# How the chronological families fit together

**Integrated working draft · C631 foundation and C632–C731 research cycle · 28 September 2026**

The chronological families fit together through a small set of operations on structured source data. Regular chronology measures begetting intervals; cumulative chronology measures lifespans. Rounded chronology retains selected spans and subdivisions. Other families preserve a set of corresponding roles, a local rail, or a phase field while changing its placement or scale. Calendar calibration explains why differently measured spans can reach compatible completions.

MT (Masoretic Text), LXX (Septuagint) and SP (Samaritan Pentateuch) identify source traditions; regular and cumulative identify what is accumulated. Actual and Rounded distinguish the realizations being compared, with Rounded governed by its admitted scaffold. Throughout, E=25/23 is the Priestly Key, P=70/69 the Prophetic Key, and J=300/299 the Enochian Key.

The emerging explanation is therefore a grammar of **source structures, retained components, and conserved measurements**. Its strength is that a few rules reconstruct whole families and explain their differences. The new Creation–Joseph rails, Creation–Terah refinement, and micro/macro completions extend the same grammar that already connects Actual and Rounded chronology. Source placements remain part of the explanation; the generated comparisons show what follows from them.

## Source rows explain why regular and cumulative chronologies differ

A biographical row contains begetting age \(b\), remaining years \(r\), and lifespan \(L=b+r\). The change

\[
(b,r)\longmapsto(b+d,r-d)
\]

changes the regular contribution while preserving the cumulative contribution. An inserted Cainan row contributes 130 to the begetting chain and 460 to the lifespan chain. These are two measurements of the source row.

The controlled comparisons reconstruct the following Creation movements relative to MT:

| Source comparison state | Regular movement | Cumulative movement |
|---|---:|---:|
| LXX, full 430 frame, native Cainan ON | +1380 | +890 |
| SP, equalized full 430 frame, Cainan OFF | +300 | −608 |

Their different directions and ratios follow from the selected row inputs and frames. They exclude a universal scalar conversion between regular and cumulative Creation while leaving local constructions available.

SP illustrates how a related family can retain distinct placements. Its Actual completion pair 13396/4414 has width 8982; the supplied Strategy rectangle 13406/4206 has width 9200. The regular coordinate changes by −215+7 and the cumulative coordinate by +10, increasing the gap by 218. Within the rectangle, \(P(2760)=2800\) and \(J(11960)=12000\), both gains of 40. The state change and the equal-gain calculation explain different parts of the relation.

The Covenant family exposes the same mechanism locally. In File60’s MT-Minimum state, Covenant 1866, Levi’s death 1567, and Conquest 1406 give

\[
299+161=460=23(13+7).
\]

Priestly expansion gives \(325+175=500\). The mixed Enochian/Priestly calculation gives \(300+175=475\), a diagnostic total without an appointed terminal. Some coefficients have a biographical cause: Levi and Joseph begin four years apart and live 137 and 110 years, so their death separation is \(137-110-4=23\). Equal 137-year lifespans likewise preserve the 161-year Ishmael–Levi birth displacement at death. The further 161 from Levi’s death to the supplied terminal is an additional placement.

*Sources: File18, controlled comparisons C483–C491; File60 §§2–6, C532–C551; SP state bridge C523, with dependency summaries C528 and C613; Research Strategy §§3.1–3.2.*

## Rounded chronology makes the retained path decisive

Latest File52c remains the controller of the Rounded study. Its decimal operation reverses an admitted duration while preserving trailing-zero placeholders. Evaluation acts on the supplied span list:

\[
V_a(s_1,\ldots,s_n)=a+\sum_i I(s_i).
\]

The principal MT paths show why the subdivision matters:

| Path to Conquest 1406 | Supplied spans | Reversed spans | Evaluated endpoint |
|---|---|---|---:|
| Regular Creation–Flood–Conquest | 1650+1050 | 5610+5010 | 12026 |
| Cumulative Creation–Flood–Conquest | 9170+3430 | 7190+3430 | 12026 |

Reversing the unsplit totals would instead give 7200 and 62100 before anchor addition. The original partition is therefore essential. Summation forgets component order, although that order still determines the intermediate roles. Admitted cumulative refinements produce 13016; the shared 1400-year Conquest–Nativity continuation gives the 14726 backbone. Its nine nonzero family members have \(X=6+115k\), with Creation-held Priestly outputs \(125k-1274\). One affine rule explains those outputs once the source members are supplied.

The clearest Actual–Rounded interface starts from cumulative Creation 14004, regular Jacob’s call 1929, and Exodus 1446:

\[
14004-1929=12075=25\times483,\qquad1929-1446=483.
\]

Applying \(E=25/23\) only to the final 483 changes it to 525. Applying \(J=300/299\) to the whole 12558 also yields 12600. With Creation held, the two routes reach generated 1404:

```mermaid
flowchart TD
    S["Source: 14004 → 1929 → 1446"] -->|"Whole J"| U["14004 → 24552/13 → 1404"]
    S -->|"E on final 483"| V["14004 → 1929 → 1404"]
    U -->|"Evaluate endpoints"| W["14004 / 1404"]
    V -->|"Evaluate endpoints"| W
    W -->|"Pair translation +2"| R["Rounded: 14006 / 1406"]
```

Their junctions differ by \(525/13\). The diagram commutes after endpoint evaluation, which discards that difference. The +2 pair translation then reaches the Rounded Creation–Conquest pair. Historical Exodus 1446 and generated terminal 1404 retain different roles.

The same cumulative shift participates in a second interface. Actual cumulative/regular Creation 14004/4114 and Rounded 14006/4106 both preserve

\[
\frac{4C+R}{5}=12026,
\qquad4\Delta C+\Delta R=4(2)-8=0.
\]

Thus endpoint agreement, weighted-point conservation, and component reversal connect the families in different ways. The derived LXX Rounded paths yield 10436 and 11426 under their respective subdivisions: the evaluator transfers, while the MT convergence remains a selected result.

*Sources: File52c §§3.3–3.4,3.11,3.14; File51a; File63 §§7.3–7.4,9.7; C495–C499, C512–C519, C570–C573, C608–C609.*

## Calendar calibration explains the shared completions

The Priestly, Prophetic, and Enochian Keys are

\[
E=25/23,\qquad P=70/69,\qquad J=300/299,
\]
\[
336E=360P=364J=K=8400/23.
\]

A span measured against calendar length \(d\) has converted value \(K(s/d)\). Equal normalized measures therefore give equal converted spans. A common endpoint additionally requires the same held anchor and direction. For example, with Covenant 1866 held, Joseph’s and Levi’s death radii satisfy \(E(276)=J(299)=300\), reaching 1566. With 12026 held, Jared’s 8372 and Noah G1’s 8970 satisfy \(E(8372)=P(8970)=9100\), reaching 2926.

Selective expansion supplies another general relation:

\[
\tfrac56S+E(\tfrac16S)=PS,
\qquad\tfrac{25}{26}S+E(\tfrac1{26}S)=JS.
\]

The source subdivision determines whether the identity has a chronological realization. For 483, the supplied \(402.5+80.5\) partition yields 490 after expanding its one-sixth portion. For 12558, the final 483 is one twenty-sixth, producing the endpoint square above. The carrier \(483=3\times161\) connects both with the Covenant’s fourteen-year Priestly gain per 161.

The common integral-input lattice is \(897n\xrightarrow{J,P,E}(900n,910n,975n)\). Its gains and gaps are consequences of one rule. The Actual 9890 bridge has its own supplied unit: Abraham 2435 to Exodus 1446 is 989, and its 2:8 arms expand from 1978/7912 to 2150/8600 under E.

The Rounded cumulative lower leg supplies another connection through its Actual anatomy: \(3430=2401+1029=7\times343+3\times343=7\times490\). The 7:3 partition explains that duration family. It does not appoint a further breakpoint for decimal reversal; that operation still needs the admitted Rounded path.

*Sources: File12 calendar definitions; File60 §§6.1–6.2,9; File62 §§8.3,8.6A–C; File63 §§7.9,9.7; File52c §§3.14–3.15; C534–C535, C567–C575, C586, C612.*

## Translated shapes connect the genealogical families

The cumulative Abraham–Moses boundary vector is

\[
C_0=(2435,2260,2080,1933,1796,1663,1526).
\]

All 42 coordinates in its six exact source rails follow from

\[
C(p,d)=C_0+p-d,\qquad p\in\{0,7/2\},\quad d\in\{0,7,14\}.
\]

Phase selects Moses/Nisan or exact Aaron/Tishri; displacement selects original, half-clutch, or full-clutch placement. Every rail preserves the lifespan sequence. Moving one boundary contracts a span; moving both translates it intact.

Joseph’s collateral intervals \((\ell,\ell-110)\) and \((q+110,q)\), attached to Levi boundary \(\ell\) and Kohath boundary \(q\), translate with those boundaries. Their shared geometry is \(27|83|27\). The nine comparisons with fixed regular Joseph retain \(a+b=27\) and \(a+b+c=110\); cyclically rotating the duration components gives the three 110-year partitions. Joshua supplies the terminal interface: the backward Nisan interval translates 430 to regular Joshua 1476–1366; the forward whole-year Aaron interval translates 460, or 460.5 in exact phase. A further 70 reaches cumulative Joshua 1406–1296.

At the upper end, translated Isaac 2246 divides the Rounded trunk into \(11760+840=12600\), or 35 copies of \(336+24=360\). The ratio 15/14 is E/P. Separately, cumulative Cainan 460 and Apparent Age 30 add one 490 to the upper leg. This joins translated genealogical boundaries, source variants, and calendar measurement through a retained junction.

File70 now shows the same principle at finer resolution. Define

\[
R(c,u)=c+u(7,1,0,-6,-7).
\]

| Source field | \(c\) | \(u\) | Five supplied positions |
|---|---:|---:|---|
| Standard Creation | 4114 | 1 | 4121,4115,4114,4108,4107 |
| Restored Creation | 4244 | 1 | 4251,4245,4244,4238,4237 |
| Terah | 2226 | 10 | 2296,2236,2226,2166,2156 |

Regular Cainan translates all five Creation positions by 130. The Creation–Terah map scales the complete \(6|1|6|1\) refinement to \(60|10|60|10\). These are structural correspondences among differently named roles. Jacob’s 1929/1922/1915 and Joseph’s 1885/1878/1871 share only the coarser \((1,0,-1)\) rail; the source supplies no matching five-position refinement for them.

The admitted Noah and Shem alternative-date fields give another translated shape, \(60|70|60\). The local Noah Gear comparison adds 2 to every position; the displayed Shem comparison subtracts 150. These entries are alternative death coordinates, so their ordered spacing is a comparison field rather than a succession of deaths. The geometry explains their relation without changing the original node classes.

*Sources: File61 §§6–7,11–12,14,17; File62 §§1,5–6,9; C552–C567. File70 §§2.3,3.2,4.2–6 and Appendices A.2–A.7; C648–C650, C656–C657.*

## The calendar-half family follows from its rails

Restored Creation \(A=(4251,4244,4237)\) and Joseph \(B=(1885,1878,1871)\) have identical 7+7 spacing. Their complete cross-rail matrix is generated by

\[
A_i-B_j=T+7(j-i),\qquad T=2366.
\]

The source’s two thirteen-year exposure offsets add the 2340 comparison:

| Relation | Geometric expression | Calendar form |
|---|---:|---:|
| Exposure | \(T-26=2340\) | \(13\times180\) |
| Minimum diagonal | \(T-14=2352\) | \(14\times168\) |
| Corresponding positions | \(T=2366\) | \(13\times182\) |
| Maximum diagonal | \(T+14=2380\) | Generated outer comparison |

Joseph’s ages 17 and 30 supply the lower thirteen-year offset. The calendar interpretation consequently has two normalized count classes:

\[
2340P=2366J=\tfrac{13}{2}K,\qquad2352E=7K.
\]

The 2340 and 2366 arms use held heads thirteen years apart; their equal converted lengths preserve that endpoint displacement. Calendar equality explains their measured shape, while the source heads determine placement.

There is also a stronger conditional result. For positive integer half-counts, fixing the 14/26 offsets and requiring

\[
T=182m,\qquad T-14=168n,\qquad T-26=180p
\]

gives \(T=2366+32760q\). The source 2366 is the least positive solution, with counts 13/14/13; 32760 is the least common multiple of 168,180,182. Removing regular Cainan preserves both rails but changes their translation to 2236. The least nonnegative correction back into the compatible class is then 130. This is a selection by the fixed geometric and divisibility conditions; it does not establish why a scribe selected those conditions or inserted Cainan.

The Enoch close 3257 supplies an additional internal cut: \(4237\to3257\to1885\) gives \(980+1372=2352\), or 5:7. Replacing close with head 3264 transfers seven between the arms. Their total survives; the ratio belongs to the supplied close role.

*Sources: File70 §6, §§5.5–5.6, Appendix D; C632–C647, C655.*

## Reflection and indexing preserve other parts of the structure

Jacob’s biography \(2006\to1936\to1929\to1859\) has symmetric gaps \(70|7|70\), centered at 1932.5. The separate death-centered object \(1936,1929,1859,1789,1782\) has offsets \(77,70,0,-70,-77\) about 1859; reflection \(x\mapsto3718-x\) reverses it. Its traditional Judah label 1789 remains secondary. Shared nodes connect the objects without identifying their centers.

File70’s cumulative 4900 body has an equally compact generator: \(x(k)=5436-70k\). Its selected indices are

| Index | Coordinate | Supplied role |
|---:|---:|---|
| 0 | 5436 | Cumulative comparison head |
| 49 | 2006 | Regular Jacob birth |
| 50 | 1936 | Whole-year cumulative Jacob/Levi boundary |
| 57 | 1446 | Exodus |
| 69 | 606 | Captivity |
| 70 | 536 | Restoration |

The cuts 49|21,50|20,69|1 and 50|7|13 are partitions of one body selected by those roles. The larger \(14011\to5436\to1936\to536\) spine is \(49|20|8\) in 175-year units. Holding outer 5436/536 fixed and moving only the central display to exact 1936.5 transfers half a year between its arms while preserving 4900. The whole-year household path \(2083\to1936\to1446\) similarly gives \(147+490=637\); its integer anatomy belongs to that display.

The corresponding cumulative first-to-last triples, \((14011,14004,13997)\) and \((1936,1929,1922)\), are one seven-year rail translated by 12075. Their middle pair is the upper arm of the Actual–Rounded square. Its \(12075=69\times175\) Prophetic completion is \(12250=70\times175\). The complete rail therefore connects the previously isolated arm with its supplied companion positions.

Files68–69 need the genealogical index retained as well. Their Enoch cells use

\[
u=g+110\rho,\quad x=(N-7)u-64,
\qquad F=(x+110,x,x+110-g,x-g),
\]

with \(N\in\{77,63,50\}\), \(g\in\{70,72\}\), and \(\rho\in\{0,1\}\). This generates all 48 coordinates. Accumulation changes with \(u\), while the local interval stays \(g\). The 49-row four-rail grid uses \(a(j)=65-(49-j)g+r\), with \(r\in\{-110,0\}\), and generates 196 interval cells. An eighteen-row attribution overlay changes occupants while its coordinates remain fixed.

A universal date dilation cannot preserve the 110 rail and simultaneously change 70 to 72. The primary weekly source also retains its residual: with \(D=600\), accumulated distances are \(10D,8D,6D+70\). Different resolutions explain the totals 3540,4200,8820; complete fifty-cell generation fields give 9000 and 9100. Their equality to \(J(8970)\) and \(P(8970)\) is a derived duration interface with the calibrated lattice.

The bounded Lukan companion adds 140 to accumulation while retaining its local 70-year interval. Replacing that local interval with 72 would change the Joshua comparison from 1406 to 1404. Accumulation, local span, and role attribution are consequently distinct parameters within the common indexed construction.

*Sources: File70 §§5.3,5.8,9.1A, Appendices A.5,A.13; C652–C664. File68 §7,§8.5, Appendices C,J; File69 §§1–4, Appendix B; C587–C607.*

## Retained cores connect the local and macro constructions

Supplement A begins with carrier 720 and flank 30. Their crossed core is \(c=690=23\times30\). Applying a Key to that core while retaining the flanks generates the whole table:

| Operation | Core | Core plus one flank | Core plus two flanks |
|---|---:|---:|---:|
| Native | 690 | 720 | 750 |
| P | 700 | 730 | 760 |
| E | 750 | 780 | 810 |

Native bracket and E-expanded core both measure 750 because \(c=23f\). They remain different component objects. Whole-bracket conversion adds \(2(k-1)f\) more than converting only the core. This is the retained-subdivision distinction already seen in the partial-E/whole-J endpoint square.

Two 720-step rails separated by 30 generate the ten-node 4326-to-1416 ladder. Its inner six-node cell has width \(2c+3f=1470\), centered at 2871. The scalar regroupings 690+780 and 720+750 follow from those components; only an actual supplied junction makes a regrouping an ordered path cut. The separate reconstructed calendar block \(4\times360+30=1470\) days has the same 48+1 month anatomy. Its ledger reaches the Julian-mean target \(6\times2190+1470=40\times365.25\), with units retained.

The central scale link is exact:

\[
(690,700,750)=20(34.5,35,37.5).
\]

The local core and the fine phase seed therefore reuse the same P/E calculation. Normalized 40 has a different role: its twentyfold value is 800.

*Sources: Supplement A §§1.1–1.5,2.2–2.3,4,6; File12 intercalation scheme; C665–C670, C686.*

## Centers and phases explain the fine structure

For endpoints \(U,L\), center \(c=(U+L)/2\) and width \(w=U-L\) separate two mechanisms. Uniform translation changes the center while preserving width; equal and opposed endpoint movement preserves the center while changing width.

This explains the source corridor sequence in Nisan coordinates:

| Endpoint pair | Width | Center |
|---|---:|---:|
| 4326/1416 | 2910 | 2871 |
| 4336/1406 | 2930 | 2871 |
| 4341/1401 | 2940 | 2871 |

The first opening applies P only to the two exterior 690 cores, gaining ten per side and retaining the inner six nodes. The final five per side comes from the source’s local 2+3 construction. Its extremities are generated comparison positions.

Phase adds another coordinate. In the Supplement’s convention, \(\operatorname{coord}(Yn)=Y\) and \(\operatorname{coord}(Yt)=Y-1/2\). The formula

\[
x=2871+\epsilon\pm w/2,\qquad\epsilon\in\{0,1/2\},\quad w\in\{34,40\}
\]

generates every before/after phase endpoint. The exact centers remain 2871 and 2871.5. The preserved half-year separation demands affine slope 1, whereas widening 34 to 40 demands 20/17: each phase must therefore retain its own center.

The source’s normalization uses \(P(34.5)=35\), \(E(34.5)=37.5\), and an additional 2.5. Because that supplied residual equals the output gap, the construction compresses to \(2E(34.5)-P(34.5)=40\). Corresponding coordinate widths increase by six through outward movements of three; the scalar 34.5→40 comparison changes measurement class and increases by 5.5.

Translating the lower 40 bracket by the supplied 80 produces a 120 envelope centered at Nisan coordinate2911 (corresponding Tishri center2911.5). The separately supplied Enoch70 body shares that center, forcing the partition \(25|35|35|25\). Its halves are P(34.5), and its margins E(23). The partition follows from the two widths and their source-appointed common center.

The source’s 1470/1475 comparison shows why measured objects remain explicit. Applying P to a scalar 724.5 gives 735 and hence a doubled 1470. Starting thirteen from the center and moving outward 724.5 instead gives a half-width of 737.5 and total 1475. Different initial offsets explain the difference directly.

*Sources: Supplement A §§3.2,4.1–4.5,5.1; C672–C682. The civil continuation in §§5.2–5.4 uses BC+AD−1; C683–C684.*

## The micro/macro completion rejoins Rounded chronology

At seed \(s=34.5\), existing calibration gives

\[
336E(s)=360P(s)=364J(s)=12600.
\]

The source also supplies \(364s=12558\). Its common conversion structure appears at two scales:

| Source object | Native measure | J completion | Further register |
|---|---:|---:|---:|
| Micro calendar volume | 34.5×364=12558 days | 35×360=12600 days | 35×364=12740 days |
| Actual cumulative Creation–Exodus | 14004−1446=12558 years | 12600 years | P(12558)=12740 years |

The governing identities are \(J(364s)=360P(s)\) and \(P(364s)=364P(s)\). Matching numerical inputs connect distinct units; the Rounded pair 14006/1406 supplies its own endpoint realization. The 12600/12740 ratio 90:91 and equal cross-product 4586400 follow from the same completed count 35. As a derived calendrical interpretation, measuring the source’s phase residual 2.5 in 336-day units adds 840 days: \(40\times336=12600+840=13440\). The macro analogue is 840 years in the declared years-of-years carrier.

These three bodies also have an explicitly supplied cumulative realization. Hold hinge 4346 and its 2940 lower arm to 1406 fixed, and change only the upper 9660:

| Upper-arm operation | Upper arm | Retained lower arm | Total | BC head |
|---|---:|---:|---:|---:|
| Native | 9660 | 2940 | 12600 | 14006 |
| P | 9800 | 2940 | 12740 | 14146 |
| E | 10500 | 2940 | 13440 | 14846 |

Here 14006 is the native Rounded head; generated companions 14146 and 14846 retain the source’s provisional/appendix-led status. The table explains how the whole register family is realized through retained components. The anchored constructions below explain the source-conditioned placement of its4346 hinge.

*Sources: Supplement A §6,§10.3; File52c cumulative scaffold; C685–C691, building on C570–C573.*

## Anchored routes explain the shared 4346 hinge

The retained macro family has a source-conditioned placement as well as a common arithmetic shape. Two supplied regular-MT routes reach 4346:

| Source state | Held anchor | Operation on radius | Generated head |
|---|---:|---|---:|
| Standard Creation 4114 | Exodus 1446 | E(2668)=2900 | 4346 |
| Creation 4304, with +60 Terah and +130 Cainan | Conquest 1406 | P(2898)=2940 | 4346 |

The source-state difference 190=60+130 equals the difference in expansion gains, 232−42. The completed radii differ by 40, exactly matching the anchor separation. Those balances explain the convergence. The shared head is a generated parent label for the two operations; its historical transmission is a separate question.

The second route also maps the complete source path 4304→2648→1406 to 4346→2666→1406. Its 1656|1242 partition becomes 1680|1260, retaining the 4:3 ratio. The internal labels are generated seams within that comparison. They do not inherit the identities of unrelated nodes with the same numerical label.

One civil 1470-step ladder then generates 4346 BC→2876 BC→1406 BC→AD65. A five-year forward translation gives 4341 BC→2871 BC→1401 BC→AD70. Enoch’s close 2876 and the Long-pair center 2871 retain different roles. In the source’s wider Danielic path, the retained junctions give

$$
490|1400|70|1470|1400|35|35=4900.
$$

Grouping the four middle parts gives 490|4340|35|35. Its tenth-scale template is 49|434|3.5|3.5; the fine and coarse paths are dependent resolutions of the same supplied object. Their arithmetic does not by itself appoint a literal historical 434-year interval.

The same retained-component law unifies the new macro completion with the earlier Actual–Rounded connection. If a fraction f of a span is converted by k while the rest remains fixed, the effective total ratio is

$$
q=1+(k-1)f.
$$

| Converted portion | Key | Effective total ratio |
|---|---|---|
| 1/6 | E | P |
| 1/26 | E | J |
| 23/30, the upper 9660 of 12600 | P | 91/90 |
| 23/30, the upper 9660 of 12600 | E | 16/15 |

The last two ratios produce 12740 and 13440. Applying those effective ratios uniformly would move the retained junction: the partial and uniform arms differ by 98/3 or 196 respectively, although their totals agree. This extends the Actual–Rounded endpoint lesson to a second complete source family. An effective total ratio describes the evaluation; the retained partition describes the path.

*Sources: Supplement A §§9.1–9.8,10.3,11.1–11.6; File22 owns the dual-route construction. C692–C695, C699–C702, C713–C715.*

## Bounded companion tests distinguish whole families from selected matches

The admitted four-row directional comparison provides a compact example. Its input magnitude is always 690. Forward E/P assignments from held heads 3626/3576 converge at 2876; reverse P/E assignments from 2216/2166 converge at 2916. The common cause is 50=(E−P)690, which balances the supplied 50 offset. Swapping the Keys breaks the pairings. The comparison keeps its existing SKL status and does not extend to new source states.

Its supplied quartet 2946/2916/2906/2876 has gaps 30|10|30 and center 2911. The declared 1470 translation preserves that shape in 1476/1446/1436/1406. Only 2916 and 2876 are direct paired-Key outputs; the other positions retain their distinct source or generated-companion roles.

Supplement A §13 supplies a separate fixed twelve-node phase field, centered at −2921 in its exact civil phase chart. Translation by 5842 sends every node onto the mirror set. The associated affine reflection pairs all twelve nodes in six transpositions. For the four supplied totals 5840/5842/5844/5846, whole-set overlap counts are 8/12/8/4. The conditional placement equation is c+5842=−c, giving c=−2921 after the template and chart have been fixed. The supplied Long center −2871 instead requires 5742. The earlier positive-BC phase chart q and this signed chart x satisfy x=1/4−q; the2922t/2921n pair brackets the January instant−2921 rather than equaling it.

This is affine reflection of a fixed phase set, distinct from Rounded digit reversal and its mirror convention. Section13 remains a verification-complete, registered working amendment, without separate author Finalization or full-file pressure testing. These bounded reconstructions retain that status. Remote extensions remain outside this cycle.

*Sources: Supplement A §§8.1–8.6,13.1–13.5,13.7,13.9; C696–C709. The source-location and status review is recorded in REVIEW_C692_C709.md.*


## What the common grammar now explains

The families share source realization, translation and reflection of labelled shapes, selective component changes, indexed carrier substitution, and compatible calendar measurement. Their conserved quantities include a lifespan \(b+r\), a center \((U+L)/2\), the weighted Creation point \((4C+R)/5\), and an endpoint span after internal information is discarded. These are specific measurements preserved by specific operations.

The normal, Rounded, and forwarded constructions add the common duration ladder

\[
k\cdot529\xrightarrow{E}k\cdot575\xrightarrow{E}k\cdot625.
\]

It generates 1058→1150→1250, the tenfold Rounded radius, and the forwarded 4232/8464/12696 families. The 8464 ladder shares 9200→10000 with the supplied SP rectangle. Their anchors still determine where those widths land. The three-axis presentation consequently maps demonstrated family relations; its current evidence does not require an orthogonal geometry or a universal rotation between events.

The strongest gain is whole-family reconstruction with fewer separately specified outputs. The new rail compatibility theorem additionally selects the least positive member under its fixed source conditions. These are mathematical results. The Strategy’s historical proposal—local inherited operations producing wider compatibility across traditions—remains a hypothesis to assess with transmission evidence. Its theological premise is separate from the arithmetic.

This draft preserves the latest File52c, the controlled MT/LXX/SP source states, distinct phase and crossing conventions, and the existing limits on local operations. It proposes no canonical amendment or new transport. Its equations are supported by the successive journals and exact checks; reconstruction counts describe dependent outputs, not independent discoveries.

*Sources: File52c and C505–C511; C613–C625; Research Strategy §§1,5–6,9; C632–C731 reconstruction ledger and independent reviews.*