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
{
"base_center": 2871,
"widths": [
34,
40
],
"phase_offsets": {
"n": 0,
"t": "1/2"
},
"BC_label_rule": "coord(tY)=Y−1/2"
}
Results
{
"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
{
"width_change": [
34,
40
],
"phase_change": [
"1/2",
"1/2"
]
}
Results
{
"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
{
"source_scalar": "69/2",
"normalization_residual": "5/2"
}
Results
{
"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
{
"corresponding_widths": [
34,
40
],
"phase_offset": "1/2"
}
Results
{
"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
{
"T_coordinates": [
"5783/2",
"5703/2"
],
"N_coordinates": [
2891,
2851
],
"withdrawal": "5/2"
}
Results
{
"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
{
"lower_bracket": [
2891,
2851
],
"translation": 80
}
Results
{
"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
{
"center": 2911,
"outer_width": 120,
"Enoch_width": 70,
"source_Enoch": [
2946,
2876
]
}
Results
{
"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
{
"center": 2871,
"start_offsets": [
-13,
13
],
"outward_arm": "1449/2",
"scalar_arm": "1449/2"
}
Results
{
"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
{
"native": [
4326,
4296,
3606,
3576,
2886,
2856,
2166,
2136,
1446,
1416
],
"outer_core_gain": 10
}
Results
{
"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
{
"pairs": [
[
4326,
1416
],
[
4336,
1406
],
[
4341,
1401
]
],
"final_outward_displacement": 5,
"source_local_components": [
2,
3
]
}
Results
{
"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?
Inputs
{
"changes": {
"translate80": [
80,
80
],
"open10": [
10,
-10
],
"open5": [
5,
-5
]
}
}
Results
{
"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
{
"civil_nodes": [
"1446BC",
"1416BC",
"726BC",
"696BC",
"6BC",
"AD25"
],
"elapsed_coordinate": "BC B -> 1−B; AD A -> A"
}
Results
{
"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
{
"civil_pairs": [
[
726,
25
],
[
724,
27
]
],
"micro_input": "69/2",
"AD_start": 27
}
Results
{
"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
{
"seed": "69/2",
"calendar_lengths": [
336,
360,
364
],
"operators": [
"E",
"P",
"J"
]
}
Results
{
"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
{
"micro_seed": "69/2",
"scale": 20,
"native_core": 690
}
Results
{
"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
{
"micro_count": "69/2",
"calendar": 364,
"Actual_pair": [
14004,
1446
],
"Rounded_pair": [
14006,
1406
]
}
Results
{
"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
{
"count": 35,
"calendar_lengths": [
360,
364
],
"root_volume": 12558
}
Results
{
"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
{
"E_count": "75/2",
"normalized_count": 40,
"Priestly_calendar": 336
}
Results
{
"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
{
"source_head": 14006,
"hinge": 4346,
"tail": 1406,
"upper": 9660,
"retained_lower": 2940
}
Results
{
"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
{
"completed_range": [
672,
691
],
"chain": [
"phase scalar",
"common calibration",
"unit-qualified12558",
"retained macro realization"
]
}
Results
{
"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.