# Rounded chronology, the shared backbone, and the calendar Keys

The Rounded, inverse, and Key families connect through source-defined paths measured in different ways. Rounded chronology accumulates rounded source intervals. File52c applies a declared decimal operation to selected complete spans, preserving their named divisions. The calendar Keys then rescale durations about a stated anchor. The shared12026/14726 BC backbone makes these connections concrete: distinct regular and cumulative paths reach the same outer measurements while retaining their internal structure.

The backbone, Actual1:4 bridge, and Key equations are **pre-existing results**. The **new source interface**, established in C1273–C1279, connects the complete reconstructed MT genealogy to File52c’s admitted regular paths. It explains where their inputs come from and why their source roles matter.

The Moses-centred reconstruction supplies Rounded Creation4106 BC and Noah3056 BC from the full regular path with Moses1526 BC held. Its rounded intervals total2580. The corresponding strict Actual intervals total2586; rounding that whole total once would give2585. Thus the published2580 comes from rounding the rows before accumulation. The six-year accumulated residual recovers strict Actual Creation4112 without moving the Rounded head. File52c separately supplies Nativity6, Conquest1406 and the selected Rounded Flood2456. Together these nodes reproduce all three admitted regular partitions. Arphaxad’s birth also occupies2456 in the reconstructed path, but its birth role remains distinct from the Flood event. [C1234–C1235; C1273–C1278]

File52c’s decimal operation reverses the significant digits of each original span while retaining its trailing-zero places. Thus1650 becomes5610,1050 becomes5010, and1400 becomes4100. A source-appointed partition is part of the operation: the regular2700-year Creation-to-Conquest span, treated whole, produces7200; its1650+1050 partition produces5610+5010=10620. The named Flood division therefore carries information that the total alone omits. [File52c §§1.2,3.3–3.4]

With Conquest1406 BC held, the regular primary path gives

\[
1406+5610+5010=12026.
\]

The cumulative primary path uses Creation14006 and Flood4836, giving9170+3430. Its one-pass images are7190+3430, also totaling10620 and rebuilding12026 BC. This convergence concerns the specified primary paths; their source breakpoints remain distinct. [File52c §3.3]

An exact register argument explains part of that agreement. Consider two spans whose three-digit cores end in nonzero digits and each have exactly one trailing zero. Let `H,T,U` be the sums of the cores’ hundreds, tens and units columns. The original and transformed totals are

\[
10(100H+10T+U),\qquad10(H+10T+100U),
\]

with `2≤H,U≤18` and `0≤T≤18`. A total2700 forces `U=10` and `10H+T=26`, hence uniquely `(H,T,U)=(2,6,10)` and transformed total10620. A total12600 instead permits `(12,5,10)` or `(11,15,10)`, giving10620 or11610. The literal cumulative cores917 and343 select the first: their units carry into the tens, but their tens produce no further carry. Regular convergence follows from total and register; cumulative convergence additionally uses the source-selected carry branch. The theorem recovers aggregate measurements, while the source supplies the actual split. Its register restriction applies to these two-component calculations. [C938–C947]

The continuation to14726 BC supplies an additional original interval: Conquest1406 to Nativity6 is1400 years, whose one-pass image is4100. Moving the held anchor from1406 to6 and including this leg changes the rebuilt endpoint by4100−1400=2700. Therefore12026 becomes14726. The explanation retains both the original tail and the anchor change. [File52c §3.4; C952]

All four admitted completed paths can be compared compactly. The divisions are listed outward from the held6 BC anchor toward earlier source dates:

| Source path after Nativity | Original spans | One-pass spans |
|---|---|---|
| Regular: Conquest–Flood–Creation | 1400 + 1050 + 1650 | 4100 + 5010 + 5610 |
| Regular: Conquest–Noah–Creation | 1400 + 1650 + 1050 | 4100 + 5610 + 5010 |
| Regular: Conquest–Flood–Noah–Creation | 1400 + 1050 + 600 + 1050 | 4100 + 5010 + 600 + 5010 |
| Cumulative primary: Conquest–Flood–Creation | 1400 + 3430 + 9170 | 4100 + 3430 + 7190 |

Every transformed row totals14720 and therefore ends at14726 BC. The Noah alternatives preserve the regular result through their particular component relations. The new genealogy interface matches the first three original rows exactly and reuses their established images; the cumulative row retains its own source measurement. The first transformed leg ends at4106 BC, numerically equal to original regular Creation, but it remains the generated endpoint of the Conquest leg. This preserves the distinction between a shared coordinate and a shared chronological role. [File52c §3.4; C1274–C1276]

The same12026 anchor also divides the selected Actual Creation bridge. Its cumulative and regular completion endpoints are14004 and4114 BC, respectively. Their9890-year separation splits into

\[
14004-12026=1978,\qquad12026-4114=7912=4\times1978.
\]

Priestly expansion by25/23 gives2150 and8600, preserving1:4 and totaling10750. These expanded parts belong to the expanded total. The Rounded endpoints14006/4106 instead divide into1980+7920=9900, again1:4. Both selected pairs satisfy

\[
12026=\frac{4C+R}{5},
\]

where `C,R` are their cumulative and regular Creation coordinates. Moving Rounded→Actual changes them by−2,+8, preserving the weighted value because `4(−2)+8=0`. The regular+8 combines the six-year row residual with the separate standard Shem+2 convention; selecting the cumulative completion endpoint supplies its−2. The matching weight is derived from these selected values. It expresses their compatibility without making the state selections disappear. [File52c §3.14; C932–C948]

The Keys explain how another kind of agreement crosses families. Write

\[
E=25/23,\quad P=70/69,\quad J=300/299.
\]

Their corresponding schematic year lengths give

\[
336E=360P=364J=8400/23.
\]

Consequently different year counts can carry the same modeled day-volume. The completed14720 duration gives16000 underE,44800/3 underP, and192000/13 underJ. Multiplying those counts by336,360 and364 respectively gives5376000 in every case. File52c appoints theE completion here; theP/J values illustrate the inherited calibration conditionally. Equal modeled volume does not itself appoint a new chronological route or a historical calendar. [Strategy §3.3; C1062]

For the appointedE route, holding6 BC converts14720 to16000 and locates the expanded endpoint at16006 BC. The intermediate positions need not become integers. Every proper prefix of the four transformed paths has a nonzero residue modulo23. For example, the shared first4100-year prefix becomes102500/23; the whole14720 becomes16000. Uniform multiplication remains additive over exact rational durations, so componentwise conversion and whole-span conversion agree at the endpoint. What fails is the further requirement that every retained internal position lie on an integer-year grid. The rational path is a conditional display of the same uniform conversion, not a universal map between all original chronological date labels. [File52c §3.6; C1059–C1062]

This distinction gives23 and529 their structural roles. The first controls integral Priestly conversion. Its square529 permits the duration ladder

\[
529k\longrightarrow575k\longrightarrow625k.
\]

For the12026 backbone and Exodus1446, the supplied interval is10580=20×529, yielding10580→11500→12500 under two successive Priestly expansions. The coefficient20 is obtained from that span; the later values follow from the same factor. Endpoint placement additionally requires the declared anchor and counting convention. [File52c §§6.1–6.2]

These connections give the Strategy a common explanatory structure: source rows generate paths; named partitions determine the admitted span operations; decimal registers explain aggregate convergence; selected states preserve the Actual1:4 bridge; and calibrated Keys relate their measures. The new interface ties the regular inverse inputs to a complete genealogy already reconstructed from its intervals. Their historical interpretation remains attached to the source roles and placements that the arithmetic faithfully carries forward.