# 490d — C832–C891: 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.

## C852 — Reconstruct all existing paired-anchor endpoints

**Question.** Does a single digit-register operation recover every original File52c endpoint?

**Sources.** model/inverse_inputs.json; latest File52c existing manifest

**Inputs**

```json
{}
```

**Results**

```json
{
  "field": {
    "path": "model/inverse_paired_field.json",
    "sha256": "eea45a0802d05e2ada4ee33ae2685da893929e1995083b0df2ebcfa2f86f2ce9",
    "bytes": 19047
  },
  "source_rows": 32,
  "existing_endpoints": 64,
  "scope": "original durations only; no target enumeration"
}
```

**Finding.** Every existing endpoint is reproduced by one digit-position evaluation with its original zero-placeholders. The whole paired field is now available for explanation without adding source or target states.

**Reassessment.** Express anchor dependence as a gain field on original durations.

## C853 — Derive the complete paired gain field

**Question.** Where does the anchor dependence enter the forward construction?

**Sources.** C852; inherited one-pass gain definition

**Inputs**

```json
{}
```

**Results**

```json
{
  "gains": {
    "path": "model/inverse_gain_field.json",
    "sha256": "093b354383add7d87f104fb0d8a32411bab1bb2465eddb6ccb4e78273938c8ba",
    "bytes": 3362
  },
  "paired_delta": [
    7200,
    990,
    990,
    990,
    990,
    990,
    990,
    990,
    1980,
    1980,
    990,
    990,
    990,
    -6930,
    -3330,
    -630,
    -6840,
    -4140,
    -1440,
    -4050,
    -3960,
    0,
    -360,
    -1800,
    -900,
    -5940,
    990,
    -6660,
    -5670,
    -2970,
    -2970,
    1980
  ],
  "identity": "A+I(D−A)=D+g(D−A), with g(s)=I(s)−s"
}
```

**Finding.** All anchor dependence resides in the gain of the original selected duration. The32 differences are one generated field, not32 adjustable translations.

**Reassessment.** Derive the long-core no-borrow and borrow branches from digit positions.

## C854 — Explain the990 and1980 branches by decimal borrowing

**Question.** Does subtraction of the1400 anchor gap force the repeated paired differences?

**Sources.** C852–853 digit registers

**Inputs**

```json
{}
```

**Results**

```json
{
  "branches": {
    "path": "model/inverse_long_core_branches.json",
    "sha256": "418a6416c1a58fb34c0422d9915ccb2ad6bd1b44ed9bcc43ccf96f0c7c7c02dd",
    "bytes": 2979
  },
  "branch_values": [
    990,
    1980
  ],
  "derivation": "the shared terminal core digit cancels after positional reversal; leading/tens changes force the remaining gap"
}
```

**Finding.** No borrow yields990; a tens borrow yields1980. These are forced branch values for every applicable source row, rather than separately chosen gaps.

**Reassessment.** Explain every shortened-core source row by the same positional rule.

## C855 — Explain every shortened-core branch

**Question.** Which positional changes produce the later nonconstant paired differences?

**Sources.** C852–854

**Inputs**

```json
{}
```

**Results**

```json
{
  "branches": {
    "path": "model/inverse_short_core_branches.json",
    "sha256": "fb9e0311663e549bcb790b83c963c12b4f8c56b5a4c775d265f84df52f81b8c0",
    "bytes": 2122
  },
  "formulas": {
    "three_to_two": "1260+90a−90b−900c",
    "three_to_one": "990−990c"
  }
}
```

**Finding.** The later variations follow core-length changes, which move the same digits into different place values. The nonlinear field therefore needs its digit register, not extra chronological corrections.

**Reassessment.** Resolve the two retained-zero source cases and complete the partition.

## C856 — Resolve retained-zero cases without changing the operation

**Question.** Do Adam and Jacob follow the same register model?

**Sources.** C852; latest File52c placeholder rule

**Inputs**

```json
{}
```

**Results**

```json
{
  "retained_zero_cases": [
    {
      "name": "Adam / Creation",
      "n": 410,
      "source_durations": [
        4100,
        2700
      ],
      "placeholder_counts": [
        2,
        2
      ],
      "endpoints": [
        1406,
        8606
      ],
      "delta": 7200
    },
    {
      "name": "Jacob",
      "n": 200,
      "source_durations": [
        2000,
        600
      ],
      "placeholder_counts": [
        3,
        2
      ],
      "endpoints": [
        2006,
        2006
      ],
      "delta": 0
    }
  ],
  "complete_case_partition": [
    14,
    16,
    2
  ]
}
```

**Finding.** Adam’s two-placeholders route gives7200 separation; Jacob’s fully retained2000/600 values give0. The exceptional-looking rows are ordinary cases of the stated zero-place rule.

**Reassessment.** Test the full paired-field law and rule out one affine column conversion.

## C857 — Test complete coverage and the affine obstruction

**Question.** Can the two original endpoint columns be related by one global affine map?

**Sources.** C852–856

**Inputs**

```json
{}
```

**Results**

```json
{
  "forced_affine": {
    "slope": 1,
    "shift": 990,
    "defining_sources": [
      "Seth",
      "Enosh"
    ]
  },
  "counterexample": {
    "source": "Lamech",
    "predicted": 4226,
    "actual": 5216
  },
  "coverage": 32
}
```

**Finding.** Two distinct990-gap rows force a translation by990, which fails on the1980 branch. The digit-register model covers the complete manifest where a single affine column map cannot.

**Reassessment.** Integrate the gain field with the source-path model without conflating it with rounding.

## C858 — Integrate the decimal module by its retained register

**Question.** How does one-pass reversal join the common grammar without becoming a chronology-wide affine operation?

**Sources.** C852–857; C831 rounding/path model

**Inputs**

```json
{}
```

**Results**

```json
{
  "module": {
    "path": "model/inverse_module_summary.json",
    "sha256": "bde0cd431d5efd716efd80d92de26b2cf52ba7dd2128c23d73168826f5c48c1f",
    "bytes": 494
  }
}
```

**Finding.** The shared model needs a digit register for decimal evaluation just as it needs categories for list measurement. The complete gain field explains route dependence while leaving source selection and historical interpretation open.

**Reassessment.** Freeze the complete NT indexed-display family and its comparison domains.

## C859 — Freeze the complete NT display domain

**Question.** Which source object can be compared without changing its claim status?

**Sources.** File43 §3.4; File51a §3.1; File54 §§6,8.6,13.2

**Inputs**

```json
{}
```

**Results**

```json
{
  "packet": {
    "path": "model/NT_inputs.json",
    "sha256": "5b5baff0eaaf68b71c1be598c1619cf840bc2da75c779d30336c822b6e8fb269",
    "bytes": 129000
  },
  "sources": [
    "File43",
    "File54",
    "File15",
    "File51a"
  ],
  "literal_tables": 41,
  "display_rows": 78
}
```

**Finding.** The new object is the entire display-only ledger, with shared-name comparisons ending at Jacob. Its schematic labels, count types and civil crossing remain explicit.

**Reassessment.** Reconstruct all 78 rows from one index rule.

## C860 — Recover the complete 78-row display

**Question.** Does one civil index reconstruct every opening and closing boundary?

**Sources.** File43 §3.4

**Inputs**

```json
{
  "civil": "BC b maps to 1−b; AD a maps to a"
}
```

**Results**

```json
{
  "display": {
    "path": "model/NT_full_display.json",
    "sha256": "d2f7053688c4b16b3671bcf84e6ce0539652095cacaa7b13561b40df7527f591",
    "bytes": 8304
  },
  "primary_span": 5390,
  "extended_span": 5460,
  "crossing_row": {
    "r": 0,
    "open": -5,
    "close": 65,
    "Luke": "Jesus",
    "Matthew": "Jesus Christ"
  }
}
```

**Finding.** One rule x(r)=−5−70r reconstructs all 78 displayed rows. The primary 77-row span is 5390; the final AD65–135 extension increases it to 5460.

**Reassessment.** Separate named generations, intervals and carrier slots before comparison.

## C861 — Separate the complete count objects

**Question.** Why do several legitimate NT spans coexist?

**Sources.** File43 §§2,3.4

**Inputs**

```json
{}
```

**Results**

```json
{
  "objects": {
    "Luke_names": 76,
    "Luke_intervals": 75,
    "Luke_birth_span": 5250,
    "Luke_names_projection": 5320,
    "Matthew_names": 41,
    "Matthew_birth_span": 2800,
    "Matthew_carrier_span": 2870
  },
  "artifact": {
    "path": "model/NT_count_objects.json",
    "sha256": "adc4fb617491fdf1f1f9fbf5f6c559dd1ff7088ce790bcfa34caf6f8b486bf38",
    "bytes": 192
  }
}
```

**Finding.** 76 Luke names measure 75 birth intervals; 41 Matthew names measure 40 birth intervals or 41 carrier slots through AD65. These are different measured objects, so none silently repairs the others.

**Reassessment.** Compare the full 22 shared Genesis names with the strict MT Rounded path.

## C862 — Construct the whole Luke–MT comparison field

**Question.** Can the highlighted bridges be located within a complete named field?

**Sources.** File43 lines674–696; File51a §3.1 strict regular table

**Inputs**

```json
{}
```

**Results**

```json
{
  "field": {
    "path": "model/NT_MT_field.json",
    "sha256": "d5868ac088ea4c4394124a7e43edfb3e05bfeba639827f5e973f7b01bec2c280",
    "bytes": 2861
  },
  "ordered_G": [
    1150,
    1210,
    1245,
    1265,
    1265,
    1260,
    1350,
    1345,
    1460,
    1570,
    2000,
    2030,
    1925,
    1885,
    1850,
    1810,
    1770,
    1730,
    1690,
    1690,
    1720,
    1710
  ]
}
```

**Finding.** The 22 common names give one complete difference field. The 1150 Adam and 1260 Jared bridges are members of that field, with the strict Actual4112 and Rounded4106 states kept distinct.

**Reassessment.** Recover all field values from local edge differences and one terminal.

## C863 — Derive the local generator of the NT–MT field

**Question.** Does a terminal value plus every local difference reconstruct the whole comparison?

**Sources.** C862 complete named field

**Inputs**

```json
{}
```

**Results**

```json
{
  "generator": {
    "path": "model/NT_MT_edge_generator.json",
    "sha256": "cd68ef6b5e218c258dc3ff30d648772d33404f38860df5085feb5f154472ad64",
    "bytes": 2884
  },
  "Cainan_contraction": {
    "from": "Arphaxad",
    "to": "Shelah",
    "NT_edge": 140,
    "MT_edge": 35,
    "difference": 105
  },
  "Noah_edge": {
    "from": "Noah",
    "to": "Shem",
    "NT_edge": 70,
    "MT_edge": 500,
    "difference": -430
  }
}
```

**Finding.** All 22 differences are generated by G_i−G_next=NT_edge−MT_edge and terminal1710. The sole 140-year NT shared-name edge records the intervening Cainan; the Noah edge explains the −430 local change.

**Reassessment.** Restore the admitted MT Cainan explicitly and compare the two edge measures.

## C864 — Compare the restored common name graph

**Question.** Does inserting the admitted Cainan equate the two measurements?

**Sources.** C831 admitted Cainan insertion; C862–863

**Inputs**

```json
{
  "MT_Cainan_interval": 130,
  "NT_slot": 70
}
```

**Results**

```json
{
  "restored": {
    "path": "model/NT_MT_Cainan_restored.json",
    "sha256": "46441b0b41bf1df994edae7b66280d7656c4698f1b5e8a807042db4b5b176e53",
    "bytes": 4037
  },
  "local_edges": [
    {
      "from": "Arphaxad",
      "to": "Cainan",
      "NT": 70,
      "MT": 35
    },
    {
      "from": "Cainan",
      "to": "Shelah",
      "NT": 70,
      "MT": 130
    }
  ],
  "local_G": [
    1900,
    1865,
    1925
  ]
}
```

**Finding.** Restoration yields the same 23-name graph but different edge weights: 70|70 in NT and 35|130 in restored MT. Topological alignment does not require equal durations.

**Reassessment.** Transfer the Actual-to-Rounded residual theorem across the whole native comparison.

## C865 — Transfer the rounding residual to the complete NT comparison

**Question.** Can the Actual and Rounded comparison fields be related without new fitting?

**Sources.** File51a §3.1; C862

**Inputs**

```json
{}
```

**Results**

```json
{
  "response": {
    "path": "model/NT_rounding_response.json",
    "sha256": "e6e2bb2979741f262f8d970c828b0198a3ae315d5382cdc57de1c22c5ff2e9e4",
    "bytes": 2788
  },
  "nonzero_support": [
    "Adam",
    "Seth",
    "Enosh",
    "Kenan",
    "Mahalalel",
    "Jared",
    "Enoch",
    "Methuselah",
    "Lamech",
    "Peleg",
    "Reu",
    "Serug",
    "Nahor"
  ]
}
```

**Finding.** Across every shared node, G_Rounded−G_Actual equals the negative MT rounding residual. The existing residual mechanism transfers unchanged to this new display object.

**Reassessment.** Classify all E returns within the finite NT index domain.

## C866 — Classify the complete finite E return family

**Question.** Which positive display indices return to display indices under E?

**Sources.** File43 complete display; File54 Enoch fork

**Inputs**

```json
{
  "domain": "r=0,...,76",
  "E": "25/23"
}
```

**Results**

```json
{
  "returns": [
    {
      "r": 0,
      "image_r": 0,
      "source": "Jesus",
      "image": "Jesus",
      "source_BC": 6,
      "image_BC": 6
    },
    {
      "r": 23,
      "image_r": 25,
      "source": "Melki",
      "image": "Cosam",
      "source_BC": 1616,
      "image_BC": 1756
    },
    {
      "r": 46,
      "image_r": 50,
      "source": "Salmon",
      "image": "Hezron",
      "source_BC": 3226,
      "image_BC": 3506
    },
    {
      "r": 69,
      "image_r": 75,
      "source": "Enoch",
      "image": "Creation of Adam",
      "source_BC": 4836,
      "image_BC": 5256
    }
  ],
  "artifact": {
    "path": "model/NT_E_returns.json",
    "sha256": "3fd2848a494b025de9c077ff77f10011cb24bf0a463fc3b1f5a7754b04e029d7",
    "bytes": 528
  }
}
```

**Finding.** Coprimality of25 and23 makes 23-divisibility the complete return rule. The four finite returns, including the fixed hinge, follow from one lattice condition; only the Enoch fork carries the supplied interpretation.

**Reassessment.** Classify P on the same fixed domain and compare its intersection with E.

## C867 — Classify P and the complete shared-return fork

**Question.** What singles out the finite Enoch fork without endpoint selection?

**Sources.** C866; File43 display; File54 §6

**Inputs**

```json
{
  "P": "70/69",
  "domain": "r=0,...,76"
}
```

**Results**

```json
{
  "returns": [
    {
      "r": 0,
      "image_r": 0,
      "source": "Jesus",
      "image": "Jesus",
      "source_BC": 6,
      "image_BC": 6
    },
    {
      "r": 69,
      "image_r": 70,
      "source": "Enoch",
      "image": "Jared",
      "source_BC": 4836,
      "image_BC": 4906
    }
  ],
  "joint_source_indices": [
    0,
    69
  ],
  "artifact": {
    "path": "model/NT_P_returns.json",
    "sha256": "57e7e768f128f3bde28682375c60890d81a29710cb6f03a930ac9d45c0306761",
    "bytes": 255
  }
}
```

**Finding.** P returns exactly at multiples of69. On the complete r0–76 domain, Enoch at69 is the only nonzero source returning under both E and P, because lcm(23,69)=69. This is a dependent domain classification, not a probability result.

**Reassessment.** Transport the full indexed result to the admitted 36BC rail.

## C868 — Transport the complete NT fork to the second rail

**Question.** Does moving both hinge and display preserve every Key radius?

**Sources.** File43 rail36; File54 Enoch fork; C866–867

**Inputs**

```json
{
  "old_hinge_BC": 6,
  "new_hinge_BC": 36
}
```

**Results**

```json
{
  "translated_returns": {
    "path": "model/NT_36BC_rail_returns.json",
    "sha256": "17f7a1fd328c63476e6ad5c6c76eda6f800ea34f63d9a0470af4b40ece161565",
    "bytes": 868
  },
  "Enoch": [
    {
      "key": "E",
      "source": "Enoch",
      "image": "Creation of Adam",
      "source_BC": 4866,
      "image_BC": 5286,
      "old_image_plus30": 5286
    },
    {
      "key": "P",
      "source": "Enoch",
      "image": "Jared",
      "source_BC": 4866,
      "image_BC": 4936,
      "old_image_plus30": 4936
    }
  ]
}
```

**Finding.** Moving the whole display and hinge by30BC preserves every radius and Key index. The source fork becomes4866→5286/4936. This is whole-frame covariance, distinct from an isolated MT apparent-age adjustment.

**Reassessment.** Construct the entire common49/70 coarsening of the primary span.

## C869 — Recover the whole49/70 common coarsening

**Question.** How do the two measurements divide one primary5390-year display?

**Sources.** File54 §§8.6,13.2; C860

**Inputs**

```json
{
  "span": 5390,
  "units": [
    49,
    70
  ]
}
```

**Results**

```json
{
  "unit": 490,
  "boundaries": {
    "path": "model/NT_49_70_coarsening.json",
    "sha256": "0f8f6b632e788373b74ddac612833439ab7771d12b0a72baef53e4076e4974ee",
    "bytes": 959
  },
  "partition_common_blocks": [
    3,
    5,
    3
  ],
  "partition_70_slots": [
    21,
    35,
    21
  ],
  "partition_49_units": [
    30,
    50,
    30
  ]
}
```

**Finding.** 70k=49j holds exactly at k7m,j10m. The whole primary span has12 common boundaries in490-year steps; the source1470|2450|1470 partition becomes3|5|3 blocks in either measurement. The extended49-grid is a comparison construction.

**Reassessment.** Test reflection on the complete common grid and its source-marked partition.

## C870 — Recover the full reflection of the primary display

**Question.** Which part of the chiasm is automatic grid closure and which part is supplied placement?

**Sources.** File54 primary chiasm; C860,869

**Inputs**

```json
{}
```

**Results**

```json
{
  "reflection": {
    "path": "model/NT_primary_reflection.json",
    "sha256": "f8d0f90c2df70d00c1860bba662e5b44f509ff956993a095f01ee1e9e0e1a460",
    "bytes": 6988
  },
  "marked": {
    "outer_pair": [
      -5325,
      65
    ],
    "BJ_pair": [
      -3855,
      -1405
    ],
    "center_civil": -2630,
    "center_BC": 2631
  }
}
```

**Finding.** Reflection x→−5260−x closes the entire primary grid and sends coarse indexm to11−m. Closure follows from uniform spacing; the placement of BJ Creation and Conquest at swapped indices21 and56 is the supplied structural fact.

**Reassessment.** Checkpoint the completed NT transfer, then test Key domains beyond the homogeneous display.

## C871 — Checkpoint the complete new display transfer

**Question.** What has the cycle established after40 actions?

**Sources.** C832–870

**Inputs**

```json
{
  "completed_before_checkpoint": 39
}
```

**Results**

```json
{
  "checkpoint": {
    "path": "deliverables/490d_Chronological_Families_Checkpoint_C871.md",
    "sha256": "c60360eafabad58481459cfa629adae3c322cc83c4481f8f717e2afc19f11a56",
    "bytes": 1328
  },
  "prefix_actions": 39
}
```

**Finding.** The complete NT family transfers to the same ordered-path and measurement grammar. Its new contribution is whole-field reconstruction, finite lattice classification and common coarsening, not additional isolated dates.

**Reassessment.** Freeze the complete operator packet and derive the general grid-preservation condition.

## C872 — Freeze the complete operator-domain packet

**Question.** Which grids and whole source fields will test the Keys?

**Sources.** Strategy §§5C,5E,5F; File12; File46; File51a; File63

**Inputs**

```json
{}
```

**Results**

```json
{
  "packet": {
    "path": "model/operator_inputs.json",
    "sha256": "fbed103853494c21cb6fa7e37ef4e47e21d9de8d9b4348cea817505001fff8e3",
    "bytes": 22303
  },
  "sources": 7,
  "quarter_rows": 5
}
```

**Finding.** The finite test uses integer, Rounded five-year and exact quarter-year grids, the complete File46 head field and the complete File63 multiplier ladder. Formal diagnostic compositions remain distinguished from source-authorized routes.

**Reassessment.** Derive the one-Key theorem for any declared lattice.

## C873 — Derive the exact lattice-preservation criterion

**Question.** When does a reduced Key p/q preserve hZ?

**Sources.** C872 declared domains

**Inputs**

```json
{
  "keys": {
    "E": "25/23",
    "P": "70/69",
    "J": "300/299"
  }
}
```

**Results**

```json
{
  "theorem": {
    "path": "model/one_key_grid_theorem.json",
    "sha256": "99ef73648f71a47a6a4c02ada19ea51bdd1db7db3bd5264862b0864719ee7a34",
    "bytes": 232
  },
  "reduced_denominators": {
    "E": 23,
    "P": 69,
    "J": 299
  }
}
```

**Finding.** A Key preserves a declared grid exactly on hqZ. This is an input-domain condition, not a demand to round rational outputs or a license to apply a Key to every chronology.

**Reassessment.** Classify integer and Rounded domains for all three Keys and their intersection.

## C874 — Classify all one-Key integer and Rounded domains

**Question.** How do same-grid conditions differ from merely integral output?

**Sources.** C873; File51a q in5Z

**Inputs**

```json
{}
```

**Results**

```json
{
  "domains": [
    {
      "key": "E",
      "ratio": "25/23",
      "integer_generator": 23,
      "five_year_generator": 115
    },
    {
      "key": "P",
      "ratio": "70/69",
      "integer_generator": 69,
      "five_year_generator": 345
    },
    {
      "key": "J",
      "ratio": "300/299",
      "integer_generator": 299,
      "five_year_generator": 1495
    }
  ],
  "joint_integer": 897,
  "joint_five_year": 4485,
  "artifact": {
    "path": "model/one_key_domains.json",
    "sha256": "db9630b099b86ca6a4011d5fdd5af153f007f4a19ee315fc73351c12bd5ead50",
    "bytes": 431
  }
}
```

**Finding.** Integer self-domains are23Z,69Z,299Z; Rounded5Z self-domains are115Z,345Z,1495Z. Their intersections are897Z and4485Z. These intersections concern separate images, not sequential authorization.

**Reassessment.** Locate the existing source examples inside these domains without counting them as new discoveries.

## C875 — Explain the existing completions by domain membership

**Question.** Do established source scalars have different compatibility profiles?

**Sources.** File46; File63; C831 inherited examples; C874

**Inputs**

```json
{
  "source_scalars": [
    460,
    690,
    483,
    12558
  ]
}
```

**Results**

```json
{
  "profiles": [
    {
      "input": 460,
      "images": {
        "E": 500,
        "P": "1400/3",
        "J": "6000/13"
      },
      "integer_keys": [
        "E"
      ],
      "input_on5Z": true
    },
    {
      "input": 690,
      "images": {
        "E": 750,
        "P": 700,
        "J": "9000/13"
      },
      "integer_keys": [
        "E",
        "P"
      ],
      "input_on5Z": true
    },
    {
      "input": 483,
      "images": {
        "E": 525,
        "P": 490,
        "J": "6300/13"
      },
      "integer_keys": [
        "E",
        "P"
      ],
      "input_on5Z": false
    },
    {
      "input": 12558,
      "images": {
        "E": 13650,
        "P": 12740,
        "J": 12600
      },
      "integer_keys": [
        "E",
        "P",
        "J"
      ],
      "input_on5Z": false
    }
  ],
  "artifact": {
    "path": "model/existing_key_profiles.json",
    "sha256": "f60bf0b288ce5e0edc4f6928749c64bacca0016149bfab9f40dbe827cccc2b6f",
    "bytes": 716
  }
}
```

**Finding.** The familiar460,690,483 and12558 completions occupy different exact domains. Their compatibility follows from denominator divisibility; the new explanation is the shared criterion, not the old scalar outputs.

**Reassessment.** Evaluate every row of the quarter-carrier family under the three Keys.

## C876 — Evaluate the complete quarter-carrier image family

**Question.** Which declared phase rows retain quarter resolution?

**Sources.** File63 §§1.1–1.3; C872 packet

**Inputs**

```json
{}
```

**Results**

```json
{
  "complete_images": {
    "path": "model/quarter_carrier_images.json",
    "sha256": "a3c62680ece3c724c3d999d4cdcc62792ad8db27c9dc0a0496e88292813c3387",
    "bytes": 1029
  },
  "membership": [
    {
      "E": true,
      "P": true,
      "J": false
    },
    {
      "E": true,
      "P": false,
      "J": false
    },
    {
      "E": true,
      "P": true,
      "J": false
    },
    {
      "E": true,
      "P": true,
      "J": false
    },
    {
      "E": true,
      "P": true,
      "J": false
    }
  ]
}
```

**Finding.** Every source multiplier is evaluated exactly. E preserves all five quarter-grid rows; P preserves four; J preserves none. Non-quarter outputs remain valid rational diagnostics, with only source-licensed routes carrying chronology status.

**Reassessment.** Derive these profiles from multiplier divisibility rather than five separate matches.

## C877 — Derive the quarter-family divisibility law

**Question.** What single condition explains each whole-family grid profile?

**Sources.** C873,876

**Inputs**

```json
{
  "seed": "161/4"
}
```

**Results**

```json
{
  "coefficients": {
    "E": "175/4",
    "P": "245/6",
    "J": "525/13"
  },
  "quarter_index_divisors": {
    "E": 1,
    "P": 3,
    "J": 13
  },
  "artifact": {
    "path": "model/quarter_multiplier_law.json",
    "sha256": "ca5bfffc2493bf2b6fdb41f0f8d6a5c203fc730d0372a1cf7cce21c17ab9afd2",
    "bytes": 229
  }
}
```

**Finding.** For source161m/4, E gives175m/4, P gives245m/6 and J gives525m/13. Quarter preservation is automatic for E, requires3|m for P and13|m for J. This explains the entire finite ladder.

**Reassessment.** Apply E to every head in File46 rather than only its outer width.

## C878 — Transform the complete File46 head field

**Question.** Does the integral outer completion imply integral internal heads?

**Sources.** File46 §6A.5

**Inputs**

```json
{
  "anchor_BC": 14006
}
```

**Results**

```json
{
  "head_field": {
    "path": "model/File46_full_head_image.json",
    "sha256": "8cba8fea255e7d925916785e2833f4e080f1d4c067198eace368b3e8951492a2",
    "bytes": 573
  },
  "rows": [
    {
      "head": 14926,
      "radius": 920,
      "E_radius": 1000,
      "image": 15006,
      "integral": true
    },
    {
      "head": 14896,
      "radius": 890,
      "E_radius": "22250/23",
      "image": "344388/23",
      "integral": false
    },
    {
      "head": 14466,
      "radius": 460,
      "E_radius": 500,
      "image": 14506,
      "integral": true
    },
    {
      "head": 14436,
      "radius": 430,
      "E_radius": "10750/23",
      "image": "332888/23",
      "integral": false
    },
    {
      "head": 14006,
      "radius": 0,
      "E_radius": 0,
      "image": 14006,
      "integral": true
    }
  ]
}
```

**Finding.** The exact whole field exists, but only apparent14926,14466 and anchor14006 map to integer heads at this anchor. The source920→1000 completion does not make the890 and430 radii integral.

**Reassessment.** Determine whether any single integer anchor could repair the full field.

## C879 — Prove the whole-field integer obstruction

**Question.** Can changing a common anchor put every image head on integer years?

**Sources.** File46 §6A.5; C878

**Inputs**

```json
{}
```

**Results**

```json
{
  "gaps": [
    430,
    30,
    430
  ],
  "image_gaps": [
    "10750/23",
    "750/23",
    "10750/23"
  ],
  "proof": {
    "path": "model/File46_integer_obstruction.json",
    "sha256": "5b51b11b5cdba7e6ceec0e04132f39ab96d0d32cbb46300872e0800f792cc495",
    "bytes": 441
  }
}
```

**Finding.** No common anchor can make every File46 head integral: the transformed430 and30 gaps are nonintegral independently of anchor. Whole-field shape is preserved, while whole-grid membership fails.

**Reassessment.** Derive the stagewise grid criterion for a fixed Key word.

## C880 — Derive the prefix-denominator theorem

**Question.** What must hold for every stage of a finite Key word to preserve a grid?

**Sources.** C873; Strategy partial actions

**Inputs**

```json
{}
```

**Results**

```json
{
  "theorem": {
    "path": "model/key_word_domain_theorem.json",
    "sha256": "9e8fc429be24813a422a5cbe2e49b519cb624e2b33a35d21147754afe10f1623",
    "bytes": 369
  }
}
```

**Finding.** The exact all-stage domain is h times the least common multiple of reduced prefix denominators. Final integrality is a weaker condition whenever cancellation occurs only later.

**Reassessment.** Evaluate the complete fixed inventory of nine two-Key words.

## C881 — Classify all nine two-Key grid domains

**Question.** Which ordered pairs differ between final and stagewise membership?

**Sources.** C880

**Inputs**

```json
{
  "inventory": "E,P,J each followed by E,P,J"
}
```

**Results**

```json
{
  "matrix": {
    "path": "model/two_key_domains.json",
    "sha256": "2dc016baa528ed002f1d353d53f91d2e5c971f0e0bb4222bf778e008349f1a58",
    "bytes": 1578
  },
  "rows": [
    {
      "execution": "E→E",
      "coefficient": "625/529",
      "final_integer_domain": 529,
      "all_stage_integer_domain": 529,
      "all_stage_5Z_domain": 2645
    },
    {
      "execution": "E→P",
      "coefficient": "1750/1587",
      "final_integer_domain": 1587,
      "all_stage_integer_domain": 1587,
      "all_stage_5Z_domain": 7935
    },
    {
      "execution": "E→J",
      "coefficient": "7500/6877",
      "final_integer_domain": 6877,
      "all_stage_integer_domain": 6877,
      "all_stage_5Z_domain": 34385
    },
    {
      "execution": "P→E",
      "coefficient": "1750/1587",
      "final_integer_domain": 1587,
      "all_stage_integer_domain": 1587,
      "all_stage_5Z_domain": 7935
    },
    {
      "execution": "P→P",
      "coefficient": "4900/4761",
      "final_integer_domain": 4761,
      "all_stage_integer_domain": 4761,
      "all_stage_5Z_domain": 23805
    },
    {
      "execution": "P→J",
      "coefficient": "7000/6877",
      "final_integer_domain": 6877,
      "all_stage_integer_domain": 20631,
      "all_stage_5Z_domain": 103155
    },
    {
      "execution": "J→E",
      "coefficient": "7500/6877",
      "final_integer_domain": 6877,
      "all_stage_integer_domain": 6877,
      "all_stage_5Z_domain": 34385
    },
    {
      "execution": "J→P",
      "coefficient": "7000/6877",
      "final_integer_domain": 6877,
      "all_stage_integer_domain": 6877,
      "all_stage_5Z_domain": 34385
    },
    {
      "execution": "J→J",
      "coefficient": "90000/89401",
      "final_integer_domain": 89401,
      "all_stage_integer_domain": 89401,
      "all_stage_5Z_domain": 447005
    }
  ]
}
```

**Finding.** Eight of nine ordered pairs have the same final and stagewise integer domain. P→J is the exception: its final domain is6877Z, but its intermediate stage requires20631Z. Scalar products commute while partial integer-grid actions can differ.

**Reassessment.** Expose the P/J asymmetry with the denominator-forced minimal example.

## C882 — Isolate the partial-action order asymmetry

**Question.** Can equal scalar endpoints conceal different intermediate grids?

**Sources.** C881 P/J domains

**Inputs**

```json
{
  "forced_input": 6877
}
```

**Results**

```json
{
  "J_then_P": [
    6877,
    6900,
    7000
  ],
  "P_then_J": [
    6877,
    "20930/3",
    7000
  ],
  "artifact": {
    "path": "model/PJ_order_witness.json",
    "sha256": "f1644a7da917c0ef52021aa12b1875a3ef07a331da682dbed989250b1edf6a0e",
    "bytes": 170
  }
}
```

**Finding.** 6877→6900→7000 stays integral under J→P;6877→20930/3→7000 under P→J has a fractional intermediate. The same final number therefore does not identify the same staged integer-grid route.

**Reassessment.** Check source-authorized File51a chains against the new theorem.

## C883 — Explain the authorized File51a Key chains

**Question.** Do the exact civil inputs satisfy the full stagewise domains?

**Sources.** File51a author clarification lines26–66; C880–881

**Inputs**

```json
{}
```

**Results**

```json
{
  "chains": [
    {
      "civil_input": 10051,
      "EE_chain": [
        10051,
        10925,
        11875
      ],
      "multiple529": 19
    },
    {
      "civil_input": 55016,
      "EE_chain": [
        55016,
        59800,
        65000
      ],
      "multiple529": 104,
      "EJ_chain": [
        55016,
        59800,
        60000
      ],
      "multiple6877": 8
    }
  ],
  "artifact": {
    "path": "model/File51a_authorized_key_chains.json",
    "sha256": "164285d8a6e23e20bbf840fa4de7a8c12dd5f5467fd650e8021a0ede0778a34c",
    "bytes": 330
  }
}
```

**Finding.** Both source E² chains satisfy the529 domain; the55016→59800→60000 E→J chain satisfies6877. The general criterion explains their intermediate integrality while preserving their civil-count input roles.

**Reassessment.** Test why the neighboring Rounded widths cannot replace those civil counts.

## C884 — Keep civil counts distinct from Rounded widths

**Question.** What arithmetic changes if neighboring width labels are substituted?

**Sources.** File51a C versus W_R; C883

**Inputs**

```json
{}
```

**Results**

```json
{
  "responses": [
    {
      "civil": 10051,
      "Rounded_width": 10050,
      "difference": 1,
      "E_width": "251250/23",
      "EE_width": "6281250/529",
      "E_difference": "25/23",
      "EE_difference": "625/529"
    },
    {
      "civil": 55016,
      "Rounded_width": 55015,
      "difference": 1,
      "E_width": "1375375/23",
      "EE_width": "34384375/529",
      "E_difference": "25/23",
      "EE_difference": "625/529"
    }
  ],
  "artifact": {
    "path": "model/civil_Rounded_input_distinction.json",
    "sha256": "2818366718815ab42ed4bfa9fed56973a9eca69ecd8a361434b9b59ddeead1ca",
    "bytes": 399
  }
}
```

**Finding.** The one-unit source-type difference propagates as25/23 after E and625/529 after E². Replacing the civil inputs by neighboring Rounded widths destroys the stated integral chains; matching near labels does not authorize type substitution.

**Reassessment.** Test the predetermined six distinct-Key orders as a bounded confirmation of the prefix theorem.

## C885 — Close the fixed three-distinct-Key inventory

**Question.** Does the P/J order condition persist in all six finite orders?

**Sources.** C880–882

**Inputs**

```json
{
  "inventory": "permutations of E,P,J only"
}
```

**Results**

```json
{
  "finite_inventory": {
    "path": "model/three_distinct_key_domains.json",
    "sha256": "396e93451a0634f82f1679a9fa96b483eab9ab4be5d902e76818129e324f851b",
    "bytes": 1012
  },
  "domains": [
    158171,
    474513
  ]
}
```

**Finding.** All six orders share final coefficient175000/158171. Stagewise domains split only by P/J order:158171Z with J before P and474513Z with P before J. This bounded confirmation closes the word test; no unlimited route search follows.

**Reassessment.** Separate grid-based order effects from different-anchor affine order.

## C886 — Derive the held-anchor order displacement

**Question.** How does anchor choice affect composition on an entire path?

**Sources.** File46 named anchors; formal diagnostic only

**Inputs**

```json
{
  "k": "25/23",
  "l": "70/69",
  "a": 14006,
  "b": 4836
}
```

**Results**

```json
{
  "constant_displacement": "18340/1587",
  "identity": {
    "path": "model/anchor_order_identity.json",
    "sha256": "35567eeb0a0f76c5555bd5cfa2aef64f2b742fde4914394a2cc772025e8c0713",
    "bytes": 349
  }
}
```

**Finding.** Different held anchors produce a constant whole-path displacement even though the scalar products agree. For the source anchors14006 and4836 with E/P, the formal composition difference is18340/1587 at every point.

**Reassessment.** Test that constant displacement over the complete frozen head field.

## C887 — Verify the anchored order law on a complete source field

**Question.** Do both compositions preserve the same interval pattern while shifting placement?

**Sources.** C878 complete File46 field; C886

**Inputs**

```json
{}
```

**Results**

```json
{
  "field": {
    "path": "model/complete_anchor_order_field.json",
    "sha256": "881c7fc4043aa06a0d2d2579462aecf169d0e70f9c8c36784dd6f1ca63c9271a",
    "bytes": 576
  },
  "number_of_heads": 5,
  "common_difference": "18340/1587"
}
```

**Finding.** The complete five-head diagnostic has the same scaled gaps in both orders and the same constant displacement at every head. Anchor order changes placement; intermediate divisibility changes domain. These are separate mechanisms.

**Reassessment.** Relate the NT return family directly to the general grid theorem.

## C888 — Join the NT lattice to the general Key domain theorem

**Question.** Why are integer years weaker than return to an NT70-year slot?

**Sources.** C866–867; C873–874

**Inputs**

```json
{
  "NT_grid_h": 70
}
```

**Results**

```json
{
  "domain_transfer": {
    "path": "model/NT_grid_domain_transfer.json",
    "sha256": "8dab8c8f0f0df5055ecf650a69e8f256eeac08bfc956fd91e82019e7da83e1f8",
    "bytes": 519
  },
  "same_slots": {
    "E": [
      0,
      23,
      46,
      69
    ],
    "P": [
      0,
      69
    ],
    "J": [
      0
    ]
  }
}
```

**Finding.** The NT return rule is the h=70 instance of the general theorem. E/P returns remain0,23,46,69 and0,69; J has only the fixed hinge in this finite domain. Integer output and return to a named slot are different requirements, even when these particular finite index sets coincide.

**Reassessment.** Integrate the operator results as grid and anchor registers.

## C889 — Integrate Keys as partial actions on measured paths

**Question.** What does the common grammar need to retain after the whole-family tests?

**Sources.** C872–888; C831 core grammar

**Inputs**

```json
{}
```

**Results**

```json
{
  "module": {
    "path": "model/operator_module_summary.json",
    "sha256": "64dbdc45c798c9244bbb69c9b15405cb4711f7ef6a942e5564b189621450624b",
    "bytes": 582
  }
}
```

**Finding.** Keys fit the shared grammar as exact anchored actions with explicit domains. The new whole-field tests explain both successful completions and failed grid closure without date repair or a universal transformation group.

**Reassessment.** Freeze the new Covenant macrohead and ordered carrier paths.

## C890 — Freeze the complete Covenant carrier lift

**Question.** Which new whole families extend the earlier scalar carrier results?

**Sources.** File63 §§7.2–7.9; C831 and C574–586 novelty controls

**Inputs**

```json
{}
```

**Results**

```json
{
  "packet": {
    "path": "model/covenant_inputs.json",
    "sha256": "c0f5cf1fab8019e064c4229d4ef2d8a4c3f59834c2b8f2cb39742c350a0fda1c",
    "bytes": 85017
  },
  "tables": 9,
  "code_blocks": 103
}
```

**Finding.** The selected new object is the six typed endpoint pairs and their ordered carrier changes, followed by the retained terminal biography. The older143520 scalar completions are inherited context.

**Reassessment.** Checkpoint60 actions, then reconstruct the six macro heads from the source-appointed multiplier.

## C891 — Checkpoint the partial-action explanation

**Question.** What does the60-action prefix add to the Strategy model?

**Sources.** C832–890

**Inputs**

```json
{}
```

**Results**

```json
{
  "checkpoint": {
    "path": "deliverables/490d_Chronological_Families_Checkpoint_C891.md",
    "sha256": "9dd481ca60ab6312e0c0017f79d58d641e75060d7ed38f9d278a71da8c450469",
    "bytes": 1380
  },
  "prefix": 59
}
```

**Finding.** The Key family is now explained as exact partial action on declared grids and anchored paths. The next whole-family test can distinguish outer expansion from retention of an internal biography.

**Reassessment.** Reconstruct the complete six-row macrohead field.