# 490d — C632–C651: 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.

## C632 — Freeze the calendar-half source family

**Question.** What complete source packet is needed before testing File70’s three calendar bridges?

**Sources.** File70 §§6.1–6.7; source state register

**Inputs**

```json
{
  "restored_Creation_rail": [
    4251,
    4244,
    4237
  ],
  "standard_Creation_rail": [
    4121,
    4114,
    4107
  ],
  "exposure_nodes": [
    4238,
    4108
  ],
  "Joseph_birth_descent_elevation": [
    1915,
    1898,
    1885
  ],
  "Joseph_rail": [
    1885,
    1878,
    1871
  ],
  "regular_Cainan_difference": 130
}
```

**Results**

```json
{
  "packet": {
    "path": "model/calendar_half_inputs.json",
    "sha256": "c3794e48f499ce37b49dfc94f7ead80771e50a487e4331f9c599e8e537886578",
    "bytes": 975
  },
  "primary_registers": {
    "Prophetic": [
      4238,
      1898,
      13,
      180
    ],
    "Priestly": [
      4237,
      1885,
      14,
      168
    ],
    "Enochian": [
      4251,
      1885,
      13,
      182
    ]
  }
}
```

**Finding.** The complete packet fixes two Creation states, Joseph’s corresponding nodes, and the three13/14/13 half-year counts before calculation. The test can now explain the family from one labelled geometry.

**Reassessment.** Generate the corresponding-node translation of the two seven-plus-seven rails.

## C633 — Generate the corresponding Creation–Joseph rail map

**Question.** Do the two source seven-plus-seven rails require more than one translation?

**Sources.** File70 §6.3

**Inputs**

```json
{
  "Creation": [
    4251,
    4244,
    4237
  ],
  "Joseph": [
    1885,
    1878,
    1871
  ]
}
```

**Results**

```json
{
  "corresponding_spans": [
    2366,
    2366,
    2366
  ],
  "rail_gaps": [
    [
      7,
      7
    ],
    [
      7,
      7
    ]
  ],
  "translation": 2366
}
```

**Finding.** One translation of2366 carries all three source nodes and preserves the7+7 partition. The three matches are dependent coordinates of one rigid relation.

**Reassessment.** Generate every cross-rail span to distinguish the special diagonals from the whole family.

## C634 — Generate the full cross-rail matrix

**Question.** Does one indexed formula account for all nine span comparisons?

**Sources.** File70 §§6.3,6.5; derived completion of source geometry

**Inputs**

```json
{
  "head_difference": 2366,
  "rail_step": 7,
  "indices": [
    0,
    1,
    2
  ]
}
```

**Results**

```json
{
  "span_matrix": [
    [
      2366,
      2373,
      2380
    ],
    [
      2359,
      2366,
      2373
    ],
    [
      2352,
      2359,
      2366
    ]
  ],
  "multiplicities": {
    "2352": 1,
    "2359": 2,
    "2366": 3,
    "2373": 2,
    "2380": 1
  },
  "formula": "D(i,j)=2366+7(j−i)",
  "source_selected_extremes": [
    2352,
    2366,
    2380
  ]
}
```

**Finding.** The complete matrix has five span values with multiplicities1,2,3,2,1. Minimum, corresponding and maximum spans are2366−14,2366,2366+14; the other entries are generated comparisons, not additional source appointments.

**Reassessment.** Explain the separate exposure span using its two thirteen-year flanks.

## C635 — Reconstruct the exposure bridge from its flanks

**Question.** How does the13+2340+13 path relate to the corresponding2366 translation?

**Sources.** File70 §§6.1,6.4

**Inputs**

```json
{
  "path": [
    4251,
    4238,
    1898,
    1885
  ]
}
```

**Results**

```json
{
  "partition": [
    13,
    2340,
    13
  ],
  "total": 2366,
  "calendar_difference": 26,
  "Joseph_ages": [
    17,
    30
  ]
}
```

**Finding.** The source path supplies13-year flanks around2340, giving2366. The26 added years are exactly6.5 copies of the360-to364 calendar difference. Joseph’s ages17 and30 provide the same13-year lower flank.

**Reassessment.** Compare all three calendar spans under their own calibrated Keys.

## C636 — Normalize the three calendar registers

**Question.** Do the three spans become one common output under their respective Keys?

**Sources.** File70 §6.7; C586 calendar calibration

**Inputs**

```json
{
  "spans": [
    2340,
    2352,
    2366
  ],
  "days": [
    360,
    336,
    364
  ],
  "Keys": [
    "70/69",
    "25/23",
    "300/299"
  ]
}
```

**Results**

```json
{
  "normalized_year_counts": [
    "13/2",
    7,
    "13/2"
  ],
  "Key_outputs": [
    "54600/23",
    "58800/23",
    "54600/23"
  ],
  "output_ratio": "13:14:13",
  "half_year_unit": "4200/23"
}
```

**Finding.** The Prophetic and Enochian arms normalize to6.5 calendar years; the Priestly arm to7. Their outputs are13,14,13 copies of one calibrated half-year. The family preserves two count classes, not one undifferentiated result.

**Reassessment.** Determine what equal transformed spans imply when their held heads differ.

## C637 — Carry normalized equality into anchored placement

**Question.** Do equal Prophetic and Enochian outputs imply an identical dated endpoint here?

**Sources.** File70 source endpoints; derived anchored-Key diagnostic

**Inputs**

```json
{
  "Prophetic_head": 4238,
  "Enochian_head": 4251,
  "common_output": "54600/23"
}
```

**Results**

```json
{
  "Prophetic_generated_endpoint": "42874/23",
  "Enochian_generated_endpoint": "43173/23",
  "endpoint_difference": 13,
  "frame_adjusted_Enochian_endpoint": "42874/23"
}
```

**Finding.** The generated endpoints differ13 because the held heads differ13. Translating the Enochian output frame by−13 aligns them. Equal normalized spans explain the shape; source anchor differences explain its placement. These fractional outputs are diagnostics, not adopted event dates.

**Reassessment.** Derive why the Priestly minimum has fourteen half-year units.

## C638 — Explain the Priestly minimum by factor structure

**Question.** Why does subtracting the14-year rail width from13×182 produce14×168?

**Sources.** File70 §§6.2–6.5

**Inputs**

```json
{
  "corresponding_count": 13,
  "Enochian_half": 182,
  "rail_width": 14
}
```

**Results**

```json
{
  "corresponding_factorization": [
    14,
    169
  ],
  "minimum_factorization": [
    14,
    168
  ],
  "Priestly_half": 168,
  "identity": "13×182−14=14(169−1)=14×168"
}
```

**Finding.** The Priestly minimum follows from182=14×13 and168=13²−1. The common14-year width turns the corresponding vector into fourteen Priestly halves. This explains the change from13 to14 calendar units through the rail geometry.

**Reassessment.** Test whether the combined half-calendar conditions select a unique count or a congruence family.

## C639 — Solve the two-calendar compatibility equation

**Question.** What count family permits an Enochian corresponding span and a Priestly minimum fourteen years shorter?

**Sources.** Derived from File70 §6 source geometry

**Inputs**

```json
{
  "equation": "182m−14=168n",
  "positive_integer_counts": true
}
```

**Results**

```json
{
  "reduced_equation": "13m−12n=1",
  "general_solution": "m=1+12q; n=1+13q",
  "examples_m_n": [
    [
      1,
      1
    ],
    [
      13,
      14
    ],
    [
      25,
      27
    ],
    [
      37,
      40
    ]
  ],
  "source_choice_q": 1
}
```

**Finding.** The pair of calendar requirements has an infinite integer family. Its source member is m=13,n=14; the smaller m=n=1 also fits the two-calendar equation. The exposure condition must supply any stronger selection.

**Reassessment.** Add the fixed thirteen-year exposure flanks and solve the complete three-calendar system.

## C640 — Solve the complete three-calendar congruence family

**Question.** Do the fixed fourteen-year rail and thirteen-year flanks select the source count triple as the smallest positive solution?

**Sources.** File70 §6 geometry; derived integer compatibility theorem

**Inputs**

```json
{
  "conditions": [
    "T=182m",
    "T−14=168n",
    "T−26=180p"
  ],
  "counts": "positive integers"
}
```

**Results**

```json
{
  "m_residue_mod180": [
    13
  ],
  "general_T": "2366+32760q",
  "general_counts": "m=13+180q; n=14+195q; p=13+182q",
  "examples": [
    {
      "q": 0,
      "T": 2366,
      "m": 13,
      "n": 14,
      "p": 13
    },
    {
      "q": 1,
      "T": 35126,
      "m": 193,
      "n": 209,
      "p": 195
    },
    {
      "q": 2,
      "T": 67886,
      "m": 373,
      "n": 404,
      "p": 377
    }
  ],
  "least_positive_solution": [
    2366,
    13,
    14,
    13
  ]
}
```

**Finding.** With the source offsets fixed, the full system has T=2366+32760q. Its least positive solution is exactly the source2366 and13/14/13 half-count triple. This is a conditional local minimality result, not a claim of a globally minimal chronology.

**Reassessment.** Relate the congruence period to the calendar lattice, then test the source no-Cainan companion.

## C641 — Identify the calendar compatibility period

**Question.** Why is the full congruence period32760, and how does it meet the earlier common-volume family?

**Sources.** C640; File12 calendar definitions; C575

**Inputs**

```json
{
  "half_calendars": [
    168,
    180,
    182
  ],
  "full_calendars": [
    336,
    360,
    364
  ]
}
```

**Results**

```json
{
  "half_calendar_lcm": 32760,
  "full_calendar_lcm": 65520,
  "previous_volume_coefficient": 327600,
  "ratios": [
    2,
    10
  ]
}
```

**Finding.** The period32760 is the least common multiple of the three half-calendars. Doubling gives65520 for full calendars; the earlier327600 common-volume coefficient is ten half-periods. The numerical link comes from the same calendar denominators, with units retained.

**Reassessment.** Execute the no-Cainan rail companion and locate which calendar divisibilities it preserves.

## C642 — Reconstruct the no-Cainan companion

**Question.** Which relations survive when the Creation rail alone moves130?

**Sources.** File70 §§6.1,6.6

**Inputs**

```json
{
  "standard_Creation": [
    4121,
    4114,
    4107
  ],
  "Joseph": [
    1885,
    1878,
    1871
  ],
  "standard_exposure": 4108
}
```

**Results**

```json
{
  "span_family": [
    2210,
    2222,
    2236,
    2250
  ],
  "corresponding_spans": [
    2236,
    2236,
    2236
  ],
  "calendar_half_counts": [
    "221/18",
    "1111/84",
    "86/7"
  ],
  "shift_from_restored": 130
}
```

**Finding.** Removing130 preserves both7+7 rails and all their relative diagonals, giving2236 as the corresponding translation. It changes the calendar counts to nonintegers, so the restored state’s13/14/13 specialization is not automatic from rail shape alone.

**Reassessment.** Ask whether the fixed standard state and three-calendar conditions select130 as the least nonnegative correction.

## C643 — Solve the source-state correction congruence

**Question.** Given the standard2236 vector, what correction first enters the complete calendar-compatible class?

**Sources.** C640,C642; File70 regular Cainan comparison

**Inputs**

```json
{
  "standard_vector": 2236,
  "compatible_class": "2366+32760q",
  "permitted_test": "nonnegative integer correction to Creation rail only"
}
```

**Results**

```json
{
  "correction_class": "130+32760q",
  "least_nonnegative": 130,
  "examples": [
    130,
    32890,
    65650
  ],
  "source_regular_Cainan": 130
}
```

**Finding.** With the standard vector and the14/26 offsets fixed, the least nonnegative correction satisfying all three calendar conditions is130. This recovers the supplied regular-Cainan increment conditionally; it does not infer an insertion history or change the frozen MT state.

**Reassessment.** Use the supplied Enoch close to reconstruct the Priestly minimum’s internal partition.

## C644 — Recover the Enoch partition of the Priestly minimum

**Question.** What additional placement does the appendix’s supplied Enoch close contribute?

**Sources.** File70 Appendix D.2–D.4

**Inputs**

```json
{
  "path": [
    4237,
    3257,
    1885
  ],
  "Enoch_status": "pre-existing secondary source comparison; no Gear transport"
}
```

**Results**

```json
{
  "partition": [
    980,
    1372
  ],
  "jubilee49_coefficients": [
    20,
    28
  ],
  "primitive196_coefficients": [
    5,
    7
  ],
  "whole": 2352
}
```

**Finding.** The supplied Enoch close partitions2352 as980+1372, or20+28 jubilee49 units. The reduced ratio is5:7. The total follows from the rail geometry; this particular internal cut depends on the separately supplied3257 role.

**Reassessment.** Test the corresponding Enoch head to identify which part of the partition is role-specific.

## C645 — Test the Enoch partition role

**Question.** Does the supplied Enoch head give the same internal partition as its close?

**Sources.** File70 Appendix D.1–D.4; C644

**Inputs**

```json
{
  "head": 3264,
  "close": 3257,
  "outer": [
    4237,
    1885
  ]
}
```

**Results**

```json
{
  "head_partition": [
    973,
    1379
  ],
  "close_partition": [
    980,
    1372
  ],
  "head_49_coefficients": [
    "139/7",
    "197/7"
  ]
}
```

**Finding.** Replacing the supplied close with the head transfers seven years between the two arms and preserves2352. The5:7 partition belongs to the close role; it is not a property of any Enoch placement.

**Reassessment.** Recover the complete two-node Enoch comparison from its seven-year rails.

## C646 — Generate the Creation–Enoch rail square

**Question.** Can one translation recover all four admitted Creation–Enoch spans?

**Sources.** File70 Appendix D.1

**Inputs**

```json
{
  "upper": [
    4244,
    4237
  ],
  "lower": [
    3264,
    3257
  ]
}
```

**Results**

```json
{
  "matrix": [
    [
      980,
      987
    ],
    [
      973,
      980
    ]
  ],
  "translation": 980,
  "rail": 7
}
```

**Finding.** The four comparisons are one seven-year rail translated by980: corresponding980s and crossed973/987. This reuses the same rail generator as the larger calendar family.

**Reassessment.** Ask whether the source’s nested14/140/980 spans share one normalized shape.

## C647 — Normalize the nested bilateral spans

**Question.** What structure is shared by the source’s14,140 and980 layers?

**Sources.** File70 Appendix D.5

**Inputs**

```json
{
  "widths": [
    14,
    140,
    980
  ],
  "halves": [
    7,
    70,
    490
  ]
}
```

**Results**

```json
{
  "normalized_halves": [
    [
      1,
      1
    ],
    [
      1,
      1
    ],
    [
      1,
      1
    ]
  ],
  "scale_from14": [
    1,
    10,
    70
  ]
}
```

**Finding.** The three layers share the bilateral1|1 shape at scales1,10,70. This establishes nested geometric scale, without creating an appointment at an unsupplied midpoint.

**Reassessment.** Test the more informative five-node Creation–Terah refinement rather than stopping at bilateral similarity.

## C648 — Recover the five-node Creation refinement

**Question.** Can one small template reproduce both standard and restored Creation refinements?

**Sources.** File70 §§2.3,3.2; A.3

**Inputs**

```json
{
  "standard": [
    4121,
    4115,
    4114,
    4108,
    4107
  ],
  "restored": [
    4251,
    4245,
    4244,
    4238,
    4237
  ],
  "template": [
    7,
    1,
    0,
    -6,
    -7
  ]
}
```

**Results**

```json
{
  "generated": [
    [
      4121,
      4115,
      4114,
      4108,
      4107
    ],
    [
      4251,
      4245,
      4244,
      4238,
      4237
    ]
  ],
  "gaps": [
    6,
    1,
    6,
    1
  ],
  "component_translation": [
    130,
    130,
    130,
    130,
    130
  ]
}
```

**Finding.** Both Creation states are the same five-position template centered at4114 or4244; regular Cainan changes every coordinate by130 and leaves the6|1|6|1 refinement intact.

**Reassessment.** Test the complete Terah refinement against this template.

## C649 — Map Creation’s entire refinement to Terah

**Question.** Does the tenfold scale preserve all four subdivisions and five roles?

**Sources.** File70 §§2.3,3.2,4.2; A.2–A.3

**Inputs**

```json
{
  "creation": [
    4121,
    4115,
    4114,
    4108,
    4107
  ],
  "terah": [
    2296,
    2236,
    2226,
    2166,
    2156
  ],
  "centers": [
    4114,
    2226
  ]
}
```

**Results**

```json
{
  "mapped": [
    2296,
    2236,
    2226,
    2166,
    2156
  ],
  "terah_gaps": [
    60,
    10,
    60,
    10
  ],
  "normalized_roles": [
    7,
    1,
    0,
    -6,
    -7
  ]
}
```

**Finding.** One affine map carries the complete Creation6|1|6|1 refinement into Terah60|10|60|10. The correspondence is between structural positions; the source’s biographical and comparison roles remain distinct.

**Reassessment.** Determine which broader three-node families are projections of this refined template.

## C650 — Locate the common coarse rail

**Question.** Which supplied triples share the same bilateral projection?

**Sources.** File70 §§3–6; A.3,A.5,A.6

**Inputs**

```json
{
  "triples": [
    [
      4121,
      4114,
      4107
    ],
    [
      4251,
      4244,
      4237
    ],
    [
      2296,
      2226,
      2156
    ],
    [
      1929,
      1922,
      1915
    ],
    [
      1885,
      1878,
      1871
    ]
  ],
  "centers": [
    4114,
    4244,
    2226,
    1922,
    1878
  ],
  "steps": [
    7,
    7,
    70,
    7,
    7
  ]
}
```

**Results**

```json
{
  "normalized": [
    [
      1,
      0,
      -1
    ],
    [
      1,
      0,
      -1
    ],
    [
      1,
      0,
      -1
    ],
    [
      1,
      0,
      -1
    ],
    [
      1,
      0,
      -1
    ]
  ],
  "refinement_projection": [
    0,
    2,
    4
  ]
}
```

**Finding.** All five supplied triples normalize to1|0|-1. Creation and Terah additionally share the five-node refinement; Jacob and Joseph establish the coarse rail only. A shared outer shape does not supply missing inner events.

**Reassessment.** Consolidate the first20 steps into an explanation of geometry versus selected calendar placement.

## C651 — Consolidate the rail and refinement families

**Question.** What is the shortest supported explanation after the first20 steps?

**Sources.** C632–C650; Research Strategy

**Inputs**

```json
{
  "families": [
    "calendar rails",
    "five-position Creation–Terah refinement",
    "coarse Jacob/Joseph rails"
  ],
  "scope": "source-conditioned geometry"
}
```

**Results**

```json
{
  "artifact": {
    "path": "deliverables/C651_Interim_Synthesis.md",
    "sha256": "ff98f9808fb24b7648ba2eeb7f8e0a1a23508c84f34e2dae154bb7f095f750ab",
    "bytes": 1119
  },
  "completed_range": [
    632,
    651
  ],
  "core_rules": [
    "Dij=T+7(j−i)",
    "R(c,u)=c+u(7,1,0,−6,−7)",
    "T=2366+32760q"
  ]
}
```

**Finding.** Twenty sequential steps now explain two whole-family generators and distinguish their inherited shapes from the source-selected placements. Further scalar boundary searches are lower priority than the prepared reflection and cumulative families.

**Reassessment.** Reconstruct the source’s Jacob-centered reflection as a complete event-role object.