How source variants connect the chronological families
Candidate explanatory passage for the C831 synthesis. This is preparation, not a numbered research action. The complete mode-profile reconstruction is already recorded in C733–C748; the finite-variant and rounding passage states the proposed integration for subsequent verified work. It does not certify those later steps in advance.
The chronological families fit together most clearly when we retain the ordered source rows. A patriarch's begetting interval determines the regular birth sequence; the full lifespan determines the cumulative sequence. Both are evaluated backward from their declared terminal. Their difference is therefore an ordered accumulation of the information that the regular construction leaves out.
Write a regular boundary as R_i, a cumulative boundary as C_i, and their difference as G_i=C_i−R_i. If the next regular boundary is separated by the effective begetting interval b_i and the next cumulative boundary by lifespan L_i, then
The effective interval includes any already declared counting or binding correction. It is not always identical to a printed cardinal age. Once the terminal difference is fixed, these local contributions reconstruct the entire mode bridge. Conversely, adjacent differences of the complete bridge recover its local contributions. The total at Creation alone cannot recover their order.
This one recurrence reconstructs all 61 retained native-Cainan node comparisons across MT, LXX and SP. In the common full430 regular comparison frame, the Creation comparisons are:
| Source state | Regular head | Lower cumulative member | Cumulative minus regular |
|---|---|---|---|
| MT, Cainan absent | 4114 | 14004 | 9890 |
| LXX, Cainan native | 5494 | 14894 | 9400 |
| SP, Cainan absent | 4414 | 13396 | 8982 |
The common comparison frame matters. Returning SP to its native215 regular head4199 changes the last bridge to9197. These are different placements of a declared family, not competing corrections to one number. Likewise, using cumulative center14006 instead of the lower completion14004 changes the MT bridge to9892 throughout that profile.
A complete shared birth sequence can retain different cumulative histories
SP and LXX make the distinction concrete. With Cainan restored in regular SP, every birth position from Noah through Abraham agrees with native LXX at matched settings. In the full430 frame the ordered positions are
The source-selected Noah and Arphaxad-bound Shem bindings remain attached to those labels. Cainan's130 restores the missing regular interval; the remaining birth-driving suffix is already shared.
Identical births do not imply identical deaths. The LXX lifespans exceed SP by27 for Arphaxad and Shelah,100 for Eber through Serug, and60 for Nahor and Terah. At a common birth, that excess shifts the LXX death label forward by the same amount. In cumulative chronology the differences accumulate instead: from Shelah through Abraham, the complete LXX-minus-SP profile is
Its successive differences recover27,100,100,100,100,60,60. Thus agreement of the entire regular birth path and divergence of the cumulative path are two consequences of the same source data. Neither needs to be treated as an exception to a universal date conversion.
Terah shows why the name of a variant is insufficient
The local Terah row supplies a particularly small example. Let b mean begetting age and r the remaining years.
| Source-supported row | b | r | Lifespan b+r |
|---|---|---|---|
| SP textual | 70 | 75 | 145 |
| MT/LXX baseline | 70 | 135 | 205 |
| The130/205 alternative | 130 | 75 | 205 |
Moving from SP textual to MT/LXX baseline adds60 only to the lifespan. Moving from that baseline to the130/205 alternative adds60 to the begetting interval while preserving the lifespan. Their combined regular/cumulative effect is
With regular Abraham1951 held, the three Terah birth/death pairs are2021/1876,2021/1816,2081/1876. With cumulative Abraham2435 held, their lifespan evaluations give2580,2640,2640. The first two cumulative heads are printed source rows. The SP architectural205 interpretation of the third row is a derived compatible biography from File18's shared Abraham/death targets; official SP cumulative chronology retains145.
A regular change of60 therefore does not, by itself, specify the cumulative change. The underlying source operation does. File51a's separate cumulative Rounded+60 boundary blocks are an additional source-admitted comparison operation; they must not be mistaken for the zero cumulative effect of MT's lifespan-preserving Terah repartition.
Finite choices operate on nested parts of the genealogy
The regular variants can be written with three binary choices: Sojourn placement s, Terah placement t, and Cainan presence c. Their supports are nested:
| Common birth rows | Sojourn215 | Terah60 | Cainan130 |
|---|---|---|---|
| Adam through Arphaxad | active | active | active |
| Shelah through Terah | active | active | fixed |
| Abraham through Joseph/Levi | active | fixed | fixed |
| Aaron/Moses onward | fixed | fixed | fixed |
Cainan himself exists only in the ON state and belongs to the four-position middle block. Kohath and Amram have no assigned regular pivots here; unassigned is not the same as a fixed coordinate. Native LXX begins ON, while MT/SP restoration is explicitly selected.
For common rows this becomes one masked formula:
It explains the8→4→2 restriction pattern and every supported span response. When both endpoints have the same masks, their interval survives the switch. When their masks differ, their interval records the intervening source hinge. These are finite switches between admitted configurations, not permission to keep adding the increments indefinitely.
Cainan uses the same support in cumulative chronology but weight460. Sojourn placement is not an added lifespan, and MT/LXX Terah repartition preserves205. The common source description therefore has distinct regular and cumulative realizations; it does not require every regular configuration to become a different cumulative chronology.
Rounded chronology preserves these particular choices through a local arithmetic law
For nearest-five rounding Q of integer source durations,
The permitted increments60,130,215 and460 are multiples of5. Where applied to their proper source components, these choices commute with rounding and preserve the existing rounding residuals. The same statement can then be carried through the ordered chronological evaluation. It concerns source durations; the scaffold is not obtained by rounding individual BC labels.
For Cainan, the newly inserted duration has zero local rounding residual. Its coordinate still inherits the downstream residual: cumulative4393/4396 becomes4853/4856 after adding460. The retained earlier coordinates move together, while their Actual–Rounded differences persist. The complete residual path carries more information than an aggregate cancellation at Creation.
This is the useful extent of the common grammar: ordered source rows, supported finite choices, separate chronological projections, and rounding compatible with those particular changes. Counting conventions, phase selections and secondary display translations remain explicitly attached. The arithmetic explains connected families without making their historical appointments, source status or theological interpretations interchangeable.
Editorial assessment and scope corrections
Strongest explanatory finding: the complete SP-with-Cainan/LXX birth-suffix agreement together with its reconstructed death and cumulative divergence. This is a whole-family realization of the Strategy's central source-row proposal, stronger than another isolated shared total. The invertible suffix-sum/adjacent-difference relation explains why intermediate boundaries provide information beyond the head.
Necessary scope corrections:
- Call the9890/9400/8982 comparison a common full430 regular frame with native Cainan states; unqualified “native SP” would conceal its+215 placement.
- Keep the cumulative lower completion and center profiles distinct. Their uniform2 difference is source-selected, not a fitted node-by-node correction.
- Keep absent Cainan separate from a zero-duration row and Kohath/Amram unassigned regular dates separate from fixed dates.
- State the SP205 Terah biography as an implied compatible local construction. Do not replace officialSP cumulative145 with it.
- Keep File51a's secondary cumulative+60 block distinct from the raw MT/LXX Terah repartition.
- Restrict the rounding interchange claim to its demonstrated integer source-duration domain and admitted multiples-of-five moves. It supplies no general additivity, invertibility, or unrestricted SP Rounded branch.
- The61 reconstructed comparisons and the matched suffix positions are dependent outputs of the common model, not independent witnesses.
Sources: File18 §§1.3.1–1.3.2,1.6,2–4,6A–6D; File51a §§2.1,3.1,16.1–16.2,23.7; Research Strategy §§5B–5G; C733–C748 and the variant preparation packet. No canonical amendment, further Gear transport, deferred primer, or second decimal inversion is proposed.