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 shared 12026/14726 BC backbone makes these connections concrete: distinct regular and cumulative paths reach the same outer measurements while retaining their internal structure.

The backbone, Actual 1:4 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 Creation at 4106 BC and Noah at 3056 BC from the full regular path with Moses at 1526 BC held. Its rounded intervals total 2580. The corresponding strict Actual intervals total 2586; rounding that whole total once would give 2585. Thus the published 2580 comes from rounding the rows before accumulation. The six-year accumulated residual recovers strict Actual Creation at 4112 without moving the Rounded head. File52c separately supplies Nativity 6, Conquest 1406 and the selected Rounded Flood 2456. Together these nodes reproduce all three admitted regular partitions. Arphaxad’s birth also occupies 2456 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. Thus 1650 becomes 5610, 1050 becomes 5010, and 1400 becomes 4100. A source-appointed partition is part of the operation: the regular 2700-year Creation-to-Conquest span, treated whole, produces 7200; its 1650+1050 partition produces 5610+5010=10620. The named Flood division therefore carries information that the total alone omits. [File52c §§1.2,3.3–3.4]

With Conquest 1406 BC held, the regular primary path gives

The cumulative primary path uses Creation 14006 and Flood 4836, giving 9170+3430. Its one-pass images are 7190+3430, also totaling 10620 and rebuilding 12026 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 original spans, each equal to ten times a three-digit core ending in a nonzero digit. Each span therefore has 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

with 2≤H,U≤18 and 0≤T≤18. A total 2700 forces U=10 and 10H+T=26, hence uniquely (H,T,U)=(2,6,10) and transformed total 10620. A total 12600 instead permits (12,5,10) or (11,15,10), giving 10620 or 11610. The literal cumulative cores 917 and 343 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 to 14726 BC supplies an additional original interval: Conquest 1406 to Nativity 6 is 1400 years, whose one-pass image is 4100. Moving the held anchor from 1406 to 6 and including this leg changes the rebuilt endpoint by 4100−1400=2700. Therefore 12026 becomes 14726. 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 held 6 BC anchor toward earlier source dates:

Every transformed row totals 14720 and therefore ends at 14726 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 at 4106 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 same 12026 anchor also divides the selected Actual Creation bridge. Its cumulative and regular completion endpoints are 14004 and 4114 BC, respectively. Their 9890-year separation splits into

Priestly expansion by 25/23 gives 2150 and 8600, preserving 1:4 and totaling 10750. These expanded parts belong to the expanded total. The Rounded endpoints 14006/4106 instead divide into 1980+7920=9900, again 1:4. Both selected pairs satisfy

where C,R are their cumulative and regular Creation coordinates. Moving Rounded→Actual changes them by −2 and +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 same division is visible in the original one-pass convergence. The regular span grows from 2700 to 10620, gaining 7920; the cumulative span falls from 12600 to 10620, losing 1980. The gain is four times the loss. Thus the weighted coordinate and the inverse construction express a shared relation among these source values. Once the inputs are fixed, the two descriptions belong to one connected calculation. The historical significance rests on why those particular intervals and their named divisions occur in the source family. [C937–C948]

The completed backbone also has a 9890-year interval to the Conquest-held cumulative Flood companion at 4836 BC: 14726−4836=9890=23×430. The Flood companion retains the 1406 anchor and unchanged 3430 span; it is a separate anchored comparison from the completed Nativity-held path. Its interval matches the Actual Creation bridge in magnitude, while the original Rounded Creation gap stays 9900. The correspondence links declared measurements without replacing their endpoints or chronological roles. [File52c §3.6]

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

Their corresponding schematic year lengths give

Consequently different year counts can carry the same modeled day-volume. The completed 14720 duration gives 16000 under E, 44800/3 under P, and 192000/13 under J. Multiplying those counts by 336, 360 and 364 respectively gives 5376000 in every case. File52c appoints the E completion here; the P/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 appointed E route, holding 6 BC converts 14720 to 16000 and locates the expanded endpoint at 16006 BC. The intermediate positions need not become integers. Every proper prefix of the four transformed paths has a nonzero residue modulo 23. For example, the shared first 4100-year prefix becomes 102500/23; the whole 14720 becomes 16000. 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 gives 23 and 529 their structural roles. The first controls integral Priestly conversion. Its square 529 permits the duration ladder

For the 12026 backbone and Exodus 1446, the supplied interval is 10580=20×529, yielding 10580→11500→12500 under two successive Priestly expansions. The coefficient 20 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 Actual 1:4 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.