# Research trajectory for the sixth handoff, C266–C479

This is a suggested main-handoff section. Conclusions below are read from the actual milestone reports; older next-step paragraphs are superseded by C479.

## From the new axis report to the 2K/17K source geometry

**C266–C276** integrated Dean's substantial 4232/8464/12696 addition. The inherited cumulative sources give the explicit sequence **5+1+1+5+1+1=14**, a local junction at January AD 2, and selected radii 5, 6 and 7. Radius 6 has seasonal component radii 23/4 and 25/4, supplying the ratio 25/23. At the declared scale 1000 these become 5750 and 6250. The fixed charts 1196 and 1300 connect that bracket to the inherited 13K chains. The global Mirror remains January AD 1.

The important limitation is now precise. Translation values and chart scales are determined **when the declared joining and calibration conditions are imposed**. The source 598 interval alone does not prescribe a 598 difference between translation parameters; requiring selected images to align supplies that extra condition. The 7K regular Noah shift follows from the retained 17K transport and held 5290 placement. These equations clarify a documented construction without treating its equivalent formulations as independent witnesses.

**C277–C310** established a complete finite source domain and the four-node normal form: 2K on each short side, cross-widths 17K/19K/15K/17K, and four complete source rows. The six duration equations have rank 3; absolute placement remains a premise. Exact 8K reflection and the pure-K rectangle are separately supported but do not coincide in the tested domain. A related fixed-anchor comparison covers seven cumulative annual positions only by combining ordinary MT Noah/Flood and alternate SP Shem-life contexts. One position remains unsupported.

**C311–C323** clarified that composite cover. The regular displacement is −5 on the SP primary rail and −6 on its companion, from 650−500+60−215−rail. A reversed annual-index correspondence supplies the 500-year binding with explicit phase reversal. Six direct OFF 600/17K rectangles and three native ON/MT primary counterparts are sourced; the full primary biography translates by 460. However, two 17K relations alone also permit a valid 1060-year rectangle. Selecting 1058 requires the two specific +2 choices. The bounded source audit found no instruction prescribing the exact annual context switch.

## Lifespans and the two expansion stages

**C324–C333** inserted Noah's actual death into the existing 2K side: **1058=950+108**. The primary chain is 460+490+108. Companion rows use actual MT Noah births and therefore give 459+491+108; missing MT person companions cannot be supplied to force 490. These death criteria select the regular side but leave the cumulative annual choice open.

**C334–C343** obtained complete Noah-held source paths about 3706 and 3708:

- 2648→2556→2456 about 3706;
- 2650→2558→2458 about 3708.

Distances from held Noah expand **1058→1150→1250**, with gains 92+100. The detailed primary partition is 460+490+108+92+100. Actual companion and 502/98 middle intervals are 1149 and 1152 respectively, so their outer squared-E equality does not provide this complete intermediate. The scalar square ladder was already known; this batch supplied a particular source realization.

**C344–C353** compared File_02's J=300/299 gap conversion with those source states. Both cumulative and regular interiors can be repaired to **458+600=1058**, preserving the original outer 2K/17K corners. Keeping the primary 460/490 partition instead gives 1060. Anchored E/J relations have precise partial source domains; a converted gap does not automatically transport an entire 950-year biography.

**C354–C363** joined the two ratios: **460E+598J=500+600=1100**, reaching an actual SP Noah→Shem-birth→Shem-death chain. This positive source biography changes an outer endpoint by 42 and therefore breaks one original 17K cross-span. The SP/MT Shem biographies also give **600+50+600=1250**. Held Shem 3206/3208 sends 2148/2150 to 1956/1958 under E², but the ordinary E intermediates 2056/2058 are absent. Later work found a different sourced factorization for the 3206 map.

## Reduced operator relationships and their limits

**C364–C373** connected the native cumulative Noah 950-year interval to regular SP by 19K, retaining a birth/death role exchange. More importantly, P=70/69 and J=300/299, at actual Flood pivots 460 apart, share a two-year gain and reach the same sourced endpoint. The primitive integer grammar is:

**69m→70m; 299m→300m; pivot separation 230m; whole side (230+299)m=529m.**

The source separation fixes m=2. The first source segment must also equal the pivot separation; a 529 side by itself does not guarantee the paired conversion. The joint application remains a derived comparison rather than a located source prescription.

**C374–C383** completed the finite common-P/J census: six cumulative and four regular source arrows, with only gain 2 supported. When pivots move by h, the two mixed composition orders differ by **(D−230h)/20631** and commute exactly when D=230h. A moving-frame normal form depends on operator counts. Fixed pivots do not commute. Continued paired movement changes cross-spans by 4, so the local structure does not close globally on 17K.

**C384–C393** supplied a sourced E/J square connecting MT Noah repair to SP/LXX Shem births. A representative square reaches 3208 either through 3246→3248→3208 or through 3246→3206→3208, with the appropriate sourced pivots. An unfiltered coherent-family census recovers the same source motif. The 500/600 geometry forces h=2, input distance 460=230h and J gap 598=299h; hence the 2K side. Transported P pivots and complete cumulative Mirror images are absent. The result explains the arithmetic dependency without adding independent witnesses.

**C394–C403** joined two complete E/P/J return graphs through the same MT Noah record 3033, yielding three vertices 3031/3033/3035. Every admitted path has an endpoint-and-operator-count normal form. Complete circuits have multipliers **1750/(3K)** and **7500/(13K)** and commute about their source vertex; their constituent operators do not commute. A denominator argument shows that no nonempty positive circuit preserves another coordinate in the frozen 373-coordinate register. This is finite algebraic structure, not a global 529-preserving action.

**C404–C413** used the existing −500 relation to transport the Noah return graphs to Shem; a direct Shem census agrees. The 502 binding works on a smaller Gear domain. The complete −500 source domain contains one additional four-step coordinate chain, but that chain is not one biography. Translating the original regular side preserves 2K while changing fixed 17K cross-spans to 8493.

## The current source grammar and factorization results

**C414–C423** found one coherent 2K SP-companion→MT person profile preserving Noah's 950 and Shem's 600 across four stations. Its fifth station uses a mixed Flood boundary; the full same-boundary Flood year does not translate coherently. No fixed 17K join exists for that profile. Held 3899 gives a sourced E² endpoint from 2841 to 2649, but its ordinary middle 2749 is absent.

**C424–C434** reduced all 26 coordinate 2K pairs to four source templates. The all-pivot E² census found the sourced path **2148→2020→1956**, at pivots **3620 then 2756**, equal as an entire affine map to E_3206². Its gains are **128+64**. Different contexts and rails do not form an 864-year Noah lifespan. The +2 partner lacks 2022, which would require Gear 4. C434 corrected C432's E source excerpt locator to File_46 lines 117–133; the arithmetic had always used 25/23.

**C435–C469** identified the source return **3206→3170→3206**, with the same pivots. Their common midpoint 3188 is already MT Noah birth. Four sourced local reflection centers have a complete finite source grammar. Combining the source paths gives a common-word interval expansion:

| Stage | Lower / upper source coordinates | Width |
|---|---|---:|
| Input | 2148 / 3206 | 1058 = 2×529 |
| After E3620 | 2020 / 3170 | 1150 = 2×575 |
| After E2756 | 1956 / 3206 | 1250 = 2×625 |

Both endpoints undergo the same operator at each stage. This is a complete source-qualified interval comparison using context changes. No mixed word connects the members of a nonzero 529 endpoint gap in the fixed menu; that is a different question from transporting both ends of an interval. The fixed Mirror menu retains known corners but does not supply complete cumulative E closure.

**C470–C479** completed the 1150 and 1250 inventories. There are **51 coordinate / 101 labelled 1150 pairs**, three admitted fixed-E expansions to 1250, and only one complete 2K ancestry. All **74 coordinate / 130 labelled 1250 pairs** fall into seven source classes. Typed +500/+499 family correspondences preserve output structure without supplying every required input or pivot. Equal whole affine maps can have different source support. Companion-held families have five source inputs—including the inherited 3899/2841 alias—but no sourced middle. Eighteen fixed-Mirror trials retain the known 17K corners and supply no cumulative E path in the frozen 48-record menu.

**Best next bounded step—C480, proposed only:** classify the existing 51-coordinate/101-label 1150 inventory into disjoint source-role and context templates, retaining aliases; explain the three admitted fixed-E expansions and the failures without a new date, pivot, phase or operator.

## Direction and correction ledger recommendations

1. Actual C266 was the redirected endpoint ledger. The superseded AD1905–1911 proposal remains unexecuted.
2. January AD 1 is the global Mirror; January AD 2 is the local cumulative join. Later source reflection centers do not create another global axis.
3. C322 corrected an initial assumption: same-Gear Noah-bound MT uses 500/100; cross-Gear Arphaxad-bound 502/98 remains valid.
4. C333 companions use actual MT births and 491, rather than invented MT companions or forced 490.
5. C354's successful SP-companion Shem repairs show that the earlier companion Noah/Flood failure is role-specific.
6. File_02 J=300/299 is a gap conversion, not a date operator or Residue Protocol. Anchored maps are research comparisons with declared source domains.
7. C432's E excerpt locator is corrected by C434 to File_46 lines 117–133; numerical results are unchanged.
8. C465's mixed-cycle 12→12 count refers to vertices only. Including operator anchors gives 48→12.
9. C476 has five companion source inputs, including 3899/2841, and zero ordinary sourced middles.
10. C429's Noah-held ladders, C434's split-family boundary, C287's four rectangles and C461's five Mirror images remain inherited results when later batches compare them.
11. The corrected LXX8464 observation and SKL appendix remain awareness-only, with their exact phases and superseded errors retained.
12. Independent-agent review occurred in some early batches, including C314–C333. Later alternate implementations are root-agent second methods. Preserve each packet's actual review provenance.
13. Source labels, translated rows, aliases and verification-check counts are not independent historical witness counts.