# 490d — C832–C851: sequential research record

Working continuation of C831 · 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.

## C832 — Select complete-family transfer tests

**Question.** Can the established path grammar explain new source-controlled families without fitting their endpoints?

**Sources.** Strategy stagesD–H; C831 synthesis; local File58

**Inputs**

```json
{
  "first_family": "File58 complete ledger matrices and declared partitions",
  "other_candidates": [
    "File43/54 full Luke display",
    "File52c complete paired-anchor field",
    "File63 ordered carrier network"
  ],
  "predecessor": "C831"
}
```

**Results**

```json
{
  "selection": {
    "path": "model/RESEARCH_SELECTION.json",
    "sha256": "57658fdf754e76e533371e0183608c0c10b33a58131b0dcfd0c92e85dfdc1709",
    "bytes": 285
  },
  "source_snapshots": 41,
  "primary": "a5ea84562101158b60d0cf296765d6eff38e7a2abda4e74ad1b353dfd13b9530"
}
```

**Finding.** The next test moves from genealogy to a bounded list with multiple source-defined measurements. Complete-family transformations, not additional endpoint searches, will guide the cycle.

**Reassessment.** Freeze File58’s complete ledger and partition data before calculating its consequences.

## C833 — Freeze the complete File58 list objects

**Question.** Which literal tables and declared orders are the inputs to the holdout?

**Sources.** File58 §§3–5,7,13

**Inputs**

```json
{}
```

**Results**

```json
{
  "packet": {
    "path": "model/list_inputs.json",
    "sha256": "9884a95572290132effce69c3a16e362d9c298c9a9b4c6bded06545f449dd2f1",
    "bytes": 14983
  },
  "literal_table_rows": {
    "category": 8,
    "leviticus": 3,
    "feast_scale": 6,
    "full_scale": 7,
    "forward": 8,
    "reverse": 8,
    "esau": 9,
    "clean_walk": 6
  }
}
```

**Finding.** The complete category, sequence and partition tables are frozen with source line locators. Subsequent tests can distinguish source inputs from generated consequences.

**Reassessment.** Reconstruct the whole seven-day Sukkot count matrix.

## C834 — Reconstruct a complete Sukkot species-by-day matrix

**Question.** Can the seven day totals be generated by one declining component and three constant components?

**Sources.** File58 §§3.1–3.2,7.1

**Inputs**

```json
{
  "uniform_daily_rams_lambs_goats": [
    2,
    14,
    1
  ],
  "status": "declared allocation reconstruction; marginal totals alone do not imply uniformity"
}
```

**Results**

```json
{
  "matrix": {
    "path": "model/Sukkot_matrix.json",
    "sha256": "2dd0f49ee2c98a58d388b4eb25ad26956588342e8058e1f8517abb1f9e4204a4",
    "bytes": 642
  },
  "species": [
    "bulls",
    "rams",
    "lambs",
    "goats"
  ],
  "day_rows": [
    [
      13,
      2,
      14,
      1
    ],
    [
      12,
      2,
      14,
      1
    ],
    [
      11,
      2,
      14,
      1
    ],
    [
      10,
      2,
      14,
      1
    ],
    [
      9,
      2,
      14,
      1
    ],
    [
      8,
      2,
      14,
      1
    ],
    [
      7,
      2,
      14,
      1
    ]
  ],
  "day_totals": [
    30,
    29,
    28,
    27,
    26,
    25,
    24
  ],
  "column_totals": [
    70,
    14,
    98,
    7
  ],
  "rank": 2,
  "generator": "row_d=(14−d,2,14,1), d=1…7"
}
```

**Finding.** The declared daily allocation has a rank-two generator: only the bull component changes. It recovers all seven totals and the complete189 species ledger. The uniform component allocation remains an input, not a deduction from margins alone.

**Reassessment.** Recover the whole Tishri category/species table as a linear measurement object.

## C835 — Recover both complete Tishri margins

**Question.** Do event-category and species totals measure the same frozen count matrix?

**Sources.** File58 §3.1 literal category table

**Inputs**

```json
{}
```

**Results**

```json
{
  "matrix": {
    "path": "model/Tishri_count_matrix.json",
    "sha256": "ccbb58ccd547dbe8e826eba227e09d3454451f998f4531f34677c06ee73375c2",
    "bytes": 665
  },
  "category_totals": [
    44,
    6,
    11,
    10,
    10,
    189,
    10
  ],
  "species_totals": [
    75,
    18,
    176,
    11
  ],
  "grand_total": 280
}
```

**Finding.** The two280 totals are different marginals of one seven-by-four matrix. Their agreement is forced by the shared ledger, while the chosen categories and entries remain source inputs.

**Reassessment.** Determine what the margins preserve and what they cannot recover.

## C836 — Test whether the margins determine the ledger

**Question.** Can preserved category/species totals recover every internal cell?

**Sources.** C835; diagnostic only

**Inputs**

```json
{}
```

**Results**

```json
{
  "diagnostic_changed_rows": {
    "NewMoon": [
      3,
      1,
      6,
      1
    ],
    "Trumpets": [
      0,
      1,
      8,
      1
    ]
  },
  "kernel_move": "a2×2 +1,−1,−1,+1 cell circulation",
  "source_status": "not an admitted offering variant"
}
```

**Finding.** Both margins survive a nonzero cell circulation. Totals compress the ledger but do not reconstruct its source cells; the complete table supplies information beyond280.

**Reassessment.** Evaluate the source-admitted calendar/corpus overlay states.

## C837 — Generate the finite calendar/corpus comparison matrix

**Question.** Do independent declared additions reconstruct all four source states?

**Sources.** File58 §4.3

**Inputs**

```json
{}
```

**Results**

```json
{
  "states": {
    "path": "model/Tishri_overlay_states.json",
    "sha256": "6b9e2c3bfcfaf70af42bf0540baf3553326d37c0998a9450fee912eadec5ac0d",
    "bytes": 387
  },
  "00": {
    "species": [
      75,
      18,
      176,
      11
    ],
    "total": 280
  },
  "01": {
    "species": [
      76,
      20,
      176,
      13
    ],
    "total": 285
  },
  "10": {
    "species": [
      75,
      18,
      178,
      11
    ],
    "total": 282
  },
  "11": {
    "species": [
      76,
      20,
      178,
      13
    ],
    "total": 287
  },
  "controls": "00 controls; fourth Sabbath and Leviticus16 are separate comparison choices"
}
```

**Finding.** The280/282/285/287 matrix is a two-choice additive configuration with independently stated supports. Calendar placement adds lambs; corpus inclusion adds a different animal vector.

**Reassessment.** Check the separate slaughtered subset and scaled overlay response.

## C838 — Separate ledger selection from scale

**Question.** How do ritual disposition and shemitah scale interact with the finite overlay?

**Sources.** File58 §§4.3,5

**Inputs**

```json
{}
```

**Results**

```json
{
  "complete_rite": 5,
  "slaughtered_subset": 4,
  "scaled_states": {
    "00": 1960,
    "01": 1995,
    "10": 1974,
    "11": 2009
  },
  "scaled_calendar_increment": 14,
  "scaled_complete_rite_increment": 35,
  "derived_slaughtered_totals": [
    284,
    286
  ]
}
```

**Finding.** Selecting displayed ritual animals or slaughtered animals changes the measured object by one goat. Scaling then carries that difference to seven; it cannot be suppressed by retaining the same ledger name.

**Reassessment.** Regenerate every forward festival boundary from its day generator.

## C839 — Regenerate the full forward festival path

**Question.** Does one day generator recover every supplied cumulative and chronological row?

**Sources.** File58 §7.1

**Inputs**

```json
{
  "anchor": 1859,
  "scale": 7
}
```

**Results**

```json
{
  "path": {
    "path": "model/festival_forward_path.json",
    "sha256": "b92c95b5b92dd58e80d9a6bbba6b1b8e0bae4b0de0da292d19d6bb68e552a67a",
    "bytes": 340
  },
  "prefix_counts": [
    0,
    30,
    59,
    87,
    114,
    140,
    165,
    189,
    199
  ],
  "BC": [
    1859,
    1649,
    1446,
    1250,
    1061,
    879,
    704,
    536,
    466
  ]
}
```

**Finding.** The complete forward path follows the descending day total and one appended Eighth-Day10. All eight source boundaries are forced once the source order, scale7, and1859 anchor are fixed.

**Reassessment.** Generate the complete declared reverse order through the same accumulation rule.

## C840 — Regenerate the complete reverse-order festival path

**Question.** What relation joins the two declared orders at every boundary?

**Sources.** File58 §7.2; C839

**Inputs**

```json
{}
```

**Results**

```json
{
  "path": {
    "path": "model/festival_reverse_path.json",
    "sha256": "2a0416f26a90c2385b0ac39c1ea69d227783bc5d0804feadd08722dc4b6e0cbc",
    "bytes": 287
  },
  "prefix_counts": [
    0,
    10,
    34,
    59,
    85,
    112,
    140,
    169,
    199
  ],
  "BC": [
    1859,
    1789,
    1621,
    1446,
    1264,
    1075,
    879,
    676,
    466
  ],
  "paired_formula": "reverse_prefix(k)=199−forward_prefix(8−k)",
  "coordinate_pair_sum": 2325
}
```

**Finding.** Every reverse boundary is the complement of a corresponding forward boundary. This is a sequence-reversal relation on two paths, not a newly licensed formal Mirror operation.

**Reassessment.** Identify exactly where the two source orders share boundaries.

## C841 — Resolve all shared boundaries of the two festival orders

**Question.** Which overlaps follow from the complete prefix sets?

**Sources.** C839–840

**Inputs**

```json
{}
```

**Results**

```json
{
  "shared_boundaries": [
    {
      "count": 0,
      "years": 0,
      "BC": 1859,
      "forward_step": 0,
      "reverse_step": 0
    },
    {
      "count": 59,
      "years": 413,
      "BC": 1446,
      "forward_step": 2,
      "reverse_step": 3
    },
    {
      "count": 140,
      "years": 980,
      "BC": 879,
      "forward_step": 5,
      "reverse_step": 6
    },
    {
      "count": 199,
      "years": 1393,
      "BC": 466,
      "forward_step": 8,
      "reverse_step": 8
    }
  ],
  "status": "complete dependent comparison of the two declared orders; no alternative-order search"
}
```

**Finding.** The two paths share only counts0,59,140,199, producing1859,1446,879,466. Their common internal boundaries occur at different step indices, so endpoint agreement does not identify the ordered paths.

**Reassessment.** Test scaling and order as separate linear operations.

## C842 — Separate scale, order and accumulation

**Question.** Which operations commute, and which change the internal chronological field?

**Sources.** File58 declared order and×7; C839–841

**Inputs**

```json
{}
```

**Results**

```json
{
  "scale_order_commutes": true,
  "scale_accumulation_commutes": true,
  "forward_prefix_years": [
    210,
    413,
    609,
    798,
    980,
    1155,
    1323,
    1393
  ],
  "reverse_prefix_years": [
    70,
    238,
    413,
    595,
    784,
    980,
    1183,
    1393
  ]
}
```

**Finding.** Scaling commutes with both declared reordering and accumulation. Reordering preserves the total but changes the intermediate prefix field; that distinction is the same one needed for genealogical and Rounded paths.

**Reassessment.** Construct the complete sex-by-purity measurement table for Esau’s ledger.

## C843 — Build the complete Esau joint-category table

**Question.** What cell structure lies beneath the equal490/60 marginals?

**Sources.** File58 §13.1 nine printed rows

**Inputs**

```json
{}
```

**Results**

```json
{
  "joint_table": {
    "path": "model/Esau_joint_measurements.json",
    "sha256": "74bd41362ddc19e287e21419635e918f8fa091f10f6f059baf43cab8a13a8fbf",
    "bytes": 1169
  },
  "joint_clean_unclean_by_female_male": [
    [
      440,
      50
    ],
    [
      50,
      10
    ]
  ],
  "purity_totals": [
    490,
    60
  ],
  "sex_totals": [
    490,
    60
  ]
}
```

**Finding.** The complete joint table is[[440,50],[50,10]]. Equal490/60 marginals arise because the two off-diagonal cells agree, while purity and sex remain different classifications.

**Reassessment.** Identify the exact constraint and freedom behind that equality.

## C844 — Derive the equal-marginal constraint

**Question.** What exactly must be true for purity and sex totals to coincide?

**Sources.** C843

**Inputs**

```json
{}
```

**Results**

```json
{
  "clean_minus_female": 0,
  "unclean_minus_male": 0,
  "fixed_margin_family": "[[490−t,t],[t,60−t]], 0≤t≤60",
  "source_t": 50,
  "constraint_count": 1,
  "independence_determinant": 1900
}
```

**Finding.** The double490/60 display imposes one cross-category balance, b=c. Given those margins, one cell parameter remains; the source selects50. Distinct category definitions and equal marginals do not establish statistical independence.

**Reassessment.** Recover the exact rank of the nine-row measurement map.

## C845 — Measure the information retained by the category projections

**Question.** How many independent totals do the four marginals supply?

**Sources.** File58 nine-row ledger; C843–844

**Inputs**

```json
{}
```

**Results**

```json
{
  "incidence_matrix": [
    [
      1,
      1,
      1,
      1,
      1,
      1,
      0,
      0,
      0
    ],
    [
      0,
      0,
      0,
      0,
      0,
      0,
      1,
      1,
      1
    ],
    [
      1,
      0,
      1,
      0,
      1,
      0,
      1,
      1,
      0
    ],
    [
      0,
      1,
      0,
      1,
      0,
      1,
      0,
      0,
      1
    ]
  ],
  "rank": 3,
  "kernel_dimension_on_unrestricted_nine_counts": 6,
  "row_relation": "clean+unclean=female+male"
}
```

**Finding.** The four totals contain three independent linear measurements of nine source counts. Six numerical directions are invisible to these marginals; species, order and individual counts remain necessary source data.

**Reassessment.** Regenerate the complete source-admitted clean category-order walk.

## C846 — Regenerate the complete clean category-order walk

**Question.** Do declared category keys determine every source landing without selecting targets?

**Sources.** File58 §13.3

**Inputs**

```json
{
  "anchor": 1896,
  "sort_keys": [
    "male before female",
    "goats sheep cattle"
  ]
}
```

**Results**

```json
{
  "ordered_counts": [
    20,
    20,
    10,
    200,
    200,
    40
  ],
  "landings": [
    1876,
    1856,
    1846,
    1646,
    1446,
    1406
  ],
  "source_rows_used": [
    [
      "Goats",
      "male"
    ],
    [
      "Sheep",
      "male"
    ],
    [
      "Cattle",
      "male"
    ],
    [
      "Goats",
      "female"
    ],
    [
      "Sheep",
      "female"
    ],
    [
      "Cattle",
      "female"
    ]
  ]
}
```

**Finding.** The predeclared male-first/species order yields all six source landings, including its intermediate boundaries. The result depends on category order in addition to the conserved490 total.

**Reassessment.** Explain the female cumulative register without turning it into one date walk.

## C847 — Resolve the female register as a two-endpoint measurement

**Question.** How do400/430/470/490 arise when the chronological endpoints also change?

**Sources.** File58 §13.2

**Inputs**

```json
{}
```

**Results**

```json
{
  "female_source_counts": [
    200,
    200,
    30,
    40,
    20
  ],
  "prefix_register": [
    200,
    400,
    430,
    470,
    490
  ],
  "chronology_heads": [
    1846,
    1876,
    1876,
    1896
  ],
  "chronology_tails": [
    1446,
    1446,
    1406,
    1406
  ],
  "span_changes": [
    {
      "head": 30,
      "tail": 0,
      "span": 30
    },
    {
      "head": 0,
      "tail": -40,
      "span": 40
    },
    {
      "head": 20,
      "tail": 0,
      "span": 20
    }
  ]
}
```

**Finding.** The30,40,20 increments come respectively from an earlier head, a later tail, and an earlier head. This is a two-endpoint span register, not one chronology walk from a fixed anchor.

**Reassessment.** Test a second list’s declared regroupings against the same measurement grammar.

## C848 — Transfer the projection model to the four-household ledger

**Question.** Do equal regrouped totals recover the ordered household inputs?

**Sources.** File58 §1.4 Genesis46 printed-household subtotal register

**Inputs**

```json
{
  "source_vector": [
    33,
    16,
    14,
    7
  ]
}
```

**Results**

```json
{
  "source_pair": [
    49,
    21
  ],
  "source_bracket": [
    40,
    30
  ],
  "source_prefix": [
    33,
    49,
    63,
    70
  ],
  "invisible_regrouping_direction": [
    1,
    -1,
    1,
    -1
  ],
  "diagnostic_prefix": [
    34,
    49,
    64,
    70
  ],
  "scope": "printed source subtotals only; no person-count repair or chronology projection"
}
```

**Finding.** The same four counts produce49/21 and40/30 through two declared partitions. Both partitions miss one direction of change that the ordered prefix register detects. Retaining order adds information beyond shared totals.

**Reassessment.** State the precise extension of the path grammar established by the list holdout.

## C849 — State the new list-family extension of the grammar

**Question.** What has transferred unchanged, and what must be added to the source object?

**Sources.** C833–848; Strategy stageH

**Inputs**

```json
{}
```

**Results**

```json
{
  "transfer": {
    "path": "model/list_transfer_summary.json",
    "sha256": "62e4c18b5e59ca75bc8be2c620ee822cd8c4d7b978eb5ec526b76817b4e3614f",
    "bytes": 762
  }
}
```

**Finding.** The grammar transfers to lists when source rows retain category annotations and declared order. The new explanation identifies the constraints behind equal totals and the information those totals lose; it does not derive the textual counts or their historical intention.

**Reassessment.** Freeze the next complete File52c paired-anchor test before execution.

## C850 — Freeze the full paired-anchor inverse field

**Question.** Can one digit-register rule explain every existing pair of File52c endpoints?

**Sources.** latest File52c §2.1 and AppendixA.2–A.3; inverse preparation audit

**Inputs**

```json
{}
```

**Results**

```json
{
  "packet": {
    "path": "model/inverse_inputs.json",
    "sha256": "dbcb35bf6cfe88b318b4c8c327e97aa0be06583d5e945c43c9aac8b7f7b80911",
    "bytes": 19844
  },
  "scope": "existing32 sources and64 endpoints; target columns excluded; original input durations only",
  "hypothesis": "anchor dependence is generated by decimal borrow, core length and retained zero-placeholders"
}
```

**Finding.** The entire original paired field is the next fixed reconstruction target. No extra dates, target counts, or second-pass reversal are admitted.

**Reassessment.** Close the first20-step checkpoint, then test the paired field.

## C851 — Close the first twenty-step explanatory checkpoint

**Question.** What did the complete list holdout add to the common grammar?

**Sources.** C832–850

**Inputs**

```json
{}
```

**Results**

```json
{
  "checkpoint": {
    "path": "deliverables/490d_Chronological_Families_Checkpoint_C851.md",
    "sha256": "18816601d66beb94848c9353dd59c9d2cbc13c64b9fad842459de3a7d3e78369",
    "bytes": 1453
  }
}
```

**Finding.** The first transfer succeeds with one explicit enrichment: category predicates and declared order remain attached to source rows. Equal totals are now traced to exact constraints rather than treated as interchangeable objects.

**Reassessment.** Reconstruct the full paired-anchor field from its original digit registers.