C09 — Seven 529 families, with the 11-fold Conquest connection integrated
21 September 2026 · Continues C08 and incorporates Dean's new 11×529 branch handoff.
The source-and-dependency map now includes the 2-, 10-, 11-, 12-, 13-, 17- and 20-fold families. The attachment's arithmetic is confirmed: the native full-week SP Creation-to-Conquest-mirror width is 5820; the proposed compression of its additional 300-year SP–MT separation to 299 makes it 5819.
A further distinction strengthens the integration: 5819 already has seven interior native endpoint pairs, and two use exactly C01's Creation origins. At those two origins, a reflected Conquest member and the reflected MT Flood start delimit a sourced 1058 interval. Thus
has an explicit contiguous source path. The full-week compression remains a separately labeled hypothesis.
Creation stays Normal/G2. Regular Gear changes remain confined to Noah/Shem/Flood; the regular MT/SP frame is full-430, Terah-shift-zero and Cainan OFF. The existing C05 apparent field retains its own +30 label. The new derived −1 Creation comparison is neither a Creation Gear nor an extension of the SP Noah/Shem companion rail. All AD targets below are reflections or comparison images of source boundaries.
The family map distinguishes shared arithmetic from each construction's endpoints. Every row follows the scalar carriers 529→575→625 under two applications of 25/23. The status column specifies what that means for its dates.
| Multiplier | Base | First expansion | Second expansion | Source role and qualification |
|---|---|---|---|---|
| 2 | 1058 | 1150 | 1250 | Retained cumulative ON-earlier to OFF-later comparison, 460+598. C09 adds a distinct sourced Conquest-to-Flood interval of the same length. Later entries are generated in the original expansion construction. |
| 10 | 5290 | 5750 | 6250 | Derived LXX OFF cumulative Flood opening coordinate. The first image also equals the retained native ON opening. The second is generated. |
| 11 | 5819 | 6325 | 6875 | Native SP Creation to interior Conquest-mirror members; full matched week conditional on the proposed 300→299 compression. Later entries are formal scalar images in this construction. |
| 12 | 6348 | 6900 | 7500 | Native cumulative LXX Shem opening 6350 to the held comparison landing 2 BC. Expansion generates left boundaries 6902 and 7502. C05 separately supplies sourced apparent-Creation intervals of widths 6900 and 7500. |
| 13 | 6877 | 7475 | 8125 | C01's two complete chains hold ordinary SP Creation fixed and reach reflected MT Flood, MT Noah and SP primary Noah. All three widths have retained endpoints in those chains. |
| 17 | 8993 | 9775 | 10625 | Sourced cumulative/regular Noah–Flood comparison, 8395+598. Later entries are generated in the original expansion construction. |
| 20 | 10580 | 11500 | 12500 | Self-Mirror width of the 5290 opening. The first image also has the native ON 5750 self-Mirror realization; the second comes from the generated 6250 opening image. |
The numerical stages are uniform, but their source mechanisms differ. In particular, 5290 is a coordinate; 10580 is its paired self-Mirror interval. C01's successive 13-fold widths compare different retained events against one Creation origin. C04 established that this is not automatically the same operation as expanding the entire original cumulative/regular configuration. A number's separate appearance elsewhere does not transfer source identity to a generated endpoint.
The main selection rules can be stated compactly:
| Construction | What selects it? |
|---|---|
| 10 and 20 | The cumulative opening member, rather than an interior week member. The 20-fold width doubles the 10-fold coordinate. |
| 2 and 17 | The retained 598 comparison and the required cumulative/regular endpoint support. C08 shows their divisibility tests are equivalent in its finite domain. |
| Complete 13-fold chain | The same 598 comparison plus the ordinary Creation-member condition that gives 7475 to MT Noah. |
| 12 | Native ON cumulative Shem opening and the declared 2 BC landing; the 6900/7500 apparent-Creation realization is separately anchored. |
| Native 11 | Creation member offset one greater than the matched Conquest-mirror offset. |
| Derived full-week 11 | The proposed 299-year SP–MT separation with the MT and Conquest fields held fixed. |
The native Conquest calculation preserves the one-year full-week residual. File_18's held fields are MT Creation 4121–4114 BC, SP Creation 4421–4414 BC, and Conquest/rest 1406–1399 BC. Reflection reverses the Conquest field, giving AD 1399t/1400n → AD 1406t/1407n.
Write the paired centers at matched displayed offsets as
Then
Advancing each native SP Creation member by 5819 instead gives
The whole translated field is therefore AD 1398t/1399n → AD 1405t/1406n, exactly as the attachment states. It is one year before the matched Conquest-mirror field. Both phases obey (C+1)+R−1=C+(R+1)−1=C+R.
Interior membership supplies a source comparison without changing either native week. For offsets i=1 through 7, the 11-fold landing is an actual Conquest-mirror member at offset i−1. There are seven matching annual boundaries, covering six elapsed years. This is an offset correspondence inside the two fields, not a complete matched seven-year overlap.
The native 13-fold landings reach MT G1/G2/G3 Flood starts at Creation offsets 0/2/4. Combining this with the interior Conquest condition leaves offsets 2 and 4, precisely the origins of C01's complete chains:
| Native SP Creation origin | Reflected Conquest member | Reflected MT Flood start | Contiguous widths |
|---|---|---|---|
| 4420t/4419n BC, offset 2 | AD 1400t/1401n | AD 2458t/2459n, G2 | 5819 + 1058 = 6877 |
| 4418t/4417n BC, offset 4 | AD 1402t/1403n | AD 2460t/2461n, G3 | 5819 + 1058 = 6877 |
These are two placements of the same comparison. At native offset 0, the 11-fold landing is AD 1398t/1399n, outside the retained Conquest field; that opening control is not promoted to a Conquest endpoint.
The new and old 1058 intervals now have an explicit endpoint correspondence. The cumulative comparison keeps its native ON/derived OFF distinction; the new interval uses reflected Conquest and MT Flood:
| Placement | Cumulative interval, width 1058 | Reflected regular interval, width 1058 | Corresponding cross-Mirror widths |
|---|---|---|---|
| N02 | 6346t/6345n BC → 5288t/5287n BC | AD 1400t/1401n → AD 2458t/2459n | 6345+1400 = 5287+2458 = 7745 |
| N04 | 6344t/6343n BC → 5286t/5285n BC | AD 1402t/1403n → AD 2460t/2461n | 6343+1402 = 5285+2460 = 7745 |
In signed paired-center coordinates, translation by 7745 carries each displayed cumulative interval onto the corresponding reflected regular interval. Both terminal pairs are sourced. This is a finite comparison of distinct node classes and states, not a global chronological action or a new Mirror axis. The 7745 width is inherited from 8395−650, the held cumulative/SP Flood Mirror width less the SP–MT displacement. It supplies the endpoint connection behind the equal 1058 lengths without making them independent numerical discoveries.
Dean's 300→299 proposal supplies exact full-week alignment on a derived field. Hold each MT Creation member and its Conquest partner fixed. Contract only the SP–MT gap:
The derived SP week is 4420–4413 BC, with paired display 4421t/4420n → 4414t/4413n BC. It satisfies
at all eight boundaries. Seven derived boundaries overlap members of the native SP field; the closing 4413 boundary lies outside it. The derived week remains a comparison construction. Its exact alignment verifies the arithmetic of the proposal; the interpretation that this particular 300-year separation should be compressed remains provisional.
The scope of the compression matters. Applying 299/300 to the entire 5820 bridge would give 5800.6, not 5819. Likewise, the ordinary 25/23 expansion of 5520 gives 6000, but expanding only that component and then adding 299 gives 6299. Expanding the entire 5819 interval gives 6325, the first 11-fold image. These are distinct declared operations.
C05 supplies a prior sourced correspondence for the same compression ratio. Inverting its established 300/299 completion, while holding each reflected right endpoint, gives:
| Retained input interval | Inverse 299/300 image | Left-boundary change |
|---|---|---|
| Apparent SP closing −4444 → MT G1 Flood 2456; width 6900 | Ordinary SP opening −4421 → same Flood; width 6877 | +23 in signed center |
| Apparent SP closing −4444 → MT G1 Noah 3056; width 7500 | Ordinary SP member −4419 → same Noah; width 7475 | +25 in signed center |
Thus the inverse ratio already has qualified endpoint realizations in the retained packet. Its action depends on the interval's held right anchor: the same apparent left boundary has different images in these two examples. Dean's new proposal instead holds each MT Creation member as the right anchor of a 300-year SP–MT interval. The prior examples strengthen the analogy, but do not by themselves establish the new application. The broader resemblance to SP's personal −1 behavior is retained as an analogy; its Noah/Shem-only scope is unchanged.
The resulting dependency map has several different kinds of equality. Let Fₖ denote the numerical k-fold family at a common carrier stage. These equations hold at carriers 529, 575 and 625, with the following source qualifications:
| Relation | What the relation establishes |
|---|---|
| F₂₀=2F₁₀ | The paired self-Mirror construction at a retained opening; later generated openings retain their generated status. |
| F₁₂=F₁₀+F₂ | The arithmetic 6350−2=5290+460+598. It does not concatenate the earlier sourced 1058 interval with the 5290 coordinate. Splitting the actual 6350→2 interval at 5290 instead gives 1060+5288. |
| F₁₃=F₁₁+F₂ | At the source stage, the new contiguous Creation→Conquest-mirror→Flood paths above. Here F₂ has the Conquest/Flood endpoints. |
| F₁₇=F₁₃+2F₂ | C08's dependent width identity. The difference between its nested 8993 and 6877 intervals comprises flanks 868 and 1248, not two 1058 flanks. |
| F₁₇=F₁₁+3F₂ | A scalar consequence of the preceding two identities; it does not establish a new three-step 1058 event path. |
This extends the original square-of-23 audit while retaining the distinction between an opening coordinate, a self-Mirror width, a selected cumulative comparison, a sourced event chain, and a derived comparison field. It also preserves C05's boundary: its sourced 6900/7500 intervals do not turn the aligned 6348 image at AD 1904 into a retained Noah/Shem/Flood event.
Evidence and checkpoint. evidence/check_c09.py passes 134 exact checks, with a byte-identical isolated replay. Ten frozen dependencies include the new branch handoff, unchanged File_18/File_09, the original Mirror handoff and Report 09 results, and C01/C04/C05/C08 records. C09_FAMILY_MAP.json supplies the seven-family register with typed dependencies and stage qualifications; C09_RESULTS.json records all eight native/derived week members, the two new 1058 correspondences, the inverse-completion controls and the operation distinctions. Canonical sources and the historical graph remain unchanged; no earlier campaign is rerun and interrupted Reports 11–25 remain unestablished as completed.
Best next bounded step: fit the native Conquest endpoints and 11-fold widths into the existing three-coordinate Mirror description, then determine whether the proposed full-week compression requires an additional parameter or only a separately qualified boundary rule. High is sufficient.