# Independent finite endpoint review

The concrete central sequence is recovered in S7 §§8–10 and 14–15. Reading only the new synthesis would leave its endpoints unassigned; the named dependency resolves that gap. These notes review only the existing finite rows and their arithmetic consequences.

## Exact coordinate convention

Write a paired point with center `x` as `(x−1/4,x+1/4)` in the supplied proof coordinate. Centers below do not replace civil date labels. At center zero the pair is `1t BC / AD 1n`; center one is `AD 1t/2n`. Same-season interval lengths equal center differences.

## Supplied central envelopes

| Source | Translated Flood F | Translated Shem H | Reflected translated Shem M(H) |
|---|---|---|---|
| MT ON | [−6,1], T5290 | [−8,−1], T5888 | [1,8] |
| SP OFF | [−1,6], T4715 | [−3,4], T5313 | [−4,3] |
| LXX OFF | [0,7], T5290 | [−2,5], T5888 | [−5,2] |

Source lifespan is 600, translation increment is 598. Hence `H=F−2` for each source, and `M(H)=J₁(F)` where `J₁(x)=2−x`. This is a derived local reflection about center one; the global Mirror remains at zero.

The original partition uses centers `−6,−1,0,1,6,7,8`: lengths `5+1+1+5+1+1=14`. Suppress interior center 6 and obtain `5+1+1+6+1`. The junction at center one is an exact co-location of MT translated Flood closing and reflected translated MT Shem opening. Interior SP/LXX boundaries subdivide this continuous union. Source labels remain distinct.

## A further finite incidence consequence

All supplied Flood and reflected-Shem endpoint centers form the set `{−6,−5,−4,−1,0,1,2,3,6,7,8}`. It is invariant under `J₁`. Outer paired centers are:

| Pair | Radius about 1 | Provenance |
|---|---:|---|
| −6 ↔ 8 | 7 | MT F opening ↔ MT M(H) closing |
| −5 ↔ 7 | 6 | LXX M(H) opening ↔ LXX F closing |
| −4 ↔ 6 | 5 | SP M(H) opening ↔ SP F closing |

Thus the selected center subset `−6,−5,−4,1,6,7,8` is relationally equivalent to `−7000,−6000,−5000,0,5000,6000,7000` under the diagnostic local chart `h(x)=1000(x−1)`. This uses existing endpoints only. It is a different traversal from the original serial partition, and its monotone increments are `1+1+5+5+1+1`. It does not identify the sources of the converted macro radii with these source events automatically.

The exact statement is a correspondence of pair-center radii and their incidence at the declared 1000:1 scale. A uniform dilation of proof coordinates would also scale each quarter-year phase offset to 250. Preserving ±1/4 while scaling centers is a separate centerlift convention. No global-axis movement or three-dimensional embedding follows.

## Dependency reduction through 529

`5290−4715=575=529×25/23`; the MT ON−SP OFF source opening gap is `5296−4716=580`, leaving `580−575=5`. LXX OFF−SP OFF has source gap `5290−4716=574`, leaving `575−574=1`. The MT−LXX gap is `6=5+1`.

Normalizing SP to ON gives source gap `5296−5176=120`, translation gap `5290−5175=115`, and the same `5=120−115`. Therefore `575=115+460` and `580=120+460`: the 529 conversion identity is a reduction of the same construction, not another independent witness.

## Finite join verdicts

| Join/comparison | Verdict | Exact reason |
|---|---|---|
| Central MT Flood closing → reflected MT Shem opening | PASS | Both center 1, identical phase pair AD 1t/2n |
| Central 5+1+1+6(+1) partition | PASS | Supplied centers −6,−1,0,1,7,8; widths 5,1,1,6,1 |
| Central source-projected 5/6/7 radii | PASS, relational | J₁ pairs above; all endpoints already supplied |
| Zero-centered macro radius origins → source-event identities of central radial points | UNASSIGNED | Center/length agreement is established; literal common event/anchor interpretation is not |
| MT and LXX regular Creation → reflected Conquest | PASS, common-terminal branches | Both end at C=[1399,1406]; raw lengths 5520/6900 |
| Cainan birth/death cumulative landings | PASS, common-anchor branches | Q7838…7845 and Q8298…8305, separation 460; common comparator Q1398…1405 |
| Flood 6000 companion envelope = primary Conquest | FAIL as identity | Comparator [1400,1407] is +1 from C |
| Cainan/Shelah companion envelope = primary Conquest | FAIL as identity | Comparator [1398,1405] is −1 from C |
| Flood and Cainan/Shelah full companion envelopes = each other | FAIL as identity | Two-year offset |
| Methuselah ON landing = normal MT Creation reflection | FAIL as identity | Corresponding centers are ten years earlier |
| Exodus-centered 350+133+7 | PASS, generated endpoint chain | P1796→P1446→P1313→P1306; source-tagged generated endpoints |
| Converted 26-carrier plus completed 490 as one serial path | UNASSIGNED | Scalar sum valid; shared terminal/start placement not supplied |
| 15 from two 7.5 central arms = full outer elapsed span | FAIL as identity | 7.5+7.5 shares half-year; union is 14.5 |
| 7500 macro radius → existing central pair-center radius 7.5 | UNASSIGNED | Existing outer pair centers have radius 7; far seasonal points radius 7.25 |

## Exact conversion distinctions

Use `κ=25/23`, `π=70/69`, and retain `K=529` elsewhere. `6440κ=7000`, `6900π=7000`, `6900κ=7500`. If the same companion anchor Q1405 were declared held, the first two would generate the same endpoint center 8405, while the last gives 8905. This conditional calculation does not place those converted endpoints unless that anchoring choice is declared. Using κ on both Cainan endpoints preserves the converted lifespan 500; using κ and π collapses the endpoint difference to zero. It is a mixed conversion, not one injective affine map.

Uniformly dilating a seven-year envelope gives `175/23` years under κ and `490/69` under π. These exceed 7 by `14/23` and `7/69`. The supplied §13 construction instead converts closing-member widths then translates each entire envelope, retaining seven years. Uniform dilation of a 7.5-year spread would give `375/46` or `175/23` years respectively.

## Year 14 / Year 15

Central matching-season arms have lengths 7 and 7, meeting exactly at the selected matching seasonal component of center one. Outer full seasonal intervals are `[−25/4,5/4]` and `[3/4,33/4]`: each 15/2, overlap 1/2, outer union 29/2. Their sum is 15 and union 14.5. Preserve the user's complementary Year 14/15 readings with this distinction. The macro ±7000 radius construction and the older 7000+4+7000 fixed-landing construction likewise retain different anchors.