File: File_15
Title: The Generational Axiom
Entity: Generational Units
Classification: Foundational Axiom / Authorized Measuring Standards
UUID: AXIOM_GENERATIONAL_UNITS
Status: Final; post-final pressure tested
Pressure-test status: Current; publication-clean verification complete; post-final pressure-tested status retained.
Register Agreement status: Checked against State_Vocabulary_Register v1.17 and active companion controls; convergence / pointer update only; no arithmetic, anchors, modal states, node-classes, operators, sign conventions, slash-pairs, ranges, envelopes, Mirror protocols, Machine Guards, claim-status labels, theological claims, or dependency boundaries changed.
Publication-cleanup status: Public-clean Markdown source; archival amendment history routed to Repository_Change_Archive.
Source-history status: Version 5.1
Primary domain: Regular; Cumulative; Theological; Foundational Axiom
Traditions: MT; NT / Matthew; patriarchal narrative chronology; Exodus / Conquest chronology
Canonical source: Markdown
Primary anchors: 2166 BC; 1886 BC; 1846 BC; 1486 BC; 1446 BC; 1406 BC; 1399 BC; 6 BC; AD 2; 14006 BC; 14466 BC
Related files: File_00; File_01; File_04; File_05; File_06; File_07; File_09; File_13; File_14; File_16; File_17; File_51a; File_54; Restart Capsule v11.15; 490d Repository Style Guide v2.5; Project Procedures v3.2; State Vocabulary Register v1.17
Major operators: post-calling lifespan authorization; Abrahamic 100; Jacobean 70; Mosaic 40; exact normalized triad `100:70:40 = 10:7:4`; Abrahamic-base fractions `10/10`, `7/10`, `4/10`; 10x macro scale; fourth-generation multiplication; LCM(40, 70, 100) = 1400; same-side BC span; civil cross-axis span `BC + AD − 1`; endpoint-pair bracket display; cumulative-state distinction; chiastic overlap; Matthew 14-generation echo
Major modal states: authorized generational-unit state; call-to-death generational-span register; biological-fractal generation register by dependency; exact normalized generational triad state; Abrahamic unit; Jacobean unit; Mosaic unit; fourth-generation multi-scale state; 1400-year LCM state; regular Christological 1400 endpoint state; cross-axis Christological endpoint state; ordered endpoint-pair bracket state; cumulative MT 14000 state; MT + Cainan / restored cumulative 2nd Cainan dependency state; chiastic-overlap state; Terah-Egypt center/doubled-center display state; Matthew 14-generations echo state; Double-7 Sabbath-completion state
Revision basis: 490d Repository Style Guide v2.5; Restart Capsule v11.15; State Vocabulary Register v1.17; Project Procedures v3.2; prior finalized source; Repository_Change_Archive (non-controlling history)
Current refresh note: Public-clean trust-status equalization aligns active control pointers to 490d Repository Style Guide v2.5, Restart Capsule v11.15, State_Vocabulary_Register v1.17, and Project Procedures v3.2. Register Agreement status is recorded in the header; no arithmetic, anchors, modal states, node-classes, operators, sign conventions, slash-pairs, ranges, envelopes, Mirror protocols, Machine Guards, claim-status labels, theological claims, or dependency boundaries are changed. Detailed pass logs, pressure-test records where applicable, pointer-refresh history, and replacement-workflow notes are preserved in Repository_Change_Archive.

# File_15 — The Generational Axiom

## Related Repository Files

| File | Function for `File_15` |
|---|---|
| `File_00` | Controls foundational axiom style, chiastic overlap, source-map discipline, Machine Guards, and Mirror / civil / inverse firewalling. |
| `File_01` | Controls Abraham biography anchors, especially Abraham birth, call, and death states where Abrahamic `100` is invoked. |
| `File_04` | Controls Jacob as Jubilee standard and Jacob’s `147 = 3 × 49` lifespan where Jacobean `70` is invoked. |
| `File_05` | Controls the primary `1446 BC` Exodus anchor and Exodus-related dependency boundaries. |
| `File_06` | Controls `1406 BC`, land-rest / Sabbath-rest envelope material, and Conquest-side state language by dependency. |
| `File_07` | Controls cumulative Creation `14006 BC`, restored cumulative 2nd Cainan Creation states, and Creation-side cumulative dependencies. |
| `File_09` | Controls cumulative architecture and restored cumulative 2nd Cainan `+460` where the source `MT + Cainan` row is active. |
| `File_13` | Controls Christological dual anchors, `6 BC`, `AD 2`, civil cross-axis Christological states, and Matthew / Luke Christological states by dependency. |
| `File_14` | Controls Biological Fractals dependency vocabulary where Moses `40` / `80`, purification, gestation, or biological-fractal state logic is opened by dependency. |
| `File_16` | Controls `1846 BC`, `1876 BC`, Affliction / Sojourn distinctions, and Christological criterion material. |
| `File_17` | Controls prophetic span-component anatomy and identifies `File_15` as a related foundational axiom file. |
| `File_51a`; `File_54` | Provide current procedural, state-register, File map, and claim-status style precedents. |

## 0. File-function

`File_15` defines the authorized generational units used in the repository: the Abrahamic `100`, the Jacobean `70`, and the Mosaic `40`.

The source argues that these units are authorized by the productive lifespan of a patriarch from divine commission to death rather than by abstract numerology. The file then applies these units to fourth-generation patterns, the `1400`-year LCM harmonic, chiastic overlap structures, and the `14000`-year cumulative Christological lock.

Publication-cleanup note:
This public-clean source preserves the Final and post-final pressure-tested file status in the header. Detailed pass logs, pressure-test records, finalization records, and replacement-workflow notes are archived rather than repeated in the public body.

Modal-state rule:
A shared date-label, number, span, range, theological phrase, or generational term does not create identity unless state, node-class, tradition, and operator agree.

Argument-control note:
Arithmetic facts and textual / source-retained chronology data carry the base layer of this file. Structural inferences explain how the authorized units, fourth-generation scales, `1400` LCM, and `14000` cumulative lock interlock. Theological and typological claims explain meaning, but they do not replace arithmetic, state, or dependency control.

Source-reconstruction scope note:
No LXX chronology layer is active in this file. The Smith / Rudd LXX source-control instructions therefore remain inactive for `File_15` unless a later pass or cross-file update explicitly opens an LXX Genesis chronology state.

### 0.1 Working state register

The following states are active in `File_15`. They are simultaneous, non-competing states unless a local section explicitly narrows the working state.

| State | Primary values / forms | Node-class / function | Handling in `File_15` |
|---|---:|---|---|
| File identity state | `File_15`; `THE_GENERATIONAL_AXIOM.md`; `AXIOM_GENERATIONAL_UNITS` | foundational axiom / authorized measuring-standards file | Preserved. |
| Source-history state | Version 5.1 | source-history label | Preserved separately from the current Final status. |
| Authorization principle state | divine commission to death | theological / structural axiom | Productive call-to-death lifespan authorizes a generational unit. This is the file’s central structural axiom, not an ordinary lifespan table by itself. |
| Call-to-death generational-span register | Abraham `100`; Jacob `70`; Moses `40` | foundational generational-unit axiom | Integrated before Arithmetic Pass. The shared metric is the span from divine call to death. |
| Exact normalized generational triad state | `100:70:40 = 10:7:4`; `10/10`; `7/10`; `4/10 = 2/5` | exact ratio / Abrahamic-base display | Added as the executable ratio state. Decimal displays remain secondary source displays only. |
| Biological-fractal generation register | `40`; `80`; Moses-related dependency | File_14 dependency state where opened | Retained as dependency vocabulary. `File_15` does not locally re-prove File_14's biological-fractal argument. |
| Abrahamic post-calling unit | `75 → 175 = 100`; unit `100` | authorized generational unit | Preserved as Abrahamic standard. Do not collapse with every other `100` in the file. |
| Jacobean post-calling unit | `77 → 147 = 70`; unit `70` | authorized generational unit | Preserved as Jacobean standard. Author clarification identifies `File_04` as the governing dependency for Jacob age `77`, using Jacob birth `2006 BC` and Flight to Haran / Bethel at `1929 BC`. |
| Mosaic post-calling unit | `80 → 120 = 40`; unit `40` | authorized generational unit | Preserved as Mosaic standard. Author clarification identifies `File_14` Biological Fractals as the controlling dependency for Moses `40` / `80` biological-fractal framing where that layer is opened; Arithmetic Pass checks only the visible File_15 span while fuller provenance remains dependency-controlled. |
| Authorized-unit triad | `100`; `70`; `40` | coordinated measuring rods | These units synchronize at `1400` but remain distinct authorized standards. |
| Call-to-death normalized fraction state | `100:70:40 = 10:7:4`; Abrahamic `10/10`; Jacobean `7/10`; Mosaic `4/10` | exact rational / normalized unit display | Author clarification integrated in the Arithmetic Pass. The exact normalized display is proof-bearing; the source decimal approximation remains source-retained only. |
| 10x macro-scale state | `1000`; `700`; `400` | macro-scale display / amplification | Scale-state only. Do not treat the 10x labels as ordinary date spans unless a local row supplies endpoints. |
| Fourth-generation multi-scale state | `4 × 100 = 400`; `4 × 70 = 280`; `4 × 40 = 160` | Genesis 15:16 fourth-generation application | Preserved as three authorized-scale applications of the fourth-generation principle. |
| Abraham → Isaac death state | `2166 BC → 1886 BC = 280` | Jacobean fourth-generation application | Source-retained; Arithmetic Pass confirms the same-side BC span. |
| Affliction state | `1846 BC → 1446 BC = 400` | Abrahamic fourth-generation / Affliction state | Source-retained; dependency-controlled by `File_16` / `File_05`. Do not collapse with the `1876 BC` Sojourn state. |
| Moses Midian state | `1486 BC → 1446 BC = 40` | single Mosaic generation | Source-retained. Author clarification assigns the fuller Moses / Midian provenance to `File_14`; Arithmetic Pass checks the visible same-side span only. |
| LCM harmonic state | `LCM(40, 70, 100) = 1400` | generational convergence cycle | Preserved as the file’s mathematical harmonic claim; Arithmetic Pass confirms the LCM computation. |
| Regular Christological 1400 endpoint state | `1406 BC → 6 BC = 1400` | Conquest-to-Christ birth endpoint pair | Same-side BC state. Distinct from the cross-axis `1399 BC → AD 2` state. |
| Cross-axis Christological endpoint state | `1399 BC → AD 2 = 1400` | first Sabbath rest / post-naming endpoint pair | Civil cross-axis state. Arithmetic Pass checks by `1399 + 2 − 1 = 1400`. |
| Ordered endpoint-pair bracket state | `1406–1399 BC → 6 BC–AD 2` | range / bracket display | Preserved as two coordinated endpoint-pair claims: front-to-front and rear-to-rear. Author clarification assigns the Conquest / rest side to `File_06` and the Christological / cross-axis side to `File_13`. It is not treated as a claim that every internal cross-pair yields `1400` unless a later author decision says so. |
| Cumulative MT 14000 state | `14006 BC → 6 BC = 14000`; `10 × 1400` | cumulative MT macro-scale lock | Preserved as cumulative state, not an ordinary regular Creation endpoint. |
| MT + Cainan / restored cumulative dependency state | `14466 BC → 6 BC = 14460` | restored cumulative 2nd Cainan dependency state | Classified as restored cumulative 2nd Cainan by dependency. It is not locally re-proved and must not be confused with regular restored 2nd Cainan `+130`. |
| Chiastic-overlap state | literal reading and chiastic reading | simultaneous overlap readings | Preserved under `File_00` dependency. Literal and chiastic results must not be collapsed. |
| Shem → Isaac chiastic state | `100 + 290 + 100`; `390 + 390` | literal / chiastic paired reading | Preserved as a chiastic-overlap example. The `100` values frame the Abrahamic standard but do not become the post-calling Abrahamic unit by themselves. |
| Terah → Egypt center / doubled-center state | center `160`; doubled center `320` | Mosaic fourth-generation center and chiastic display | Preserve `160 = 4 × 40` as center value and `160 × 2 = 320` as doubled-center display. |
| Abraham lifespan chiastic state | `75 + 25 + 75`; `100 + 100` | literal lifespan / chiastic reading | Preserve literal lifespan and chiastic reading separately. |
| Generational hierarchy state | `100 = 2.5 × 40`; `70 = 1.75 × 40`; `100 ≈ 1.43 × 70`; normalized triad `10:7:4` | hierarchy / ratio display | Decimal displays are retained as source display labels. Author clarification allows the exact normalized fractions `10/10`, `7/10`, and `4/10` relative to the Abrahamic unit. Exact-ratio review is completed in Arithmetic Pass. |
| Synchronization-at-1400 state | `1400 ÷ 100`; `1400 ÷ 70`; `1400 ÷ 40` | tri-unit synchronization display | Preserved as generational convergence table. |
| Matthew 14-generations echo state | `14 + 14 + 14 = 42`; `14000 = 14 × 1000`; `1400 = 14 × 100` | NT / Matthew comparison state | Preserve Matthew’s genealogical `14` and the chronological `14` echoes as related but non-identical uses. |
| Double-7 Sabbath-completion state | `14 = 7 + 7`; `1400 = 7 × 200`; `14000 = 7 × 2000` | theological / structural interpretation | Preserved as source theological meaning. Does not replace arithmetic. |

### 0.2 Node-class and display-state register

| Date / value / display | Node-class | Active state | Do not collapse with |
|---|---|---|---|
| 2166 BC | Abraham birth | Abraham → Isaac death verification; Abrahamic dependency | Abraham call at age `75`; `2091 BC` unless a dependency opens it |
| 2091 BC | Abraham call at age `75` | Abrahamic post-calling unit dependency | Abraham birth `2166 BC`; Abraham death `1991 BC` |
| 1991 BC | Abraham death at age `175` | Abrahamic post-calling unit dependency | adjusted / SP / LXX death-state `1776 BC` unless that state is active |
| 2006 BC | Jacob birth | File_04 dependency anchor for Jacob age `77` | other Jacob biography states unless state label agrees |
| 1929 BC | Jacob Flight to Haran / Bethel | File_04 dependency anchor for Stairway-to-Heaven age `77` | Jacob return from Haran `1909 BC`; Jacob Entry `1876 BC` |
| 1886 BC | Isaac death | Jacobean fourth-generation endpoint | Joseph elevation or rounded scaffold `1886 BC` states unless explicitly opened by dependency |
| 1846 BC | Affliction start | Affliction / `400` state | `1876 BC` Sojourn / Entry state controlled by `File_16` and `File_04` |
| 1486 BC | Moses in Midian start | Mosaic generation state | `1446 BC` Exodus anchor |
| 1446 BC | Exodus | primary Exodus anchor by dependency | subordinate Ramesside `1231 BC` state; Conquest `1406 BC` |
| 1406 BC | Conquest | front member of Christological endpoint bracket | `1399 BC` first Sabbath rest / land-rest endpoint |
| 1399 BC | first Sabbath rest / land-rest endpoint | rear member of Christological endpoint bracket | `1406 BC` Conquest |
| 6 BC | Christ’s birth | birth endpoint and cumulative MT target | AD 2 post-naming endpoint; `1n BC → AD 1n` File_13 covenant anchor unless explicitly opened |
| AD 2 | post-naming endpoint | cross-axis Christological endpoint | `6 BC` birth endpoint |
| 14006 BC | cumulative MT Creation | cumulative MT macro-scale state | regular Creation endpoints; restored cumulative `14466 BC`; rounded Scaffold `4106 BC` |
| 14466 BC | MT + Cainan cumulative Creation | restored cumulative 2nd Cainan dependency state | regular restored Cainan `+130`; ordinary regular Creation endpoint |
| `100` | Abrahamic unit | post-calling Abrahamic standard | Shem/Abraham begetting age `100` unless the local chiastic section states that frame |
| `70` | Jacobean unit | post-calling Jacobean standard | nations / Daniel / Matthew uses unless the local section opens them |
| `40` | Mosaic unit | post-calling Mosaic standard | generic testing `40`; purification `40`; File_17 component state unless opened |
| `160` | center value | Terah → Egypt Mosaic fourth-generation center | doubled-center display `320` |
| `320` | doubled-center display | chiastic center doubled | ordinary elapsed chronology unless chiastic operator is active |
| `1400` | LCM convergence cycle | generational harmonic | generic span family or prophetic component unless state is named |
| `14000` | cumulative MT 10x lock | `10 × 1400` macro-scale state | `14460` MT + Cainan span |
| `14460` | MT + Cainan span display | restored cumulative 2nd Cainan dependency row | `14000` standard MT cumulative span |
| `14` | Matthew count / Abrahamic generation count / Double-7 value | related but distinct displays | one generic `14` operator. Restate whether Matthew, `1400 ÷ 100`, or `7 + 7` is active. |

### 0.3 File map

| Section | Function | Active state / operator |
|---|---|---|
| Related Repository Files | Names controlling dependencies and boundary files | dependency tracking |
| §0 | Defines file function, working states, node-classes, File map, operators, range controls, Machine Guards, arithmetic controls, claim-status controls, argument controls, and pass scope | file-level modal-state controls; Arithmetic Pass ledger; Argument Pass ledger |
| §1 | States the authorization principle | post-calling lifespan authorization |
| §2 | Records the three patriarchal standards and their 10x macro scale | call-to-death generational-span register; Abrahamic `100`; Jacobean `70`; Mosaic `40`; exact normalized triad `10:7:4`; `10x` scale-state |
| §3 | Applies the fourth-generation principle | `4 × 100`; `4 × 70`; `4 × 40`; same-side BC source checks |
| §4 | Records the `1400`-year harmonic | `LCM(40, 70, 100) = 1400`; endpoint-pair bracket state; same-side and cross-axis endpoint states |
| §5 | Records generational patterns in chiastic structures | chiastic overlap; literal / chiastic readings; center / doubled-center display |
| §6 | Records the generational hierarchy | synchronization at `1400`; hierarchy ratio displays |
| §7 | Records the `14000`-year Christological lock | cumulative MT; restored cumulative Cainan dependency; Matthew 14 echo; Double-7 theological note |
| §8 | Preserves cross-reference and dependency index | legacy source references and current dependency handling |
| Audit notes | Records retained audit and dependency notes | audit notes; dependency boundaries |
| §0.12 | Records the post-final pressure-test result | pressure-test status; source-hygiene confirmation |
| Audit notes | Records retained audit and dependency notes | audit notes; dependency boundaries |

### 0.4 Operator and sign-convention register

| Operator / sign convention | Active form | Scope / guard |
|---|---|---|
| Post-calling lifespan authorization | age at death minus age at calling | Defines Abrahamic, Jacobean, and Mosaic units. Arithmetic checks cleanly in §0.8 and §2.1. |
| Fourth-generation multiplication | `unit × 4` | Applies separately to Abrahamic `100`, Jacobean `70`, and Mosaic `40`. |
| Lowest common multiple | `LCM(40, 70, 100)` | Produces the `1400` convergence state. Arithmetic Pass checks by prime-factor LCM. |
| Same-side BC span | higher BC value minus lower BC value | Active for BC-to-BC claims such as `1406 BC → 6 BC`, `14006 BC → 6 BC`, and `14466 BC → 6 BC`. |
| Civil cross-axis span | `BC + AD − 1` | Active for `1399 BC → AD 2`. This is civil arithmetic, not Mirror protocol. |
| Endpoint-pair bracket display | front endpoint maps to front endpoint; rear endpoint maps to rear endpoint | Active for `1406–1399 BC → 6 BC–AD 2`. Does not create every-internal-pair equality. |
| 10x macro projection | `1400 × 10`; `100 × 10`; `70 × 10`; `40 × 10` | Scale-state operator. Do not treat 10x as Mirror scaling or ordinary date subtraction unless endpoints are supplied. |
| Cumulative-state distinction | cumulative MT; restored cumulative 2nd Cainan | Cumulative states must not be used as regular Creation endpoints. |
| Chiastic overlap | literal reading plus chiastic / overlapped reading | Active only in §5. Literal and chiastic results are simultaneous readings. |
| Generational division | `1400 ÷ unit` | Synchronization display in §6.2; arithmetic checked in §0.8. |
| Exact normalized triad | `100:70:40 = 10:7:4`; Abrahamic-base fractions `10/10`, `7/10`, `4/10 = 2/5` | Executable ratio state added in the Arithmetic Pass. |
| Approximate decimal display | `100 ≈ 1.43 × 70` | Source display only. The exact executable form is `100/70 = 10/7 ≈ 1.43`; the decimal is retained only as an approximate display. |
| Mirror protocol | inactive | No formal Mirror target-generation protocol is active in the main `File_15` argument. Civil cross-axis span counting remains distinct from Mirror. |
| LXX source-reconstruction witness | inactive | No active LXX Genesis chronology layer appears in `File_15`; Smith / Rudd source-control is not invoked locally. |

### 0.5 Slash-pair, range, and display-state register

| Form | State type | Final handling |
|---|---|---|
| `1406–1399 BC` | ordered endpoint range / Conquest-to-rest bracket | Preserve as a bracket. Do not collapse to one year and do not average. |
| `6 BC–AD 2` | ordered Christological endpoint bracket | Preserve as a birth-to-post-naming bracket. Do not collapse `6 BC` and AD 2 into one Christological endpoint. |
| `1406–1399 BC → 6 BC–AD 2` | ordered endpoint-pair display | Treat as two paired endpoint checks: `1406 − 6 = 1400` and `1399 + 2 − 1 = 1400`. Author decision required only if the intended claim is stronger than endpoint-pairing. |
| `Call from Ur/Haran` | source-geography compression | Preserve as source wording. This is not a date slash-pair. |
| `4 × 80 or 8 × 40` | equivalent chiastic generation-type display | Preserve both as display forms for `320`; do not treat them as competing chronological states. |
| `10x` | macro-scale display | Preserve as scale-state marker. Do not convert to `×10` table arithmetic unless the row supplies the scale relation. |
| Calendar suffixes | none active | No `n` / `t` suffix-state is active in this file. |
| Mirror slash-pairs | none active | No Mirror slash-pair or phase-resolved Mirror state is active in this file. |

### 0.6 Machine Guards for `File_15`

Machine Guard [AUTHORIZED GENERATIONAL UNITS]:

The Abrahamic `100`, Jacobean `70`, and Mosaic `40` are authorized generational units in this file. They must not be treated as interchangeable spans merely because they synchronize at `1400`.

Machine Guard [SAME-SIDE / CROSS-AXIS FIREWALL]:

Same-side BC spans use ordinary BC subtraction. Cross-axis civil spans use `BC + AD − 1`. The source’s `1406 BC → 6 BC` and `1399 BC → AD 2` endpoint claims remain distinct and check under their proper operators in the Arithmetic Pass.

Machine Guard [ENDPOINT-PAIR BRACKET]:

The bracket `1406–1399 BC → 6 BC–AD 2` is preserved as an ordered endpoint-pair display. The front endpoint pairs with the front endpoint, and the rear endpoint pairs with the rear endpoint. Do not treat the bracket as a claim that every internal cross-pair equals `1400` unless a later author decision explicitly states that stronger claim.

Machine Guard [CUMULATIVE CAINAN DEPENDENCY]:

The `MT + Cainan` row preserves the source value `14466 BC → 6 BC = 14460`. In this pass it is classified as restored cumulative 2nd Cainan by dependency. It remains dependency-controlled until the active Cainan state is confirmed by `File_07`, `File_09`, `File_16`, Restart Capsule v11.15, or another controlling source.

Machine Guard [CHIASTIC OVERLAP]:

Literal and chiastic readings are simultaneous readings. Do not collapse the literal span into the chiastic result, and do not treat the chiastic result as ordinary elapsed chronology unless the local section explicitly opens that operator.

Machine Guard [TERAH-EGYPT CENTER / DOUBLED-CENTER]:

In the Terah → Egypt chiasm, `160` is the center value and `320` is the doubled-center display. Preserve both node-classes. Do not treat `320` as replacing the `160 = 4 × 40` fourth-generation center.

Machine Guard [MATTHEW 14 NON-COLLAPSE]:

Matthew’s `14 + 14 + 14`, the `1400 ÷ 100 = 14` Abrahamic-generation count, and the Double-7 statement `14 = 7 + 7` are related but distinct states. Restate the active state before using `14` in argument.

Machine Guard [ARGUMENT BURDEN]:

Arithmetic facts and textual / source-retained chronology data carry the proof burden in `File_15`. Structural inferences may support the argument when the active state and operator are named. Theological, typological, and Christological notes explain meaning and coherence but must not be used as substitutes for arithmetic verification or dependency control.

Machine Guard [CANONICAL MARKDOWN / DERIVED AI FORMATS]:

This file remains canonical Markdown source. The file header, File map, state register, tables, Machine Guards, and audit notes are source controls, not JSONL, prompt/response, or generic AI-summary format.

### 0.7 Pre-Arithmetic audit clarification register

This register preserves author-supplied clarifications received before the Arithmetic Pass. It resolves dependency provenance; arithmetic verification is recorded in §0.8 and in the section-level arithmetic checks.

| Clarification | Applied handling after Arithmetic Pass |
|---|---|
| `1406–1399 BC → 6 BC–AD 2` bracket | Treat as an ordered endpoint-pair display governed by two dependencies: `File_06` controls the `1406 BC` Conquest and `1399 BC` land-rest side; `File_13` controls the Christological side, including `6 BC`, AD 2, dual Christ anchors, civil cross-axis states, and Mirror / civil / inverse firewalling. No author-decision blocker remains unless the author later intends every internal cross-pair to equal `1400`. |
| Jacob age `77` | Treat as dependency-resolved by `File_04`: Jacob birth `2006 BC`; Flight to Haran / Bethel at `1929 BC`; visible same-side derivation `2006 − 1929 = 77`. `File_15` may cite this as the provenance of the Jacobean call-state without re-proving the whole Jacob file. |
| Abrahamic `100` | Treat as dependency-controlled by `File_01`: MT Regular Abraham biography uses birth `2166 BC`, call `2091 BC`, and death `1991 BC`; the authorized unit is the post-calling span from call at age `75` to death at age `175`. |
| Moses / Midian `40` | Treat as source-retained in `File_15` and fuller-provenance controlled by `File_14` where Biological Fractals / Moses `40` / `80` state logic is active. The Arithmetic Pass checks the visible `1486 BC → 1446 BC = 40` same-side span without importing the entire File_14 argument. |
| Triad ratio display | Preserve the source decimal display `100 ≈ 1.43 × 70`; Arithmetic Pass adds exact ratio language: `100:70:40 = 10:7:4`, or fractions of the Abrahamic base as Abrahamic `10/10`, Jacobean `7/10`, Mosaic `4/10`. |
| Call-to-death axiom restatement | Integrate the author-supplied axiom as canonical File_15 prose before executing the Arithmetic Pass. Use exact rational notation and claim-status control. |

### 0.8 Arithmetic-control summary

Arithmetic Pass scope:
This pass checks visible same-side BC spans by subtraction, the active civil cross-axis span by `BC + AD − 1`, exact integer multiplications and divisions, exact ratio reductions, LCM computation, and displayed chiastic sums. It does not re-prove dependency-controlled provenance in `File_01`, `File_04`, `File_06`, `File_13`, `File_14`, `File_16`, `File_17`, `File_07`, or `File_09`.

Arithmetic-control result:
All visible arithmetic in `File_15` checks under the stated operators. Remaining notes are dependency-boundary or legacy-cross-reference notes, not arithmetic blockers.

| Field | Calculation | Result | Arithmetic status | Claim-status |
|---|---|---:|---|---|
| Abrahamic call-to-death span | `175 − 75` | 100 | checks | Arithmetic fact; File_01 dependency for biography provenance. |
| Jacobean call-to-death span | `147 − 77` | 70 | checks | Arithmetic fact; File_04 dependency for Jacob age `77`. |
| Mosaic call-to-death span | `120 − 80` | 40 | checks | Arithmetic fact; File_14 / Moses dependency for fuller biological-fractal provenance where opened. |
| Abraham date-level call-to-death span | `2091 − 1991` | 100 | checks | Arithmetic fact; File_01 dependency. |
| Jacob birth to Bethel / Flight to Haran | `2006 − 1929` | 77 | checks | Arithmetic fact; File_04 dependency. |
| Jacob Bethel / Flight to Haran to death | `1929 − 1859` | 70 | checks | Arithmetic fact; File_04 dependency. |
| Moses Exodus / call-transition to death / Conquest-begins state where active | `1446 − 1406` | 40 | checks | Dependency-controlled arithmetic fact; File_14 / File_06 node-class boundary. |
| Abrahamic macro scale | `100 × 10` | 1000 | checks | Arithmetic fact; 10x scale-state. |
| Jacobean macro scale | `70 × 10` | 700 | checks | Arithmetic fact; 10x scale-state. |
| Mosaic macro scale | `40 × 10` | 400 | checks | Arithmetic fact; 10x scale-state. |
| Fourth-generation Abrahamic scale | `4 × 100` | 400 | checks | Arithmetic fact; Genesis 15:13 / 15:16 application by source argument. |
| Fourth-generation Jacobean scale | `4 × 70` | 280 | checks | Arithmetic fact. |
| Fourth-generation Mosaic scale | `4 × 40` | 160 | checks | Arithmetic fact. |
| Abraham birth to Isaac death | `2166 − 1886` | 280 | checks | Arithmetic fact; same-side BC span; dependency-controlled endpoint provenance. |
| Affliction span | `1846 − 1446` | 400 | checks | Arithmetic fact; same-side BC span; File_16 / File_05 dependency. |
| Moses in Midian | `1486 − 1446` | 40 | checks | Arithmetic fact; same-side BC span; Moses / File_14 provenance remains dependency-controlled. |
| LCM convergence | `LCM(40, 70, 100)` | 1400 | checks | Arithmetic fact. |
| LCM prime-factor check | `40 = 2^3 × 5`; `70 = 2 × 5 × 7`; `100 = 2^2 × 5^2` | `2^3 × 5^2 × 7 = 1400` | checks | Arithmetic fact. |
| Conquest to Christ birth endpoint | `1406 − 6` | 1400 | checks | Arithmetic fact; same-side BC span; File_06 / File_13 dependencies. |
| Land-rest endpoint to post-naming endpoint | `1399 + 2 − 1` | 1400 | checks | Arithmetic fact; civil cross-axis span. |
| Ordered bracket display | front pair `1406 → 6`; rear pair `1399 → AD 2` | 1400 / 1400 | checks as endpoint-pair display | Arithmetic fact; not an every-internal-cross-pair claim. |
| Conquest / land-rest bracket width | `1406 − 1399` | 7 | checks | Arithmetic fact; ordered range display. |
| Christological bracket width | `6 + 2 − 1` | 7 | checks | Arithmetic fact; civil cross-axis ordered range display. |
| Cumulative MT Creation to Christ birth | `14006 − 6` | 14000 | checks | Arithmetic fact; same-side BC span; cumulative MT state. |
| Cumulative 10x projection | `1400 × 10` | 14000 | checks | Arithmetic fact; macro-scale state. |
| Restored cumulative Cainan row | `14466 − 6` | 14460 | checks | Arithmetic fact; same-side BC span; Cainan mechanism dependency-controlled. |
| Restored cumulative Cainan visible offset | `14466 − 14006` | 460 | checks | Arithmetic fact; mechanism remains dependency-controlled. |
| Shem → Isaac literal chiasm | `100 + 290 + 100` | 490 | checks | Arithmetic fact; literal reading. |
| Shem → Isaac Jacobean factorization | `70 × 7` | 490 | checks | Arithmetic fact. |
| Shem → Isaac chiastic reading | `390 + 390` | 780 | checks | Arithmetic fact; chiastic reading. |
| Curse multiple display | `130 × 6` | 780 | checks | Arithmetic fact. |
| Terah → Egypt literal chiasm | `130 + 160 + 130` | 420 | checks | Arithmetic fact; literal reading. |
| Terah → Egypt doubled center | `160 × 2` | 320 | checks | Arithmetic fact; doubled-center display. |
| Terah → Egypt Mosaic equivalent | `4 × 80`; `8 × 40`; `4 × 40` | 320; 320; 160 | checks | Arithmetic fact; center and doubled-center node-classes preserved. |
| Abraham lifespan literal structure | `75 + 25 + 75` | 175 | checks | Arithmetic fact; literal lifespan structure. |
| Abraham lifespan chiastic reading | `100 + 100`; `2 × 100` | 200 | checks | Arithmetic fact; chiastic reading. |
| Abrahamic-to-Mosaic relation | `100 ÷ 40` | `5/2 = 2.5` | checks | Arithmetic fact; exact ratio with decimal display. |
| Jacobean-to-Mosaic relation | `70 ÷ 40` | `7/4 = 1.75` | checks | Arithmetic fact; exact ratio with decimal display. |
| Abrahamic-to-Jacobean relation | `100 ÷ 70` | `10/7 ≈ 1.43` | checks | Arithmetic fact; exact ratio governs, decimal retained as approximate display. |
| Normalized generational triad | `100:70:40` | `10:7:4` | checks | Arithmetic fact; exact ratio state. |
| Abrahamic-base fractions | `100/100`; `70/100`; `40/100` | `10/10`; `7/10`; `4/10 = 2/5` | checks | Arithmetic fact; exact normalized display. |
| Abrahamic generations in 1400 | `1400 ÷ 100` | 14 | checks | Arithmetic fact. |
| Jacobean generations in 1400 | `1400 ÷ 70` | 20 | checks | Arithmetic fact. |
| Mosaic generations in 1400 | `1400 ÷ 40` | 35 | checks | Arithmetic fact. |
| Matthew count | `14 + 14 + 14` | 42 | checks | Arithmetic fact; NT / Matthew textual datum by source. |
| Cumulative span as Abrahamic millennial | `14 × 1000` | 14000 | checks | Arithmetic fact. |
| Regular span as Abrahamic | `14 × 100` | 1400 | checks | Arithmetic fact. |
| Double-7 expression | `7 + 7` | 14 | checks | Arithmetic fact; theological meaning remains source claim. |
| Regular Double-7 span | `7 × 200`; `7 × (2 × 100)` | 1400 | checks | Arithmetic fact. |
| Cumulative Double-7 span | `7 × 2000`; `7 × (2 × 1000)` | 14000 | checks | Arithmetic fact. |

Arithmetic note:
The source approximation `1 Abrahamic (100) ≈ 1.43 Jacobean (70)` is retained as a display, but the executable ratio is `100/70 = 10/7`. For internal repository use, the exact normalized triad is preferred: `100:70:40 = 10:7:4`, with Abrahamic-base fractions `10/10`, `7/10`, and `4/10 = 2/5`.

### 0.9 Claim-status register for `File_15`

| Claim-status label | Function in `File_15` |
|---|---|
| Textual datum | Direct biblical source data cited by the file, especially Genesis 15:13, Genesis 15:16, Matthew 1:17, and the call-event passages named in the patriarchal table. |
| Source-retained chronology datum | Date anchors and event labels preserved from dependency files, such as `2166 BC`, `1886 BC`, `1846 BC`, `1486 BC`, `1446 BC`, `1406 BC`, `1399 BC`, `6 BC`, AD 2, `14006 BC`, and `14466 BC`. |
| Arithmetic fact | Direct calculations from stated ages, dates, spans, ratios, or multiplications, including the call-to-death spans, `LCM(40, 70, 100) = 1400`, endpoint-pair checks, cumulative spans, chiastic sums, and Matthew / Double-7 expressions. |
| Foundational axiom | The file-level claim that the Abrahamic `100`, Jacobean `70`, and Mosaic `40` function as authorized repository measuring standards. |
| Structural inference | The claim that the three call-to-death units form an exact normalized triad, that the fourth-generation principle operates across these scales, and that the `1400` LCM functions as the complete generational convergence state. |
| Typological reading | The association of Abraham with promise / faith / covenant, Jacob with prophecy / nations, and Moses with testing / wilderness / judgment where the file uses those themes to explain the generational standards. |
| Theological note | Christological, Sabbath-completion, and Double-7 meaning attached to the checked arithmetic. These notes preserve the source’s theological claims but do not replace arithmetic proof. |
| Dependency-controlled | Claims whose full provenance is governed by another file, including Jacob age `77` by `File_04`, Abraham biography by `File_01`, the Moses / File_14 biological-fractal layer, Conquest / land-rest by `File_06`, Christological endpoints by `File_13`, Affliction / Sojourn by `File_16`, and cumulative Cainan by `File_07` / `File_09` / `File_16`. |
| Source-history note | Legacy section labels, compressed source headings, and retained source displays such as the approximate decimal `1.43`. |
| Audit note | Material retained for future author decision or mapping, especially any stronger reading of the endpoint bracket and legacy cross-reference section labels. |

Claim-status rule:
`File_15` may use theological or typological language to explain why the generational standards matter, but the executable argument rests on textual data, source-retained chronology data, arithmetic facts, and explicitly named structural inferences.

### 0.10 Argument-control summary

Argument Pass scope:
This pass clarifies the evidential role of each claim. It does not re-check arithmetic, alter endpoints, change dependency assignments, open new modal states, or promote theological readings into arithmetic proof.

Argument-control result:
The file’s argument now proceeds in four layers.

| Layer | Function | Controls |
|---|---|---|
| Base layer | Textual data and source-retained chronology data supply the dates, ages, and passages used by the file. | Preserve dependencies and node-classes. |
| Arithmetic layer | Direct calculations establish the authorized units, fourth-generation products, LCM, endpoint-pair spans, cumulative spans, chiastic sums, exact ratios, and Matthew / Double-7 expressions. | Arithmetic facts carry the proof burden. |
| Structural layer | The file interprets the checked arithmetic as an authorized generational architecture: `100`, `70`, and `40`; normalized `10:7:4`; synchronized at `1400`; scaled to `14000`. | Structural inference is permitted only when active state and operator are named. |
| Theological / typological layer | Patriarchal themes, Christological fulfillment, Matthew echo, and Sabbath completion explain meaning. | These claims support interpretation but do not replace arithmetic or dependency control. |

Argument Pass note:
The phrase “complete generational cycle” is retained as a structural-inference label for the `1400` LCM state, not as a generic claim about every `1400`-year span. The `14000` Christological lock is retained as a cumulative MT macro-scale inference from checked arithmetic. The Matthew `14` echo is retained as a textual and structural comparison, not as a claim that Matthew explicitly stated the `14000`-year cumulative architecture.

## 1. The authorization principle

Active state:
Authorization principle state; call-to-death generational-span register.

Generational units in biblical chronology are not arbitrary. Each is authorized by calling, established by a patriarch’s productive lifespan from divine commission to death.

Structural inference:
This grounds the file’s chronological mathematics in narrative events rather than in abstract number selection.

### 1.1 Axiom: the call-to-death generational spans

Active state:
Call-to-death generational-span register. Biological-fractal generation register is active only where the Moses / File_14 dependency is explicitly opened.

Axiom:
The repository’s generational architecture uses three exact units: Abrahamic `100`, Jacobean `70`, and Mosaic `40`. Each unit is structurally derived from the same metric: the span from divine call to death.

| Authorized unit | Controlling figure | Call state | Death state | Call-to-death span | Abrahamic-base fraction |
|---|---|---|---|---:|---:|
| Abrahamic | Abraham | Call at age `75` | Death at age `175` | `175 − 75 = 100` | `100/100 = 10/10` |
| Jacobean | Jacob | Bethel / Stairway call-state at age `77` | Death at age `147` | `147 − 77 = 70` | `70/100 = 7/10` |
| Mosaic | Moses | Burning Bush call-state at age `80` | Death at age `120` | `120 − 80 = 40` | `40/100 = 4/10 = 2/5` |

Exact normalized triad:

$$100:70:40=10:7:4.$$

Claim-status note:
The unifying call-to-death rationale is a structural inference and foundational axiom of `File_15`. The exact spans `100`, `70`, and `40` are arithmetic facts from stated age pairs, with provenance controlled by the relevant entity or dependency files: `File_01` for Abraham, `File_04` for Jacob, and `File_14` / Moses chronology controls where the Mosaic biological-fractal layer is opened. These exact rational standards replace generalized decimal approximations as executable notation. The source decimal display remains retained only where marked as approximate.

## 2. The three patriarchal standards

Active state:
Authorized generational-unit triad.

### 2.1 Authorization by post-calling lifespan

| Patriarch | Age at calling | Event | Age at death | Years after calling | Arithmetic check | Unit authorized |
|---|---:|---|---:|---:|---|---|
| Abraham | 75 | Call from Ur/Haran (Genesis 12:1–4) | 175 | 100 years | `175 − 75 = 100` | Abrahamic |
| Jacob | 77 | Stairway to Heaven (Genesis 28:10–22) | 147 | 70 years | `147 − 77 = 70` | Jacobean |
| Moses | 80 | Burning Bush (Exodus 3:1–10) | 120 | 40 years | `120 − 80 = 40` | Mosaic |

Modal-state note:
The ages `75`, `77`, and `80` are age-at-calling states. The death ages `175`, `147`, and `120` are death-age states. The resulting `100`, `70`, and `40` values are authorized generational units, not generic lifespan totals detached from their calling events.

Exact fractional architecture:

| Unit | Exact value | Normalized ratio | Abrahamic-base fraction |
|---|---:|---:|---:|
| Abrahamic | 100 | 10 | `10/10 = 1` |
| Jacobean | 70 | 7 | `7/10` |
| Mosaic | 40 | 4 | `4/10 = 2/5` |

The executable triad is therefore `100:70:40 = 10:7:4`.

Dependency clarification:
`File_01` controls Abraham’s MT Regular birth / call / death biography state. `File_04` controls Jacob’s age `77` at Flight to Haran / Bethel. `File_14` controls the fuller Moses biological-fractal layer where Moses `40` / `80` is opened. `File_15` retains the triad as authorized generational units and does not replace those entity files.

### 2.2 The three authorized units

| Standard | Unit | Primary scale | Macro scale |
|---|---:|---|---:|
| Abrahamic | 100 years | Patriarchal epochs | 1000 years; millennial |
| Jacobean | 70 years | Prophetic periods | 700 years |
| Mosaic | 40 years | Testing / wilderness cycles | 400 years; Affliction |

Structural axiom:
These three units, and their 10x multiples, function in this file as the authorized measuring rods for biblical chronology.

Modal-state note:
The macro-scale values `1000`, `700`, and `400` are 10x scale-state displays. They do not erase the primary unit values `100`, `70`, and `40`.

## 3. The fourth-generation principle

Active state:
Fourth-generation multi-scale state.

Textual datum:

> “In the fourth generation they shall return here” (Genesis 15:16).

Structural inference:
The fourth-generation principle operates at each authorized scale in the source argument.

| Standard | Unit | `× 4` | Biblical application |
|---|---:|---:|---|
| Abrahamic | 100 | 400 | Years of affliction (Genesis 15:13) |
| Jacobean | 70 | 280 | Abraham → Isaac’s death (`2166 BC → 1886 BC`) |
| Mosaic | 40 | 160 | Center of Terah → Egypt chiasm |

Modal-state note:
The phrase “fourth generation” is not one fixed chronological length in this file. It is a scale-dependent operator applied to the active authorized unit.

Argument-control note:
Genesis 15:16 supplies the textual datum. The multi-scale application is a structural inference from the three authorized units and their checked `×4` products.

### 3.1 Source arithmetic verification

| Verification | Source calculation | Arithmetic Pass check | Modal-state handling |
|---|---|---|---|
| Abraham birth to Isaac death | `2166 BC → 1886 BC = 280 years = 4 × 70` | `2166 − 1886 = 280`; `4 × 70 = 280`; checks | Source-retained Jacobean fourth-generation state. |
| Affliction span | `1846 BC → 1446 BC = 400 years = 4 × 100` | `1846 − 1446 = 400`; `4 × 100 = 400`; checks | Source-retained Abrahamic fourth-generation / Affliction state; arithmetic checks cleanly in §0.8; dependency-controlled by `File_16` / `File_05`. |
| Moses in Midian | `1486 BC → 1446 BC = 40 years = 1 × 40` | `1486 − 1446 = 40`; `1 × 40 = 40`; checks | Source-retained single Mosaic-generation state; fuller provenance dependency-controlled by Moses / `File_14` chronology. |

## 4. The 1400-year harmonic

Active state:
Generational LCM harmonic state.

### 4.1 Mathematical foundation

The lowest common multiple of the three authorized units is:

$$LCM(40, 70, 100)=1400.$$

Arithmetic check:

$$40=2^3 \times 5.$$

$$70=2 \times 5 \times 7.$$

$$100=2^2 \times 5^2.$$

Therefore:

$$LCM(40, 70, 100)=2^3 \times 5^2 \times 7=1400.$$

Claim-status note:
The LCM value `1400` is an arithmetic fact. The phrase “complete generational cycle” is the file’s structural inference from the convergence of all three authorized units.

Modal-state note:
The `1400` value is the LCM convergence state for the authorized-unit triad. It must not be treated as a generic `1400` span without naming the active endpoints and operator.

Argument-control note:
The LCM is an arithmetic fact. The claim that it functions as the complete generational cycle is a structural inference from that arithmetic fact.

### 4.2 Christological application

Active states:
Regular Christological 1400 endpoint state; cross-axis Christological endpoint state; ordered endpoint-pair bracket state.

| From | To | Span | Significance |
|---|---|---:|---|
| 1406 BC; Conquest | 6 BC; Christ’s birth | 1400 years | LCM cycle complete in source argument |
| 1399 BC; first Sabbath rest | AD 2; post-naming | 1400 years | Alternate endpoint-pair in source argument |

Source bracket statement:

```markdown
1406–1399 BC → 6 BC–AD 2 = 1400 years across the full bracket.
```

Modal-state handling:
The bracket statement is preserved as an ordered endpoint-pair display. The front endpoint state is `1406 BC → 6 BC`. The rear endpoint state is `1399 BC → AD 2`. The front pair is a same-side BC state and checks as `1406 − 6 = 1400`. The rear pair is a civil cross-axis state and checks as `1399 + 2 − 1 = 1400`. The Arithmetic Pass does not treat the statement as a claim that every internal pairing across the two brackets equals `1400`.

Audit-resolution note:
Author clarification assigns this bracket to `File_13` on the Christological / cross-axis side and `File_06` on the Conquest / land-rest side. No author-decision blocker remains under the endpoint-pair reading. Author decision would be required only if the intended source claim is stronger than endpoint-pairing.

Argument-control note:
The endpoint-pair table uses arithmetic facts to support a Christological structural inference. The Christological meaning is preserved as a theological application, not as a substitute for the same-side and cross-axis checks.

Arithmetic-control note:
The endpoint-pair checks are clean under their stated operators:

$$1406-6=1400.$$

$$1399+2-1=1400.$$

The first equation is a same-side BC span. The second equation is a civil cross-axis span. The bracket is not executed as an every-internal-cross-pair claim.

### 4.3 Cumulative projection at 10x scale

Active state:
Cumulative MT 10x macro-scale state.

| From | To | Span | Derivation |
|---|---|---:|---|
| 14006 BC; Cumulative Creation | 6 BC; Christ’s birth | 14000 years | `10 × 1400` |

Claim-status note:
As an arithmetic fact, `14006 BC → 6 BC = 14000`, and `14000 = 10 × 1400`. As a structural inference, the source treats this as the macro-scale completion of the three generational standards multiplied.

Source arithmetic statement:

$$14006-6=14000.$$

Source note:
Since both dates are BC, no year-zero adjustment is needed.

Arithmetic-control note:
The same-side BC span and macro-scale relation both check exactly: `14006 − 6 = 14000`; `10 × 1400 = 14000`.

Modal-state note:
`14006 BC` is a cumulative MT state. It must not be treated as an ordinary regular Creation endpoint. The `10x` relation is a macro-scale state, not Mirror scaling and not inverse-number reversal.

Argument-control note:
The checked span `14006 − 6 = 14000` is an arithmetic fact. Its role as a Christological lock is a structural and theological inference within the cumulative MT state.

## 5. Generational patterns in chiastic structures

Active state:
Chiastic-overlap state.

The source reads the authorized units as appearing within chiastic overlaps.

Cross-reference note:
The source points to `File_00`, Section 2.J. Current `File_00` dependency language should be confirmed in a later pass; `File_00` now carries chiastic-overlap controls in the supplied precedents.

Modal-state note:
Each example preserves two simultaneous readings: a literal reading and a chiastic / overlapped reading. The chiastic reading does not overwrite the literal chronology.

Argument-control note:
The chiastic tables present arithmetic facts and structural readings. Chiastic readings are not ordinary elapsed spans unless the local row explicitly opens the chiastic-overlap operator.

### 5.1 Shem → Isaac

Source structure:

$$100+290+100=490.$$

| Reading | Calculation | Result | Generation type |
|---|---|---:|---|
| Literal | `100 + 290 + 100` | `490 = 70 × 7` | Jacobean basis |
| Chiastic | `390 + 390` | `780 = 130 × 6` | Curse multiple |

Structural-inference note:
Both Shem and Abraham beget at `100` years. The source identifies the Abrahamic standard as framing the structure, but this framing remains chiastic / typological unless the local section explicitly opens a post-calling Abrahamic-unit operator.

Arithmetic-control note:
The literal reading checks as `100 + 290 + 100 = 490`, and `70 × 7 = 490`. The chiastic reading checks as `390 + 390 = 780`, and `130 × 6 = 780`.

Modal-state note:
The begetting-age `100` values are Abrahamic framing values inside the chiastic state. They are related to, but not identical with, Abraham’s post-calling `100` authorized unit unless the local section explicitly makes that connection.

### 5.2 Terah → Egypt

Source structure:

$$130+160+130=420.$$

| Reading | Calculation | Result | Generation type |
|---|---|---:|---|
| Literal | `130 + 160 + 130` | 420 | — |
| Chiastic center doubled | `160 × 2` | 320 | `4 × 80` or `8 × 40`; Mosaic |

The center `160` years equals `4 × 40`, the Mosaic fourth-generation scale.

Arithmetic-control note:
The literal reading checks as `130 + 160 + 130 = 420`. The doubled-center display checks as `160 × 2 = 320`; `4 × 80 = 320`; `8 × 40 = 320`; and the center value checks as `4 × 40 = 160`.

Modal-state note:
The center value is `160`. The doubled-center display is `320`. Preserve both node-classes: `160` carries the Mosaic fourth-generation center, while `320` displays the chiastic center doubled.

### 5.3 Abraham’s lifespan

Source structure:

$$75+25+75=175.$$

| Reading | Calculation | Result | Generation type |
|---|---|---:|---|
| Literal | `75 + 25 + 75` | 175 | Actual lifespan |
| Chiastic | `100 + 100` | 200 | `2 ×` Abrahamic standard |

Typological note:
The chiastic reading yields `100 + 100 = 200`, which the source interprets as two complete Abrahamic-generation units. This does not revise Abraham’s literal lifespan of `175`.

Arithmetic-control note:
The literal lifespan reading checks as `75 + 25 + 75 = 175`. The chiastic reading checks as `100 + 100 = 200`, and `2 × 100 = 200`.

Modal-state note:
The literal lifespan state remains `175`. The chiastic reading produces `100 + 100 = 200`. The chiastic value is not a correction of Abraham’s actual lifespan.

## 6. The generational hierarchy

Active state:
Generational hierarchy and synchronization state.

### 6.1 Exact normalized triad and nesting structure

The three units nest within one another, but the executable comparison state is exact-rational rather than decimal.

| Relation | Exact calculation | Result |
|---|---|---:|
| Abrahamic to Mosaic | `100/40` | `5/2 = 2.5` Mosaic generations |
| Jacobean to Mosaic | `70/40` | `7/4 = 1.75` Mosaic generations |
| Abrahamic to Jacobean | `100/70` | `10/7 ≈ 1.43` Jacobean generations |
| Normalized triad | `100:70:40` | `10:7:4` |
| Abrahamic-base fractions | `100/100`; `70/100`; `40/100` | `10/10`; `7/10`; `4/10 = 2/5` |

Arithmetic Pass result:
The source decimal relation `1 Abrahamic (100) ≈ 1.43 Jacobean (70)` checks as an approximation because `100/70 = 10/7 ≈ 1.43`. For executable repository notation, use the exact forms `10/7`, `100:70:40 = 10:7:4`, and Abrahamic-base fractions `10/10`, `7/10`, and `4/10 = 2/5`.

### 6.2 Synchronization at 1400 years

At `1400` years, all three standards synchronize.

| Calculation | Result |
|---|---:|
| `1400 ÷ 100` | 14 Abrahamic generations |
| `1400 ÷ 70` | 20 Jacobean generations |
| `1400 ÷ 40` | 35 Mosaic generations |

Arithmetic-control note:
The synchronization values check exactly: `1400 ÷ 100 = 14`; `1400 ÷ 70 = 20`; `1400 ÷ 40 = 35`.

Modal-state note:
The synchronization table does not make `100`, `70`, and `40` interchangeable. It shows their convergence at the LCM state. Arithmetic Pass checks are recorded in §0.8.

Claim-status note:
The division table is arithmetic fact. The theological and typological meanings attached to Abraham, Jacob, and Moses remain interpretive support. They do not make one unit replace another.

### 6.3 Theological significance

| Patriarch | Standard | Associated theme |
|---|---:|---|
| Abraham | 100 | Promise; faith; covenant |
| Jacob | 70 | Prophecy; nations; 70 nations; Daniel 9 |
| Moses | 40 | Testing; wilderness; judgment |

Theological note:
The generational unit carries the theological character of its authorizing patriarch in the source argument. This note preserves the theological meaning without making it replace arithmetic or dependency control.

Argument-control note:
The patriarchal themes are typological readings. They explain why the three standards matter inside the repository’s theological architecture, but the authorized units themselves are grounded in the call-to-death arithmetic and dependency-controlled chronology.

## 7. The 14000-year Christological lock

Active state:
Cumulative MT 14000 state; MT + Cainan / restored cumulative 2nd Cainan dependency state; Matthew 14-generations echo state.

### 7.1 Convergence from cumulative Creation to Christ

| Tradition | Cumulative Creation | To Christ | Span |
|---|---:|---:|---:|
| MT | 14006 BC | 6 BC | 14000 years |
| MT + Cainan | 14466 BC | 6 BC | 14460 years |

Structural inference:
The standard MT Cumulative aligns with the `10 × 1400` harmonic in the source argument. Arithmetic Pass checks are recorded in §0.8.

Arithmetic-control note:
The MT row checks as `14006 − 6 = 14000`, and `14 × 1000 = 14000`. The `MT + Cainan` row checks as `14466 − 6 = 14460`. The Cainan row is arithmetic-clean as a same-side BC span, but it is not the standard MT `10 × 1400` lock.

Dependency note:
The `MT + Cainan` row is classified in this pass as a restored cumulative 2nd Cainan dependency state. This classification follows the controlling distinction between regular restored 2nd Cainan `+130` and cumulative restored 2nd Cainan `+460` in the repository precedents. `File_15` does not locally re-prove the Cainan mechanism.

### 7.2 The “14 generations” echo

Textual datum:

> “So all the generations from Abraham to David are fourteen generations, from David to the deportation to Babylon fourteen generations, and from the deportation to Babylon to the Christ fourteen generations” (Matthew 1:17).

Source claims:

| Claim | Source expression |
|---|---|
| Matthew’s count | `14 + 14 + 14 = 42` generations |
| Cumulative span | `14000 years = 14 × 1000`; Millennial Abrahamic |
| Regular span | `1400 years = 14 × 100`; Abrahamic |

Structural comparison:
The number `14` recurs in both the genealogical count and the chronological span in the source argument.

Arithmetic-control note:
The Matthew count checks as `14 + 14 + 14 = 42`. The cumulative and regular echo expressions check as `14 × 1000 = 14000` and `14 × 100 = 1400`.

Modal-state note:
Matthew’s `14 + 14 + 14` genealogical state, the `1400 ÷ 100 = 14` Abrahamic-generation state, and the Double-7 `14 = 7 + 7` state are related but distinct. The file preserves the echo without collapsing the states.

Argument-control note:
Matthew 1:17 is the textual datum. The relation between Matthew’s `14` structure and the `1400` / `14000` chronological spans is a structural comparison inside `File_15`, not a claim that Matthew locally states the cumulative chronology.

### 7.3 The Double-7 structure

Source expressions:

$$14=7+7.$$

$$1400=7 × 200 = 7 × (2 × 100).$$

$$14000=7 × 2000 = 7 × (2 × 1000).$$

Arithmetic-control note:
The Double-7 expressions check exactly: `14 = 7 + 7`; `1400 = 7 × 200`; `200 = 2 × 100`; `14000 = 7 × 2000`; `2000 = 2 × 1000`.

Theological note:
The source interprets these spans as Sabbath completion, `7`, applied to doubled Abrahamic units.

Argument-control note:
The Double-7 arithmetic checks directly. The Sabbath-completion meaning is theological interpretation of the checked arithmetic.

Modal-state note:
The Double-7 state explains theological meaning. It does not replace the Matthew textual datum or the LCM arithmetic state.

## 8. Cross-reference and dependency index

### 8.1 Source cross-reference index

| File | Source section label | Relevance | Final handling |
|---|---|---|---|
| `File_00` | Section J | Chiastic Overlap Principle | Legacy reference retained; current section mapping should be confirmed. |
| `File_00` | Section B; Adam `+30` | 30-year apparent age | Legacy reference retained; current section mapping should be confirmed. |
| `File_01` | Section 5 | Abrahamic 2160 Harmonic | Retained as source dependency. |
| `File_04` | Logic Engine | Jacob’s 147-year lifespan, `3 × 49` | Retained as source dependency. |
| `File_05` | Section 1 | Exodus `1446 BC` anchor | Retained as source dependency. |
| `File_07` | Section 1 | Creation `14006 BC`; cumulative | Retained as source dependency; restored cumulative Cainan classification remains dependency-controlled. |
| `File_13` | Section 1 | Christological dual anchors | Retained as source dependency for `6 BC` and AD 2 endpoint language. |
| `File_16` | Section 5B | `1876 BC` Christological Criterion | Retained as source dependency. |

### 8.2 Additional current dependency controls

| File | Function added by current repository controls |
|---|---|
| `File_06` | Controls Conquest / land-rest envelope language where `1406 BC` and `1399 BC` are active. |
| `File_09` | Controls cumulative architecture and restored cumulative 2nd Cainan `+460` where the source `MT + Cainan` row is active. |
| `File_14` | Controls Biological Fractals vocabulary where Moses `40` / `80`, purification, gestation, or biological-fractal structures are opened by dependency. |
| `File_17` | Controls span-component terminology and already identifies `File_15` as a related foundational axiom file. |
| `File_51a` | Controls state-register, File map, cumulative / rounded dependency style, and pass-structure precedent. |
| `File_54` | Controls later-file claim-status restraint and theological / typological note preservation. |

## Audit notes

Audit note [FILE_15 ENDPOINT-PAIR BRACKET]:
The phrase `1406–1399 BC → 6 BC–AD 2 = 1400 years across the full bracket` is retained as source wording under ordered endpoint-pair handling. `File_06` controls the Conquest / land-rest side, and `File_13` controls the Christological / civil cross-axis side. The executable endpoint-pair checks remain `1406 − 6 = 1400` and `1399 + 2 − 1 = 1400`. A stronger claim that every internal cross-pair equals `1400` would require a future author decision.

Dependency note [FILE_15 RESTORED CUMULATIVE CAINAN]:
The `MT + Cainan` row gives `14466 BC → 6 BC = 14460` by same-side BC subtraction. The row remains a restored cumulative 2nd Cainan dependency state; `File_15` does not locally re-prove the Cainan mechanism.

Dependency note [FILE_15 JACOB / MOSES PROVENANCE]:
Jacob's age `77` at the stairway event remains governed by `File_04`. Moses' `1486 BC → 1446 BC` Midian span remains source-retained in `File_15`, with fuller Moses `40` / `80` biological-fractal provenance controlled by `File_14` where that layer is opened.

Audit note [FILE_15 EXACT RATIO DISPLAY]:
The decimal approximation `1 Abrahamic (100) ≈ 1.43 Jacobean (70)` is retained only as an approximate source display. The executable ratio is `100/70 = 10/7`; the active normalized triad is `100:70:40 = 10:7:4`, with Abrahamic-base fractions `10/10`, `7/10`, and `4/10 = 2/5`.

Dependency note [FILE_15 LEGACY SECTION LABELS]:
Legacy cross-reference labels such as `File_00`, Section J and Section B remain source-history labels until current conformed section mappings are confirmed.

Scope note [FILE_15 INACTIVE STATES]:
No formal Mirror protocol, inverse-number operator, Residue Protocol, calendar suffix state, or LXX source-reconstruction witness is active in the main argument of `File_15`.

## Unresolved issues

No blocking unresolved issues remain. Retained items are dependency-boundary or source-history controls, not publication blockers.

## Revision and archive note

This public-clean Markdown source removes only archival amendment history, pass logs, pressure-test repetition, pointer-refresh records, and obsolete replacement-workflow notes.

Detailed revision history is preserved in the `Repository_Change_Archive`.

No arithmetic, anchors, modal states, node-classes, sign conventions, operators, slash-pairs, ranges, envelopes, Mirror protocols, Machine Guards, claim-status labels, theological claims, or dependency boundaries are changed.

Final pressure-test status is preserved in the file header. Detailed pressure-test records are archived rather than repeated in the public file body.