# 490d — C832–C911: 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.

## C892 — Reconstruct all six twelvefold macro heads

**Question.** Does one source-appointed anchored rule recover the complete head field?

**Sources.** File63 §§7.7–7.9

**Inputs**

```json
{
  "multiplier": 12
}
```

**Results**

```json
{
  "whole_field": {
    "path": "model/six_macro_heads.json",
    "sha256": "14d47cc4de24cd1211d60820e90a6ea65a31289a8872c771c73730132a329a3d",
    "bytes": 1417
  },
  "heads": [
    146006,
    146066,
    146836,
    146833,
    145601,
    145571
  ]
}
```

**Finding.** M=R+12(C−R)=12C−11R recovers all six published macro heads. The twelvefold multiplier and endpoint roles are source premises; the head field is their exact consequence.

**Reassessment.** Derive and test the complete source-pair response field.

## C893 — Recover the complete macrohead response field

**Question.** Why do60,770,−3 and−30 arise from one twelvefold rule?

**Sources.** C892 six typed pairs

**Inputs**

```json
{}
```

**Results**

```json
{
  "response": [
    {
      "from": "regular_Jacob_whole",
      "to": "regular_Jacob_jubilee",
      "delta_C": 5,
      "delta_R": 0,
      "delta_body": 60,
      "delta_M": 60,
      "predicted": 60
    },
    {
      "from": "regular_Jacob_jubilee",
      "to": "cumulative_Levi_upper",
      "delta_C": 0,
      "delta_R": -70,
      "delta_body": 840,
      "delta_M": 770,
      "predicted": 770
    },
    {
      "from": "cumulative_Levi_upper",
      "to": "cumulative_Levi_lower",
      "delta_C": -3,
      "delta_R": -3,
      "delta_body": 0,
      "delta_M": -3,
      "predicted": -3
    },
    {
      "from": "apparent_Covenant",
      "to": "Creation_offering",
      "delta_C": -30,
      "delta_R": -30,
      "delta_body": 0,
      "delta_M": -30,
      "predicted": -30
    }
  ],
  "artifact": {
    "path": "model/macrohead_response_field.json",
    "sha256": "9edec1dac947076ed0688da232cb8f1acf8a6fd011474e04730fe212388cbfd1",
    "bytes": 716
  }
}
```

**Finding.** The whole response is ΔM=12ΔC−11ΔR. Moving only the target by−70 adds840 to the body but moves the target by−70, giving770. Matched endpoint shifts of−3 or−30 pass through unchanged.

**Reassessment.** Determine exactly what information is lost when only the expanded body is kept.

## C894 — Identify the information lost by a body-only comparison

**Question.** Can the expanded span alone recover a macro head?

**Sources.** C892–893

**Inputs**

```json
{}
```

**Results**

```json
{
  "equal_body_pairs": {
    "144900": [
      {
        "id": "cumulative_Levi_upper",
        "R": 1936,
        "M": 146836
      },
      {
        "id": "cumulative_Levi_lower",
        "R": 1933,
        "M": 146833
      }
    ],
    "143520": [
      {
        "id": "apparent_Covenant",
        "R": 2081,
        "M": 145601
      },
      {
        "id": "Creation_offering",
        "R": 2051,
        "M": 145571
      }
    ]
  },
  "artifact": {
    "path": "model/carrier_body_information_loss.json",
    "sha256": "330ec0621328d7de77209faf09a0003e9b22151cd0be27619f1d8443a7a90c3a",
    "bytes": 652
  }
}
```

**Finding.** Identical144900 or143520 bodies occupy different source-appointed positions. A body-only match loses a translation coordinate; retaining the target restores the numeric pair, while source labels still need their own record.

**Reassessment.** Reconstruct the entire ordered carrier chain from endpoint changes.

## C895 — Generate the full ordered carrier chain by endpoint incidence

**Question.** How do the three differently supported changes produce40|5|70?

**Sources.** File63 §§7.7–7.9; C890 packet

**Inputs**

```json
{}
```

**Results**

```json
{
  "source_spans": [
    11960,
    12000,
    12005,
    12075
  ],
  "changes": [
    {
      "delta_C": -5,
      "delta_R": -45,
      "delta_span": 40,
      "delta_twelvefold": 480
    },
    {
      "delta_C": 5,
      "delta_R": 0,
      "delta_span": 5,
      "delta_twelvefold": 60
    },
    {
      "delta_C": 0,
      "delta_R": -70,
      "delta_span": 70,
      "delta_twelvefold": 840
    }
  ],
  "artifact": {
    "path": "model/ordered_carrier_incidence.json",
    "sha256": "5078e79f05db4043c8ebf78ced5540cea7b974e7cd2d096237e5e42b8460841e",
    "bytes": 570
  }
}
```

**Finding.** The complete11960→12000→12005→12075 chain has endpoint-supported changes40|5|70, lifted to480|60|840. This is an ordered comparison of typed spans, not a single historical timeline through the four scalar values.

**Reassessment.** Relate the total115 change to its source-appointed mixed-mode hinge.

## C896 — Connect the complete carrier change to the typed115 hinge

**Question.** What does the source115|115 symmetry explain and what remains a premise?

**Sources.** File63 §§7.7–7.9

**Inputs**

```json
{}
```

**Results**

```json
{
  "hinge": [
    2166,
    2051,
    1936
  ],
  "gaps": [
    115,
    115
  ],
  "carrier_total_change": 115,
  "artifact": {
    "path": "model/carrier_115_hinge.json",
    "sha256": "2e3792102917bad5494997444a37f55b4bf8e797d126cf6f176528f5ad8f77dc",
    "bytes": 337
  }
}
```

**Finding.** The source2166→2051→1936 hinge supplies two115 intervals, and the full carrier path increases by115. This connects the objects structurally while retaining the proposed offering date and the regular/cumulative change of role.

**Reassessment.** Reconstruct the complete144900 terminal Key branch and its gain vector.

## C897 — Recover the full terminal carrier branch

**Question.** How do the144900 branch outputs and gains fit one Key family?

**Sources.** File63 §7.8.2

**Inputs**

```json
{}
```

**Results**

```json
{
  "input": 144900,
  "P": 147000,
  "E": 157500,
  "gains": {
    "P": 2100,
    "E": 12600,
    "output_gap": 10500
  },
  "artifact": {
    "path": "model/terminal_carrier_branch.json",
    "sha256": "866fcc58d1e8ed8a8bde2ff8571cb8181004b7caafd4e3d320c80070bbf78f53",
    "bytes": 314
  }
}
```

**Finding.** 144900=900×161 maps to147000=1000×147 under P and157500=900×175 under E. The gains and output gap are2100×(1,6,5). Their lifespan interpretation remains source-supplied.

**Reassessment.** Test whether the retained147-year terminal biography is the uniform P image of its original interval.

## C898 — Distinguish retained biography from uniform transport

**Question.** Does the outer147000 span force every internal boundary to be P-scaled?

**Sources.** File63 §7.8.2 terminal thousandth

**Inputs**

```json
{}
```

**Results**

```json
{
  "comparison": {
    "source_nodes": [
      148936,
      2083,
      1936
    ],
    "retained_partition": [
      146853,
      147
    ],
    "uniform_P_terminal_length": "3430/23",
    "uniform_P_birth": "47958/23",
    "retained147_preimage": "1449/10",
    "rule": "expand outer body; retain supplied biography as appointed terminal subdivision"
  },
  "artifact": {
    "path": "model/retained_biography_distinction.json",
    "sha256": "89799e51d7dbd8c7b4228907deaed57baea933a12ddc12884fd2349eda927ff0",
    "bytes": 314
  }
}
```

**Finding.** The source retains2083→1936 as the final147 of147000, giving999×147|147. Uniform P would move2083 to2085+3/23 and make the interval149+3/23. The outer agreement therefore supports retention, not transport of the original internal147.

**Reassessment.** Check the complete paired civil squares with their external endpoints.

## C899 — Recover the two complete civil comparison squares

**Question.** Do the paired30 shifts preserve each entire source-appointed arm?

**Sources.** File63 §7.9

**Inputs**

```json
{}
```

**Results**

```json
{
  "squares": [
    {
      "civil_nodes": [
        -145600,
        -2080,
        0
      ],
      "body": 143520,
      "terminal": 2080,
      "total": 145600,
      "source_total": 145600
    },
    {
      "civil_nodes": [
        -145570,
        -2050,
        30
      ],
      "body": 143520,
      "terminal": 2080,
      "total": 145600,
      "source_total": 145600
    }
  ],
  "artifact": {
    "path": "model/paired_civil_squares.json",
    "sha256": "f93729bd50b8e39a077d0776e95f581e7288413a6c66c4468be2912ad4f82e8b",
    "bytes": 330
  }
}
```

**Finding.** Both complete arms decompose145600 as143520+2080. Their corresponding civil nodes all shift by30, including1BC→AD30. The equality uses civil crossing and preserves the source-appointed comparison roles.

**Reassessment.** Integrate the carrier lift and retained-subdivision result into the common grammar.

## C900 — Integrate the complete carrier network

**Question.** Which added information makes the whole carrier family reconstructible?

**Sources.** C892–899

**Inputs**

```json
{}
```

**Results**

```json
{
  "module": {
    "path": "model/carrier_module_summary.json",
    "sha256": "695f2584682366ab20d84fa38c95e1a3cc51b1fed9c4aeb0da838b61b9dccda9",
    "bytes": 649
  }
}
```

**Finding.** The complete carrier family requires the source pair and the treatment of internal boundaries. One anchored rule explains the six heads and their changes; retained biographies require an explicitly different interior construction.

**Reassessment.** Join interpolation and extrapolation in one reversible endpoint-coordinate description.

## C901 — Unify endpoint interpolation and extrapolation

**Question.** Can one span and one generated coordinate retain the entire source pair?

**Sources.** C892; C831 weighted Creation bridge

**Inputs**

```json
{}
```

**Results**

```json
{
  "basis": {
    "path": "model/endpoint_coordinate_basis.json",
    "sha256": "47b0713d83431b05bb871bd6fd461cc179ef1d3d5de57403a84c8f817a06e7d7",
    "bytes": 391
  },
  "symbolic_coefficients": {
    "R_recovered": [
      0,
      1
    ],
    "C_recovered": [
      1,
      0
    ]
  }
}
```

**Finding.** The pair (D,W_lambda) is an exact coordinate system for (C,R), for every fixed lambda. The old weighted bridge and new twelvefold head use the same endpoint algebra with different source-appointed coefficients and roles.

**Reassessment.** Recover every macro source pair and both inherited weighted Creation pairs from this basis.

## C902 — Recover whole source-pair families from span and placement

**Question.** Does the endpoint-coordinate basis reconstruct all supplied pairs exactly?

**Sources.** C892 complete six; C831 Actual/Rounded weighted bridge

**Inputs**

```json
{}
```

**Results**

```json
{
  "recovery": {
    "path": "model/endpoint_basis_recovery.json",
    "sha256": "db5f0def0e97416526798219789a2f562c5cecb648b744c5c98d3ed0d1fddf39",
    "bytes": 1504
  },
  "weighted_rows": [
    {
      "id": "Actual_weighted",
      "lambda": "4/5",
      "D": 9890,
      "W": 12026,
      "recovered_C": 14004,
      "recovered_R": 4114,
      "source_C": 14004,
      "source_R": 4114
    },
    {
      "id": "Rounded_weighted",
      "lambda": "4/5",
      "D": 9900,
      "W": 12026,
      "recovered_C": 14006,
      "recovered_R": 4106,
      "source_C": 14006,
      "source_R": 4106
    }
  ]
}
```

**Finding.** All six macro pairs and both weighted Creation pairs are recovered exactly. The Actual and Rounded weighted coordinates both equal12026, but their different spans9890 and9900 recover different original pairs. A shared coordinate alone does not merge them.

**Reassessment.** Compare whole-frame motion with source-fixed anchor changes across the new modules.

## C903 — Contrast whole-frame motion with source-fixed anchor change

**Question.** Why are some paired fields rigid translations while File52c is not?

**Sources.** C852–858; C868; C893; C839

**Inputs**

```json
{}
```

**Results**

```json
{
  "comparison": {
    "path": "model/frame_vs_anchor_change.json",
    "sha256": "c480a3cf3b943f4779a68c30de4e713baba009c64cc59d481640dea0af4a0ecf",
    "bytes": 398
  },
  "original_spans_preserved": 64,
  "different_inverse_gaps": 19
}
```

**Finding.** Whole-frame covariance applies across the new objects because their relative inputs remain fixed. File52c changes only the anchor while holding the source date, so its measured span and digit branch change. This explains rigid NT/carrier translations versus the nonuniform paired inverse field.

**Reassessment.** Classify every festival prefix against the49/70 common measure.

## C904 — Connect the complete festival walks to the49/70 interface

**Question.** Which scaled prefix durations belong to the shared490 measure?

**Sources.** C839–841; C869

**Inputs**

```json
{
  "scale": 7,
  "grids": [
    49,
    70,
    490
  ]
}
```

**Results**

```json
{
  "classification": {
    "path": "model/festival_common_measure.json",
    "sha256": "bb2701e6a04d267a6ee3826881a79e4dfdb4e929f7bbbc3016d8fb69ac02b0f9",
    "bytes": 2156
  },
  "shared490": {
    "forward": [
      0,
      140
    ],
    "reverse": [
      0,
      140
    ]
  }
}
```

**Finding.** For countc,7c lies in49Z,70Z or490Z exactly when7,10 or70 dividesc. Both complete walks meet the common490 grid only at count0 and140. Their other shared prefixes59 and199 are outside it. Equal duration membership does not identify events across anchors.

**Reassessment.** Extend the insertion comparison across the source-specific Cainan measurements.

## C905 — Separate common topology from source-specific measurement

**Question.** What does the full NT insertion response add to the earlier Cainan comparisons?

**Sources.** C864; C831 insertion and BJ deletion results

**Inputs**

```json
{}
```

**Results**

```json
{
  "transfer": {
    "path": "model/topology_measure_transfer.json",
    "sha256": "13776a4ab8d0a07e54ce29312bdf2eda35338b9278b39a4f75b1eb5616ebd907",
    "bytes": 1698
  },
  "upstream_delta": -130,
  "downstream_delta": 0
}
```

**Finding.** The complete NT/MT field changes by−130 at every common node through Arphaxad and by0 from Shelah onward. This makes the insertion support explicit. The130,460,70 and BJ57 measures retain their source-specific roles even when name bookkeeping aligns.

**Reassessment.** Generalize the retained-biography distinction as a subdivision rule.

## C906 — Derive the retained-subdivision criterion

**Question.** When can two constructions share an outer span but disagree at an internal biography?

**Sources.** C898; C831 complete-path criterion

**Inputs**

```json
{
  "S": 144900,
  "l": 147,
  "k": "70/69"
}
```

**Results**

```json
{
  "criterion": {
    "path": "model/subdivision_retention_criterion.json",
    "sha256": "f99e9c904c85e66da56fa237f9701db460678c4fd4ceb1744d71bd1a91e452fb",
    "bytes": 325
  },
  "partitions": {
    "uniform": [
      "3377570/23",
      "3430/23"
    ],
    "retained": [
      146853,
      147
    ]
  },
  "cut_difference": "49/23"
}
```

**Finding.** Uniform expansion and retained biography have the same outer span but internal cuts differing by(k−1)l. They coincide only for unit scale or zero-length biography. The complete path, rather than its total, distinguishes these constructions.

**Reassessment.** Compare the distinct mechanisms behind apparent operation-order effects.

## C907 — Separate the finite mechanisms of order and retention

**Question.** Which differences would a vague statement that order matters conceal?

**Sources.** C842; C853–857; C881–887; C898

**Inputs**

```json
{}
```

**Results**

```json
{
  "taxonomy": {
    "path": "model/operation_difference_taxonomy.json",
    "sha256": "57cbc1ccc6e7c7f81d5e0889b3d3a318681086b608a49b3551ffc073a9f549ce",
    "bytes": 1212
  },
  "mechanisms": 5
}
```

**Finding.** The five mechanisms preserve different information. A common final product cannot stand in for equality of ordered prefixes, original digit inputs, intermediate domains, held anchors or retained internal cuts.

**Reassessment.** Separate forced identities from additional source constraints across all new domains.

## C908 — Build the explanatory constraint ledger

**Question.** Which agreements follow automatically and which require particular source inputs?

**Sources.** C832–907 complete-family results

**Inputs**

```json
{}
```

**Results**

```json
{
  "ledger": {
    "path": "model/explanatory_constraint_ledger.json",
    "sha256": "f8eee370994066bfd9f3fb220940c564a163f6bd1555f6020d9fc139c7a0c40d",
    "bytes": 1279
  }
}
```

**Finding.** The model separates algebraic consequences from source-dependent placements, weights and roles. Regenerated coordinates are linked consequences of shared inputs, not independent witnesses to the source choices.

**Reassessment.** Identify the finite necessity evidence for the information retained by the grammar.

## C909 — Justify the retained information by complete-object witnesses

**Question.** Why does each additional source register belong in the model?

**Sources.** C836,840,844–848,857,861,879,882,894,898

**Inputs**

```json
{}
```

**Results**

```json
{
  "necessity_evidence": {
    "path": "model/finite_register_necessity.json",
    "sha256": "20d499d369f930b8c000392f332ef9d557d78b415487f95aa813cfb7cb8c4026",
    "bytes": 1092
  }
}
```

**Finding.** Each retained information type has a concrete witness showing what its removal loses. This supports a compact grammar over the tested families; it does not prove a unique or globally minimal model.

**Reassessment.** State the unified source-object schema and the distinct operations it admits.

## C910 — State the common measurement-and-construction schema

**Question.** How do the families fit together without erasing their source distinctions?

**Sources.** C831 baseline; C849,858,870,889,900,909

**Inputs**

```json
{}
```

**Results**

```json
{
  "schema": {
    "path": "model/unified_family_schema.json",
    "sha256": "39e20f67fe16cbacc00a3775111c47182d0a37065864794b2ffdd42d4a953691",
    "bytes": 1586
  }
}
```

**Finding.** The larger picture is a shared measurement-and-construction grammar. Regular, cumulative, Rounded, list, NT, inverse, Key and Covenant families differ in the source structure or measurement they retain, while reusing a small set of explicit operations.

**Reassessment.** Checkpoint80 actions, then make the explanation constructive and reader-facing.

## C911 — Checkpoint the integrated80-action model

**Question.** What is ready for the final explanatory deliverables?

**Sources.** C832–910

**Inputs**

```json
{}
```

**Results**

```json
{
  "checkpoint": {
    "path": "deliverables/490d_Chronological_Families_Checkpoint_C911.md",
    "sha256": "114e219469ec824f67fc2d1f953dfcc3d10c940ab5c637b6ecfc2b70d5b8146a",
    "bytes": 1616
  },
  "prefix": 79
}
```

**Finding.** The80-action model answers how the families fit by reconstructing complete objects and explaining their differences. The remaining work will make this result readable, source-traceable and portable.

**Reassessment.** Build one compact source packet that regenerates a complete family without stored output dates.