# 490d — C732–C771: sequential research record

Working continuation of C731 · 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.

## C732 — Select the ordered cross-tradition family

**Question.** Can one row-to-boundary rule explain the full MT/LXX/SP displacement fields and their rounded residuals?

**Sources.** Research Strategy §§3,5B–G; C731 checkpoint; File18; File51a

**Inputs**

```json
{
  "source_manifest_sha256": "60159f91fa858a5651da5b097c2e7b5523835ded3757374fed59943b67f4727f",
  "predecessor": "C731",
  "comparison": "whole ordered boundary fields; not repeated endpoint totals"
}
```

**Results**

```json
{
  "selection": {
    "path": "model/RESEARCH_SELECTION.json",
    "sha256": "f839f7769dfae2e351c0a66baeadae3b71bee524b05302bb3a933706e12b8bbc",
    "bytes": 397
  },
  "source_snapshots": 41,
  "primary_File52c": "a5ea84562101158b60d0cf296765d6eff38e7a2abda4e74ad1b353dfd13b9530"
}
```

**Finding.** The new cycle will test ordered support and operation compatibility across entire source families. Previous aggregate sums are retained as baseline evidence; the new target is reconstruction of every intermediate boundary and residual.

**Reassessment.** Freeze the shared source-row alignment and choose the first complete displacement field.

## C733 — Derive the full mode-bridge recurrence

**Question.** What row quantity generates the change in cumulative-minus-regular displacement between adjacent named boundaries?

**Sources.** File18 MT regular/cumulative rows; File61 root vector; Research Strategy §5B

**Inputs**

```json
{
  "definition": "G_i=C_i−R_i",
  "regular_edge": "R_i−R_(i+1)=b_eff_i",
  "cumulative_edge": "C_i−C_(i+1)=L_i",
  "local_MT_example": {
    "regular": [
      2166,
      2066,
      2006
    ],
    "cumulative": [
      2435,
      2260,
      2080
    ],
    "lifespans": [
      175,
      180
    ],
    "effective_begetting": [
      100,
      60
    ]
  }
}
```

**Results**

```json
{
  "mode_bridge": [
    269,
    194,
    74
  ],
  "bridge_increments": [
    75,
    120
  ],
  "row_remainders": [
    75,
    120
  ],
  "general_rule": "G_i−G_(i+1)=L_i−b_eff_i"
}
```

**Finding.** The mode bridge has a local generator: lifespan minus the effective regular edge. Telescoping these row contributions reconstructs the whole bridge once the terminal displacement is supplied. Declared binding corrections belong in b_eff rather than being hidden in a global conversion.

**Reassessment.** Apply this recurrence to every shared MT boundary, retaining source roles and terminal frame.

## C734 — Reconstruct the entire MT mode-bridge field

**Question.** Does one terminal displacement and the row remainder rule recover every Adam-to-Abraham boundary?

**Sources.** File18 §§2.1.3,2.2,6A

**Inputs**

```json
{
  "packet_sha256": "37dd27bbe2eb9d0b7606d75620de499b0ab3493d7188ec814f4f443adb6af9b2",
  "regular_frame": "+215 full430",
  "cumulative_frame": "Moses/Nisan centers; AdamYear6 role retained",
  "Noah_edge_correction": 2
}
```

**Results**

```json
{
  "names": [
    "Adam",
    "Seth",
    "Enosh",
    "Kenan",
    "Mahalalel",
    "Jared",
    "Enoch",
    "Methuselah",
    "Lamech",
    "Noah",
    "Shem",
    "Arphaxad",
    "Shelah",
    "Eber",
    "Peleg",
    "Reu",
    "Serug",
    "Nahor",
    "Terah",
    "Abraham"
  ],
  "center_mode_bridge": [
    9892,
    9092,
    8285,
    7470,
    6630,
    5800,
    5000,
    4700,
    3918,
    3323,
    2875,
    2375,
    1972,
    1569,
    1139,
    930,
    723,
    523,
    404,
    269
  ],
  "row_generators": [
    800,
    807,
    815,
    840,
    830,
    800,
    300,
    782,
    595,
    448,
    500,
    403,
    403,
    430,
    209,
    207,
    200,
    119,
    135
  ],
  "terminal_bridge": 269,
  "head_role": "9892 pairs cumulativeYear6 with regularCreation-completion; not the established9890 completion bridge"
}
```

**Finding.** The same remainder-plus-binding rule reconstructs all20 MT mode comparisons from terminal269. The mixed Adam-label displacement9892 is explicitly a center-profile quantity; it is not silently substituted for the established completion bridge9890.

**Reassessment.** Change the entire cumulative profile to its source lower-envelope members and test what is preserved.

## C735 — Transport the whole bridge to the lower envelope

**Question.** Does the chosen completion profile preserve the row-generating law?

**Sources.** File18 §6A envelopes; C734; C731 Actual bridge

**Inputs**

```json
{
  "cumulative_profile_shift": -2,
  "regular_frame": "+215",
  "scope": "all selected cumulative lower-envelope members, not newGeartransport"
}
```

**Results**

```json
{
  "lower_profile": [
    9890,
    9090,
    8283,
    7468,
    6628,
    5798,
    4998,
    4698,
    3916,
    3321,
    2873,
    2373,
    1970,
    1567,
    1137,
    928,
    721,
    521,
    402,
    267
  ],
  "center_to_lower": [
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2,
    -2
  ],
  "terminal": 267,
  "completion_head": 9890
}
```

**Finding.** Selecting the lower-envelope member translates the entire MT bridge by−2 and preserves every adjacent generator. It recovers9890 at Creation and267 at Abraham. This is a source-defined profile choice, not an adjustment fitted independently at each node.

**Reassessment.** Transfer the same full-field recurrence to nativeLXX, retaining its insertedCainan row.

## C736 — Transfer the full bridge to native LXX

**Question.** Does the MT row-to-boundary rule reconstruct the nativeLXX profile without removingCainan?

**Sources.** File18 §§4.1,6D; C733–C735

**Inputs**

```json
{
  "source_names": [
    "Adam",
    "Seth",
    "Enosh",
    "Kenan",
    "Mahalalel",
    "Jared",
    "Enoch",
    "Methuselah",
    "Lamech",
    "Noah",
    "Shem",
    "Arphaxad",
    "Cainan2",
    "Shelah",
    "Eber",
    "Peleg",
    "Reu",
    "Serug",
    "Nahor",
    "Terah"
  ],
  "Cainan": "nativeON",
  "terminal_lower_pair": [
    2433,
    2166
  ],
  "Lamech": "182/753"
}
```

**Results**

```json
{
  "names": [
    "Adam",
    "Seth",
    "Enosh",
    "Kenan",
    "Mahalalel",
    "Jared",
    "Enoch",
    "Methuselah",
    "Lamech",
    "Noah",
    "Shem",
    "Arphaxad",
    "Cainan2",
    "Shelah",
    "Eber",
    "Peleg",
    "Reu",
    "Serug",
    "Nahor",
    "Terah",
    "Abraham"
  ],
  "lower_mode_bridge": [
    9400,
    8700,
    7993,
    7278,
    6538,
    5808,
    5008,
    4808,
    4026,
    3455,
    3007,
    2507,
    2177,
    1847,
    1517,
    1147,
    938,
    731,
    531,
    402,
    267
  ],
  "row_generators": [
    700,
    707,
    715,
    740,
    730,
    800,
    200,
    782,
    571,
    448,
    500,
    330,
    330,
    330,
    370,
    209,
    207,
    200,
    129,
    135
  ],
  "source_center_checks": {
    "Adam": 14896,
    "Lamech": 8048,
    "Cainan2": 5280,
    "Peleg": 3856
  }
}
```

**Finding.** The identical rule generates all21 nativeLXX comparisons, including theCainan330 remainder contribution. Four separately printed cumulative nodes confirm the reconstructed chain. The resulting Creation bridge9400 belongs to the nativeLXX state, rather than being forced toMT9890.

**Reassessment.** Transfer the same rule toSP while keeping its inclusive and primaryNoah correction explicit.

## C737 — Transfer the full bridge to SP

**Question.** Does the same recurrence preserve SP’s local counting conventions throughout the field?

**Sources.** File18 §§3.1–3.2,6C; C733–C736

**Inputs**

```json
{
  "Cainan": "OFF",
  "terminal_lower_pair": [
    2433,
    2166
  ],
  "edge_corrections": {
    "Noah": 2,
    "Lamech": -1
  },
  "Lamech_count": "53rd year52completed;653inclusive lifespan ledger"
}
```

**Results**

```json
{
  "names": [
    "Adam",
    "Seth",
    "Enosh",
    "Kenan",
    "Mahalalel",
    "Jared",
    "Enoch",
    "Methuselah",
    "Lamech",
    "Noah",
    "Shem",
    "Arphaxad",
    "Shelah",
    "Eber",
    "Peleg",
    "Reu",
    "Serug",
    "Nahor",
    "Terah",
    "Abraham"
  ],
  "lower_mode_bridge": [
    8982,
    8182,
    7375,
    6560,
    5720,
    4890,
    4105,
    3805,
    3152,
    2551,
    2103,
    1603,
    1300,
    997,
    727,
    618,
    511,
    411,
    342,
    267
  ],
  "row_generators": [
    800,
    807,
    815,
    840,
    830,
    785,
    300,
    653,
    601,
    448,
    500,
    303,
    303,
    270,
    109,
    107,
    100,
    69,
    75
  ],
  "source_center_checks": {
    "Adam": 13398,
    "Peleg": 3436,
    "Terah": 2580
  }
}
```

**Finding.** All20 SP comparisons follow from the same recurrence with its two declared edge corrections. The Creation bridge8982 arises from the full source profile. The653 inclusive lifespan ledger and52 completed regular interval remain distinct inputs, rather than an unexplained one-year repair.

**Reassessment.** Compare the three complete profiles on their shared named boundaries and locate their source differences.

## C738 — Unify the three complete mode profiles

**Question.** Can one implementation reconstruct all native profiles while preserving their distinct row membership?

**Sources.** C734–C737; File18

**Inputs**

```json
{
  "traditions": [
    "MT",
    "LXX",
    "SP"
  ],
  "common_terminal": {
    "regular": 2166,
    "cumulative_center": 2435,
    "lower": 2433
  }
}
```

**Results**

```json
{
  "artifact": {
    "path": "model/native_mode_fields.json",
    "sha256": "bc9bbd34af7478bd21d03d23dba6c926c4f741271bc6ebb761337a71e06652b4",
    "bytes": 10042
  },
  "native_row_counts": {
    "MT": 20,
    "LXX": 21,
    "SP": 20
  },
  "head_G": {
    "MT": 9890,
    "LXX": 9400,
    "SP": 8982
  }
}
```

**Finding.** A single source-aware constructor now carries all61 native node comparisons. TheLXX-onlyCainan row stays explicit; an absent row is not replaced with a zero-life patriarch. These fields support direct cross-tradition comparisons without another global fit.

**Reassessment.** Compute the entire inter-tradition displacement profiles on the common named nodes.

## C739 — Recover the complete inter-tradition displacement fields

**Question.** Where do regular and cumulative differences actually change along the shared genealogy?

**Sources.** File18 source fields; C738

**Inputs**

```json
{
  "shared_nodes": [
    "Adam",
    "Seth",
    "Enosh",
    "Kenan",
    "Mahalalel",
    "Jared",
    "Enoch",
    "Methuselah",
    "Lamech",
    "Noah",
    "Shem",
    "Arphaxad",
    "Shelah",
    "Eber",
    "Peleg",
    "Reu",
    "Serug",
    "Nahor",
    "Terah",
    "Abraham"
  ],
  "reference": "MT; commonfull430regular andlowercumulativeprofile"
}
```

**Results**

```json
{
  "artifact": {
    "path": "model/tradition_displacement_profiles.json",
    "sha256": "a30d1703e4ddaa7670451a1e32597fde8825da57aace1430526d3a8f688fb4f2",
    "bytes": 2056
  },
  "profiles": {
    "LXX": {
      "R": [
        1380,
        1280,
        1180,
        1080,
        980,
        880,
        880,
        780,
        780,
        780,
        780,
        780,
        550,
        450,
        350,
        250,
        150,
        50,
        0,
        0
      ],
      "C_lower": [
        890,
        890,
        890,
        890,
        890,
        890,
        890,
        890,
        890,
        914,
        914,
        914,
        427,
        400,
        360,
        260,
        160,
        60,
        0,
        0
      ],
      "G": [
        -490,
        -390,
        -290,
        -190,
        -90,
        10,
        10,
        110,
        110,
        134,
        134,
        134,
        -123,
        -50,
        10,
        10,
        10,
        10,
        0,
        0
      ]
    },
    "SP": {
      "R": [
        300,
        300,
        300,
        300,
        300,
        300,
        400,
        400,
        520,
        650,
        650,
        650,
        550,
        450,
        350,
        250,
        150,
        50,
        0,
        0
      ],
      "C_lower": [
        -608,
        -608,
        -608,
        -608,
        -608,
        -608,
        -493,
        -493,
        -244,
        -120,
        -120,
        -120,
        -120,
        -120,
        -60,
        -60,
        -60,
        -60,
        -60,
        0
      ],
      "G": [
        -908,
        -908,
        -908,
        -908,
        -908,
        -908,
        -893,
        -893,
        -764,
        -770,
        -770,
        -770,
        -670,
        -570,
        -410,
        -310,
        -210,
        -110,
        -60,
        0
      ]
    }
  }
}
```

**Finding.** The complete displacement fields expose plateaus and change points hidden byCreation totals. At every shared node, the bridge displacement equals cumulative displacement minus regular displacement; each mode retains its own source-driven shape.

**Reassessment.** Use adjacent differences to recover the exact support of theLXX row changes, including thecollapsedCainanedge.

## C740 — Recover LXX local changes from its boundary profile

**Question.** Can adjacent differences identify the contributing rows and distinguish aninsertedrowfromalifespanchange?

**Sources.** C739; File18 §§2,4,6D

**Inputs**

```json
{
  "shared_edge": "Arphaxad→Shelah spansLXXCainan2",
  "profile": "LXXminusMT"
}
```

**Results**

```json
{
  "nonzero_edge_changes": {
    "R": {
      "Adam": 100,
      "Seth": 100,
      "Enosh": 100,
      "Kenan": 100,
      "Mahalalel": 100,
      "Enoch": 100,
      "Arphaxad": 230,
      "Shelah": 100,
      "Eber": 100,
      "Peleg": 100,
      "Reu": 100,
      "Serug": 100,
      "Nahor": 50
    },
    "C_lower": {
      "Lamech": -24,
      "Arphaxad": 487,
      "Shelah": 27,
      "Eber": 40,
      "Peleg": 100,
      "Reu": 100,
      "Serug": 100,
      "Nahor": 60
    },
    "G": {
      "Adam": -100,
      "Seth": -100,
      "Enosh": -100,
      "Kenan": -100,
      "Mahalalel": -100,
      "Enoch": -100,
      "Lamech": -24,
      "Arphaxad": 257,
      "Shelah": -73,
      "Eber": -60,
      "Nahor": 10
    }
  },
  "collapsed_Arphaxad_components": {
    "regular": [
      100,
      130
    ],
    "cumulative": [
      27,
      460
    ],
    "bridge": [
      -73,
      330
    ]
  }
}
```

**Finding.** Finite differences recover the entireLXX change support. At the sharedArphaxad→Shelah edge, regular230 splits into100+130 and cumulative487 into27+460. KeepingCainan as a native row distinguishes insertion from the adjacent biography change; collapsing them would lose that explanation.

**Reassessment.** Recover theSP support with its separateLamech completed-year correction.

## C741 — Recover SP local changes from its boundary profile

**Question.** Which source rows and counting terms generateSP’s entire displacement staircase?

**Sources.** C739; File18 §§3.1–3.2,6C

**Inputs**

```json
{
  "profile": "SPminusMT",
  "Lamech_regular": "53counted versus52completed"
}
```

**Results**

```json
{
  "nonzero_edge_changes": {
    "R": {
      "Jared": -100,
      "Methuselah": -120,
      "Lamech": -130,
      "Arphaxad": 100,
      "Shelah": 100,
      "Eber": 100,
      "Peleg": 100,
      "Reu": 100,
      "Serug": 100,
      "Nahor": 50
    },
    "C_lower": {
      "Jared": -115,
      "Methuselah": -249,
      "Lamech": -124,
      "Eber": -60,
      "Terah": -60
    },
    "G": {
      "Jared": -15,
      "Methuselah": -129,
      "Lamech": 6,
      "Arphaxad": -100,
      "Shelah": -100,
      "Eber": -160,
      "Peleg": -100,
      "Reu": -100,
      "Serug": -100,
      "Nahor": -50,
      "Terah": -60
    }
  },
  "Lamech_regular_decomposition": [
    -129,
    -1
  ]
}
```

**Finding.** TheSP profile localizes its regular130-yearLamech difference as129 from the nominal age plus one completed-year correction. Cumulative support remains the five lifespan changes. The whole staircase therefore explains both the shared plateaus and the different mode response.

**Reassessment.** Test the entire post-ShelahLXX/SP family where births agree but cumulative coordinates differ.

## C742 — Explain equal births with unequal cumulative boundaries

**Question.** Why does the entireLXX/SP birthsuffix coincide while itscumulativeprofilediverges?

**Sources.** File18 §§3–4,6C–D; C738

**Inputs**

```json
{
  "nodes": [
    "Shelah",
    "Eber",
    "Peleg",
    "Reu",
    "Serug",
    "Nahor",
    "Terah",
    "Abraham"
  ],
  "native_Cainan": [
    "LXXON",
    "SPOFF"
  ],
  "regular_frame": "commonfull430"
}
```

**Results**

```json
{
  "regular_differences": [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
  ],
  "cumulative_differences": [
    547,
    520,
    420,
    320,
    220,
    120,
    60,
    0
  ],
  "lifespan_changes": [
    27,
    100,
    100,
    100,
    100,
    60,
    60
  ]
}
```

**Finding.** All eightLXX/SP birthpositionsShelah throughAbraham coincide. Their cumulative differences547→520→420→320→220→120→60→0 are exactly generated by the local lifespan changes. This is a complete positive example of shared regular geometry with divergent cumulative realization.

**Reassessment.** Insert the single admittedCainan row intoSP and test how far the shared birthsuffix extends.

## C743 — Recover the shared Cainan-ON birthsuffix

**Question.** Does theadmittedSP insertion align the completeNoah-to-Abraham birthpath with nativeLXX?

**Sources.** File18 Cainan restoration§1.3; §§3–4; C742

**Inputs**

```json
{
  "names": [
    "Noah",
    "Shem",
    "Arphaxad",
    "Cainan2",
    "Shelah",
    "Eber",
    "Peleg",
    "Reu",
    "Serug",
    "Nahor",
    "Terah",
    "Abraham"
  ],
  "SP_regular_insertion": 130,
  "scope": "single source-authorized insertion aboveShelah; noGeartransport"
}
```

**Results**

```json
{
  "SP_ON_births": [
    3838,
    3336,
    3236,
    3101,
    2971,
    2841,
    2707,
    2577,
    2445,
    2315,
    2236,
    2166
  ],
  "native_LXX_births": [
    3838,
    3336,
    3236,
    3101,
    2971,
    2841,
    2707,
    2577,
    2445,
    2315,
    2236,
    2166
  ],
  "all_differences": [
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0,
    0
  ]
}
```

**Finding.** The single admittedSPCainan insertion aligns every one of the12Noah-to-Abraham birthpositions with nativeLXX. This is a whole-path agreement produced by shared downstream begetting rows and the inserted130 interval, while the source lifespans remain different.

**Reassessment.** Compare deaths across the newly shared birthpath to identify exactly what the agreementdoesnotpreserve.

## C744 — Explain death differences within identical birthpaths

**Question.** What remains different afterSPCainan-ON births align with nativeLXX?

**Sources.** File18 source rows; C743

**Inputs**

```json
{
  "shared_births": "C743",
  "Cainan_inserted_lifespan": 460,
  "nodes": [
    "Noah",
    "Shem",
    "Arphaxad",
    "Cainan2",
    "Shelah",
    "Eber",
    "Peleg",
    "Reu",
    "Serug",
    "Nahor",
    "Terah"
  ]
}
```

**Results**

```json
{
  "death_comparison": [
    {
      "name": "Noah",
      "SP_death": 2888,
      "LXX_death": 2888,
      "LXX_minus_SP_death": 0
    },
    {
      "name": "Shem",
      "SP_death": 2736,
      "LXX_death": 2736,
      "LXX_minus_SP_death": 0
    },
    {
      "name": "Arphaxad",
      "SP_death": 2798,
      "LXX_death": 2771,
      "LXX_minus_SP_death": -27
    },
    {
      "name": "Cainan2",
      "SP_death": 2641,
      "LXX_death": 2641,
      "LXX_minus_SP_death": 0
    },
    {
      "name": "Shelah",
      "SP_death": 2538,
      "LXX_death": 2511,
      "LXX_minus_SP_death": -27
    },
    {
      "name": "Eber",
      "SP_death": 2437,
      "LXX_death": 2337,
      "LXX_minus_SP_death": -100
    },
    {
      "name": "Peleg",
      "SP_death": 2468,
      "LXX_death": 2368,
      "LXX_minus_SP_death": -100
    },
    {
      "name": "Reu",
      "SP_death": 2338,
      "LXX_death": 2238,
      "LXX_minus_SP_death": -100
    },
    {
      "name": "Serug",
      "SP_death": 2215,
      "LXX_death": 2115,
      "LXX_minus_SP_death": -100
    },
    {
      "name": "Nahor",
      "SP_death": 2167,
      "LXX_death": 2107,
      "LXX_minus_SP_death": -60
    },
    {
      "name": "Terah",
      "SP_death": 2091,
      "LXX_death": 2031,
      "LXX_minus_SP_death": -60
    }
  ]
}
```

**Finding.** With the whole birthpath fixed, each death difference is exactly the negative lifespan difference. Noah,Shem and insertedCainan deaths coincide; the other differences follow locally. Birthpath agreement therefore transfers a specific projection of the sources, not every chronological measure.

**Reassessment.** Compare matchedCainan cumulativefields to locate the maximal translation blocks.

## C745 — Locate matched-Cainan translation blocks

**Question.** Does the familiar430 cumulative alignment extend through the whole matched-state genealogy?

**Sources.** File18 §§6B,6D; C739

**Inputs**

```json
{
  "comparison": "nativeLXXON minus admittedMTON",
  "insertion_support": "Adam throughArphaxad",
  "cumulative_insertion": 460
}
```

**Results**

```json
{
  "matched_profile": [
    430,
    430,
    430,
    430,
    430,
    430,
    430,
    430,
    430,
    454,
    454,
    454,
    427,
    400,
    360,
    260,
    160,
    60,
    0,
    0
  ],
  "constant_blocks": [
    {
      "from": "Adam",
      "through": "Lamech",
      "displacement": 430
    },
    {
      "from": "Noah",
      "through": "Arphaxad",
      "displacement": 454
    },
    {
      "from": "Shelah",
      "through": "Shelah",
      "displacement": 427
    },
    {
      "from": "Eber",
      "through": "Eber",
      "displacement": 400
    },
    {
      "from": "Peleg",
      "through": "Peleg",
      "displacement": 360
    },
    {
      "from": "Reu",
      "through": "Reu",
      "displacement": 260
    },
    {
      "from": "Serug",
      "through": "Serug",
      "displacement": 160
    },
    {
      "from": "Nahor",
      "through": "Nahor",
      "displacement": 60
    },
    {
      "from": "Terah",
      "through": "Abraham",
      "displacement": 0
    }
  ]
}
```

**Finding.** The430 alignment is exactly theAdam-throughLamech plateau. Noah-throughArphaxad differs454, and later blocks follow their own source rows. MatchingCainan removes insertion from the comparison but does not turn the entire genealogy into one translation.

**Reassessment.** Test pointwise composition of thethree native comparisonfields with all source modes retained.

## C746 — Verify composition on the full shared genealogy

**Question.** Do MT→SP→LXX comparisons compose node-by-node in both modes and the modebridge?

**Sources.** C738–C745; Research Strategy §5D

**Inputs**

```json
{
  "shared_nodes": [
    "Adam",
    "Seth",
    "Enosh",
    "Kenan",
    "Mahalalel",
    "Jared",
    "Enoch",
    "Methuselah",
    "Lamech",
    "Noah",
    "Shem",
    "Arphaxad",
    "Shelah",
    "Eber",
    "Peleg",
    "Reu",
    "Serug",
    "Nahor",
    "Terah",
    "Abraham"
  ],
  "modes": [
    "R",
    "C_lower",
    "G"
  ],
  "excluded_from_shared_grid": "nativeLXX-onlyCainan retained in itsownpath"
}
```

**Results**

```json
{
  "composition_residuals": {
    "R": [
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0
    ],
    "C_lower": [
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0
    ],
    "G": [
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0
    ]
  },
  "typed_comparisons": 60
}
```

**Finding.** All60 typed shared-node comparisons compose exactly. This is consistency of a common source-coordinate representation, not independent confirmation from60matches. The native inserted row remains visible rather than being invented in the othertraditions.

**Reassessment.** Test why matchingCreation totals alone cannot recover the ordered source structure.

## C747 — Test the information supplied by intermediate boundaries

**Question.** Could the same aggregate lifespan difference conceal a different source-row order?

**Sources.** C742; Research Strategy §5B

**Inputs**

```json
{
  "source_deltas": [
    27,
    100,
    100,
    100,
    100,
    60,
    60
  ],
  "diagnostic": "swap firsttwo row differences; no newsource state"
}
```

**Results**

```json
{
  "source_profile": [
    547,
    520,
    420,
    320,
    220,
    120,
    60,
    0
  ],
  "diagnostic_profile": [
    547,
    447,
    420,
    320,
    220,
    120,
    60,
    0
  ],
  "profile_difference": [
    0,
    -73,
    0,
    0,
    0,
    0,
    0,
    0
  ]
}
```

**Finding.** Swapping just two row contributions preserves the547total and terminal but changes theEber comparison by73. The supplied intermediate boundary therefore contains information that theCreation or outer total cannot recover. This is a diagnostic of explanatory sufficiency, not an admitted variant.

**Reassessment.** State the exact reconstruction relation between local row changes and thewholeboundaryprofile.

## C748 — Identify the ordered reconstruction operator

**Question.** Does the complete boundary profile determine itslocalcontributionsuniquely once theterminal is fixed?

**Sources.** C742,C747; Research Strategy §5F

**Inputs**

```json
{
  "local_vector": [
    27,
    100,
    100,
    100,
    100,
    60,
    60
  ],
  "terminal": 0,
  "operator": "uppertriangularsuffixsum"
}
```

**Results**

```json
{
  "suffix_matrix": [
    [
      1,
      1,
      1,
      1,
      1,
      1,
      1
    ],
    [
      0,
      1,
      1,
      1,
      1,
      1,
      1
    ],
    [
      0,
      0,
      1,
      1,
      1,
      1,
      1
    ],
    [
      0,
      0,
      0,
      1,
      1,
      1,
      1
    ],
    [
      0,
      0,
      0,
      0,
      1,
      1,
      1
    ],
    [
      0,
      0,
      0,
      0,
      0,
      1,
      1
    ],
    [
      0,
      0,
      0,
      0,
      0,
      0,
      1
    ]
  ],
  "profile": [
    547,
    520,
    420,
    320,
    220,
    120,
    60,
    0
  ],
  "recovered_rows": [
    27,
    100,
    100,
    100,
    100,
    60,
    60
  ],
  "determinant": 1,
  "rank": 7
}
```

**Finding.** The suffix-sum matrix is triangular with unit diagonal. With terminal fixed, every ordered boundary determines exactly one local row vector, recovered by adjacent differences. This gives theStrategy a small linear mechanism for whole-family explanation, while total-only evaluation loses theordering.

**Reassessment.** Test thewhole rounded-row family for compatibility between lifespan projection androunding.

## C749 — Freeze the three rounding domains

**Question.** Which row operations and domains may be compared?

**Sources.** File51a §§3.4,16.1–16.2; File18 regular tables

**Inputs**

```json
{}
```

**Results**

```json
{
  "packet": {
    "path": "model/rounding_inputs.json",
    "sha256": "1c34bdf733d5822070359212b1fc1bedd09747e2ea47ae613a45570546f1aaf4",
    "bytes": 50661
  },
  "row_counts": {
    "MT_native_OFF": 26,
    "LXX_native_ON": 27,
    "SP_native_OFF": 26
  },
  "operations": [
    "Q(b) regular births",
    "Q(b)+Q(r) theoretical regular lifespan",
    "Q(L) cumulative lifespan"
  ],
  "restrictions": [
    "SP inclusive counts retain count type",
    "Jacob b=91 belongs to Joseph collateral",
    "four final b inputs withheld"
  ]
}
```

**Finding.** Rounded is three typed row operations, followed by path accumulation. A missing or collateral begetting input cannot become a trunk edge.

**Reassessment.** Reconstruct both MT lifespan operators over every admitted row.

## C750 — Reconstruct every MT rounded lifespan

**Question.** Where do the two lifespan operations diverge?

**Sources.** File51a §3.4 and §16.1 literal tables

**Inputs**

```json
{}
```

**Results**

```json
{
  "parts": [
    930,
    910,
    905,
    910,
    895,
    960,
    365,
    965,
    775,
    950,
    600,
    440,
    435,
    465,
    240,
    235,
    230,
    150,
    205,
    175,
    180,
    145
  ],
  "whole": [
    930,
    910,
    905,
    910,
    895,
    960,
    365,
    970,
    775,
    950,
    600,
    440,
    435,
    465,
    240,
    240,
    230,
    150,
    205,
    175,
    180,
    145,
    135,
    135,
    135,
    120
  ],
  "nonzero_defects": {
    "Methuselah": -5,
    "Reu": -5
  }
}
```

**Finding.** Methuselah and Reu both give parts-minus-total = −5. The source tables agree; the unrestricted uniqueness sentence about Methuselah needs qualification.

**Reassessment.** Record the first 20-step checkpoint, including the common regular comparison frame.

## C751 — First twenty-step explanatory checkpoint

**Question.** What generalizes, and what frame must remain explicit?

**Sources.** C733–750

**Inputs**

```json
{
  "regular_frame": "common full430 comparison",
  "native_Cainan": {
    "MT": "OFF",
    "LXX": "ON",
    "SP": "OFF"
  }
}
```

**Results**

```json
{
  "checkpoint": {
    "path": "deliverables/490d_Chronological_Families_Checkpoint_C751.md",
    "sha256": "6865f3a55c73e0a3b0d2d6964597a26f6c4cbe5a46484ffd85a542ca18784464",
    "bytes": 1313
  },
  "SP_frame_check": {
    "full430": 8982,
    "native215": 9197
  }
}
```

**Finding.** The comparison now has an explicit frame and a whole-field explanation. Proceed with the residual operator rather than surveying more endpoints.

**Reassessment.** Derive the complete finite rounding defect law.

## C752 — Derive the finite rounding defect law

**Question.** Can one residue table explain every parts-versus-total defect?

**Sources.** C749–750; File51a rounding rules

**Inputs**

```json
{}
```

**Results**

```json
{
  "table_b_rows_r_columns": [
    [
      0,
      0,
      0,
      0,
      0
    ],
    [
      0,
      0,
      -5,
      0,
      0
    ],
    [
      0,
      -5,
      -5,
      0,
      0
    ],
    [
      0,
      0,
      0,
      5,
      5
    ],
    [
      0,
      0,
      0,
      5,
      0
    ]
  ],
  "formula": "K(b,r)=Q(b)+Q(r)-Q(b+r)",
  "periodicity": "Q(x+5k)=Q(x)+5k"
}
```

**Finding.** Every compatible integer row belongs to one of 25 residue classes; defects are only −5,0,+5. Methuselah and Reu share class(2,2).

**Reassessment.** Check whether the three traditions add new defect classes within the admitted domains.

## C753 — Transfer the defect law across traditions

**Question.** Which ordinary rows have nonzero defects in each tradition?

**Sources.** File18 rows; File51a rules

**Inputs**

```json
{}
```

**Results**

```json
{
  "defects": {
    "MT_native_OFF": {
      "Methuselah": {
        "K": -5,
        "residues": [
          2,
          2
        ]
      },
      "Reu": {
        "K": -5,
        "residues": [
          2,
          2
        ]
      }
    },
    "LXX_native_ON": {
      "Methuselah": {
        "K": -5,
        "residues": [
          2,
          2
        ]
      },
      "Lamech": {
        "K": -5,
        "residues": [
          2,
          1
        ]
      },
      "Reu": {
        "K": -5,
        "residues": [
          2,
          2
        ]
      }
    },
    "SP_native_OFF": {
      "Reu": {
        "K": -5,
        "residues": [
          2,
          2
        ]
      }
    }
  },
  "inclusive_rows_not_ordinary_biographies": {
    "MT_native_OFF": [],
    "LXX_native_ON": [],
    "SP_native_OFF": [
      "Jared",
      "Methuselah",
      "Lamech"
    ]
  }
}
```

**Finding.** LXX adds main Lamech(182+571=753), class(2,1), with defect−5. SP ordinary Reu also has−5. Inclusive SP rows remain ledger diagnostics.

**Reassessment.** Reconstruct the full MT cumulative rounding field.

## C754 — Reconstruct the complete MT residual field

**Question.** Do all 26 literal cumulative boundaries follow row rounding?

**Sources.** File51a §16.1–16.2

**Inputs**

```json
{
  "terminal": 1406
}
```

**Results**

```json
{
  "field": {
    "path": "model/MT_rounding_field.json",
    "sha256": "58f0d98b8bcc3918c2dc9c79ce018326db0166ed7d1efbc21773b62e63ecdcde",
    "bytes": 1359
  },
  "D": [
    0,
    0,
    2,
    2,
    2,
    2,
    4,
    4,
    3,
    5,
    5,
    5,
    3,
    1,
    0,
    -1,
    -2,
    -2,
    -4,
    -4,
    -4,
    -4,
    -2,
    0,
    -2,
    0
  ]
}
```

**Finding.** All 26 printed MT boundaries follow one suffix-sum residual field. Endpoint cancellation coexists with internal displacements from−4 to+5.

**Reassessment.** Recover the local residuals from the boundary field and identify unchanged intervals.

## C755 — Invert and classify the MT residual field

**Question.** Which entire interval families survive rounding exactly?

**Sources.** C754

**Inputs**

```json
{}
```

**Results**

```json
{
  "recovered_e": [
    0,
    -2,
    0,
    0,
    0,
    -2,
    0,
    1,
    -2,
    0,
    0,
    2,
    2,
    1,
    1,
    1,
    0,
    2,
    0,
    0,
    0,
    -2,
    -2,
    2,
    -2,
    0
  ],
  "equal_residual_classes": {
    "-4": [
      "Terah",
      "Abraham",
      "Isaac",
      "Jacob"
    ],
    "-2": [
      "Serug",
      "Nahor",
      "Levi",
      "Amram"
    ],
    "-1": [
      "Reu"
    ],
    "0": [
      "Adam",
      "Seth",
      "Peleg",
      "Kohath",
      "Moses",
      "terminal1406"
    ],
    "1": [
      "Eber"
    ],
    "2": [
      "Enosh",
      "Kenan",
      "Mahalalel",
      "Jared"
    ],
    "3": [
      "Lamech",
      "Shelah"
    ],
    "4": [
      "Enoch",
      "Methuselah"
    ],
    "5": [
      "Noah",
      "Shem",
      "Arphaxad"
    ]
  },
  "interval_rule": "rounded(i,j)-actual(i,j)=D_i-D_j"
}
```

**Finding.** Every pair within one residual class preserves its interval. These are consequences of one field, not independent numerical hits. Adjacent differences recover every row residual.

**Reassessment.** Transfer the full residual operator to LXX and isolate changed row support.

## C756 — Transfer the cumulative residual field to LXX

**Question.** Which changed rows explain every LXX-versus-MT rounding difference?

**Sources.** File18 LXX main rows; File51a rounding rule

**Inputs**

```json
{}
```

**Results**

```json
{
  "fields": {
    "path": "model/all_rounding_fields.json",
    "sha256": "6ce58f06e5687e621578a52663a0d6f32b50d3b1cd1ad74ff2cc4225a359c6fe",
    "bytes": 7609
  },
  "row_support": {
    "Lamech": 4,
    "Arphaxad": -2,
    "Shelah": -2
  },
  "nonzero_boundary_differences": {
    "Noah": -4,
    "Shem": -4,
    "Arphaxad": -4,
    "Shelah": -2
  }
}
```

**Finding.** Three LXX residual changes generate the complete comparison: Lamech+4, Arphaxad−2, Shelah−2. Their total cancels, but the Noah–Arphaxad plateau changes−4 and Shelah−2.

**Reassessment.** Test SP support and explain its changed head residual.

## C757 — Transfer the residual field to SP

**Question.** Why does SP gain three at the cumulative head?

**Sources.** File18 SP inclusive lifespan ledger; C756

**Inputs**

```json
{}
```

**Results**

```json
{
  "row_support": {
    "Methuselah": -1,
    "Lamech": 4
  },
  "boundary_differences": {
    "Adam": 3,
    "Seth": 3,
    "Enosh": 3,
    "Kenan": 3,
    "Mahalalel": 3,
    "Jared": 3,
    "Enoch": 3,
    "Methuselah": 3,
    "Lamech": 4
  },
  "head_actual_rounded": [
    13398,
    13401
  ]
}
```

**Finding.** SP rounding changes the cumulative head by+3, entirely from Methuselah−1 and Lamech+4 relative to MT residuals. The inclusive ledger remains the input type.

**Reassessment.** Determine cross-tradition interval families preserved by rounding.

## C758 — Classify cross-tradition interval preservation

**Question.** When does rounding preserve a difference between traditions?

**Sources.** C754–757

**Inputs**

```json
{}
```

**Results**

```json
{
  "comparison_classes": {
    "LXX_native_ON": {
      "-4": [
        "Noah",
        "Shem",
        "Arphaxad"
      ],
      "-2": [
        "Shelah"
      ],
      "0": [
        "Adam",
        "Seth",
        "Enosh",
        "Kenan",
        "Mahalalel",
        "Jared",
        "Enoch",
        "Methuselah",
        "Lamech",
        "Eber",
        "Peleg",
        "Reu",
        "Serug",
        "Nahor",
        "Terah",
        "Abraham",
        "Isaac",
        "Jacob",
        "Levi",
        "Kohath",
        "Amram",
        "Moses"
      ]
    },
    "SP_native_OFF": {
      "0": [
        "Noah",
        "Shem",
        "Arphaxad",
        "Shelah",
        "Eber",
        "Peleg",
        "Reu",
        "Serug",
        "Nahor",
        "Terah",
        "Abraham",
        "Isaac",
        "Jacob",
        "Levi",
        "Kohath",
        "Amram",
        "Moses"
      ],
      "3": [
        "Adam",
        "Seth",
        "Enosh",
        "Kenan",
        "Mahalalel",
        "Jared",
        "Enoch",
        "Methuselah"
      ],
      "4": [
        "Lamech"
      ]
    }
  },
  "criterion": "DeltaD_i=DeltaD_j iff the cross-tradition interval difference is unchanged by rounding"
}
```

**Finding.** Equal comparison-residual plateaus identify complete preserved interval families. Cross-tradition compatibility follows from two local field values, without a new list of isolated matches.

**Reassessment.** Test whether admitted Cainan insertion preserves these residual fields.

## C759 — Test Cainan residual neutrality

**Question.** Does insertion change existing rounding discrepancies?

**Sources.** File18 Cainan; C754–758

**Inputs**

```json
{
  "Cainan": {
    "b": 130,
    "r": 330,
    "L": 460
  }
}
```

**Results**

```json
{
  "results": {
    "MT_native_OFF": {
      "all_common_residuals_equal": true,
      "new_node_D": 3,
      "Shelah_D": 3
    },
    "LXX_native_ON": {
      "all_common_residuals_equal": true,
      "new_node_D": 1,
      "Shelah_D": 1
    },
    "SP_native_OFF": {
      "all_common_residuals_equal": true,
      "new_node_D": 3,
      "Shelah_D": 3
    }
  },
  "duration_residuals": [
    0,
    0,
    0
  ]
}
```

**Finding.** Cainan insertion changes admitted durations but preserves every corresponding residual. Its new boundary inherits the local plateau; zero row residual does not imply zero boundary displacement.

**Reassessment.** Prove rounding compatibility with admitted multiples-of-five changes.

## C760 — Prove rounding equivariance for admitted shifts

**Question.** Which state changes commute with componentwise rounding?

**Sources.** File51a variants; C752

**Inputs**

```json
{
  "admitted_amounts": [
    60,
    130,
    215,
    460
  ]
}
```

**Results**

```json
{
  "identity": "For k=5m, Q(n+k)=5 floor((n+5m+2)/5)=Q(n)+5m. Thus row residuals and corresponding boundary residuals persist under compatible admitted changes.",
  "checked_amounts": {
    "60": true,
    "130": true,
    "215": true,
    "460": true
  },
  "scope": "compatible coordinates or rows; insertion retains topology; finite source states only"
}
```

**Finding.** Rounding compatibility is an equivariance under multiples of five. It explains why several variants preserve residual profiles without making all chronology operations interchangeable.

**Reassessment.** Test counted-versus-completed input order where the shift is not a multiple of five.

## C761 — Resolve SP counting before rounding

**Question.** How does the local 53rd-year versus 52-completed distinction affect rounding?

**Sources.** File18 §3.1; File51a §17

**Inputs**

```json
{
  "counted": 53,
  "completed": 52
}
```

**Results**

```json
{
  "Q53": 55,
  "Q52": 50,
  "raw_separation": 1,
  "rounded_separation": 5,
  "round_then_subtract_one": 54
}
```

**Finding.** The raw one-unit distinction becomes a five-unit rounding distinction:55 versus50. Rounding then subtracting one gives54, a different construction. Counting convention must be resolved at the input.

**Reassessment.** Test the complete SP regular diagnostic while keeping its source status conditional.

## C762 — Test the conditional SP lower-block reconstruction

**Question.** Can one local input distinction generate the transmitted five-unit separation?

**Sources.** File51a §17; File18 SP begetting rows

**Inputs**

```json
{
  "terminal_Jacob": 2006,
  "Noah_additional_binding": "omitted in this strict begetting diagnostic"
}
```

**Results**

```json
{
  "diagnostics": {
    "counted53": {
      "raw": 4413,
      "rounded": 4411
    },
    "completed52": {
      "raw": 4412,
      "rounded": 4406
    }
  },
  "status": "conditional reconstruction, not established derivation of File51a blocks"
}
```

**Finding.** The strict branch calculation gives4411 versus4406, matching the transmitted and secondary blocks. The match identifies a possible mechanism; it does not promote this diagnostic to the primary SP path.

**Reassessment.** Separate terminal branch selection from row rounding in the SP cumulative head.

## C763 — Separate SP rounding from terminal branch selection

**Question.** Can the Strategy13406 head be reconstructed by explicit typed increments?

**Sources.** File51a §16.2 Aaron variant; Strategy cumulative SP/J rectangle; C757

**Inputs**

```json
{
  "SP_lower_actual": 13396,
  "Moses_center": 13398,
  "rounded_Moses": 13401,
  "rounded_Aaron_increment": 5
}
```

**Results**

```json
{
  "chain": [
    13396,
    13398,
    13401,
    13406
  ],
  "increments": [
    2,
    3,
    5
  ],
  "status": "conditional cross-tradition application of the source rounding/branch rules",
  "distinction": "rounded Aaron125-minus-Moses120 is not exact Actual Tishri phase+3.5"
}
```

**Finding.** The Strategy head13406 has a transparent conditional reconstruction: +2 endpoint selection, +3 SP row-rounding residual, +5 rounded Aaron branch. The arithmetic explains the supplied+10 without proving that derivation was intended.

**Reassessment.** Check the complete SP/J rectangle with this head and label its generated status.

## C764 — Attach the residual decomposition to the Strategy rectangle

**Question.** What part of the SP equal-gain rectangle is now explained?

**Sources.** Strategy §4.3; C763

**Inputs**

```json
{
  "regular_head": 4206,
  "cumulative_head": 13406,
  "origin": 1446,
  "target": 1406
}
```

**Results**

```json
{
  "native_spans": [
    2760,
    11960
  ],
  "expanded_spans": [
    2800,
    12000
  ],
  "gains": [
    40,
    40
  ],
  "preserved_gap": [
    9200,
    9200
  ],
  "new_explanation": "conditional cumulative-head provenance+2/+3/+5",
  "remaining_input": "the regular4206 branch and application status remain source-controlled"
}
```

**Finding.** The existing rectangle survives exactly. This cycle adds a row-based explanation for its cumulative-head offset, not a new independent rectangle or permission to tune other heads.

**Reassessment.** Test how lifespan splitting changes the whole accumulated comparison.

## C765 — Accumulate the splitting defect as a field

**Question.** What happens if rounded theoretical lifespans are accumulated instead of rounded whole lives?

**Sources.** C750–753; diagnostic operator comparison only

**Inputs**

```json
{}
```

**Results**

```json
{
  "fields": {
    "path": "model/splitting_defect_fields.json",
    "sha256": "4321928895655ea15e0338966a6568fec38a15d73b1a07847a5736d4d60f8590",
    "bytes": 1740
  },
  "head_defects": {
    "MT_native_OFF": -10,
    "LXX_native_ON": -15
  },
  "scope": "operator diagnostic over local biographies; not a licensed replacement cumulative chronology"
}
```

**Finding.** Parts-rounding and total-rounding differ over whole fields: MT head−10, LXX−15 on the common admitted biography domain. A constant endpoint correction cannot reproduce their internal jumps.

**Reassessment.** Derive how two rounding fields modify the regular–cumulative bridge.

## C766 — Join rounding to the mode bridge

**Question.** Does bridge rounding follow the difference of the two boundary residuals?

**Sources.** File51a §3.1 strict regular and §16.2 cumulative tables

**Inputs**

```json
{}
```

**Results**

```json
{
  "source_packet": {
    "path": "model/variant_inputs.json",
    "sha256": "637ee29c7bcb34fbdf313690535d3f2035837e6f77940bc0580e5a90295f8cf0",
    "bytes": 187357
  },
  "bridge": {
    "path": "model/MT_rounding_bridge.json",
    "sha256": "8b8c6c130a8aaf156453fe1b74b3ffa72d29a9529134efc1678b7f8234a8732f",
    "bytes": 2560
  },
  "common_nodes": 22,
  "Adam": {
    "name": "Adam",
    "G_actual": 9894,
    "G_rounded": 9900,
    "delta_G": 6,
    "D_C_minus_D_R": 6
  }
}
```

**Finding.** On the literal strict regular comparison, rounded G−actual G equals D_C−D_R at every common node. Rounding modifies the mode bridge through two fields, not one date offset.

**Reassessment.** Check the local bridge-generator identity from adjacent differences.

## C767 — Recover rounded bridge changes locally

**Question.** Do changes in adjacent bridge gaps equal row rounding differences?

**Sources.** C766; File51a strict comparison

**Inputs**

```json
{}
```

**Results**

```json
{
  "local_changes": [
    {
      "name": "Adam",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Seth",
      "field_difference": -2,
      "row_prediction": -2
    },
    {
      "name": "Enosh",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Kenan",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Mahalalel",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Jared",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Enoch",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Methuselah",
      "field_difference": 3,
      "row_prediction": 3
    },
    {
      "name": "Lamech",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Noah",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Shem",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Arphaxad",
      "field_difference": 2,
      "row_prediction": 2
    },
    {
      "name": "Shelah",
      "field_difference": 2,
      "row_prediction": 2
    },
    {
      "name": "Eber",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Peleg",
      "field_difference": 1,
      "row_prediction": 1
    },
    {
      "name": "Reu",
      "field_difference": 3,
      "row_prediction": 3
    },
    {
      "name": "Serug",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Nahor",
      "field_difference": 1,
      "row_prediction": 1
    },
    {
      "name": "Terah",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Abraham",
      "field_difference": 0,
      "row_prediction": 0
    },
    {
      "name": "Isaac",
      "field_difference": 0,
      "row_prediction": 0
    }
  ],
  "scope": "21 consecutive strict regular edges Adam→Jacob; Jacob/Joseph collateral is not continued to Levi"
}
```

**Finding.** The rounded bridge itself is generated locally by e(L)−e(b). This joins source rows, mode choice, and rounding in one recurrence.

**Reassessment.** Test direct rounding of date labels against the source-derived field.

## C768 — Compare row rounding with date-grid rounding

**Question.** Can rounding already calculated dates reproduce the source chronology?

**Sources.** File51a §16.2; C754–757

**Inputs**

```json
{
  "grid_anchor": 1406
}
```

**Results**

```json
{
  "diagnostic": {
    "path": "model/date_grid_diagnostic.json",
    "sha256": "ac040f2ee4e2718dc6af24e48a39d46edbff3525e3a215e0f5ab2e1a1b8bd876",
    "bytes": 2216
  },
  "mismatch_counts": {
    "MT_native_OFF": 11,
    "LXX_native_ON": 7,
    "SP_native_OFF": 17
  },
  "MT_examples": {
    "Noah": {
      "grid": 6381,
      "row": 6386
    },
    "Abraham": {
      "grid": 2436,
      "row": 2431
    }
  }
}
```

**Finding.** A nearest1/6 date-grid projection fails at11 of26 MT boundaries. Rounded chronology depends on the source path, not just the final Actual date.

**Reassessment.** Test subdivision dependence over an informative complete source subpath.

## C769 — Measure the effect of retained subdivisions

**Question.** Does whole-chain cancellation justify rounding arbitrary subpath totals?

**Sources.** File51a cumulative Enoch and Terah boundaries

**Inputs**

```json
{}
```

**Results**

```json
{
  "subpath": "Enoch→Terah",
  "actual_span": 5852,
  "row_rounded_span": 5860,
  "once_rounded_total": 5850,
  "defect": 10,
  "field_span_change": 8
}
```

**Finding.** The source subpath5852 becomes5860 by row rounding but5850 by rounding its total. Retained subdivision is part of the construction, even when the full12600 total cancels.

**Reassessment.** State the minimal recoverable object linking Actual and Rounded.

## C770 — Identify a recoverable shared source object

**Question.** What must accompany Rounded to preserve the original family?

**Sources.** C754–769

**Inputs**

```json
{}
```

**Results**

```json
{
  "reconstruction": "Actual boundary = Rounded boundary − D; actual row = rounded row − e",
  "affine_obstruction": {
    "fixed_dates": [
      1406,
      1526
    ],
    "moved_date": [
      1663,
      1661
    ]
  },
  "retained_data": [
    "ordered source rows",
    "typed path and terminal",
    "mode selector",
    "rounding residual field"
  ]
}
```

**Finding.** Rounded plus its residual field recovers Actual exactly. Rounded alone loses information; no one affine date map works, since two fixed dates force identity while Amram moves.

**Reassessment.** Integrate the 40-step checkpoint and move to finite variant configurations.

## C771 — Forty-step synthesis checkpoint

**Question.** Which common mechanism now explains the Rounded family?

**Sources.** C732–770

**Inputs**

```json
{}
```

**Results**

```json
{
  "checkpoint": {
    "path": "deliverables/490d_Chronological_Families_Checkpoint_C771.md",
    "sha256": "4c032c5d12943f45cc600e2db12f4d16ad34f83fea9776731e994d301a974820",
    "bytes": 1493
  }
}
```

**Finding.** The general object is an ordered source path with mode-specific row measurements and a recoverable rounding field. Finite variants are the next discriminating test.

**Reassessment.** Reconstruct the finite variant masks against literal source examples.