{"id": "c434", "title": "Source Derivation and the Six-Case Gear/Terah Boundary of the 128 Plus 64 Factorization", "filename": "490d_C434_Source_Derivation_and_Six_Case_Boundary_20260925.md", "role": "Research action", "step": 434, "sha256": "0d0160a0b29069dca5191fad27d61bc55c6645d51b3e1487857e63c887073839", "bytes": 9391, "origins": ["C01–C479/packets/C434/490d_C434_Source_Derivation_and_Six_Case_Boundary_20260925.md"], "download": "/sources/0d0160a0b29069dca5191fad27d61bc55c6645d51b3e1487857e63c887073839.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Mirror / 529"], "excerpt": "490d C434 — Source derivation and the six-case boundary of the 128 + 64 factorization\n\n25 September 2026. One authorized bounded step after C433. C434 complete; C435 unexecuted.\n\nThe factorization now has a compact source explanation. The existing sojourn, Cainan and SP companion shifts give 215−130+1=86, and\n\n\\\n\\boxed", "content": "# 490d C434 — Source derivation and the six-case boundary of the 128 + 64 factorization\n\n25 September 2026. One authorized bounded step after C433. **C434 complete; C435 unexecuted.**\n\nThe factorization now has a compact source explanation. The existing sojourn, Cainan and SP companion shifts give `215−130+1=86`, and\n\n\\[\n\\boxed{25(86)+23(950)=48(500).}\n\\]\n\nThis places the effective double-expansion pivot exactly at the matched Shem-birth coordinate. All six declared Gear/Terah comparisons retain this whole-map identity. **Only G1 with Terah-base retains a fully sourced 2K input, intermediate and output in the frozen Noah/Shem/Flood register:** `2148→2020→1956 BC`.\n\nThis is an explanation of the previously found arithmetic using existing source roles and shifts. It does not establish a source-prescribed combined chronology or historical design intent.\n\n## 1. Derivation from the source roles\n\nSelect an SP primary Noah birth in the full-430 comparison frame, Cainan OFF:\n\n\\[\nN=3706+2(g-1)+60T,\\qquad g\\in\\{1,2,3\\},\\quad T\\in\\{0,1\\}.\n\\]\n\nHere `T=0/1` means Terah-base / Terah +60. The reference Cainan and sojourn settings remain fixed. Source File_18 §3.2.1 gives matched Noah lifespan 950 and matched Noah–Shem births 500. Therefore\n\n\\[\nH=N-500,\\qquad q=N-950.\n\\]\n\n`H` is SP primary Shem birth and `q` is SP primary Noah death in that same reference context. The primary Noah-death coordinate retains its inclusive/harmonic qualification.\n\nThe first pivot is another source-labelled Noah birth. Change the reference sojourn setting by −215, insert the explicitly opened Cainan +130 state, and select the SP local lower companion:\n\n\\[\np=N-215+130-1=N-86.\n\\]\n\nThis is a source-context change, with Gear and Terah held. It does not transport the companion rail into MT or LXX. Both actual Noah biographies retain 950: `N→q` and `p→p−950`. The pivot separation `p−q=864` crosses contexts and is not Noah's lifespan.\n\nFor a BC numerical coordinate `b`, write the existing gap expansion as\n\n\\[\nE_h(b)=h-\\frac{25}{23}(h-b)=\\frac{25b-2h}{23}.\n\\]\n\nComposition gives\n\n\\[\nE_q(E_p(b))=\\frac{625b-50p-46q}{529}.\n\\]\n\nIt equals \\((E_H)^2\\) for every `b` precisely when `25p+23q=48H`. Substituting the source relations gives\n\n\\[\n25p+23q=48N-[25(86)+23(950)]=48(N-500)=48H.\n\\]\n\nThe C431 integer parameter 18 is consequently explained by these source shifts:\n\n\\[\np-H=414=18\\times23,\\qquad q-H=-450=-18\\times25.\n\\]\n\nNo additional pivot was fitted to obtain this result. The first and second pivot records are the ones already authenticated in C432, transported through the declared six-case family.\n\n## 2. Why the gains are 128 and 64\n\nRetain the existing `2K=1058` input gap, with `K=529`:\n\n\\[\nx=H-1058=N-1558.\n\\]\n\nThe source-derived first pivot gives\n\n\\[\np-x=1472=64\\times23,\n\\]\n\nso the first expansion produces a 1600-year gap and moves `x` forward by 128. Its intermediate is `m=N−1686=H−1186`. The second pivot then has\n\n\\[\nq-m=736=32\\times23,\n\\]\n\nwhich expands to 800 and moves the intermediate forward by another 64. The output is `y=N−1750=H−1250`.\n\nFor G1/Terah-base this is exactly:\n\n| Leg | Pivot BC | Input BC | Output BC | Gap expansion | Gain |\n|---|---:|---:|---:|---|---:|\n| First | 3620 | 2148 | 2020 | 64×23 → 64×25 | 128 |\n| Second | 2756 | 2020 | 1956 | 32×23 → 32×25 | 64 |\n\nThe three MT stations explain the same partition directly:\n\n| Station | MT source state |\n|---|---|\n| 2148 | Shem death G2, Cainan ON, Terah +60, full-430 |\n| 2020 | Shem death G3, Cainan OFF, Terah +60, full-430 |\n| 1956 | Shem death G1, Cainan OFF, Terah-base, full-430 |\n\nThus **128 = 130−2**: remove Cainan and move one Gear upward. And **64 = 60+4**: remove Terah +60 and move from G3 to G1. Together, `192=130+60+2`. These are comparisons of source states, not successive events in one unchanged Shem biography.\n\nThe fixed-H double expansion still uses `2148→2056→1956`, whose intermediate 2056 is absent from the frozen register. The sourced factorization moves the intermediate by 36 while preserving the complete affine map.\n\n## 3. Complete six-case Gear/Terah census\n\nEvery displayed reference Noah, effective Shem pivot, first pivot and second pivot is present in the frozen register. A dagger marks a path station **absent from that register**.\n\n| Gear | Terah | Reference N | Effective H | Actual pivots p / q | Required path, BC | Complete? |\n|---|---|---:|---:|---|---|---|\n| G1 | Base | 3706 | 3206 | 3620 / 2756 | 2148 → 2020 → 1956 | **Yes** |\n| G2 | Base | 3708 | 3208 | 3622 / 2758 | 2150 → 2022† → 1958 | No |\n| G3 | Base | 3710 | 3210 | 3624 / 2760 | 2152† → 2024† → 1960 | No |\n| G1 | +60 | 3766 | 3266 | 3680 / 2816 | 2208† → 2080† → 2016 | No |\n| G2 | +60 | 3768 | 3268 | 3682 / 2818 | 2210† → 2082† → 2018 | No |\n| G3 | +60 | 3770 | 3270 | 3684 / 2820 | 2212† → 2084† → 2020 | No |\n\nThe negative result is restricted to the **528 labelled records at 373 coordinates in the C286 Noah/Shem/Flood register**. A missing coordinate here is not a claim that the number never appears elsewhere in the repository. No other event role from outside that register was admitted to repair a path.\n\nAll six compositions preserve the source-derived whole-map identity and the arithmetic 128+64 split. Only two inputs and one intermediate are in the register; all six outputs are present. The sole complete coordinate path has two actual-pivot label realizations because its second pivot can be SP primary or the coordinate-equal, explicitly theoretical LXX Cainan-OFF Noah death. These are not independent witnesses. The first pivot remains specifically SP companion.\n\n## 4. The boundary follows from the MT source formula\n\nWrite an MT Shem-death coordinate as\n\n\\[\nS(g,C,T,F)=1741+2(g-1)+130C+60T+215F.\n\\]\n\nThe three required stations are algebraically\n\n\\[\nx=S(g+1,1,T+1,1),\\quad\nm=S(g+2,0,T+1,1),\\quad\ny=S(g,0,T,1).\n\\]\n\nThese expressions are diagnostic formulas; they do not authorize G4/G5 or two Terah increments. Because source Gears are 1–3 and `T` is 0–1, the input requires `g≤2, T=0`; the intermediate requires `g=1, T=0`. The full-register membership check confirms that no other admitted role or alias supplies the missing coordinates. Hence the sole complete case is forced within this family.\n\nTwo short controls preserve the source precision:\n\n- Dropping the SP companion's −1 changes the effective pivot from `H` to `H+25/48`. The same whole-map identity fails.\n- Relabelling the effective relation as `N−502`, while keeping `q=N−950`, would require `p−N=−2246/25`, a noninteger displacement. This result uses the valid matched 500 binding. It does not reject the separately valid 502 binding or claim that every possible 502 construction has been tested.\n\n## 5. Source provenance, verification and limits\n\nControlling passages are File_18 §3.2.1 (SP person field, 950/500 relations, rails), §1.3.1 (215/60/130 variants), §2.2.1 (MT/LXX/SP person states), §3.1.4 (Flood Gear scope), the LXX theoretical-removal table, File_02 §6.2 (MT minimum person bases and transfers), and File_46 L3 (25/23). Frozen excerpts, full source files, every source record used, and all six failed/passing cases are in the evidence packet.\n\n**Provenance correction:** C432's card labelled `E_RULE` selected File_46 lines 97–113, which describe 70/69. The correct 25/23 passage is L3, lines 117–133 in these frozen bytes. C432's calculation already used 25/23 and its numerical result is unchanged. C434 records the corrected excerpt without rewriting the earlier sealed packet.\n\nC433's 75 payload hashes and 15 certificate bindings were authenticated before this step. The current File_18, File_09, File_02 and File_46 bytes match their inherited hashes. The C265 checkpoint remains available. The standing graph and source controls remain unchanged.\n\nThe C434 replay passes **97 calculation checks and 38 source/predecessor authentication checks**. An alternate implementation regenerates the 528-record domain directly from source tables and verifies the six cases by integer cross-multiplication; its **24 checks** agree. Main and alternate result files reproduce byte for byte. These are checks by the same agent, not an independent-agent review or additional historical witnesses.\n\nThe result remains a source-qualified affine comparison. The prior seasonal/component limits remain in force. Regular Gears stay at Noah/Shem/Flood; SP companions remain local; native LXX retains Cainan ON; OFF is a declared theoretical state. No source point, phase, fitted axis, cumulative/upstream Gear, primer reopening or canonical edit is introduced. The SKL appendix is retained as supplemental awareness and did not redirect this step.\n\n## Best next bounded step\n\n**C435, proposed and unexecuted:** Keep these six sourced pivot pairs fixed and test each of the 373 existing regular coordinates as an input. Require a sourced intermediate and output, and retain their event roles and context labels. The finite menu is 6×373 = 2,238 input/map comparisons.\n\nC434 explains the unique complete 2K path and its finite boundary. The next useful question is whether these same maps preserve any wider source paths, without choosing new pivots or adding coordinates. That would distinguish a single supported 2K comparison from a larger finite source relation. **No C435 trials have been executed.**\n", "evidenceLinks": [{"id": "d-68301ab760bfbe2e", "title": "File 18"}, {"id": "d-18e8c54ff1663fb1", "title": "File 46"}, {"id": "d-45851004c5864229", "title": "File 09"}, {"id": "d-24203a4c3b282f9b", "title": "File 02"}]}