490d — C512–C521: sequential research record
27 September 2026. Working continuation of C481. File52c’s latest supplied draft controls the rounded investigation. No canonical sources are changed. Second-order inversion remains deferred.
Each numbered entry records its question, declared inputs, exact results, finding, and reassessment. These are bounded research actions, not claims of independent discoveries. Parallel agents supplied source extraction and critical review; the main sequence was evaluated in order.
C512 — Reconstruct all nine nonzero14726 inputs
Question. Can every member be regenerated from its supplied source date and first reversal anchor?
Source basis. File52c §§3.11.4,8.3
Frozen inputs
{
"members": [
[
"Mahalalel",
3711,
1,
1731
],
[
"Shem",
2556,
6,
5526
],
[
"Shelah",
2421,
1,
2421
],
[
"Eber",
2391,
1,
9321
],
[
"Terah",
2236,
6,
3226
],
[
"Abraham",
2166,
1406,
2076
],
[
"Moses",
1526,
1406,
1616
],
[
"Flood_countdown",
2576,
1406,
8516
],
[
"Affliction_hub",
1846,
1406,
1846
]
],
"backbone": 14726
}
Results
{
"members": [
{
"member": "Mahalalel",
"source": 3711,
"anchor": 1,
"span": 3710,
"inverse_span": 1730,
"inverse_coordinate": 1731
},
{
"member": "Shem",
"source": 2556,
"anchor": 6,
"span": 2550,
"inverse_span": 5520,
"inverse_coordinate": 5526
},
{
"member": "Shelah",
"source": 2421,
"anchor": 1,
"span": 2420,
"inverse_span": 2420,
"inverse_coordinate": 2421
},
{
"member": "Eber",
"source": 2391,
"anchor": 1,
"span": 2390,
"inverse_span": 9320,
"inverse_coordinate": 9321
},
{
"member": "Terah",
"source": 2236,
"anchor": 6,
"span": 2230,
"inverse_span": 3220,
"inverse_coordinate": 3226
},
{
"member": "Abraham",
"source": 2166,
"anchor": 1406,
"span": 760,
"inverse_span": 670,
"inverse_coordinate": 2076
},
{
"member": "Moses",
"source": 1526,
"anchor": 1406,
"span": 120,
"inverse_span": 210,
"inverse_coordinate": 1616
},
{
"member": "Flood_countdown",
"source": 2576,
"anchor": 1406,
"span": 1170,
"inverse_span": 7110,
"inverse_coordinate": 8516
},
{
"member": "Affliction_hub",
"source": 1846,
"anchor": 1406,
"span": 440,
"inverse_span": 440,
"inverse_coordinate": 1846
}
],
"tenth_member": {
"backbone": 14726,
"nonzero_span_to6": 14720,
"Creation_held_self_span": 0
}
}
Finding. All nine supplied inverse coordinates reconstruct directly. The tenth member is the completed backbone itself: its informative expansion uses the14720 span to6, not a tenth nonzero Creation-held conversion. No member scan or new inverse operation is introduced.
Reassessment. Compress the nine Creation-held expansions into one exact coordinate rule.
C513 — One rule generates the14726 family
Question. Can the complete set of Creation-held results be compressed into one affine formula?
Source basis. File52c §§3.7.3,3.11.4; C509,C512
Frozen inputs
{
"inverse_coordinates": {
"Mahalalel": 1731,
"Shem": 5526,
"Shelah": 2421,
"Eber": 9321,
"Terah": 3226,
"Abraham": 2076,
"Moses": 1616,
"Flood_countdown": 8516,
"Affliction_hub": 1846
},
"held_anchor": 14726
}
Results
{
"normal_form": "X=6+115k;14726=6+115*128;Creation-held result=125k−1274;Nativity-held result=125k+6",
"members": {
"Mahalalel": {
"k": 15,
"complement_k": 113,
"Creation_held": 601,
"Nativity_held": 1881
},
"Shem": {
"k": 48,
"complement_k": 80,
"Creation_held": 4726,
"Nativity_held": 6006
},
"Shelah": {
"k": 21,
"complement_k": 107,
"Creation_held": 1351,
"Nativity_held": 2631
},
"Eber": {
"k": 81,
"complement_k": 47,
"Creation_held": 8851,
"Nativity_held": 10131
},
"Terah": {
"k": 28,
"complement_k": 100,
"Creation_held": 2226,
"Nativity_held": 3506
},
"Abraham": {
"k": 18,
"complement_k": 110,
"Creation_held": 976,
"Nativity_held": 2256
},
"Moses": {
"k": 14,
"complement_k": 114,
"Creation_held": 476,
"Nativity_held": 1756
},
"Flood_countdown": {
"k": 74,
"complement_k": 54,
"Creation_held": 7976,
"Nativity_held": 9256
},
"Affliction_hub": {
"k": 16,
"complement_k": 112,
"Creation_held": 726,
"Nativity_held": 2006
}
}
}
Finding. Nine outputs follow from one formula and the supplied member coefficients. The1280 anchor-change difference is uniform. This compresses the calculation, while the nine source selections and their k values remain inputs needing explanation.
Reassessment. Test a whole three-node relation under the formula: the Abraham–affliction–Moses midpoint.
C514 — Preserved midpoint versus external role agreement
Question. Which Abraham–Moses relationships are automatic under the common map?
Source basis. File52c §§3.7.2,3.10
Frozen inputs
{
"inverse_Abraham": 2076,
"affliction": 1846,
"inverse_Moses": 1616,
"adopted_Moses_birth": 1526,
"Entry_Exodus_Conquest": [
1876,
1446,
1406
]
}
Results
{
"original_equal_gaps": [
230,
230
],
"transformed_triple": [
976,
726,
476
],
"transformed_equal_gaps": [
250,
250
],
"mixed_original_generated_Moses_midpoint": 1526,
"Entry_to_generated_Abraham_partition": [
430,
40,
430
]
}
Finding. The230+230 midpoint becomes250+250 automatically. By contrast, the agreement of726 with the schematic Samaria role and of the mixed2076/976 midpoint with adopted Moses1526 uses additional source roles. The430|40|430 partition likewise uses externally supplied anchors.
Reassessment. Reconstruct Shem’s whole partition to distinguish two-anchor presentations of the same family.
C515 — Shem’s partition is one affine family
Question. How do the10000+6000 and held-Creation Shem presentations fit together?
Source basis. File52c §§3.6,3.8.2
Frozen inputs
{
"Creation": 14726,
"inverse_Shem": 5526,
"Nativity": 6
}
Results
{
"original_parts": [
9200,
5520
],
"expanded_parts": [
10000,
6000
],
"Nativity_held_triple": [
16006,
6006,
6
],
"Creation_held_Shem": 4726,
"partition_ratio": "5/3"
}
Finding. The9200+5520 split becomes10000+6000 under one Nativity-held dilation. Holding14726 instead places Shem at4726. The two Shem results differ1280 by the anchor-change law; they are related presentations, not independent outcomes.
Reassessment. Apply the same whole-family reasoning to the Mahalalel–Shelah–Eber trio.
C516 — The Cainan-associated trio as one transformed object
Question. What does one common dilation explain about the complete trio?
Source basis. File52c §3.12.1–3.12.2
Frozen inputs
{
"inverse_Eber_Shelah_Mahalalel": [
9321,
2421,
1731
],
"anchor": 14726
}
Results
{
"input_triple": [
9321,
2421,
1731
],
"output_triple": [
8851,
1351,
601
],
"input_gaps": [
6900,
690
],
"output_gaps": [
7500,
750
],
"outer_widths": [
7590,
8250
],
"radius_23_coefficients": [
235,
535,
565
],
"P_integral": [
false,
false,
false
],
"J_integral": [
false,
false,
false
]
}
Finding. The6900:690 partition becomes7500:750, preserving10:1. Its7590 outer width becomes8250. All three E results are integral, while P and J are fractional on these radii. The trio is one structured object with dependent displays.
Reassessment. Test the mixed original/generated Cainan midpoint, which is not forced by transforming the trio alone.
C517 — A genuine junction with the restored Cainan companion
Question. What extra constraint links the transformed trio to the separately supplied Cainan row?
Source basis. File52c §3.12; companion outside frozen MT manifest
Frozen inputs
{
"Cainan_birth": 2551,
"Cainan_begetting": 2421,
"Cainan_death": 2091,
"generated_Mahalalel": 601,
"reflected_generated_Shelah_AD": 1351
}
Results
{
"rounded_q": [
-2550,
-600,
1350
],
"adjacent_widths": [
1950,
1950
],
"linear_constraint": "2551=2*601+1351−2",
"Cainan_row_parts": [
130,
330
],
"civil_second_width": 1951
}
Finding. The separately supplied Cainan birth lies1950 before generated601, which lies1950 before reflectedAD1351 in rounded coordinates. This adds an actual placement constraint beyond the trio’s affine ratio preservation. The companion remains a local comparison, not a re-dating of the frozen MT chain.
Reassessment. Relate the330 remainder to both the cross-mode Cainan bridge and the trio’s width.
C518 — The330 interface between source row and rounded geometry
Question. What does the shared330 explain, and what does it leave as source placement?
Source basis. Strategy §§3.2,4.4; File52c §3.12; C488,C516–C517
Frozen inputs
{
"Cainan_b": 130,
"Cainan_remainder": 330,
"Cainan_lifespan": 460,
"trio_outer_width": 7590
}
Results
{
"cross_mode_bridge_increment": 330,
"trio_width_in_remainders": 23,
"expanded_width": 8250,
"gain_in_remainders": 2,
"relations": [
"inserting the row changes C−R by330",
"supplied inverse trio has width23*330",
"E sends23*330 to25*330"
]
}
Finding. The same remainder measures the difference between Cainan’s two chronological projections and the trio’s23→25 width unit. This is a concrete interface between genealogical and rounded/calendar modules. The source inversions supplying23 copies are an additional placement fact; the row identity alone cannot predict them.
Reassessment. Test the countdown’s460/430 pair as another reuse of the calendar module.
C519 — The countdown connects460 and430
Question. Can both countdown branches be generated without a branch-specific adjustment?
Source basis. File52c §3.11.2–3.11.3
Frozen inputs
{
"pillar": 14726,
"inverse_countdown": 8516,
"Moses_birth": 1526,
"call": 2091,
"Exodus": 1446
}
Results
{
"input_radius": 6210,
"outputs": {
"P": 8426,
"E": 7976
},
"Moses_comparisons": {
"P": 6900,
"E": 6450
},
"branch_separation": 450,
"call_Exodus_scaled": 6450,
"matched90": [
90,
90
]
}
Finding. The fixed6210 radius generates8426 and7976. Their Moses comparisons are15×460 and15×430; the450 separation follows from(E−P)6210. Matching the already adopted Moses birth and call/Exodus roles is additional source information, while the two branch outputs share one calculation family.
Reassessment. Explain the Temple/Moses/backbone convergence by identifying its smallest supplied junctions.
C520 — The Temple/Moses/backbone junction
Question. Which supplied equality generates the connected526/476 arrangement?
Source basis. File52c §§3.8–3.9
Frozen inputs
{
"BC_pillar": 14726,
"AD_pillar": 12026,
"inverse_Temple_BC": 696,
"inverse_Moses_BC": 1616,
"reflected_Exodus_AD": 1446,
"coordinate": "Rounded"
}
Results
{
"pillar_width": 26750,
"sum_original_radii": 24610,
"E_sum_original_radii": 26750,
"Temple_and_backbone_q": [
525,
525
],
"Moses_and_second_backbone_q": [
-475,
-475
],
"source_gap": 920,
"expanded_gap": 1000,
"forward_reflected_Moses_q": 1525
}
Finding. The first convergence is E(14030+10580)=26750, the declared outer-pillar width. The source920 gap then forces the two476BC routes and the1000+1000 reflected arrangement. These connected consequences share the first junction and are not additional independent trials.
Reassessment. Separate civil comparisons in the actual field from these rounded Mirror relationships.
C521 — Civil links from the actual field
Question. How do the AD25 comparisons connect to the actual generated field without borrowing rounded counts?
Source basis. File52c §3.15.5
Frozen inputs
{
"generated_Noah_BC": 2276,
"generated_Flood_BC": 2426,
"Exodus_BC": 1446,
"external_AD": 25
}
Results
{
"Noah_to_AD25_civil": 2300,
"Flood_to_AD25_civil": 2450,
"Flood_Exodus_AD25_parts": [
980,
1470
],
"rounded_countercounts": [
2299,
2449
]
}
Finding. The source’s2300 and2450 use civil BC+AD−1. The980+1470 decomposition follows from the same three dates. These are valid links to an external AD25 comparison, but they do not license changing the rounded Mirror metric used by the526/476 family.
Reassessment. Compress the actual12026 field into a parameter basis and identify its genuinely separate placement assumptions.
Evidence
The accompanying evidence checkpoint contains the sequential journal, exact-arithmetic step scripts, source snapshots and SHA256 manifest. Latest step: C521. Assertions verify arithmetic or explicit scope decisions; they are not independent historical or statistical witnesses.