460
}

Results

{
  "input_radii": [
    9660,
    10120
  ],
  "output_radii": [
    10500,
    11000
  ],
  "generated_heads": [
    14846,
    15346
  ],
  "output_difference": 500
}

Finding. The source cumulative increment460 becomes500 under E, producing the declared14846/15346 pair. This is the same affine pair-difference law operating on cumulative lifespans; it does not replace the regular Cainan130 used in the earlier calendar placement.

Reassessment. Summarize the operator types now required before testing the fixed phase-reflection set.

C703 — Apply source-role and unit review

Question. Which wording distinctions are needed before integrating the micro/macro result?

Sources. Reviewer C677–C690; Supplement A §10.3

Inputs

{
  "reviewed_steps": [
    689,
    690,
    691
  ],
  "source_locator": "Supplement A §10.3",
  "residual_count": "5/2",
  "calendar_length": 336
}

Results

{
  "corrections": {
    "C689": "The840-day interpretation is derived by applying the336-day calendar to the source-defined2.5 phase residual.",
    "C690": "14006 is the native Rounded head;14146 and14846 are generated comparison heads. The matrix locator is§10.3.",
    "C691": "Read2.5 additional336-day calendar units, totaling840days; not840 such units. The macro analogue has its own declared year units.",
    "phase_centers": "Bare2871/2911 here are Nisan coordinates; corresponding Tishri coordinates are0.5 greater."
  },
  "residual_days": 840,
  "macro_year_gain": 840
}

Finding. Independent review confirms the arithmetic and identifies three source-role/unit wording refinements. These corrections supersede the affected interim prose; all final explanations will distinguish native14006, generated companion heads, and the derived residual-volume interpretation.

Reassessment. Test the literal twelve-node source phase field through its declared affine reflection.

C704 — Reconstruct the fixed twelve-node phase field

Question. Can one declared template reproduce every literal source node?

Sources. Supplement A §13.1; amendment-local status

Inputs

{
  "literal_BC_pairs": [
    [
      2939,
      2938
    ],
    [
      2937,
      2936
    ],
    [
      2935,
      2934
    ],
    [
      2909,
      2908
    ],
    [
      2907,
      2906
    ],
    [
      2905,
      2904
    ]
  ],
  "chart": "t(B)=−B+3/4; n(B)=−B+1/4",
  "scope": "literal existing set only"
}

Results

{
  "literal_coordinates": [
    "-11753/4",
    "-11751/4",
    "-11745/4",
    "-11743/4",
    "-11737/4",
    "-11735/4",
    "-11633/4",
    "-11631/4",
    "-11625/4",
    "-11623/4",
    "-11617/4",
    "-11615/4"
  ],
  "center": -2921,
  "width": "69/2",
  "template": "−2921+15r+2g+p/4"
}

Finding. The declared six phase pairs form exactly the supplied twelve-node template centered at−2921 with width34.5. The template reconstructs a fixed admitted field; its local indices do not authorize new Gear or remote SKL states.

Reassessment. Test the whole set under its declared affine reflection and translation.

C705 — Verify the whole-set affine mirror

Question. Does the declared5842 translation match the mirror for every fixed node?

Sources. Supplement A §13.2; C704

Inputs

{
  "center": -2921,
  "template_coefficients": [
    15,
    2,
    "1/4"
  ],
  "translation": 5842
}

Results

{
  "transpositions": [
    [
      "-11753/4",
      "-11615/4"
    ],
    [
      "-11751/4",
      "-11617/4"
    ],
    [
      "-11745/4",
      "-11623/4"
    ],
    [
      "-11743/4",
      "-11625/4"
    ],
    [
      "-11737/4",
      "-11631/4"
    ],
    [
      "-11735/4",
      "-11633/4"
    ]
  ],
  "fixed_points": 0,
  "translated_equals_mirrored": true
}

Finding. The5842 translation sends the entire fixed field onto its mirror; the associated reflection permutes it in six pairs with no fixed node. This is affine reflection of phase coordinates, distinct from decimal digit reversal and from File52c’s rounded mirror chart.

Reassessment. Compare all four source-listed cycle totals against the same fixed set.

C706 — Recover the fixed-set overlap inventory

Question. Which of the source-listed cycle totals closes the whole declared field?

Sources. Supplement A §13.3; C704–C705

Inputs

{
  "cycle_totals": [
    5840,
    5842,
    5844,
    5846
  ],
  "fixed_field_center": -2921
}

Results

{
  "columns": [
    "total",
    "overlap",
    "outside"
  ],
  "rows": [
    [
      5840,
      8,
      4
    ],
    [
      5842,
      12,
      0
    ],
    [
      5844,
      8,
      4
    ],
    [
      5846,
      4,
      8
    ]
  ],
  "fixed_source_set": true
}

Finding. The complete fixed-set inventory is8/12/8/4 overlaps for5840/5842/5844/5846. Only5842 covers every node. Selected-route matches at other totals remain valid local comparisons but do not establish whole-field closure.

Reassessment. Derive the conditional placement theorem from centers rather than searching more candidate dates.

C707 — Derive the fixed-template placement condition

Question. What does whole-field mirror closure determine once the shape is fixed?

Sources. Supplement A §13.4

Inputs

{
  "translation": 5842,
  "symmetric_template_center": 0,
  "shape": "fixed literal twelve-node template"
}

Results

{
  "required_center": -2921,
  "center_equation": "c+L=−c",
  "template_symmetric": true,
  "scope": "unique placement conditional on fixed shape, chart and L"
}

Finding. Whole-field closure forces c=−L/2=−2921, and the fixed symmetric template makes that condition sufficient. The period determines placement only after shape, phase chart and closure requirement have been supplied; it does not independently determine them.

Reassessment. Compare the source’s anchored calendar-body placement packet with this fixed-template result.

C708 — Generate the anchored calendar-body quartet

Question. Can the source’s ordinary and schematic four-year bodies explain all four anchors?

Sources. Supplement A §13.7; fixed placement packet

Inputs

{
  "Exodus_anchor": 1446,
  "ordinary_body": 1460,
  "schematic_body": 1440,
  "rail": 30
}

Results

{
  "quartet": [
    2936,
    2906,
    2886,
    2856
  ],
  "gaps": [
    30,
    20,
    30
  ],
  "Short_center_label": 2921,
  "two_anchor_offset": 50
}

Finding. The anchored packet generates2936/2906/2886/2856 and30|20|30 directly from1446, two declared calendar bodies and the30 rail. Its Short center2921 agrees with the fixed phase placement, but both constructions retain their supplied premises and distinct charts.

Reassessment. Check whether the supplied Long-centered field could share the same whole-set translation.

C709 — Separate Short and Long whole-field translations

Question. Does a50-shifted center preserve the5842 whole-field closure requirement?

Sources. Supplement A §13.9; C707

Inputs

{
  "Short_center": -2921,
  "Long_center": -2871,
  "shape": "same fixed phase template"
}

Results

{
  "self_mirror_lengths": [
    5842,
    5742
  ],
  "length_difference": 100,
  "Long_center_residual_under5842": 100
}

Finding. The supplied Long center requires5742 for its own mirror closure,100 less than the Short field’s5842. A50 center displacement doubles in the required translation. No adjacent state or remote extension is introduced to erase this distinction.

Reassessment. Compare the conserved quantities across the now-completed chronological mechanisms.

C710 — Unify the conserved measurements

Question. Which simple linear quantities explain the family-preserving changes?

Sources. C631 integrated synthesis; C681–C682; Research Strategy

Inputs

{
  "changes": {
    "biography": [
      100,
      -100
    ],
    "symmetric_opening": [
      10,
      -10
    ],
    "Actual_to_Rounded_C_R": [
      2,
      -8
    ]
  },
  "weights": {
    "biography": [
      1,
      1
    ],
    "symmetric_opening": [
      1,
      1
    ],
    "Actual_to_Rounded_C_R": [
      4,
      1
    ]
  }
}

Results

{
  "conservation_checks": {
    "biography": 0,
    "symmetric_opening": 0,
    "Actual_to_Rounded_C_R": 0
  },
  "functionals": [
    "b+r",
    "U+L",
    "4C+R"
  ],
  "Actual_Rounded_values": [
    12026.0,
    12026.0
  ]
}

Finding. Three previously separate mechanisms share a precise explanation: their allowed change vectors preserve a specified measurement. Biography preserves total lifespan, opposed opening preserves center, and the Actual/Rounded displacement preserves4C+R. The weights differ because the measured objects differ.

Reassessment. Consolidate the anchored and fixed-set results, then assemble the complete explanatory deliverables.

C711 — Consolidate the anchored and fixed-set families

Question. How do the new placement results serve the Strategy’s whole-family goal?

Sources. C692–C710; Research Strategy

Inputs

{
  "completed_steps": 80,
  "families": [
    "dual4346 routes",
    "directional690 matrix",
    "refined4900 path",
    "fixed12-phase field",
    "conserved measurements"
  ]
}

Results

{
  "artifact": {
    "path": "deliverables/C711_Interim_Synthesis.md",
    "sha256": "c094ab8c8f22085d4a6965928f66a0ab04494b577d5c271a7a7bc40fdc4ccb4b",
    "bytes": 1473
  },
  "completed_steps": 80
}

Finding. Eighty sequential steps now connect whole source families through a small set of operations and conservation laws. The remaining work will compress that explanation, test its cross-family statements and complete the reviewable deliverables.

Reassessment. Build a compact mode-and-mechanism map that includes MT/LXX/SP and the Round chronology.