# 490d — C672–C691: sequential research record

Working continuation of C631 · packet date20260928.

Each completed step retains its question, declared inputs, exact results, finding, and reassessment. These are bounded research actions, not a count of independent discoveries. Canonical sources remain unchanged; second decimal inversion remains deferred.

## C672 — Generate the complete phase bracket

**Question.** Can center, width and suffix offset recover all eight supplied endpoints?

**Sources.** Supplement A §§4.1,4.3

**Inputs**

```json
{
  "base_center": 2871,
  "widths": [
    34,
    40
  ],
  "phase_offsets": {
    "n": 0,
    "t": "1/2"
  },
  "BC_label_rule": "coord(tY)=Y−1/2"
}
```

**Results**

```json
{
  "phase_rows": [
    {
      "width": 34,
      "phase": "n",
      "coordinates": [
        2888,
        2854
      ],
      "labels": [
        2888,
        2854
      ]
    },
    {
      "width": 34,
      "phase": "t",
      "coordinates": [
        "5777/2",
        "5709/2"
      ],
      "labels": [
        2889,
        2855
      ]
    },
    {
      "width": 40,
      "phase": "n",
      "coordinates": [
        2891,
        2851
      ],
      "labels": [
        2891,
        2851
      ]
    },
    {
      "width": 40,
      "phase": "t",
      "coordinates": [
        "5783/2",
        "5703/2"
      ],
      "labels": [
        2892,
        2852
      ]
    }
  ],
  "formula": "x=C+epsilon±w/2"
}
```

**Finding.** One center/width/phase formula generates all eight before/after positions. Corresponding suffix widths are34 and40; Tishri and Nisan centers remain separated by half a year in exact coordinates.

**Reassessment.** Test whether one global affine date map can perform this entire phase change.

## C673 — Test one-map compression of the phase family

**Question.** Can one affine function change34 to40 while preserving every half-year phase separation?

**Sources.** C672; Supplement A §4.3

**Inputs**

```json
{
  "width_change": [
    34,
    40
  ],
  "phase_change": [
    "1/2",
    "1/2"
  ]
}
```

**Results**

```json
{
  "required_width_slope": "20/17",
  "required_phase_slope": 1,
  "false_global_phase_gap": "10/17",
  "role_specific_rule": "C+epsilon+(20/17)(x−C−epsilon)"
}
```

**Finding.** A single affine date map is impossible: width demands slope20/17, while the preserved phase separation demands slope1. The valid expression dilates around each phase’s own center. This explains why the full object needs a phase coordinate.

**Reassessment.** Compare the scalar34.5 Key input with the coordinate34-to40 construction.

## C674 — Unify the micro Key outputs

**Question.** How do35,37.5 and40 relate to the source’s34.5 input?

**Sources.** Supplement A §§4.1,4.3

**Inputs**

```json
{
  "source_scalar": "69/2",
  "normalization_residual": "5/2"
}
```

**Results**

```json
{
  "P_output": 35,
  "E_output": "75/2",
  "output_gap": "5/2",
  "normalized": 40,
  "affine_continuation": 40
}
```

**Finding.** The normalization residual2.5 equals the gap between E and P outputs. The source-defined40 therefore equals2E(34.5)−P(34.5). This compresses the two-stage calculation without promoting it to a new universal Key.

**Reassessment.** Relate the scalar completion to the whole before/after phase envelopes.

## C675 — Recover all phase-envelope measurements

**Question.** Which widths belong to each suffix or cross-phase measurement?

**Sources.** Supplement A §4.3; C672–C674

**Inputs**

```json
{
  "corresponding_widths": [
    34,
    40
  ],
  "phase_offset": "1/2"
}
```

**Results**

```json
{
  "columns": [
    "inner_cross",
    "same_suffix",
    "outer_cross"
  ],
  "rows": [
    [
      "67/2",
      34,
      "69/2"
    ],
    [
      "79/2",
      40,
      "81/2"
    ]
  ],
  "coordinate_width_gain": 6,
  "scalar_normalization_gain": "11/2"
}
```

**Finding.** The full phase table is33.5/34/34.5 before and39.5/40/40.5 after. Outward endpoint motion adds6 to every matched measurement, whereas34.5→40 adds5.5 because it changes measurement class. Those two statements are compatible once the measured objects are named.

**Reassessment.** Resolve the source’s recovered35 row using exact phase coordinates.

## C676 — Reconstruct the recovered35 phase pair

**Question.** What exact coordinates result from withdrawing2.5 at both normalized endpoints?

**Sources.** Supplement A §4.3; derived phase-resolution test

**Inputs**

```json
{
  "T_coordinates": [
    "5783/2",
    "5703/2"
  ],
  "N_coordinates": [
    2891,
    2851
  ],
  "withdrawal": "5/2"
}
```

**Results**

```json
{
  "from_T_coordinates": [
    2889,
    2854
  ],
  "from_T_labels": [
    "2889n",
    "2854n"
  ],
  "from_N_coordinates": [
    "5777/2",
    "5707/2"
  ],
  "from_N_labels": [
    "2889t",
    "2854t"
  ]
}
```

**Finding.** Withdrawing2.5 from both normalized endpoints recovers width35 and changes the displayed suffix: the Tishri coordinates land on Nisan labels, and vice versa. The apparent integer-label ambiguity is resolved by the exact half-year convention.

**Reassessment.** Generate the two40 brackets and their common120 field.

## C677 — Generate the120 field from two40 brackets

**Question.** Does the supplied80 translation determine the entire enclosing field?

**Sources.** Supplement A §4.4

**Inputs**

```json
{
  "lower_bracket": [
    2891,
    2851
  ],
  "translation": 80
}
```

**Results**

```json
{
  "upper": [
    2971,
    2931
  ],
  "ordered_field": [
    2971,
    2931,
    2891,
    2851
  ],
  "gaps": [
    40,
    40,
    40
  ],
  "center": 2911,
  "center_shift": 40
}
```

**Finding.** Translating the40 bracket by80 generates three consecutive40 intervals across a120 envelope. Its center2911 is40 above the lower bracket’s center2871. The repeated40s are consequences of the supplied width and translation.

**Reassessment.** Insert the separately supplied70-wide Enoch body at this common center.

## C678 — Derive the nested25|35|35|25 field

**Question.** What follows once the source70 and120 bodies share center2911?

**Sources.** Supplement A §4.5

**Inputs**

```json
{
  "center": 2911,
  "outer_width": 120,
  "Enoch_width": 70,
  "source_Enoch": [
    2946,
    2876
  ]
}
```

**Results**

```json
{
  "nodes": [
    2971,
    2946,
    2911,
    2876,
    2851
  ],
  "gaps": [
    25,
    35,
    35,
    25
  ],
  "margin": 25,
  "Key_decomposition": [
    25,
    35,
    35,
    25
  ]
}
```

**Finding.** The two source widths and their common center force25|35|35|25. Each35 is P(34.5), each25 is E(23), but their sum is a retained-component construction rather than a newly defined whole120 Key image.

**Reassessment.** Distinguish the source’s scalar1470 closure from its coordinate1475 envelope.

## C679 — Resolve the1470 versus1475 closure

**Question.** Why do the source’s scalar and coordinate closures differ by five?

**Sources.** Supplement A §4.2

**Inputs**

```json
{
  "center": 2871,
  "start_offsets": [
    -13,
    13
  ],
  "outward_arm": "1449/2",
  "scalar_arm": "1449/2"
}
```

**Results**

```json
{
  "scalar_total": 1470,
  "coordinate_half_width": "1475/2",
  "coordinate_endpoints": [
    "7217/2",
    "4267/2"
  ],
  "coordinate_total": 1475,
  "Tishri_labels": [
    3609,
    2134
  ]
}
```

**Finding.** P converts the scalar724.5 arm to735, giving1470. The coordinate construction instead starts13 away from center before adding724.5, giving737.5 per side and1475 overall. The five-year difference is fully accounted for by different constructions.

**Reassessment.** Reconstruct the exterior-core opening of the ten-node ladder.

## C680 — Generate the exterior Prophetic opening

**Question.** Does converting only the two outer cores recover the supplied2930 ladder?

**Sources.** Supplement A §3.2

**Inputs**

```json
{
  "native": [
    4326,
    4296,
    3606,
    3576,
    2886,
    2856,
    2166,
    2136,
    1446,
    1416
  ],
  "outer_core_gain": 10
}
```

**Results**

```json
{
  "opened_ladder": [
    4336,
    4306,
    3606,
    3576,
    2886,
    2856,
    2166,
    2136,
    1436,
    1406
  ],
  "gaps": [
    30,
    700,
    30,
    690,
    30,
    690,
    30,
    700,
    30
  ],
  "widths": [
    2910,
    2930
  ],
  "centers": [
    2871,
    2871
  ]
}
```

**Finding.** Two ten-year core gains produce the complete source2930 ladder while preserving the six inner nodes, every30 flank and midpoint2871. This is selective opening, not P applied to the whole2910 span.

**Reassessment.** Add the source’s final five-year boundary displacements and identify the conserved quantity.

## C681 — Complete the conserved-center2940 corridor

**Question.** What invariant joins all three source-selected outer endpoint pairs?

**Sources.** Supplement A §5.1

**Inputs**

```json
{
  "pairs": [
    [
      4326,
      1416
    ],
    [
      4336,
      1406
    ],
    [
      4341,
      1401
    ]
  ],
  "final_outward_displacement": 5,
  "source_local_components": [
    2,
    3
  ]
}
```

**Results**

```json
{
  "endpoint_sums": [
    5742,
    5742,
    5742
  ],
  "centers": [
    2871,
    2871,
    2871
  ],
  "widths": [
    2910,
    2930,
    2940
  ],
  "width_increments": [
    20,
    10
  ]
}
```

**Finding.** Opposed endpoint changes preserve U+L=5742 while widening2910→2930→2940. The final five-year displacement is the source’s local2+3 construction; it supplies generated extremities4341/1401, not new canonical Creation/Conquest anchors or upstream Gear permission.

**Reassessment.** Express translation and symmetric opening in center/width coordinates.

## C682 — Separate translation from opening

**Question.** Can one coordinate rule explain both the120 construction and the2940 closure?

**Sources.** C677,C680,C681

**Inputs**

```json
{
  "changes": {
    "translate80": [
      80,
      80
    ],
    "open10": [
      10,
      -10
    ],
    "open5": [
      5,
      -5
    ]
  }
}
```

**Results**

```json
{
  "effects": {
    "translate80": {
      "center_change": 80,
      "width_change": 0
    },
    "open10": {
      "center_change": 0,
      "width_change": 20
    },
    "open5": {
      "center_change": 0,
      "width_change": 10
    }
  },
  "general_rule": "delta_c=(delta_U+delta_L)/2; delta_w=delta_U−delta_L"
}
```

**Finding.** Center and width provide a common language for two different mechanisms: translation moves the center while retaining width; symmetric opening widens the body while retaining center. The new sources instantiate both, so no single undifferentiated date shift explains the family.

**Reassessment.** Continue the admitted alternating ladder across the BC/AD boundary using civil elapsed time.

## C683 — Reconstruct the civil alternating continuation

**Question.** Does the native30|690 cell close consistently across the missing year zero?

**Sources.** Supplement A §5.2

**Inputs**

```json
{
  "civil_nodes": [
    "1446BC",
    "1416BC",
    "726BC",
    "696BC",
    "6BC",
    "AD25"
  ],
  "elapsed_coordinate": "BC B -> 1−B; AD A -> A"
}
```

**Results**

```json
{
  "elapsed_coordinates": [
    -1445,
    -1415,
    -725,
    -695,
    -5,
    25
  ],
  "gaps": [
    30,
    690,
    30,
    690,
    30
  ],
  "total": 1470,
  "direct_civil": 1470
}
```

**Finding.** The admitted continuation preserves30|690|30|690|30 and totals1470 under civil BC+AD−1. Its comparison terminals retain their source roles. The rounded mirror convention is a separate coordinate rule and is not substituted here.

**Reassessment.** Relate the source’s750 civil brackets and final three-year completion.

## C684 — Connect civil brackets with the micro gain

**Question.** How do the supplied750 brackets and AD27→30 completion arise?

**Sources.** Supplement A §§5.3–5.4

**Inputs**

```json
{
  "civil_pairs": [
    [
      726,
      25
    ],
    [
      724,
      27
    ]
  ],
  "micro_input": "69/2",
  "AD_start": 27
}
```

**Results**

```json
{
  "civil_widths": [
    750,
    750
  ],
  "coordinate_translation": 2,
  "E_micro_gain": 3,
  "completed_AD_label": 30
}
```

**Finding.** Both supplied civil pairs have width750 and differ by a uniform two-year forward translation. The separate AD27→30 completion uses the micro E-gain3. Equal durations and later completion are connected operations with separately supplied anchors.

**Reassessment.** Close the three-calendar calibration triangle at the micro input34.5.

## C685 — Close the three-calendar micro triangle

**Question.** Do all three existing Keys yield one calibrated volume at34.5?

**Sources.** Supplement A §6; C631 common calibration

**Inputs**

```json
{
  "seed": "69/2",
  "calendar_lengths": [
    336,
    360,
    364
  ],
  "operators": [
    "E",
    "P",
    "J"
  ]
}
```

**Results**

```json
{
  "converted_counts": [
    "75/2",
    35,
    "450/13"
  ],
  "calendar_volumes": [
    12600,
    12600,
    12600
  ],
  "common_volume": 12600
}
```

**Finding.** At the micro seed34.5, all three calibrated routes meet at12600:336×E(s)=360×P(s)=364×J(s). The Priestly route completes the source’s P/J comparison using the already established common-calibration identity.

**Reassessment.** Explain the exact twentyfold relation between the micro seed and the native690 core.

## C686 — Unify the690 core and34.5 micro seed

**Question.** Which parts of the two source constructions are exact scalar copies?

**Sources.** Supplement A §§1,4,6

**Inputs**

```json
{
  "micro_seed": "69/2",
  "scale": 20,
  "native_core": 690
}
```

**Results**

```json
{
  "micro_native_P_E": [
    "69/2",
    35,
    "75/2"
  ],
  "twentyfold": [
    690,
    700,
    750
  ],
  "twentyfold_normalized40": 800,
  "retained_bracket810_over20": "81/2"
}
```

**Finding.** The native/P/E core sequence690→700/750 is exactly twenty times34.5→35/37.5. The normalized40 would scale to800, so it remains a separate completion. The810/20=40.5 comparison is numerical and does not establish an endpoint map.

**Reassessment.** Trace the micro seed’s364-volume into the existing Actual/Rounded macro interface.

## C687 — Connect micro volume to the Actual/Rounded span

**Question.** Which equation explains the matching12558→12600 structure across scales?

**Sources.** Supplement A §6; File52c latest; C631 Actual/Rounded interface

**Inputs**

```json
{
  "micro_count": "69/2",
  "calendar": 364,
  "Actual_pair": [
    14004,
    1446
  ],
  "Rounded_pair": [
    14006,
    1406
  ]
}
```

**Results**

```json
{
  "micro_volume_days": 12558,
  "Actual_elapsed_years": 12558,
  "J_completed_numeric": 12600,
  "Rounded_elapsed_years": 12600,
  "identity": "J(364s)=360P(s)"
}
```

**Finding.** The micro volume34.5×364 and Actual cumulative interval share the numerical input12558. Their12600 completion follows from364J=360P. This is a common conversion structure across distinct units; the Rounded14006/1406 pair supplies its own endpoint realization.

**Reassessment.** Determine why the alternative12740 register is a different calendar choice.

## C688 — Explain the two completed volume registers

**Question.** What operation distinguishes12600 from12740 at the same35 count?

**Sources.** Supplement A §6

**Inputs**

```json
{
  "count": 35,
  "calendar_lengths": [
    360,
    364
  ],
  "root_volume": 12558
}
```

**Results**

```json
{
  "volumes": [
    12600,
    12740
  ],
  "difference": 140,
  "calendar_ratio": "91/90",
  "cross_products": [
    4586400,
    4586400
  ],
  "root_outputs": [
    12600,
    12740
  ]
}
```

**Finding.** The12600/12740 pair uses the same completed count35 with360/364 calendar lengths. Their ratio91/90 and shared cross-product follow from this construction; the product is a register equality, not an additional chronology or independent witness.

**Reassessment.** Account for the PhaseNorm40 volume and its residual relative to12600.

## C689 — Explain the normalized13440 register

**Question.** What added quantity separates PhaseNorm40 from the calibrated micro completion?

**Sources.** Supplement A §§4.3,6

**Inputs**

```json
{
  "E_count": "75/2",
  "normalized_count": 40,
  "Priestly_calendar": 336
}
```

**Results**

```json
{
  "calibrated_volume": 12600,
  "normalized_volume": 13440,
  "residual_volume": 840,
  "identity": "40×336 = E(34.5)×336 + 2.5×336"
}
```

**Finding.** The normalized13440 body adds exactly840 to12600, because the source’s2.5 residual is measured in336-day units. This explains the extra completion without assigning it to a new event or applying another Key.

**Reassessment.** Compare these scalar registers with the source’s retained2940 cumulative partition.

## C690 — Reconstruct the three cumulative completion paths

**Question.** Can one retained2940 component generate the12600/12740/13440 family?

**Sources.** Supplement A §10.2; inherited source family

**Inputs**

```json
{
  "source_head": 14006,
  "hinge": 4346,
  "tail": 1406,
  "upper": 9660,
  "retained_lower": 2940
}
```

**Results**

```json
{
  "paths": [
    {
      "upper": 9660,
      "lower": 2940,
      "total": 12600,
      "head": 14006
    },
    {
      "upper": 9800,
      "lower": 2940,
      "total": 12740,
      "head": 14146
    },
    {
      "upper": 10500,
      "lower": 2940,
      "total": 13440,
      "head": 14846
    }
  ],
  "calendar_bodies": [
    12600,
    12740,
    13440
  ],
  "upper_gain": [
    0,
    140,
    840
  ]
}
```

**Finding.** Holding4346→1406 fixed while converting only the9660 upper arm generates all three source totals12600/12740/13440. The generated heads14006/14146/14846 retain their existing status. This realizes the micro register family through retained macro components, not one whole-span conversion.

**Reassessment.** Consolidate the micro-to-Rounded bridge and decide which anchored routes best explain its placement.

## C691 — Consolidate phase and calendar completion

**Question.** What causal chain now connects fine phase structure to the Round chronology?

**Sources.** C672–C690; Supplement A §§4–6,10

**Inputs**

```json
{
  "completed_range": [
    672,
    691
  ],
  "chain": [
    "phase scalar",
    "common calibration",
    "unit-qualified12558",
    "retained macro realization"
  ]
}
```

**Results**

```json
{
  "artifact": {
    "path": "deliverables/C691_Interim_Synthesis.md",
    "sha256": "87a846f67f4e545ba4a1bbaf0dbddfe3ad0804a57346e5ee646055e5a63e463a",
    "bytes": 1253
  },
  "completed_steps": 60
}
```

**Finding.** Sixty steps now connect the local calendar geometry to the Rounded cumulative family through a common calibrated structure and a retained-component realization. The next explanatory need is the source-conditioned placement of the4346 hinge.

**Reassessment.** Reconstruct both supplied regular-MT routes to4346 as anchored maps.