# File61/62 preparation memo

This is source preparation for the parent agent, not a completed numbered research step. Exact data and source hashes are in `file61_62_tables.json`; `extract_file61_62.py` reproduces the tables.

## Best explanatory compression

The frozen C531 grammar already admits scoped translations, cumulative projections, phase states, and anchored Keys. File61/62 can be reconstructed almost entirely with those operations. The useful extension is an explicit **local domain** for a seven-year translation and its second permitted application, not a new global arithmetic operation.

Let

\[
C_0=(2435,2260,2080,1933,1796,1663,1526),\quad C(p,d)=C_0+p-d.
\]

The seven boundaries are Abraham, Isaac, Jacob, Levi, Kohath, Amram, and the Moses-side terminal interface. Use exact phase offset `p=0` or `7/2`, and permitted clutch displacement `d=0,7,14`. This reconstructs all **42 source coordinates** in File62 §§1.2–1.3. Adjacent spans are always `175,180,147,137,133,137`. The terminal at `1519/1512` is derived; Moses' birth stays `1526`. The Aaron-side terminal likewise does not move historical Aaron.

File61 §§6.3–6.6 supplies `14` as `487−473 = 444−430 = 350−336`. These three displays share the same compression: removing a fixed leading segment changes both columns equally. Source §7.1 calls the measurement heads independent, but that should not become a claim of independent numerical evidence.

## Typed operations and source coverage

| Operation | Inputs and output | Source | Reconstruction caveat |
|---|---|---|---|
| `T14(C)=C−14` | Local cumulative boundary → derived full-clutch boundary | File61 §7.3 | Moving parent only contracts a lifespan; moving both restores it. |
| `H7(C)=C−7` | Same source domain → midpoint boundary | File61 §7.3A; File62 §1 | Half of the declared displacement; not a historical variant. |
| `Jf(L)=(L,L−110)` | Levi boundary → collateral forward Joseph interval | File61 §11.3 | Joseph is not inserted into the genealogical trunk. |
| `Jb(K)=(K+110,K)` | Kohath boundary → collateral backward Joseph interval | File61 §11.2 | Same restriction. |
| Shift whole Joseph interval | Commutation with fixed `±110` construction | File61 §11.3A; File62 §§5.1–5.2 | Does not shift Joseph's regular biography `1915–1805`. |
| Parent-only clutch | Lifespan `L→L−14` | File61 §7.4 | Complete six-block output is `161,166,133,123,119,123`; retain the uninterpreted `166/119`. |

`H7` applied twice equals `T14` on these coordinates, but the three admitted rails do not form a closed group under indefinite iteration. They are a finite portion of a translation action with a declared domain.

## Joseph as an entire family

Set `L=1933+p−d`, `K=1796+p−d`. The two Joseph intervals are `(L,L−110)` and `(K+110,K)`. Their four ordered endpoints have gaps `27,83,27`; twelve source intervals follow from two portions × two exact phases × three clutch states (File61 §11.3A; File62 §5.2).

With regular Joseph fixed at `1915–1805`, a more informative duration family is

\[
a=18+p-d,\quad b=9-p+d,\quad c=83,\quad a+b=27,\quad a+b+c=110.
\]

For `p=0,3,3.5` and `d=0,7,14`, this gives all nine source-defined Nisan, Aaron whole-year-display, and Aaron exact-phase comparisons. The forward, regular, and backward lives have cyclic partitions `(a,b,c)`, `(b,c,a)`, `(c,a,b)`. File62 §§6.1–6.4 explicitly provides the full-clutch cycles and original/half/full flank transfer. A genuine `C3` permutation of **duration components** appears here, but it must not be misdescribed as an unqualified permutation of historical person nodes.

Whole-year crossed geometry uses `p=3`, giving `30,80,30` and total `140`; exact phase uses `p=3.5`, giving derived component gaps `30.5,79.5,30.5` and total `140.5`. Source File61 §12.3 prints the two exact outer diagonals `137±3.5 = 140.5,133.5`, centered on137 and seven apart. This is one parameterized family, not competing arithmetic.

## Joshua closure: essential state distinction

| Source Joseph interval | Source state | Translation to regular Joshua `1476–1366` |
|---|---|---:|
| `1906–1796` | Original backward Moses/Nisan | `430` |
| `1936–1826` | Original forward Aaron whole-year display | `460` |
| `1933–1823` | Original forward Moses/Nisan | `457` (derived diagnostic) |
| `1936.5–1826.5` | Original forward Aaron exact phase | `460.5` (derived diagnostic) |

File61 §§14.1–14.3 explicitly mixes the first two declared states for the `430/460` pair. Do not treat460 as phase-independent. Translation from regular Joshua `1476–1366` to cumulative `1406–1296` is70; total backward/forward translations are500/530. Fixed Joshua1296 yields `1526−1296=230` and `1512−1296=216`; only the derived start changes (File61 §14.3A; File62 §9).

The double Adam square uses base/backward `14006→1906→1796` and Apparent/forward whole-year `14036→1936→1826`. All three corresponding nodes differ30, so equality of the `12100/12210` spans follows from rigid translation (File61 §§12.4–12.6). The second row is source-dependent cross-branch geometry. Simply replacing1936 with1936.5 while retaining head14036 breaks the square by0.5; this diagnoses a state mismatch rather than a repaired anchor.

## Abraham, Isaac, and phase-scale inputs

File61 §§17.2–17.7 supplies an especially direct bridge to C531's calendar grammar:

- Cumulative Abraham–Isaac175 contracts to161 by moving Abraham only, then returns to175 when Isaac also moves. Moving Isaac alone while holding Abraham2435 gives189, not161.
- `E(161)=175`; exact half and quarter versions are `E(80.5)=87.5` and `E(40.25)=43.75`. These are the same linear rule with `u=7,3.5,1.75` (File62 §8.3).
- Fixed cumulative head14006 and translated Isaac2246 give11760; Isaac2246 to Conquest1406 gives840; total12600. These are `35×336`, `35×24`, `35×360` respectively. This uses the calendar ratio `360/336=15/14`, the ratio between E and P already present at C481/C531.
- Exact phase shift of **both** upper endpoints preserves11760 (`14009.5−2249.5`). The840 dated lower leg belongs to Nisan; holding1406 and moving only Isaac to2249.5 gives843.5, not840.
- Cainan+460 and Apparent Age+30 remain separate inputs. Together they take11760 to12250=`25×490`, and head14006 to14496. Their combined value490 does not identify the two operators.

File62 §§8.1,8.6A gives **proposed** phase pairs `Moses1527t/1526n`, `Aaron1530t/1529n`, `Jacob-call1930t/1929n`. They remain conditional, not promoted primary dates. Source §§8.6A–C uses a twelve-quarter carrier `483=402.5+80.5=12×40.25`; expanding only80.5 byE gives490, while expanding all483 byP also gives490. Expanding all483 byE gives525; `525×336=490×360=176400` exactly. These duration identities can be tested without reintroducing the quarantined1486 event node. Source phase decomposition includes1486 Jan1; leave that endpoint application deferred under current user scope.

## Recommended whole-family tests

1. Reconstruct all42 root coordinates, all invariant spans, and all parent-only contraction outputs from the frozen seven inputs and phase/clutch parameters. Report coordinate matches and preserved intervals separately.
2. Reconstruct all12 exact Joseph intervals and the nine cyclic-partition states from two trunk boundaries plus lifespan110; prove commutation and identify dependent outputs.
3. Recover both declared430/460 routes and their phase diagnostic failures without fitting a new translation or head. Recover70,500/530,230/216 as consequences of fixed targets.
4. Reconstruct the entire11760/840/12600 and12250 Isaac family using the same translations and calendar calibration; explicitly fail the mismatched phase lower leg.
5. Treat `23u→25u` full/half/quarter and the two483→490 routes as a duration-family compression. Preserve proposed phase status and defer1486 endpoint/primer use.
6. Ask whether Joseph's C3 component rotations and the translation commutators provide the next useful limited symmetry object; do not claim a global group acting on every chronology family.

## Exceptions and restrictions worth retaining

File62 §4.4 explicitly keeps the Amram/Famine field different from ordinary male-form landings. The source's detailed birth-law typology, BJ witnesses, and calendar ladder are not needed to establish these arithmetic maps. Source §10.4 and guards acknowledge dependent outcomes. The sources' theological interpretation is not proof of scribal engineering or a numerical probability result.

No new primary anchor, broad gear transport, primer reopening, second decimal inversion, globalH7/T14, SKL transport, or second-order inverse closure is authorized by this memo. Historical Moses1526, Exodus1446, Conquest1406, Joshua regular1366 and cumulative1296 stay fixed. All recommendations are reconstruction tests of a familiar source family, not untouched statistical holdouts.