# Rounded residual transfer: source packet and new whole-family tests

Prepared for C732–C831. This is preparatory work only; no numbered step, journal entry, canonical edit, decimal reversal, Gear transport, or primer research was performed.

The strongest next test is to reconstruct **all boundary displacements from the finite support of the row residuals**. The MT endpoint cancellation is already known from C493; the new content is the entire residual field and how that field changes across source traditions. A second strong test explains why theoretical regular-death rounding and cumulative rounding are different operators on the same rows.

Files:

- `rounded_transfer_inputs.json`: source-owned values, 26 MT / 27 native LXX / 26 SP rows; exact source excerpts with paths, line ranges, and source hashes; MT literal rounded lifespan and cumulative-date checks; rules and source qualifications.
- `rounded_transfer_diagnostics.json`: preparatory calculated row residuals, full boundary fields, regular begetting diagnostics, and 25-case residue table.
- `prepare_rounded_transfer.py`: reproducible preparation. No journal access or canonical mutation.

## What is already completed elsewhere

C485–C491 explain raw row differences, repartition, Cainan130/460, LXX cumulative890, SP−608, and the two LXX inverse endpoints. C492 explains the regular+6 rounding plus selected+2 convention; C493 establishes MT12600 cancellation; C494 distinguishes the Actual14004 completion choice from rounding. These should be premises, not new numbered discoveries. Searches of C482–C531, C532–C631, and C632–C731 journals found no Reu commutator diagnosis or full row-residual support/propagation analysis.

## 1. Source rules and a concrete correction to one source generalization

Let `R(n)=5 floor((n+2)/5)` for nonnegative integer measures, with residual `e(n)=R(n)−n`.

| Layer | Source rule | Status |
|---|---|---|
| Regular births | Round begetting intervals, retain source anchor and separate boundary terms | File51a §§2–3.3 |
| Regular theoretical deaths | Round begetting and remaining years separately, sum to a theoretical lifespan | File51a §3.4; retain MOD-5 THEORETICAL |
| Cumulative | Round each total lifespan, then accumulate backward from1406 | File51a §§16.1–16.2 |

File51a §3.4, Machine Guard [METHUSELAH BIFURCATION], says: **“Both rules are applied uniformly to all patriarchs; Methuselah is the unique case where they diverge.”** That sentence has no antediluvian-only restriction. Its own tables supply two MT cases:

| Row | Source b+r=L | Rounded parts | Rounded total | Parts−total |
|---|---|---|---|---:|
| Methuselah |187+782=969|185+780=965|970|−5|
| Reu |32+207=239|30+205=235|240|−5|

Reu235 is explicitly printed in §3.4, and Reu240 is explicitly printed in §16.1. Preserve both values and their different domains. The finding qualifies the uniqueness sentence; it does not require harmonizing the values or editing the source.

The main LXX source values additionally give a derived Lamech case:182+571=753, rounded parts180+570=750 versus total755. Main182/753 remains operative,777 appendix,188 nonoperative. This is a derived cross-tradition diagnostic, not an independently source-adopted LXX death-date table.

## 2. The exact finite rule explaining all such defects

For a compatible source row, define

`K(b,r)=R(b)+R(r)−R(b+r)=e(b)+e(r)−e(b+r)`.

It depends only on `(b mod5,r mod5)`, and takes values−5,0,+5. Its complete table is:

| b mod5 / r mod5 |0|1|2|3|4|
|---|---:|---:|---:|---:|---:|
|0|0|0|0|0|0|
|1|0|0|−5|0|0|
|2|0|−5|−5|0|0|
|3|0|0|0|+5|+5|
|4|0|0|0|+5|0|

MT Methuselah and Reu both occupy(2,2). LXX Lamech occupies(2,1). LXX Reu again occupies(2,2). All nonzero defects in the source-defined ordinary-row domain are thus explained without per-patriarch adjustments.

`R(x+5k)=R(x)+5k`. Consequently source shifts divisible by5 preserve the rounding residual, and repartitions `(b,r)→(b+5k,r−5k)` preserve the defect. This explains why centenary/half-century begetting changes preserve most source relations. Cainan130+330=460 has zero residual in both components and total; inserting or removing that one row preserves residual profiles at every corresponding named boundary, although it changes durations by the established130/460 amounts.

## 3. Whole cumulative residual fields

For rows ordered Adam→Moses, the boundary displacement at row i is the suffix sum

`D_i = Σ_{j≥i} e(L_j)`.

Its inverse is the local difference `e(L_i)=D_i−D_{i+1}`. Hence a zero-residual row propagates a constant plateau; a nonzero row changes that plateau. Equal residuals at two boundaries are exactly the condition that their intervening duration is unchanged by rounding. This gives all internal interval invariants at once.

MT full residual vector:

`[0,−2,0,0,0,−2,0,+1,−2,0,0,+2,+2,+1,+1,+1,0,+2,0,0,0,−2,−2,+2,−2,0]`.

MT full boundary vector:

`[0,0,2,2,2,2,4,4,3,5,5,5,3,1,0,−1,−2,−2,−4,−4,−4,−4,−2,0,−2,0]`.

The terminal1406 has displacement0. The JSON supplies the associated named boundaries and both actual/rounded positions, and reproduces all26 literal §16.2 rows.

The cross-tradition change of this field has very small support:

| Source comparison | Rows where e_L differs from MT | Consequence |
|---|---|---|
| LXX native ON |Lamech+4; Arphaxad−2; Shelah−2|Sum0; Creation residual remains0|
| SP native OFF |Methuselah−1; Lamech+4|Sum+3; Creation residual becomes+3|

The corresponding boundary-field differences are:

| Comparison | Nonzero boundary plateaus, at common names |
|---|---|
| LXX−MT |Noah/Shem/Arphaxad−4; Shelah−2; all other common rows0|
| SP−MT |Adam through Methuselah+3; Lamech+4; Noah onward0|

Native LXX's extra Cainan row has zero residual; it produces an extra named boundary on the same local plateau. The whole profile, not merely its total, is unchanged by explicit native/leveled row insertion at corresponding common names.

Preparatory totals (not the principal novelty): MT12600→12600; LXX13490→13490; SP11992→11995. Thus the LXX–MT cumulative differential stays890, while SP−MT changes−608→−605. All changes follow from the support table above. The LXX cancellation is+10/−10 rather than MT+12/−12; SP is+13/−10.

Selected Moses-line node diagnostics provide concrete checks:

| Tradition | Creation Actual→Rounded | Arphaxad boundary Actual→Rounded |
|---|---|---|
| MT |14006→14006|4831→4836|
| LXX native ON |14896→14896|5745→5746|
| SP native OFF |13398→13401|4711→4716|

These are direct row-rounding outputs at the held1406 terminal. MT is source adopted. Cross-tradition outputs must be labeled derived comparison until checked against an available controlling Rounded table. They do not replace Actual completion endpoints14004/14894/13396, and the Arphaxad row should not silently inherit every tradition's distinct chief Flood-marker convention.

## 4. SP counted versus completed input: one unit can become five

File18 §3.1 distinguishes Lamech's53rd counted year,52 completed units to the primary Noah station, and a separate53-full-unit companion path. Its653 lifespan remains inclusive. The JSON preserves all of these types.

`R(53)=55`, whereas `R(52)=50`. Thus interpreting the count and rounding it do not commute through an assumed one-year correction: the raw difference1 becomes a rounded difference5.

Using the otherwise fixed begetting-row input and Jacob2006 as a purely diagnostic terminal:

| Lamech input | Raw strict begetting head | Rounded diagnostic head |
|---|---:|---:|
|53 counted/full-input branch|4413|4411|
|52 completed-input branch|4412|4406|

The4411 result matches File51a's transmitted lower block;4406 is its expressly secondary alternate. This calculation identifies a possible mechanism for the separation. File51a §17 does not explicitly derive its block by this exact path, so it must be called a **conditional reconstruction**, not a proved source derivation. No primary SP state is replaced, and no companion or Gear is transported upstream.

For inclusive SP Jared/Methuselah/Lamech, `L−b` in the packet is an algebraic count difference, not an ordinary elapsed-year remainder. Their calculated regular-death diagnostics therefore must not be promoted to biographies. The ordinary SP Reu row132+107=239 gives130+105=235 versus240, a legitimate ordinary-row −5 defect.

## 5. Suggested bounded research sequence, with reassessment

1. Freeze domain rules and literal MT b/r/L data, preserving withheld begetting ages and Jacob's Joseph-collateral role.
2. Reconstruct all MT regular-theoretical and cumulative rounded row outputs; isolate both nonzero defects.
3. Derive the25-cell residue law and explain those defects structurally.
4. Reconstruct all26 MT boundary displacements and invert the suffix sum back to row support.
5. Identify all unchanged inter-boundary intervals by equality of residual field values; present useful source-labeled classes rather than a long independent-hit list.
6. Transfer the residual rule to all27 native LXX rows and explain the three-row difference support and local plateaus.
7. Test source main LXX Lamech182/753 as a new defect; keep this derived death operator separate from adopted births and cumulative inputs.
8. Transfer the cumulative residual rule to all26 SP counts and explain its two-row support/+3 head effect.
9. Test counted/completed SP input order, with the4411/4406 coincidence explicitly conditional.
10. Prove residual invariance under admitted5k row shifts and zero-residual Cainan insertion; compare native and leveled profiles at all common names.
11. Reconstruct cross-tradition interval preservation from differences of the residual fields, rather than recomputing unrelated pairs.
12. Reassess the larger synthesis: Rounded is a componentwise projection with a residual field and selected path conventions, not one global displacement or one rule shared by births, deaths, cumulative sums, and count interpretation.

These are candidate bounded tasks, not a requested fixed schedule. The first substantive novelty should be the whole row/field mechanism and its explanation of the families.

## Minor source note

File51a §16.1's speculative probability paragraph reverses the counts: its literal list has14 nonmultiples of5 and12 multiples, not12 nonmultiples and14 multiples. The+12/−12 residual cancellation remains correct. No probability estimate was computed, and this need not distract from the main structural work.