490d C433 — Four source templates and the 128 plus 64 factorization
24 September 2026. Step 10 of10, C424–C433.
The batch adds two structural results: four source-context templates account for the complete regular2K census, and one new sourced factorization realizes the held-Shem3206 squared-E map. The second result emerged from the wider pivot search and changed the final research step to a focused source audit.
The new 128 plus64 path
The exact source path is 2148→2020→1956, using E pivots 3620 then2756. Its two legs are:
| E leg | Actual pivot | Input gap | Expanded gap | Coordinate change |
|---|---|---|---|---|
| First | 3620 | 1472 =64×23 | 1600 =64×25 | 2148→2020, gain128 |
| Second | 2756 | 736 =32×23 | 800 =32×25 | 2020→1956, gain64 |
The total change is192, the same as the92+100 split for a double expansion of1058 to1250. The whole affine map agrees, not merely the value at one input:
E_2756 ∘ E_3620 = (E_3206)²,
because (25×3620+23×2756)/48=3206. In coefficients, both sides are (625t−307776)/529.
The source labels are essential:
| Function | Actual source |
|---|---|
| 2148 input | MT Shem death G2/C1/T1/F1 |
| 3620 first pivot | SP companion Noah birth G1/C1/T0/F0 |
| 2020 intermediate | MT Shem death G3/C0/T1/F1 |
| 2756 second pivot | SP primary or LXX-OFF Noah death G1/C0/T0/F1 |
| 1956 output | MT Shem death G1/C0/T0/F1 |
| 3206 effective pivot | SP primary or LXX-OFF Shem birth G1/C0/T0/F1 |
There are two labelled path versions, choosing SP primary or LXX-OFF for the second pivot. They are not independent witnesses. The three stations are different source states of MT Shem death, not three events in one unchanged biography.
Nor do the actual pivots form an864-year Noah lifespan. SP companion birth3620 has its own death2670, preserving950. The second-pivot death2756 belongs with birth3706, preserving the SP-primary qualified950 or the LXX-OFF950. The864 separation crosses context and rail settings.
The fixed-pivot intermediate2056 remains absent. The new path uses a different, sourced intermediate2020 while preserving the complete composed map. This extends the earlier restricted return-graph search; its result does not contradict the earlier negative conclusion about those particular graph paths.
The+2 partner still fails:3622/2758 and2150/1958 are sourced, but intermediate2022 is absent and would require an unavailable Gear4 MT Shem-death state. All eight fixed input/intermediate/output quarter-component assignments fail for the two unshifted pivots. Jointly shifted comparisons remain exact, but change both pivots. The result is a sourced affine factorization with context changes, not a source-prescribed calendar operation or one fixed seasonal action.
Four templates for the full2K census
C414's existing46 labelled pairs at26 coordinate pairs all obey one source accounting formula. The six MT G1/C0/T0/F0 role baselines are Noah birth2841, Shem birth2341, Flood start2241, Flood close2240, Noah death1891 and Shem death1741. Add the tradition baseline, variants and Gear offset; subtract one only for the local SP person companion.
| Template | Context difference | Role/local correction | Labels | Coordinate cases |
|---|---|---|---|---|
| T1 | 650−130−60−2 =458 | +600, or+601−1 | 20 | 12 |
| T2 | 650+60−2 =708 | +350 | 16 | 8 |
| T3 | 650+130+60+215+2 =1057 | +1: Flood start→close | 4 | 2 |
| T4 | 650+130+60+215+4 =1059 | −1: close→start or SP person companion | 6 | 5 |
Each row reaches1058. The LXX baseline is normalized to its explicitly theoretical CainanOFF setting for arithmetic comparison; native LXX remains CainanON. Equal SP/LXX coordinates retain distinct source identities.
C424–C425 established the T2 class by checking every one of6912 Flood-start/Noah-death record pairs. Exactly16 labels survive, all with the same equation1058=350+708. None has another2K station match among the768 station-pair comparisons within its fixed contexts. Same-role primary/native profiles differ708; SP companion person stations differ707.
The table's coordinate counts total27 because exactly one pair is shared:3298/2240 belongs to T2 and T3. At2240, MT Noah death G3/C1/T0/F1 coincides with MT Flood close G1/C0/T0/F0. The equation is349=130+215+4. The full576-case MT coincidence audit also finds2300 at the common+60 setting, but its required2K upper3358 is absent. These are coordinate coincidences between distinct histories.
Complete Mirror and harmonic coverage
The26 regular2K pairs crossed with all six cumulative Shem/Flood2K pairs give156 fixed17K trials. The only four coordinate rectangles are the existing C287 i2–i5:3705/2647,3706/2648,3707/2649 and3708/2650. All belong to T1. Their six labelled regular realizations include companion i2/i4 and primary/LXX i3/i5. Later operator qualifications that select primary i3/i5 remain separate from rectangle membership. No new Mirror branch, source point or axis was fitted.
The complete104-case fixed-upper E/E²/P/J inventory has13 sourced E outputs and13 sourced E² outputs. P and J outputs are nonintegral. Four E² maps have their ordinary sourced intermediate: the known primary Noah ladders held at3491,3493,3706 and3708. The remaining nine were initially endpoint-only.
C431 then tested every one of373 possible first pivots for all13 sourced E² targets. The4849 trials give39 sourced pivot-pair solutions,26 with distinct pivots. Exactly five have sourced intermediates: the four existing ladders and the new3206 factorization. Eight initially endpoint-only maps remain without a path in this generous menu. The held3899 map still requires its absent2749 intermediate.
The proof extends to every finite positive E/P/J word equal as a complete affine map to one of these targets. Prime7 forces P count0, prime13 forces J count0, and prime5 forces exactly two E factors. This does not exclude inverse operators or different-slope maps that agree at only one endpoint.
The Flood-held E strip
T2 and T3 supply nine distinct Flood-held E arrows with20 labelled realizations. T2 holds SP/LXX Flood start with Terah+60, taking next-Gear MT Noah death/Terah0 to same-Gear MT Shem death/Terah+60 with matching Cainan and sojourn flags. It exists at upper G1–G2.
At CainanON/full430, T3 supplies a Flood-close input route at upper G2–G3. Their overlap produces:
| Upper Flood Gear | Pivot | Actual input | MT Shem-death output |
|---|---|---|---|
| 1 | 3296 | 2238: Noah death G2 | 2146 |
| 2 | 3298 | 2240: Noah death G3 or Flood close G1 | 2148 |
| 3 | 3300 | 2242: Flood close G2 | 2150 |
Thus two two-Gear families meet at a source-coordinate alias. There is no single input role across all three Gears and no admitted Gear4 Noah. The triples advance by2 and satisfy the changing-pivot identity E_(H+2)(x+2)=E_H(x)+2, with source roles kept explicit.
Step sequence and verification
| Step | Completed bounded result |
|---|---|
| C424 | Fixed-context audit of the16 Flood-start/Noah-death labels |
| C425 | Unfiltered6912-pair source equation and forced708+350 pattern |
| C426 | Four-template reduction of all46 labelled2K pairs |
| C427 | Exact MT2240/2300 alias family and its2K boundary |
| C428 | Complete fixed17K join census; no new Mirror branch |
| C429 | Complete E/E²/P/J endpoint inventory |
| C430 | Source rules and input-role boundary of the Flood-held E strip |
| C431 | All-pivot E² path census; one new distinct-pivot source path |
| C432 | Direct source, biography, Gear and component audit of the new path |
| C433 | Synthesis, separate implementation and recovery verification |
Each next question was recorded after its predecessor's result. Reports, checkpoints and evidence were published before the next research step. C433 freezes all nine predecessor certificates, DATA/RESULTS files and next-step decisions. The first nine sealed packets passed integrity checks and18 isolated replay-output comparisons before synthesis.
The separate root implementation imports none of the main calculation modules. It regenerates the full528-record source grid directly from current table baselines, enumerates all labelled2K pairs, verifies all four templates and Mirror joins, computes integer affine outputs, and independently searches the Cartesian product of all regular pivot coordinates. Its719 checks reproduce the new factorization and its source/component controls. The result is byte-identical against the C433 frozen inputs. This is a second implementation by the same agent, not an independent-agent review or an independent historical witness.
The actual File_18 and File_09 bytes retain their controlling hashes; File_02, File_46, File_11 and the cumulative register match the C423 source manifest. The C265 checkpoint remains readable and its certificate matches the retained hash. Standing source and canonical controls are unchanged.
Use the C433 compact restart kit as the active packet after publication. The reading index locates all ten reports, ten checkpoints and ten evidence packages. The continuation archive preserves the complete batch and provenance. C434 remains unexecuted.
Verification and scope
75 calculation checks and 63 source/predecessor authentication checks passed. A separate main replay is byte-identical. Root-agent analysis and replay; no new independent-agent review is claimed. Verification counts are not independent historical witnesses.
The four templates and five source paths are finite results in the frozen register. The new varying-pivot map is not a reviewed prescription for a combined chronology, a single Noah biography or one fixed seasonal operation. No new phase, global529/Mirror action, source endpoint or canonical edit is established.
Mirror/529 focus: K=529, E=25/23, P=70/69; J=300/299 remains File02 gap conversion, not a calendar date operator or Residue Protocol. Regular Gears are Noah/Shem/Flood only. SP person companions are local to Noah/Shem; Flood close is an independent event boundary. Both500/100 and502/98 bindings remain valid; SP primary Noah death retains its inclusive/harmonic qualification. LXX native Cainan is ON; OFF is theoretical removal. Global paired Mirror January AD1; local junction January AD2; no civil year zero. No new source point, phase, fitted axis, upstream/cumulative Gear, primer reopening or canonical/graph edit. Capture now; canonize later. Coordinates, source labels, context aliases and derived comparisons remain distinct. Counts are not independent historical witnesses.
Decision after this result
C434, proposed and unexecuted: Test whether one source formula built from Noah950, the Noah–Shem500 relation, and the existing sojourn/Cainan/SP-companion context shift explains the new128+64 factorization. Then census its finite Gear/Terah family while requiring every pivot and intermediate to remain sourced, without fitting another anchor.
The positive factorization is now authenticated and its immediate limits are known. Explaining why these specific source roles and variant shifts produce its effective pivot is the next useful reduction; that source formula and family census have not been executed. The current authorization ends at C433.