# 490d — C652–C671: 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.

## C652 — Generate Jacob’s death-centered reflection

**Question.** Does one reflection recover every supplied paired node?

**Sources.** File70 §5.3; A.5 JB001–016

**Inputs**

```json
{
  "center": 1859,
  "path": [
    1936,
    1929,
    1859,
    1789,
    1782
  ],
  "secondary_role": "1789 traditional Judah death"
}
```

**Results**

```json
{
  "reflected": [
    1782,
    1789,
    1859,
    1929,
    1936
  ],
  "gaps": [
    7,
    70,
    70,
    7
  ],
  "centered": [
    77,
    70,
    0,
    -70,
    -77
  ]
}
```

**Finding.** Reflection about Jacob1859 reverses the full source path, with offsets77,70,0,−70,−77. The70|70 body and seven-year wings are one symmetric object; the traditional Judah role remains secondary.

**Reassessment.** Compare this death-centered object with Jacob’s biographical70|7|70 path.

## C653 — Separate Jacob’s two symmetric objects

**Question.** How do the biographical and death-centered patterns connect?

**Sources.** File70 §5.3; A.5

**Inputs**

```json
{
  "biography": [
    2006,
    1936,
    1929,
    1859
  ],
  "death_centered": [
    1936,
    1929,
    1859,
    1789,
    1782
  ]
}
```

**Results**

```json
{
  "biography_gaps": [
    70,
    7,
    70
  ],
  "biography_center": "3865/2",
  "reflection": [
    1859,
    1929,
    1936,
    2006
  ],
  "centers_separation": "147/2"
}
```

**Finding.** Jacob’s lifespan path is a different symmetric object:70|7|70 around1932.5, while the later reflection is centered at his death1859. They share sourced nodes1936/1929/1859 but cannot be collapsed into one center.

**Reassessment.** Test the30/40 subdivision family to see what its junction equality depends on.

## C654 — Reconstruct the secondary30/40 junction

**Question.** Which equation generates the source’s30|30|40|40 arrangement?

**Sources.** File70 §5.4; secondary schematic comparison

**Inputs**

```json
{
  "H": 1929,
  "D": 1859,
  "a": 30,
  "b": 40,
  "source_path": [
    1959,
    1929,
    1899,
    1859,
    1819
  ]
}
```

**Results**

```json
{
  "path": [
    1959,
    1929,
    1899,
    1859,
    1819
  ],
  "junction_from_below": 1899,
  "gaps": [
    30,
    30,
    40,
    40
  ],
  "whole": 140
}
```

**Finding.** The shared1899 junction follows exactly from H−D=a+b=70. The whole path then has width2(a+b)=140. Its schematic30-day premise supplies the junction; choosing29 would give1900 instead.

**Reassessment.** Reconstruct Joseph’s complete supplied age rail and determine what is retained from Creation.

## C655 — Generate Joseph’s age-position family

**Question.** Does one birth anchor and the supplied ages recover the economic rail?

**Sources.** File70 §§5.5–5.6; A.6 P001–011

**Inputs**

```json
{
  "birth": 1915,
  "ages": [
    17,
    30,
    37,
    44
  ],
  "source_dates": [
    1898,
    1885,
    1878,
    1871
  ]
}
```

**Results**

```json
{
  "dates": [
    1898,
    1885,
    1878,
    1871
  ],
  "gaps": [
    13,
    7,
    7
  ],
  "creation_components": [
    13,
    1
  ],
  "Joseph_components": [
    13,
    14
  ]
}
```

**Finding.** Joseph’s17/30/37/44 ages generate1898/1885/1878/1871 and13|7|7. The13-year opening is retained in the Creation comparison while the following1 becomes14; the complete paths are not uniform scalar copies.

**Reassessment.** Use the authorized local Noah variant rows to test the same separation of shape and placement.

## C656 — Reconstruct the local Noah/Shem variant field

**Question.** Which spacings survive the admitted local gear and lifespan comparisons?

**Sources.** File70 §§4.3–4.5; A.7

**Inputs**

```json
{
  "Noah_G1": [
    2296,
    2236,
    2166,
    2106
  ],
  "Noah_G2": [
    2298,
    2238,
    2168,
    2108
  ],
  "Shem_G1": [
    2146,
    2086,
    2016,
    1956
  ],
  "role": "alternative death coordinates, not successive deaths"
}
```

**Results**

```json
{
  "gaps": [
    [
      60,
      70,
      60
    ],
    [
      60,
      70,
      60
    ],
    [
      60,
      70,
      60
    ]
  ],
  "G2_minus_G1": [
    2,
    2,
    2,
    2
  ],
  "Shem_minus_Noah": [
    -150,
    -150,
    -150,
    -150
  ]
}
```

**Finding.** The admitted alternative-date rows have one60|70|60 shape. The local Noah gear change translates it by2; the Shem comparison translates it by−150. No operation is extended to Creation or the later patriarchs.

**Reassessment.** Determine how the supplied Joseph endpoint completes the local Noah400 comparison.

## C657 — Complete the admitted Noah–Joseph comparison

**Question.** How much of the400 comparison follows from the variant field?

**Sources.** File70 §§4.3–4.5; A.7

**Inputs**

```json
{
  "G2_variants": [
    2298,
    2238,
    2168,
    2108
  ],
  "Joseph_age17": 1898
}
```

**Results**

```json
{
  "comparison_gaps": [
    60,
    70,
    60,
    210
  ],
  "total": 400,
  "variant_width": 190,
  "endpoint_arm": 210
}
```

**Finding.** The source’s400 comparison is the190-wide alternative-date field plus the210 arm to Joseph1898. Its equality to a400 Affliction duration does not identify their event pairs, and the variant positions are not successive Noah deaths.

**Reassessment.** Move to the cumulative4900 body and recover all its supplied cuts with one index.

## C658 — Generate the cumulative4900 cuts

**Question.** Can one70-year index recover the principal source junctions?

**Sources.** File70 §9.1A; A.13.3 CM019–027

**Inputs**

```json
{
  "head": 5436,
  "tail": 536,
  "indices": [
    0,
    49,
    50,
    69,
    70
  ],
  "source_nodes": [
    5436,
    2006,
    1936,
    606,
    536
  ]
}
```

**Results**

```json
{
  "indexed_nodes": {
    "0": 5436,
    "49": 2006,
    "50": 1936,
    "69": 606,
    "70": 536
  },
  "cut_pairs": [
    [
      49,
      21
    ],
    [
      50,
      20
    ],
    [
      69,
      1
    ]
  ],
  "duration": 4900
}
```

**Finding.** One rule x(k)=5436−70k recovers the source’s regular Jacob, cumulative Jacob/Levi, captivity and restoration positions. The49|21,50|20 and69|1 divisions are alternate cuts of one70-unit body.

**Reassessment.** Test the1400 rectangle embedded in this indexed body.

## C659 — Generate the cumulative1400 rectangle

**Question.** Are the four source spans independent or generated by matching70-year rails?

**Sources.** File70 A.13.4 CM028–030; C658

**Inputs**

```json
{
  "upper": [
    2006,
    1936
  ],
  "lower": [
    606,
    536
  ]
}
```

**Results**

```json
{
  "matrix": [
    [
      1400,
      1470
    ],
    [
      1330,
      1400
    ]
  ],
  "formula": "1400+70(j−i)",
  "rail_indices": [
    [
      49,
      50
    ],
    [
      69,
      70
    ]
  ]
}
```

**Finding.** The1400 rectangle is the existing rail generator at spacing70. Its crossed1330/1470 spans follow from matching rails; the second1400 is a dependent corresponding comparison.

**Reassessment.** Recover the source’s490|910 lower subdivision and locate Exodus within the same index.

## C660 — Insert Exodus into the indexed cumulative body

**Question.** What does the supplied Exodus1446 add to the4900 index?

**Sources.** File70 §9.1A; A.13.3–4

**Inputs**

```json
{
  "path": [
    5436,
    1936,
    1446,
    536
  ],
  "unit": 70
}
```

**Results**

```json
{
  "indices": [
    0,
    50,
    57,
    70
  ],
  "gaps": [
    3500,
    490,
    910
  ],
  "unit_gaps": [
    50,
    7,
    13
  ]
}
```

**Finding.** Exodus is index57 in the same source-conditioned lattice. The lower1400 is7|13 units, embedded in the complete50|7|13 partition of4900. The supplied Exodus role selects the cut; the other measures follow.

**Reassessment.** Reconstruct the larger cumulative spine in its175-year unit.

## C661 — Reconstruct the77-unit cumulative spine

**Question.** Does one175-year path reproduce the entire supplied Creation-to-restoration spine?

**Sources.** File70 §§1.4,5.8,9.1A; A.13.1 CM001–010

**Inputs**

```json
{
  "nodes": [
    14011,
    5436,
    1936,
    536
  ],
  "coefficients": [
    49,
    20,
    8
  ],
  "unit": 175
}
```

**Results**

```json
{
  "generated": [
    14011,
    5436,
    1936,
    536
  ],
  "durations": [
    8575,
    3500,
    1400
  ],
  "total": 13475,
  "49_units": 275
}
```

**Finding.** The full macro spine is49|20|8 in175-year units, totaling77 units or13475. The shared175 unit connects its anatomy to the earlier Abraham/covenant work without requiring every intermediate event to occupy the same role.

**Reassessment.** Test what changes when the Jacob junction uses its exact phase instead of its whole-year display.

## C662 — Resolve phase at the cumulative Jacob junction

**Question.** Which macro properties survive replacing the displayed1936 with exact1936.5?

**Sources.** File70 §§5.8,9.1A; exact-phase caution

**Inputs**

```json
{
  "outer": [
    5436,
    536
  ],
  "display": 1936,
  "exact": "3873/2"
}
```

**Results**

```json
{
  "display_arms": [
    3500,
    1400
  ],
  "exact_arms": [
    "6999/2",
    "2801/2"
  ],
  "change": [
    "-1/2",
    "1/2"
  ]
}
```

**Finding.** The half-year phase transfers0.5 from the upper arm to the lower arm. The4900 body survives, but its20|8 split in175 units belongs to the whole-year display. A conserved total does not authorize discarding phase.

**Reassessment.** Recover the source’s complete first-to-last triple correspondence.

## C663 — Recover the cumulative first-to-last triple

**Question.** Does the12075 correspondence apply to the whole supplied triple?

**Sources.** File70 §§1.4,5.8; A.13.1

**Inputs**

```json
{
  "creation": [
    14011,
    14004,
    13997
  ],
  "household": [
    1936,
    1929,
    1922
  ],
  "companion_role": "13997 source lower comparison node"
}
```

**Results**

```json
{
  "corresponding_spans": [
    12075,
    12075,
    12075
  ],
  "cross_matrix": [
    [
      12075,
      12082,
      12089
    ],
    [
      12068,
      12075,
      12082
    ],
    [
      12061,
      12068,
      12075
    ]
  ],
  "P_output": 12250,
  "175_coefficients": [
    69,
    70
  ]
}
```

**Finding.** The supplied first-to-last triples form one7-year rail translated by12075. Its Prophetic conversion69→70 in175 units recovers12250, now within a complete coordinate family rather than an isolated duration.

**Reassessment.** Fit Jacob’s147|490 biography and household boundary into the retained-path explanation.

## C664 — Recover the13-jubilee household path

**Question.** How does Jacob’s lifespan partition the cumulative path to Exodus?

**Sources.** File70 §§5.8,9.1A; A.13

**Inputs**

```json
{
  "nodes": [
    2083,
    1936,
    1446
  ],
  "roles": [
    "cumulative Jacob birth",
    "whole-year Jacob-death/Levi-birth boundary",
    "Exodus"
  ]
}
```

**Results**

```json
{
  "gaps": [
    147,
    490
  ],
  "49_coefficients": [
    3,
    10
  ],
  "total": 637
}
```

**Finding.** The source household boundary partitions637 as147+490, or3|10 jubilee units. Its corporate reading concerns the boundary of Jacob’s household; it does not date every son’s birth to1936. File70’s macro and biographical families are linked through retained junction roles.

**Reassessment.** Open Supplement A’s native720/30 family as a test of the same retained-component mechanism.

## C665 — Derive the native core from carrier and flank

**Question.** What minimal source inputs generate the720/30 family?

**Sources.** Supplement A §§1.1–1.2

**Inputs**

```json
{
  "carrier": 720,
  "flank": 30
}
```

**Results**

```json
{
  "core": 690,
  "flank_coefficients": {
    "core": 23,
    "carrier": 24,
    "native_bracket": 25,
    "expanded_core": 25
  }
}
```

**Finding.** The crossed core690 is23 flanks of30. The carrier is24 flanks, and both the native two-flank bracket and E-expanded core are25 flanks. Their shared750 is forced by core=23×flank, while their component roles differ.

**Reassessment.** Generate all native, P and E core/flank/bracket states together.

## C666 — Generate the complete retained-flank table

**Question.** Can one formula produce every native and converted bracket column?

**Sources.** Supplement A §§1.1–1.2

**Inputs**

```json
{
  "core": 690,
  "flank": 30,
  "operators": [
    "identity",
    "P",
    "E"
  ]
}
```

**Results**

```json
{
  "columns": [
    "core",
    "gain",
    "one_flank",
    "two_flanks"
  ],
  "rows": [
    [
      690,
      0,
      720,
      750
    ],
    [
      700,
      10,
      730,
      760
    ],
    [
      750,
      60,
      780,
      810
    ]
  ]
}
```

**Finding.** Applying each Key to the core while retaining30-unit flanks generates the full source table:690/720/750,700/730/760 and750/780/810. The gain belongs to the core; a whole-bracket conversion is a different operation.

**Reassessment.** Measure the exact difference between retained-flank and whole-bracket conversion.

## C667 — Explain retained-part conversion algebraically

**Question.** What term distinguishes converting a core from converting its entire bracket?

**Sources.** C666; C631 retained-part synthesis

**Inputs**

```json
{
  "core": 690,
  "flanks": [
    30,
    30
  ],
  "operators": [
    "P",
    "E"
  ]
}
```

**Results**

```json
{
  "rows": [
    {
      "retained": 760,
      "whole": "17500/23",
      "excess": "20/23",
      "flank_gain": "20/23"
    },
    {
      "retained": 810,
      "whole": "18750/23",
      "excess": "120/23",
      "flank_gain": "120/23"
    }
  ],
  "general_difference": "2(k−1)f"
}
```

**Finding.** Whole conversion adds exactly2(k−1)f more than core conversion with retained flanks. This is the same retained-subdivision distinction that explains the earlier partial-E and whole-J routes: matching totals never establish identical internal paths.

**Reassessment.** Reconstruct the native alternating ladder from its two rail parameters.

## C668 — Generate the whole alternating720/30 ladder

**Question.** Can the source’s ten coordinates be recovered without separate date fitting?

**Sources.** Supplement A §1.3; admitted source ladder only

**Inputs**

```json
{
  "head": 4326,
  "core": 690,
  "flank": 30,
  "source": [
    4326,
    4296,
    3606,
    3576,
    2886,
    2856,
    2166,
    2136,
    1446,
    1416
  ]
}
```

**Results**

```json
{
  "generated_nodes": [
    4326,
    4296,
    3606,
    3576,
    2886,
    2856,
    2166,
    2136,
    1446,
    1416
  ],
  "gaps": [
    30,
    690,
    30,
    690,
    30,
    690,
    30,
    690,
    30
  ],
  "same_rail_steps": [
    [
      720,
      720,
      720,
      720
    ],
    [
      720,
      720,
      720,
      720
    ]
  ],
  "width": 2910
}
```

**Finding.** The ten supplied positions are two720-step rails separated by30. Their interleaving generates the alternating30|690 path and2910 outer width. Source, dependency and generated-companion roles remain attached to the original positions; no new SKL transport is inferred.

**Reassessment.** Extract the six-node1470 cell and explain its alternative partitions.

## C669 — Explain every native1470 partition

**Question.** Why do the source’s690+780 and720+750 measures agree?

**Sources.** Supplement A §§1.4–1.5,2.3

**Inputs**

```json
{
  "cell": [
    3606,
    3576,
    2886,
    2856,
    2166,
    2136
  ],
  "core": 690,
  "flank": 30
}
```

**Results**

```json
{
  "total": 1470,
  "partitions": [
    [
      690,
      780
    ],
    [
      720,
      750
    ]
  ],
  "flank_units": 49,
  "center": 2871,
  "central_rail_center": 2871
}
```

**Finding.** All three1470 expressions reduce to2c+3f=49f. The six-node cell and its central30 rail share midpoint2871. These are decompositions of one supplied object, not independent appearances of1470.

**Reassessment.** Compare the cell’s48+1 month anatomy with the supplied intercalation ledger.

## C670 — Connect the1470 cell with calendrical intercalation

**Question.** Which part of the shared1470 is the same anatomy, and which part retains distinct units?

**Sources.** Supplement A §2.2; File12 calendrical physics

**Inputs**

```json
{
  "cell_units": [
    48,
    1
  ],
  "month_days": 30,
  "ledger": [
    6,
    2190,
    1470
  ],
  "civil_year_days": "1461/4"
}
```

**Results**

```json
{
  "final_block": 1470,
  "ledger_days": 14610,
  "year_equivalent": "1461/4",
  "cell_30_units": 49
}
```

**Finding.** The supplied intercalation final block has the same48+1 anatomy:4×360+30=1470 days. Its full ledger6×2190+1470=40×365.25 is a distinct calendrical realization; it does not turn the chronological1470-year cell into days.

**Reassessment.** Consolidate File70’s indexed paths and Supplement A’s retained-flank family before phase analysis.

## C671 — Consolidate the retained-path explanation

**Question.** What new mechanism connects biography, cumulative paths and calendar flanks?

**Sources.** C652–C670; Reviewer C652–C664

**Inputs**

```json
{
  "completed_range": [
    652,
    671
  ],
  "precision": [
    "phase test holds outer nodes fixed",
    "2083/1936/1446 path uses whole-year displays"
  ]
}
```

**Results**

```json
{
  "artifact": {
    "path": "deliverables/C671_Interim_Synthesis.md",
    "sha256": "2a41c2f375f92b22f0b043d23cd38cd2a3e48eaaa15807d6e901f8eda3ebf527",
    "bytes": 988
  },
  "mechanisms": [
    "reflection",
    "indexed path",
    "retained core and flanks"
  ]
}
```

**Finding.** Forty steps now provide a connected explanatory grammar. The next productive question is how phase and center coordinates account for the source’s40/70/120 fields and Actual/Rounded closure.

**Reassessment.** Derive the complete micro phase table from center, width and phase offsets.