C08 — The 13-, 17- and 2-fold families share a source-qualified 598 selection

21 September 2026 · Bounded continuation of C07 and original Report 09.

The joint test selects exactly C01's two sourced constructions: 6877=13×529, 8993=17×529 and 1058=2×529. Both use the 598 comparison. The strongest new relation is

This relation belongs to a common formula, not just the three selected numbers. It explains why the three square factors are dependent. Across the full tested Creation-member domain there are two independent residue conditions; once C07's 7475 width is held, only one remains.

The domain is deliberately finite: the same nine MT/SP Noah-birth/Flood-start Gear pairs tested in C07, crossed with the eight boundary members of the fixed ordinary SP Creation week. Creation remains Normal/G2. Regular three-Gear movement remains confined to Noah/Shem/Flood. The regular frame is full-430, Terah-shift-zero and MT/SP Cainan OFF; SP Noah uses its primary station. Cumulative LXX native ON and derived OFF fields supply explicitly distinguished comparison endpoints. Their use does not activate the older ON/+60 regular MT profiles attached to some historical certificate labels.

The three widths can be expressed with one shared selected span. Let i be Noah Gear, j Flood Gear, and use positive Nisan coordinates:

• Fⱼ: MT Flood start; Hⱼ: SP Flood start.
• Nᵢ: MT Noah birth; Sᵢ: SP primary Noah birth.
• C: a member of ordinary SP Creation, 4421 through 4414 BC.

File_18 gives

The retained Flood Mirror width is 8395, originally calibrated by cumulative LXX OFF opening 5290 and reflected SP G1 Flood close 3105. To retain that width with each admitted SP Flood start, select the cumulative OFF later member

For Flood G1/G2/G3, D is 5289/5287/5285, all inside the fixed OFF Flood/Shem-death comparison field 5290–5283. The equal-span earlier member and its native ON counterpart would be

These must be checked against the retained cumulative Shem-birth/Noah-death fields 5890–5883 OFF and 6350–6343 ON. They are cumulative comparison nodes, not regular Noah-birth labels.

The three tested widths are

A and B are cross-Mirror widths, with BC coordinate C represented by signed center −C and reflected regular coordinate R by +R. E is a same-side BC subtraction between separately typed cumulative states. The paired Mirror axis remains January AD 1, with no civil year zero.

The full Creation-member test resolves the dependence without assuming the desired upper width. Define

Then, throughout all 72 member/Gear combinations,

Their residues modulo 529 are therefore

Here −5≤q≤6, while ε∈{−2,0,2,4,6}. Joint divisibility consequently requires exactly

The first condition is L=598, selecting Noah G1/Flood G2 or Noah G2/Flood G3. The second selects Creation 4419 or 4417, respectively. These are exactly C01's two complete 6877→7475→8125 chains. Both have all required cumulative members inside their retained fields.

B and E always differ by

Their divisibility tests are therefore equivalent. A additionally depends on the Creation-member condition. These are two algebraic conditions on this finite comparison packet, not three independent numerical discoveries or a count of independent historical witnesses.

On C07's held-7475 comparison, all three families reduce to one parameter. Setting q=0 gives

This table gives formulas; source membership is checked separately below. Its complementary identities are

Thus the selected multiplier relation is 13+2=17−2=15, equivalently 17=13+2×2. The two outer widths change in opposite directions when L changes, while the 2-fold comparison changes with the 17-fold width. The 529 condition selects their common zero-residual position.

The finite width triples have affine rank 2 when Creation members vary independently, and rank 1 after 7475 is held. These ranks concern only the three widths in this diagnostic. They do not revise the original three-coordinate Mirror geometry.

Both joint selections have explicit retained endpoints.

The 1058 decomposition is 6345−5885+5885−5287=460+598, or 6343−5883+5883−5285=460+598. Its ON earlier coordinates agree with the cumulative coordinates retained under N07/N08. Only those cumulative coordinates are used here; the old regular profiles remain separately scoped.

The two 8993 diagonals agree because U−D=Sᵢ−Hⱼ=L. At the selected cases, this is 598 on both sides. The ordinary Creation-to-Noah widths remain 7475 and 8125, and the cumulative-to-Creation gap is 868, retaining C06's 8993=868+8125 attachment.

The new identity 8993−6877=2×1058 is a relation between widths. In the source display, the 6877 interval lies inside the 8993 interval; their difference consists of two unequal flanks, 868+1248=2116. Neither flank is 1058. The existing 1058 witness is the separate cumulative cross-state interval in the table. Likewise, 7935 and 15870 are exact width identities here, without an added claim that they form new contiguous event paths.

Neighboring controls distinguish arithmetic availability from endpoint support. The earlier cumulative members required by the nine Gear pairs are:

All nine cases have sourced ordinary Creation, regular Noah/Flood and cumulative later endpoints for A and B. Six also have the earlier cumulative endpoints needed for E and the equal opposite B side. In the other three, those particular earlier-member constructions remain formal. The table does not search other node classes for replacement endpoints.

The three same-Gear cases and the supported 602 case are meaningful negative controls: their endpoints exist, yet their widths fail joint 529 divisibility. The selection is therefore not merely an artifact of excluding unsupported neighbors.

6877 alone does not select 598. Keeping only C+Fⱼ=6877 selects Creation 4421/4419/4417 for Flood G1/G2/G3, irrespective of Noah Gear. These are three distinct Creation-to-Flood intervals appearing in nine configurations. For example, ordinary SP opening 4421, MT G1 Flood 2456, MT G1 Noah 3056 and SP G1 primary Noah 3706 give

with gains 600 and 650. The corresponding joint widths are 6877,8995,1060. All their required endpoints are retained, but neither the complete 13-fold expansion nor the joint three-family selection survives. The result concerns the full qualified configuration, not a rule that every occurrence of 6877 must involve 598.

Historical source-support counts remain distinct from these controls. Original Report 09's 598 layer still contains 16 records: nine complete retained joins and seven Flood-only records. C01's SP primary Creation attachment occupies four of the complete chart placements—J02/J03/J04 over N02, and J05 over N04—representing two Creation chains. The six field-supported Gear controls above are newly evaluated endpoint comparisons; they do not change the old graph's 66 records, support labels or join counts.

Evidence and checkpoint. evidence/check_c08.py passes 174 exact checks, with a byte-identical isolated replay. Ten frozen dependencies include unchanged File_18/File_09, the original Mirror handoff and selection certificate, original Reports 06/09 result records, and C01/C03/C07 records. The evidence enumerates all 72 Creation-member/Gear combinations, distinguishes the six field-supported controls, records both selected endpoint witnesses, and checks the residue relations and finite affine ranks. No old campaign is rerun, no canonical source changes, and interrupted Reports 11–25 remain unestablished as completed.

Best next bounded step: consolidate the 2-, 10-, 12-, 13-, 17- and 20-fold constructions into one source-and-dependency map, distinguishing opening selection, the shared 598 selection, and generated expansion images. This would integrate the new 6877 construction into the original square-of-23 audit. High is sufficient.