{"id":"study-report-iv","title":"Rounded MT–SP–LXX Creation–Flood Harmonic Engine — Report IV","filename":"Rounded_MT-SP-LXX_Creation-Flood_Harmonic_Engine_Report_IV_20260904.md","role":"Study document","step":null,"sha256":"589c89260bede3fc72388f6d48c94a96ca581e0055ebeafa6721c8e7271325d6","bytes":87263,"origins":["User-supplied supporting study edition; inclusion does not change its original claim level."],"download":"/study-documents/files/Rounded_MT-SP-LXX_Creation-Flood_Harmonic_Engine_Report_IV_20260904.md","format":"md","derived":false,"tags":["Rounded","Regular","Cumulative","Mirror / 529","Calendar Keys"],"excerpt":"Rounded regular/cumulative comparisons, state operations, Mirror connections, and SP–Jubilees relationships.","collection":"study-documents","date":"2026-09-04","dateLabel":"September 4, 2026 · revised September 5","storage":"study-files","downloadFilename":"Rounded_MT-SP-LXX_Creation-Flood_Harmonic_Engine_Report_IV_20260904.md","content":"# Rounded MT–SP–LXX Creation–Flood Harmonic Engine — Report IV\n\n## The `115|120|125` Hinge, the `430/460/30` Redistribution, the Native/Generated Cainan Mirror, the `1150/1152` Aggregate, the SP `−215` / `9200` Resolution, and the SP–BJ Jubilee Bridge\n\n**Discussion synthesis — September 4, 2026; SP–BJ revision — September 5, 2026**  \n**Series:** Fourth report in the comparative MT–SP–LXX search for minimal chronological harmonization equations.  \n**Parent reports:** Reports I, II, and III listed in the source register below.  \n**Primary table and rounding controls:** File_18; File_51a.  \n**Primary cumulative, operator, and interpretive controls:** Files 09, 11, 17, 22, 46, 54, 63, 69, 70, and File_70 Supplement A.  \n**Status:** Discussion synthesis and continuation ledger; not a replacement for any canonical repository file.  \n**Scope:** Rounded regular and cumulative Creation–Noah–Shem/Flood across MT, SP, and LXX; comparison with the actual chronology; the `+5`, `+30`, Cainan, and `−215` operations; the SP–Book of Jubilees (`BJ`) Creation/Fall/Flood and Abrahamic comparison; same-side and Mirror connections to Abraham, Moses, Conquest, and Christ.  \n**Future use:** Open a fresh Project chat with Reports I–IV and continue reducing the combined evidence to the fewest equations that preserve the greatest amount of structure.\n\n---\n\n# 0. Purpose, relation to the previous reports, and reading order\n\nThe author identifies the four reports as the most useful concentrated working core for explaining the harmony of the Unified Chronology, within a repository of approximately 75 files. That describes their role in this research program, not an independently audited inventory of the repository. Other repository files remain essential for source data, operator definitions, textual distinctions, and larger-scale comparisons.\n\nReport I established the regular macro-engine: the `60|70|60` variant field, the `190|25|190` extension, the Triple-`1656` spine, the `1250/650` Creation–Flood geometry, the `625/325` half-gaps, and the external scale comparisons.\n\nReport II traced the regular mechanism patriarch by patriarch: the Creation-week descent, Cainan-leveled post-Flood countdown, `22+1=23` inventory, the SP `120+480+120` Methuselah–Noah–Flood field, and the Adam/Seth–Noah/Shem biographical correspondences.\n\nReport III opened the actual cumulative comparison. Its principal MT-centered vectors, with Cainan removed symmetrically, were:\n\n\\[\nC_{\\mathrm{cum,actual}}=(+430,0,-608)_{\\mathrm{LXX,MT,SP}},\n\\]\n\n\\[\nN/S/F_{\\mathrm{cum,actual}}=(+454,0,-120)_{\\mathrm{LXX,MT,SP}}.\n\\]\n\nIt also established the actual `9890` Creation bridge, `2880/2875` Shem bridges, the `430/720` connector family, and the explicitly analytical—not chronological—flattening procedure. [R1–R3]\n\n**Report IV adds the rounded resolution.** It begins with six tables, restores the cumulative Moses/Aaron `+5` rail, and follows the resulting geometry until the SP's native `−215` state supplies a particularly coherent internal explanation.\n\nThe argument should be read in two complementary ways. The leveled comparison exposes shared manuscript structure. Native/favored-state execution explains how each tradition uses that structure in its own configuration. Neither should displace the other.\n\n## 0.1 Principal results in one display\n\nThe elementary rounded-Key unit is:\n\n\\[\n\\boxed{u=115=\\operatorname{lcm}(5,23).}\n\\]\n\nThe elementary cell is:\n\n\\[\n\\boxed{115\\mid120\\mid125,\\qquad115\\times\\frac{25}{23}=125.}\n\\]\n\nThe main redistribution identities are:\n\n\\[\n\\boxed{430+30=460,\\qquad605-30=575,}\n\\]\n\n\\[\n\\boxed{430+605=460+575=1035=9u.}\n\\]\n\nThe maximum widths within the fixed Creation comparison states—not across every repository variant—complete to one common value:\n\n\\[\n\\boxed{1380=12u\\xrightarrow{25/23}1500,\n\\qquad1495=13u\\xrightarrow{300/299}1500.}\n\\]\n\nThe paired gains are:\n\n\\[\n\\boxed{1500-1380=120,\\qquad1500-1495=5,\\qquad120-5=u.}\n\\]\n\nThe overlapping cumulative comparison-cell sums give:\n\n\\[\n\\boxed{3450/3=1150\\quad\\text{rounded},\\qquad3456/3=1152\\quad\\text{actual}.}\n\\]\n\nThe LXX Flood Mirror supplies:\n\n\\[\n\\boxed{10580\\xrightarrow{25/23}11500\\xrightarrow{25/23}12500,}\n\\]\n\nwhere `11500` is both a generated output and the independently available native-Cainan Mirror span.\n\nThe SP's favored `−215` regular state supplies:\n\n\\[\n\\boxed{13406-4206=9200=23\\times400\\xrightarrow{25/23}10000,}\n\\]\n\nwith the dual wilderness completion:\n\n\\[\n\\boxed{2760=69\\times40\\to2800,\n\\qquad11960=299\\times40\\to12000.}\n\\]\n\nBoth expansions add the same `40`, preserving the `9200` cumulative-to-regular separation.\n\nThe SP–BJ comparison adds a Jubilee-scaled companion:\n\n\\[\n\\boxed{4206/4199-350=3856/3849,\\qquad350=5\\times70,}\n\\]\n\n\\[\n\\boxed{4199-3856=2893-2550=343=7^3,}\n\\]\n\nand, on the rounded Mirror cross-pair:\n\n\\[\n\\boxed{8050=23\\times350.}\n\\]\n\nThe corrected BJ Creation week is `3863–3856 BC`: both SP Creation-week endpoints translate uniformly by `343` years. The Flood comparison separately retains its `344` Flood-to-Flood and `343` Flood-to-Ark-year distinction.\n\n## 0.2 Evidential language\n\nThe report preserves four levels without conflating them:\n\n| Level | Meaning here |\n|---|---|\n| Source datum / source-controlled state | A lifespan, begetting age, anchor, variant, or operation already supplied by the repository or the parent reports. |\n| Arithmetic fact | An exact consequence of the declared input coordinates and operation. |\n| Structural inference | The proposed relationship between several exact constructions, including repetition, symmetry, or finite fractal-like correspondence. |\n| Theological / providential interpretation | The author's reading of the received multi-tradition field as a coherent divine overpattern, potentially exceeding demonstrable intention by any one ancient writer or scribe. |\n\nExact arithmetic does not by itself establish historical causation, statistical rarity, or independence between observations. Conversely, a generated center need not be a historical event-date to function as a mathematical center. The aim is neither to discard the patterns nor to overstate what each one establishes.\n\n---\n\n# Part I — Source and state controls\n\n## 1. Portable source register\n\nReferences in square brackets are portable source keys, not external web research. They identify the attached reports and repository files used in preparing this synthesis.\n\n| Key | Source and principal use |\n|---|---|\n| R1 | *Regular MT–SP–LXX Creation-to-Noah Engine*, August 31, 2026. Macro geometry, `1656`, `625`, SP `2760→2800`, and `345→350`. |\n| R2 | *Regular MT–SP–LXX Comparative Genealogical Engine — Report II*, September 2, 2026. Patriarchal comparison, SP `120+480+120`, Adam/Noah biographical fields, the SP/BJ Enoch relation, and the BJ Fall-to-Flood `1300`. |\n| R3 | *Cumulative MT–SP–LXX Creation–Shem/Flood Engine — Report III*, September 2, 2026. Actual cumulative states, `430/454/608/120`, `9890`, `2880`, and flattening. |\n| F18 | `File_18.Chronological_Data_Tables.md`, especially §§3, 4A, 6A–6D. Raw inputs, SP boundary distinctions, cumulative Cainan, and LXX Lamech state. Mounted edition: September 2, 2026. |\n| F51a | `File_51a.Rounded_Scaffold_Mod5_Architecture.md`, especially §§2–3, 7A, 11.3, 16.1–16.2, 16.10, 17.4, and 23.7. Rounding rules, MT tables, SP rounded state, `1650`, and the Aaron `+5` branch. |\n| F09 | `File_09.Cumulative_Architecture.md`, especially §§0.5–0.6, 3, and the cumulative Creation-cell treatment. Actual `2+3+2`, phase-resolved Mirror, `5520→6000`, and `15870`. |\n| F11 | `File_11.Three_Gear_Protocol_Harmonic_Engine.md`, especially §§8–9. SP `1656` and the established `13398−4198=9200→10000` bridge. |\n| F17 | `File_17.Prophetic_Time_Span_Anatomy.md`. Exact `25/23`, `70/69`, `300/299` ratios and prophetic/calendar vocabulary. |\n| F22 | `File_22.Cumulative_MT_Harmonics.md`, especially the `9890` bridge and §9A. Cumulative Apparent Adam; `12635`; Revelation-11 comparison; source/generated-node distinctions. |\n| F46 | `File_46.Harmonic_Expansion_Protocols.md`, especially §§1–3 and 6A. `4830`, dual Keys, double-expansion licensing, and cumulative LXX/MT overlays. |\n| F54 | `File_54.Luke_70_Year_Genealogical_Lattice.md`. `4836/4906`, `5516/5286/4136`, Luke/Rounded comparisons, and interpretive limits. |\n| F63 | `File_63.Scale_Neutral_480_483_490_Carrier.md`. `11960→12000`, `14041→1406`, and scale-neutral carrier comparisons. |\n| F69/F70/SA | Files 69, 70, and File_70 Supplement A. Enoch-centered and *Toledot* interpretations, finite recursive geometry, the `720/30` rail, and generated-center discipline. |\n| F01/F04/F05 | Abraham, Jacob, and Exodus controllers. SP `1951/1876/1776`, `215+215`, primary `1446`, and the separately controlled LXX `33+397` family-unit interpretation. |\n| BJ/DISC | Author-supplied Book of Jubilees reconstruction used in the September 5 revision: corrected initial Creation `3863–3856`, Eve/Fall week `3856–3849`, Ark year `2550`, Flood `2549`, and Abram `1981 BC`. This report verifies the comparative arithmetic but does not independently rederive every BJ coordinate from the primary text. |\n| CTRL | Restart Capsule v11.50, State Vocabulary Register v1.51, Style Guide v2.5, and Project Procedures v3.5. State, Mirror, terminology, and source boundaries. |\n| DISC | The present discussion, September 3–4, 2026. New comparisons, selected rails, rejected approaches, and author-directed interpretations. |\n\nFiles 51b, 51c, 55, and 56 are relevant named dependencies. Their broader theories are not rederived here. Where F18 or the active controls explicitly reproduce a state belonging to those files, that routed state may be used without claiming a fresh review of the entire dependency.\n\n## 2. Regular and cumulative are different engines\n\nRegular chronology accumulates **begetting ages**. Cumulative chronology stacks **lifespans**. The same patriarch therefore has different node-classes in the two systems.\n\nCainan must retain the operator appropriate to the engine:\n\n\\[\n\\boxed{c_r=130\\quad\\text{regular},\\qquad c_c=460\\quad\\text{cumulative}.}\n\\]\n\nLXX carries second Cainan natively in the adopted repository reconstruction. MT/SP restorations are explicit comparison states. Removing native Cainan from LXX is likewise a declared comparison operation, not a claim that the LXX text lacks him. [F18; R2–R3]\n\n## 3. Two different rounding formulas\n\nLet `R₅(x)` be nearest-multiple-of-five rounding. No half-way ties occur in the integer lifespan and begetting-age inputs used here.\n\nFor regular births:\n\n\\[\nB_{i+1}=B_i-R_5(g_i).\n\\]\n\nFor the secondary regular death layer:\n\n\\[\nL_i^{r}=R_5(g_i)+R_5(L_i-g_i).\n\\]\n\nFor cumulative propagation:\n\n\\[\nL_i^{c}=R_5(L_i),\\qquad\nC_i=A+\\sum_{j=i}^{N}L_j^{c}.\n\\]\n\nThis distinction matters particularly for Methuselah:\n\n\\[\n187\\to185,\\quad782\\to780,\\quad185+780=965\n\\]\n\nin the regular remaining-years method, whereas:\n\n\\[\n969\\to970\n\\]\n\nin cumulative whole-lifespan rounding. Neither result corrects the other. The regular rounded death layer remains `[MOD-5 THEORETICAL]`. [F51a §§3.4, 16.1–16.2]\n\n## 4. Fixed comparison states\n\nThe main regular tables use the common `+215` comparison plane, no `+60 Terah`, the declared rounded exclusion of the localized Shem `+2`, transmitted SP rounded Creation `4416/4411`, and native Cainan in LXX unless a column explicitly removes him.\n\nThe cumulative chains terminate at `1406 BC` for MT/LXX. SP includes Joshua's `110` and terminates at `1296 BC`; these together preserve the `1406` Moses/Joshua joint.\n\nThe adopted cumulative LXX chain uses Lamech `753` and Nahor `208`. Regular LXX uses Lamech `182/777`. The cumulative `777` Lamech overlay is row-local and does not propagate. Lamech `188` is a textual-corruption audit value, not an operative alternative in this report. SP uses Terah `145` and Eber `404`. [F18 §§4A, 6C–6D]\n\n## 5. The five-year rail is generated, not freely added\n\nAaron's `123` rounds to `125`; Moses remains `120`. Replacing the rounded Moses terminal with the rounded Aaron terminal therefore adds five years uniformly to earlier cumulative nodes:\n\n\\[\n\\boxed{125-120=5.}\n\\]\n\nThus `x+5/x` is a declared Moses/Aaron branch pair. It is not permission to move any desired event by five years. F51a directly controls the MT execution; the same branch rule is applied explicitly to the reconstructed SP and LXX cumulative tables. [F51a §16.2]\n\nRegular rounded Creation also has a five-year boundary block. Regular Noah, Flood, and their warning points remain single coordinates in the compact comparison unless another source-defined variant is expressly opened.\n\nThe actual cumulative envelope has the controlled whole-year anatomy:\n\n\\[\n\\boxed{2+3+2=7.}\n\\]\n\nThe rounded five-year presentation is a compressed boundary state, not a replacement for the actual seven-year envelope. The equality `5+2=7` will be important below, but does not by itself supply a universal mapping of every actual endpoint to a rounded endpoint. [F09; F51a §23.7]\n\n## 6. Coordinate arithmetic and Mirror conventions\n\nFor two BC labels, elapsed forward time is the earlier/higher BC label minus the later/lower label. For cross-axis spans:\n\n\\[\n\\boxed{S(B\\text{ BC},\\mathrm{AD}\\ A)=B+A-1.}\n\\]\n\nThe rounded Protocol-1 counterparts used here include:\n\n\\[\n5291\\text{ BC}\\leftrightarrow\\mathrm{AD}\\ 5290,\n\\quad5286\\text{ BC}\\leftrightarrow\\mathrm{AD}\\ 5285.\n\\]\n\nThe counterpart-generation rule and the elapsed-span formula are distinct operations. Actual phase-resolved Mirror states remain controlled separately by F09. Mirror reversal is not decimal digit reversal; no File_52 inverse-number calculation is established by this report.\n\nA slash-pair is ordered. Each member must be expanded explicitly where crossing, reversal, or different node-classes could cause ambiguity. “Residual” below means an expansion **gain** unless the repository's separate physical Residue Protocol is expressly named; that physical protocol is not being executed here.\n\n---\n\n# Part II — The rounded register and its actual counterpart\n\n## 7. Six-table result and source status\n\nThe MT regular and cumulative rounded tables are directly controlled by F51a. The complete pre-Flood SP/LXX rounded tables used in this discussion were generated from the declared F18 inputs under those two rounding rules. They are reproduced in Appendix A so that a fresh chat need not regenerate them before continuing.\n\nThe cumulative lifespan sums are:\n\n| Tradition and terminal | Actual sum | Rounded sum | Net change | Actual / rounded Creation spine |\n|---|---:|---:|---:|---|\n| MT to Moses, anchored at `1406` | `12600` | `12600` | `0` | `14006 / 14006` |\n| LXX native Cainan to Moses, anchored at `1406` | `13490` | `13490` | `0` | `14896 / 14896` |\n| SP through Joshua, anchored at `1296` | `12102` | `12105` | `+3` | `13398 / 13401` |\n\n**Important correction to a shorthand used during the discussion:** cumulative MT/LXX Creation spine coordinates remain stationary under rounding. Regular MT/LXX Creation coordinates do not: their standard Year-7 endpoints are eight years above the lower rounded members. Their mutual gap survives, not the exact regular dates.\n\n\\[\n4114\\to4106,\\qquad5494\\to5486;\n\\qquad5494-4114=5486-4106=1380.\n\\]\n\nFor SP, actual `4414` becomes rounded `4411`; this changes the SP-related regular gaps by five relative to the MT/LXX shift.\n\n## 8. Full rounded comparison register\n\nAll figures in this table are BC. LXX Cainan-OFF columns subtract `130` in regular space and `460` in cumulative space.\n\n| Node / state | LXX, Cainan OFF | MT | SP | LXX native Cainan |\n|---|---:|---:|---:|---:|\n| Regular Creation | `5361/5356` | `4111/4106` | `4416/4411` | `5491/5486` |\n| Regular Noah birth | `3706` | `3056` | `3706` | `3836` |\n| Regular warning, `120` before Flood | `3226` | `2576` | `3226` | `3356` |\n| Regular Shem birth | `3206` | `2556` | `3206` | `3336` |\n| Regular Flood / Arphaxad comparison seam | `3106` | `2456` | `3106` | `3236` |\n| Regular Noah death | `2756` | `2106` | `2756` | `2886` |\n| Regular Shem death | `2606` | `1956` | `2606` | `2736` |\n| Cumulative Creation | `14441/14436` | `14011/14006` | `13406/13401` | `14901/14896` |\n| Cumulative Noah | `6841/6836` | `6391/6386` | `6271/6266` | `7301/7296` |\n| Cumulative Shem | `5891/5886` | `5441/5436` | `5321/5316` | `6351/6346` |\n| Cumulative warning | `5411/5406` | `4961/4956` | `4841/4836` | `5871/5866` |\n| Cumulative Flood / Arphaxad seam | `5291/5286` | `4841/4836` | `4721/4716` | `5751/5746` |\n\nThe equality of a regular Flood coordinate and an Arphaxad comparison seam belongs to the declared rounded convention. It does not erase the actual SP Flood-year and post-Flood boundary distinction. Likewise, cumulative names designate serial lifespan boundaries, not biological birth dates in a simultaneous historical population.\n\n## 9. Actual cumulative envelope register retained from Report III\n\n| Node | LXX, Cainan OFF | MT | SP | LXX native |\n|---|---:|---:|---:|---:|\n| Creation | `14441–14434` | `14011–14004` | `13403–13396` | `14901–14894` |\n| Noah | `6840–6833` | `6386–6379` | `6266–6259` | `7300–7293` |\n| Shem | `5890–5883` | `5436–5429` | `5316–5309` | `6350–6343` |\n| Flood / Arphaxad | `5290–5283` | `4836–4829` | `4716–4709` | `5750–5743` |\n\nThe internal actual Nisan spine coordinates are respectively `14436/14006/13398` at Creation and `5285/4831/4711` at Flood on the Cainan-OFF plane. The rounded lower members become `14436/14006/13401` and `5286/4836/4716`.\n\nThus the MT-centered comparison is:\n\n| Field | Actual | Rounded |\n|---|---|---|\n| Regular Creation endpoints | `(+1250,0,+300)` | `(+1250,0,+305)` |\n| Regular Noah/Flood, declared leveled state | `(+650,0,+650)` | `(+650,0,+650)` |\n| Cumulative Creation | `(+430,0,−608)` | `(+430,0,−605)` |\n| Cumulative Noah/Shem/Flood | `(+454,0,−120)` | `(+450,0,−120)` |\n\nThe actual regular `650` row retains the inclusive/Gear-qualified comparison of R1–R2; it is not a claim that every raw SP Noah/Flood boundary is identical to its LXX comparator.\n\n## 10. The cumulative contraction ladder\n\nThe rounded Cainan-OFF LXX–SP gaps descend as follows:\n\n| Zone | Rounded outer gap |\n|---|---:|\n| Adam through Jared | `1035` |\n| Enoch through Methuselah | `920` |\n| Lamech | `670` |\n| Noah through Flood | `570` |\n\nThe losses are:\n\n\\[\n115+250+100=465.\n\\]\n\nThey follow directly from rounded lifespan differences:\n\n\\[\n960-845=115,\\quad970-720=250,\\quad755-655=100.\n\\]\n\nTherefore:\n\n\\[\n1035-570=465.\n\\]\n\nThe actual counterpart is:\n\n\\[\n1038-574=464=115+249+100.\n\\]\n\nThese reproduce MT Eber's actual/rounded lifespan `464/465`. The arithmetic is exact; Eber's proposed mediating role is retained as a structural lead, not a demonstrated textual encoding mechanism.\n\n---\n\n# Part III — The `1650` backbone and the elementary `115` cell\n\n## 11. The rounded continuation of the Triple-`1656` spine\n\nF51a already establishes:\n\n\\[\n14006-4106=9900=6\\times1650,\n\\]\n\n\\[\n4106-2456=1650,\n\\]\n\n\\[\n14006-2456=11550=7\\times1650.\n\\]\n\nThe rounded LXX Cainan-OFF side gives:\n\n\\[\n5356-3706=1650\n\\]\n\nto Noah's birth. SP instead gives:\n\n\\[\n4411-2756=1655\n\\]\n\nto Noah's rounded death.\n\nA `1650` projection from SP Creation reaches the generated point `2761`:\n\n\\[\n4411-1650=2761.\n\\]\n\nFrom SP Flood:\n\n\\[\n3106-2761=345,\\qquad345\\times\\frac{70}{69}=350,\n\\]\n\nand:\n\n\\[\n3106-350=2756.\n\\]\n\nThus the final gain is exactly five years:\n\n\\[\n\\boxed{1650+5=1655,\\qquad345+5=350.}\n\\]\n\nThis does not alter SP's adopted `4411` Creation or `2756` death. It explains how its different endpoint participates in the rounded backbone. The relation `345→350` also occurs in R1.\n\n## 12. Why `115`, `345`, and `460` recur\n\n\\[\n115=5\\times23=\\operatorname{lcm}(5,23),\n\\]\n\n\\[\n345=3\\times115=5\\times69=\\operatorname{lcm}(5,69),\n\\]\n\n\\[\n460=4\\times115=23\\times20.\n\\]\n\nThe exact Keys act as:\n\n\\[\n115\\xrightarrow{25/23}125,\n\\]\n\n\\[\n345\\xrightarrow{70/69}350,\n\\qquad345\\xrightarrow{25/23}375,\n\\]\n\n\\[\n460\\xrightarrow{25/23}500.\n\\]\n\nThe gains are respectively `10`, `5`, `30`, and `40`. These are arithmetic consequences of the same denominator structure, not four independent statistical witnesses.\n\n## 13. The `115|120|125` Flood–Moses/Aaron cell\n\n\\[\n\\boxed{115\\mid120\\mid125=5(23\\mid24\\mid25).}\n\\]\n\nAt the rounded MT Flood:\n\n\\[\n2576_{\\mathrm{warning}}\\to2461_{\\mathrm{Lamech\\ death}}\\to2456_{\\mathrm{Flood}}\n\\]\n\nhas:\n\n\\[\n115+5=120.\n\\]\n\nThe actual standard MT counterpart is:\n\n\\[\n2578\\to2463\\to2458=115+5.\n\\]\n\nAt the cumulative terminal:\n\n\\[\n120_{\\mathrm{Moses}}+5=125_{\\mathrm{rounded\\ Aaron}}.\n\\]\n\nThe two ends therefore employ the same cell in different roles. Aaron—not Levi—is the `123→125` patriarch. Levi is `137→135` in cumulative rounding. [F18; F51a]\n\n## 14. SP's `120+480+120` is chronological, not merely numerical\n\nThe actual SP begetting links are:\n\n\\[\n67+53=120.\n\\]\n\nTheir rounded equivalents preserve the sum:\n\n\\[\n65+55=120.\n\\]\n\nTherefore Methuselah's birth to the Flood can be displayed:\n\n\\[\n\\boxed{120+480+120=720,}\n\\]\n\nwith the joints at Noah's birth and the proposed `120`-year pre-Flood warning point. The last two terms are Noah's `600` to the Flood. [R2 §44]\n\nAt the cumulative level, the SP warning and MT Flood co-register:\n\n\\[\n4841/4836_{\\mathrm{MT\\ Flood}}\n=\n4841/4836_{\\mathrm{SP\\ warning}},\n\\]\n\nfollowed by:\n\n\\[\n4841/4836\\to4721/4716_{\\mathrm{SP\\ Flood}}=120.\n\\]\n\nF18 supplies the same relationship in actual cumulative form: SP warning `4831` and Flood `4711`, with warning envelope `4836–4829` matching the MT Flood envelope. This is the clearest chronological meaning of the persistent MT–SP `120` displacement.\n\n## 15. Narrative rationale and its limits\n\nThe discussion reads Genesis 6:3's `120` as a countdown to the Flood. That is the working interpretation used to generate warning coordinates, not an assertion that the verse explicitly dates an ark-construction commencement 120 years earlier.\n\nThe literary bridge is the shared Hebrew **תֵּבָה**, “ark/chest,” in Noah's preservation narrative and Moses' infancy narrative: Genesis 6–9 and Exodus 2:3, 5. Moses' life of `120` is explicit in Deuteronomy 34:7; Aaron's `123` is explicit in Numbers 33:39. The typological comparison is:\n\n\\[\n120\\to\\text{Noah's ark/judgment threshold},\n\\]\n\nversus:\n\n\\[\n\\text{Moses preserved in an ark}\\to120\\to\\text{death/Conquest threshold}.\n\\]\n\nNoah's household and the family of Israel supply the corporate-preservation theme. Canaan's judgment supplies a further narrative parallel, not a claim that all inhabitants historically died at one instant.\n\nRevelation 11:3, 7, 9–12 supplies `1260` testimony followed by the terminal `3.5`-day death/restoration/ascent scene. Revelation 12:6, 14–16 explicitly places the woman in wilderness preservation. These are typological comparanda; the report does not equate the literal durations `40 years`, `1260 days`, and `12600 years` without declaring the scale and operation. [F17; F22 §9A; DISC]\n\n---\n\n# Part IV — `450/120`: from gap arithmetic to dated constructions\n\n## 16. Adam and Noah share a `500` complementary arm\n\nThe rounded cumulative Cainan-OFF field gives:\n\n\\[\n14436_{\\mathrm{LXX\\ Adam}}\\to14006_{\\mathrm{MT\\ Adam}}\\to13506_{\\mathrm{LXX\\ Seth}}\n=430+500=930,\n\\]\n\nand:\n\n\\[\n6836_{\\mathrm{LXX\\ Noah}}\\to6386_{\\mathrm{MT\\ Noah}}\\to5886_{\\mathrm{LXX\\ Shem}}\n=450+500=950.\n\\]\n\nThus:\n\n\\[\n\\boxed{930-430=950-450=500.}\n\\]\n\nThe change `430→450` equals the lifespan change `930→950`. The `500` also corresponds to the Key-completed Cainan `460→500` and to Noah's son-group age in the declared `500`-year regular state. A strict `502` Shem-birth state instead gives a `448` complement at Noah's death; those biographical phases must not be collapsed.\n\n## 17. Complete cross-straddle gap field\n\nFor the rounded cumulative Flood row:\n\n\\[\n5291/5286_{L}\\mid4841/4836_{M}\\mid4721/4716_{S},\n\\]\n\nall same- and cross-rail gaps are:\n\n| Relationship | Narrow cross | Same rail | Wide cross |\n|---|---:|---:|---:|\n| LXX-off to MT | `445` | `450` | `455` |\n| MT to SP | `115` | `120` | `125` |\n| LXX-off to SP | `565` | `570` | `575` |\n\nNative cumulative Cainan increases the final row to:\n\n\\[\n1025\\mid1030\\mid1035.\n\\]\n\nThe wide cross is:\n\n\\[\n5291-4716=575=5u,\n\\]\n\nand with native Cainan:\n\n\\[\n5751-4716=1035=9u.\n\\]\n\nCreation had already supplied `1035` as a Cainan-OFF, same-rail LXX–SP gap. Flood supplies it as native-Cainan, cross-rail. This is a change in operator/state composition, not identity of the underlying events.\n\n## 18. The `690` correction must be preserved\n\nAn early response added `570+120` as though both were contiguous intervals already present in the three-manuscript Flood row. That was misleading: `570` already contains `450+120`.\n\nThe arithmetic sum is true, but a chronological interpretation must supply an additional, explicitly identified `120` warning interval. The later comparison does so:\n\n\\[\n(5291-3106)-(4721-3226)\n\\]\n\n\\[\n=(5291-4721)+(3226-3106)\n=570+120=690.\n\\]\n\nThe first diagonal is regular Flood to cumulative LXX-off Flood; the second uses the regular warning and cumulative SP Flood. Thus the additional `120` has a stated location.\n\nA separate regular Noah–Mosaic execution gives:\n\n\\[\n3706-1406=2300,\n\\qquad3056-1446=1610,\n\\]\n\n\\[\n2300-1610=(3706-3056)+(1446-1406)=650+40=690.\n\\]\n\nHence:\n\n\\[\n\\boxed{570+120=650+40=690=23\\times30=6u.}\n\\]\n\nThe compensating differences are:\n\n\\[\n650-570=120-40=80.\n\\]\n\nThis is a comparison of two dated constructions, not a claim that the `570` and `650` are the same manuscript pair in the same chronological mode.\n\n## 19. Secondary `690/460/500` algebra retained with chronology attached\n\nOnce `690` has been given the above endpoint derivations:\n\n\\[\n690+460=1150,\n\\]\n\nand:\n\n\\[\n(690+460)\\frac{25}{23}=750+500=1250.\n\\]\n\nThe rectangle is:\n\n\\[\n\\begin{array}{ccc}\n690&\\xrightarrow{+460}&1150\\\\\n\\downarrow25/23&&\\downarrow25/23\\\\\n750&\\xrightarrow{+500}&1250.\n\\end{array}\n\\]\n\nOther exact but secondary observations were:\n\n\\[\n450-120=460-130=330,\n\\]\n\n\\[\n450+10=460,\\quad120+10=130,\n\\]\n\n\\[\n450+250=700,\\quad450+350=800,\\quad450+460=910.\n\\]\n\nThey remain useful algebraic comparisons with R2's Adam remainders, but are not assigned extra proof weight merely because they can be formed from already available numbers.\n\n## 20. Lower-weight `605/600` comparison\n\nThe cumulative Creation MT–SP gap family is `600|605|610`, while the Flood family is `115|120|125`.\n\nThe observation:\n\n\\[\n605-5=600=480+120=5\\times120\n\\]\n\nwas explicitly assessed in the discussion as weak by itself. It is retained as background to the stronger dated warning structures, not as a principal theorem.\n\n\n---\n\n# Part V — Christ-side windows, `490` cycles, and Mirror completion\n\n## 21. Three adjoining five-year windows\n\nThe cumulative rounded Flood rails translate to the following Christ-side comparison windows:\n\n| Source | Target pair | Common span | In `115` units |\n|---|---|---:|---:|\n| MT `4841/4836 BC` | `11/6 BC` | `4830` | `42u` |\n| SP `4721/4716 BC` | `6/1 BC` | `4715` | `41u` |\n| LXX-off `5291/5286 BC` | `1 BC / AD 5` | `5290` | `46u` |\n| LXX native `5751/5746 BC` | `1 BC / AD 5` | `5750` | `50u` |\n\nFor example:\n\n\\[\n5291-1=5286+5-1=5290.\n\\]\n\nThe targets form:\n\n\\[\n\\boxed{11\\text{ BC}\\to6\\text{ BC}\\to1\\text{ BC}\\to\\mathrm{AD}\\ 5,\\qquad[5\\mid5\\mid5].}\n\\]\n\nSP occupies the middle window. These are chronological-model targets, not a proposal for four literal Nativity dates.\n\nThe actual-envelope companion discussed was:\n\n\\[\n\\mathrm{MT}:6\\text{ BC}\\to\\mathrm{AD}\\ 2,\n\\quad\\mathrm{SP}:1\\text{ BC}\\to\\mathrm{AD}\\ 7,\n\\quad\\mathrm{LXX}:\\mathrm{AD}\\ 1\\to\\mathrm{AD}\\ 8.\n\\]\n\nEach is seven years. Ordering their boundary labels gives:\n\n\\[\n6\\text{ BC}\\to1\\text{ BC}\\to\\mathrm{AD}\\ 1\\to\\mathrm{AD}\\ 2\n\\to\\mathrm{AD}\\ 7\\to\\mathrm{AD}\\ 8,\n\\]\n\nwith intervals:\n\n\\[\n\\boxed{5+1+1+5+1=13.}\n\\]\n\nThe boundary anatomy is exact. Its complete actual-state `115` derivations were mentioned as context, not comprehensively executed in this discussion, and should not be assumed proven by the rounded table.\n\n## 22. Why the wide LXX–SP cross is `575`\n\nThe wide cross uses two macro endpoints whose translations share the same target `1 BC`:\n\n\\[\n5291-1=46u,\\qquad4716-1=41u.\n\\]\n\nThus:\n\n\\[\n5291-4716=(46-41)u=575.\n\\]\n\nRestoring native LXX Cainan gives:\n\n\\[\n5751-4716=(50-41)u=1035.\n\\]\n\nLikewise, the same-rail gaps decompose through the five-year target offsets:\n\n\\[\n\\boxed{120=u+5,\\qquad450=4u-10,\\qquad570=5u-5.}\n\\]\n\nThese are not new independent constants; they expose the target-phase content of the original `450/120` vector.\n\n## 23. The warning advances one `115` cycle\n\nFor the regular shared SP/LXX Cainan-OFF Flood:\n\n\\[\n3106-1=3105=27u.\n\\]\n\nThe warning is `3226 BC`, and:\n\n\\[\n3226-6=3220=28u.\n\\]\n\nThe difference is:\n\n\\[\n\\boxed{120-5=115.}\n\\]\n\nThe countdown changes the early endpoint by `120`, while the Christ-side target moves by `5`; the net span advances one `u`. The coefficients `27=3^3` and `28=4\\times7` were noticed, but their further interpretation remains open.\n\n## 24. Regular-to-cumulative Flood and warning bridges\n\nThe strongest SP rectangle is:\n\n\\[\n\\begin{array}{ccc}\n4836_{\\mathrm{MT\\ cum\\ Flood/SP\\ warning}}&\\xrightarrow{120}&4716_{\\mathrm{SP\\ cum\\ Flood}}\\\\\n\\downarrow1610&&\\downarrow1610\\\\\n3226_{\\mathrm{regular\\ warning}}&\\xrightarrow{120}&3106_{\\mathrm{regular\\ Flood}}.\n\\end{array}\n\\]\n\nThus:\n\n\\[\n4836-3226=4716-3106=1610=14u.\n\\]\n\nAdjacent cross-rail bridges are:\n\n\\[\n4721-3226=1495=13u,\n\\]\n\n\\[\n5291-3106=2185=19u,\n\\]\n\n\\[\n5751-3106=2645=23u.\n\\]\n\nNative cumulative Cainan therefore changes `19u` to `23u`, and:\n\n\\[\n2645\\times\\frac{25}{23}=2875=25u.\n\\]\n\nTwo rejected arithmetic statements must not return: `4836−2576=2260`, not `2500`; and `4716−3226=1490`, not `1610`. The correct equal `1610` legs are the two vertical sides above.\n\n## 25. The `490` chain linking LXX Shem, SP Creation, MT Flood, and Christ\n\nSP rounded Creation gives:\n\n\\[\n4416-6=4411-1=4410=9\\times490.\n\\]\n\nMT regular rounded Flood gives:\n\n\\[\n2456-6=2450=5\\times490.\n\\]\n\nLXX-off cumulative rounded Shem gives:\n\n\\[\n5891-11=5886-6=5880=12\\times490.\n\\]\n\nThe MT cumulative Flood span to `6 BC` is:\n\n\\[\n4836-6=4830=69\\times70\\xrightarrow{70/69}4900.\n\\]\n\nHolding `6 BC` fixed moves the generated head to `4906 BC`. It follows that:\n\n\\[\n4836-1406=3430=7\\times490,\n\\]\n\nwhereas:\n\n\\[\n4906-1406=3500=50\\times70.\n\\]\n\nThe expansion is applied to `4830`, not directly to `3430`.\n\nThe combined chain is:\n\n\\[\n\\boxed{\n5886\\xrightarrow{2\\times490}4906\n\\xrightarrow{490}4416\n\\xrightarrow{4\\times490}2456\n\\xrightarrow{5\\times490}6\\text{ BC}.\n}\n\\]\n\nTherefore `2+1+4+5=12`, and the subsection from `5886` to `2456` is `3+4=7` cycles. The `5886→4416` link deliberately crosses lower and upper five-year members. [F46; F54; DISC]\n\nThe associated Creation span is:\n\n\\[\n14006-4906=9100=25\\times364=130\\times70.\n\\]\n\nThe discussion's initial `35×364` description was corrected: `35×364=12740`, not `9100`.\n\n## 26. Mirror Flood bridges and the decimal neighbor\n\nThe following civil cross-axis executions are exact:\n\n| BC start | AD end | Span | Factorization |\n|---:|---:|---:|---|\n| `4716` | `5285` | `10000` | decimal completion |\n| `4716` | `5290` | `10005` | `87u` |\n| `4721` | `5285` | `10005` | `87u` |\n| `4721` | `5290` | `10010` | opposite outer cross |\n| `5291` | `4830` | `10120` | `88u` |\n| `5751` | `4830` | `10580` | `92u=23×460` |\n| `3106` | `5290` | `8395` | `73u=23×365` |\n| `3226` | `5285` | `8510` | `74u=23×370` |\n\nThe `8395→8510` increase is again `120−5=115`. The `10000|10005|10010` family distinguishes the decimal cross-diagonal from the two equal `87u` spans. No one member should be substituted for another without showing the endpoints.\n\n## 27. Cainan insertion equals the LXX Flood-to-Christ Key gain\n\n\\[\n5291-1=5290=23\\times230=10\\times23^2.\n\\]\n\nThus:\n\n\\[\n5290\\times\\frac{25}{23}=5750,\n\\qquad5750-5290=460.\n\\]\n\nHolding the target `1 BC` fixed gives the source shift:\n\n\\[\n5291\\to5751.\n\\]\n\nThis is exactly the native cumulative Cainan addition. The lower member works identically against `AD 5`:\n\n\\[\n5286+5-1=5290,\n\\qquad5746+5-1=5750.\n\\]\n\nTherefore:\n\n\\[\n\\boxed{46u\\xrightarrow{+4u\\;=\\;460}50u}\n\\]\n\nis simultaneously a Cainan-state change and a Priestly expansion of the specified Christ-directed span.\n\n## 28. The native/generated Mirror junction at `11500`\n\nThe Cainan-OFF rounded LXX upper Flood member and its Protocol-1 counterpart give:\n\n\\[\n5291+5290-1=10580=23\\times460.\n\\]\n\nThe Key yields:\n\n\\[\n10580\\times\\frac{25}{23}=11500,\n\\]\n\nwith gain:\n\n\\[\n11500-10580=920=460+460.\n\\]\n\nIndependently, native cumulative Cainan already supplies:\n\n\\[\n5751+5750-1=11500.\n\\]\n\nThus `11500` has two provenances: it is a native-Cainan Mirror span and a generated completion of the Cainan-OFF Mirror span. These are two constructions of one value, not two statistically independent observations.\n\nThe next conversion is:\n\n\\[\n11500=25\\times460=23\\times500\n\\xrightarrow{25/23}12500.\n\\]\n\nIts gain is:\n\n\\[\n1000=2\\times500=920\\times\\frac{25}{23}.\n\\]\n\nThe complete relation is:\n\n\\[\n\\boxed{\n23\\times460\\to25\\times460=23\\times500\\to25\\times500.\n}\n\\]\n\nFrom the native `11500` state, the step to `12500` is a first conversion. From the Cainan-OFF `10580` state, it is a second conversion. The gain transforms coherently from `920` to `1000`. The initial `23²` factor and the independently available intermediate native state are both part of the rationale; the report does not license arbitrary repeated ratio application everywhere.\n\n## 29. Why this supports the `625+625` Creation half-hub\n\nR1 had generated the midpoint of the leveled actual regular Creation gap:\n\n\\[\n5364\\to4739\\to4114=625+625,\n\\qquad625=25^2.\n\\]\n\nThe midpoint was not a transmitted event-date. The new Mirror sequence reaches:\n\n\\[\n12500=6250+6250=10(625+625).\n\\]\n\nIf the Mirror expansion is realized bilaterally about the same center, the first step adds `460` at each end; the second adds `500` at each end. A final Protocol-1 realization is `6251 BC / AD 6250`, spanning `12500`, with equal `6250` arms about `1 BC` in the civil-coordinate representation.\n\nThis preserves the author's insight: the larger Mirror construction gives a reason to investigate the smaller `625` geometric half-unit. It does not turn `4739 BC` into a historical event, and the bilateral decomposition is a declared realization of the expansion rather than a result forced by the scalar `12500` alone. [R1; F69/F70/SA; DISC]\n\n## 30. Creation–Moses/Aaron Mirror: `49×115` to `6000`\n\nThe rounded opposed pair gives:\n\n\\[\n4111+1525-1=4106+1530-1=5635=49\\times115.\n\\]\n\nThen:\n\n\\[\n5635\\times\\frac{25}{23}=6125=49\\times125.\n\\]\n\nThe midpoint is:\n\n\\[\n5880=49\\times120=12\\times490,\n\\]\n\nso the scaled cell is:\n\n\\[\n\\boxed{5635\\mid5880\\mid6125=49(115\\mid120\\mid125).}\n\\]\n\nThe rounded terminal comparisons are:\n\n\\[\n1531-1406=1526-1401=125.\n\\]\n\nSubtracting that declared terminal `125` after expansion leaves:\n\n\\[\n6125-125=6000=48\\times125.\n\\]\n\nThis is an expanded-span operation, not an unexpanded date subtraction. To make the geometry explicit, hold the AD Moses/Aaron targets fixed during the first expansion: `4111→4601 BC` against `AD 1525`, and `4106→4596 BC` against `AD 1530`. Moving the AD targets back by `125` to `AD 1400/1405` then gives:\n\n\\[\n4601+1400-1=4596+1405-1=6000.\n\\]\n\nThe existing actual, phase-resolved route is:\n\n\\[\n4115+1406-1=4114+1407-1=5520,\n\\]\n\n\\[\n5520\\times\\frac{25}{23}=6000.\n\\]\n\nAt scalar level the two routes are related by:\n\n\\[\n(5635-115)\\frac{25}{23}=6125-125=6000.\n\\]\n\nF09 controls the actual Tishri/Nisan pairing; it must not be replaced by the incorrect unsuffixed civil sum `4114+1406−1`, which is `5519`. The rounded terminal `125` for the Moses-side comparison extends to the lower rounded Conquest companion, not to a revised literal lifespan of Moses. [F09; F51a; DISC]\n\n---\n\n# Part VI — Creation cells, Apparent Adam, and scale repetition\n\n## 31. The cumulative LXX–MT four-state cell\n\nSet `M=14006`, the lower cumulative MT Creation spine. The four states are:\n\n\\[\nM,\\quad M+430,\\quad M+460,\\quad M+430+460.\n\\]\n\nIn descending BC order:\n\n\\[\n14896\\to14466\\to14436\\to14006\n\\]\n\ngives:\n\n\\[\n\\boxed{430\\mid30\\mid430,\\qquad460-430=30.}\n\\]\n\nThese coordinates survive both actual and rounded cumulative whole-lifespan calculation. The surrounding envelopes remain different. [F18 §6D; F46 §6A]\n\nThe actual MT bridge is `9890`, not the rounded `9900`:\n\n\\[\n14006-4116=9890=23\\times430.\n\\]\n\nAdding restored cumulative Cainan and the actual `5520` Mirror component gives:\n\n\\[\n460+9890+5520=15870=69\\times230.\n\\]\n\nA point-consistent execution uses:\n\n\\[\n14464\\to14004\\to4114\\to\\mathrm{AD}\\ 1407,\n\\]\n\nwith intervals `460`, `9890`, and `5520`; its total is `14464+1407−1=15870`.\n\nThe known completions are:\n\n\\[\n15870\\times\\frac{70}{69}=16100,\n\\qquad15870\\times\\frac{25}{23}=17250.\n\\]\n\nThese source-controlled actual-state results should not be silently reclassified as rounded simply because some cumulative head coordinates co-register. [R3; F09; F22]\n\n## 32. The regular LXX–MT four-state counterpart\n\nThe lower rounded regular coordinates are:\n\n\\[\n5486\\to5356\\to4236\\to4106,\n\\]\n\nwith:\n\n\\[\n\\boxed{130\\mid1120\\mid130=1380=3\\times460=23\\times60.}\n\\]\n\nThe actual Year-7 endpoints are eight years higher:\n\n\\[\n5494\\to5364\\to4244\\to4114,\n\\]\n\nwith the same intervals. Thus regular Cainan `130` forms the flanks; the total resolves into three cumulative-Cainan units `460`, without changing the actual regular operator.\n\nAdding `30` to the LXX-off Apparent-Adam member, while retaining the MT+Cainan comparator, gives:\n\n\\[\n(5356+30)-4236=1150,\n\\]\n\nand:\n\n\\[\n1150\\times\\frac{25}{23}=1250.\n\\]\n\nHence:\n\n\\[\n\\boxed{1120+30=1150\\to1250.}\n\\]\n\n## 33. The three cumulative pairwise cells with explicit nodes\n\nOn the lower rounded cumulative rail:\n\n| Cell | Ordered coordinates, BC | Intervals | Width |\n|---|---|---|---:|\n| LXX native → SP without Cainan | `14896 → 14436 → 13861 → 13401` | `460+575+460` | `1495` |\n| LXX native → MT without Cainan | `14896 → 14466 → 14436 → 14006` | `430+30+430` | `890` |\n| MT+Cainan → SP without Cainan | `14466 → 14006 → 13861 → 13401` | `460+145+460` | `1065` |\n\nHere `13861=13401+460` is SP with cumulative Cainan; the `575` and `145` are therefore exact inner intervals, not freely inserted flanking complements.\n\nThis table resolves an ambiguity in an earlier discussion response: the symmetric `460|575|460` belongs directly to the Creation four-state cell. It is not obtained by silently adding a second native Cainan to an already native LXX Flood date.\n\n## 34. The Apparent-Age `30` redistributes an invariant width\n\nWithout Apparent Adam:\n\n\\[\n14436_{L- C}\\to14006_M\\to13401_S=430+605=1035.\n\\]\n\nAdd `30` to the two outer Creation comparison states while holding MT fixed:\n\n\\[\n14466_{L-C+30}\\to14006_M\\to13431_{S+30}=460+575=1035.\n\\]\n\nTherefore:\n\n\\[\n\\boxed{(430,605)\\to(460,575)\\quad\\text{by}\\quad(+30,-30).}\n\\]\n\nThe shared `14466` label is both MT+Cainan and LXX-off+Apparent Adam, because:\n\n\\[\n\\boxed{430+30=460.}\n\\]\n\nLikewise:\n\n\\[\n605=460+145=490+115=430+175.\n\\]\n\nThus `±30` opens `430/490` from `460` and `115/175` from `145` while preserving the selected total. Different redistributions may require different declared endpoint states; algebraic accessibility is not a license to shift a single historical event arbitrarily.\n\n## 35. Apparent Adam joins the regular and cumulative LXX fields\n\nThe regular Apparent-Adam rails are:\n\n\\[\n5521/5516_{L},\\qquad4141/4136_M.\n\\]\n\nTo `1 BC / AD 5` they give:\n\n\\[\n5520=23\\times240,\n\\qquad4140=23\\times180.\n\\]\n\nThe LXX-off cumulative Flood rail `5291/5286` gives the intermediate:\n\n\\[\n5290=23\\times230.\n\\]\n\nSo the lower BC coordinates form:\n\n\\[\n\\boxed{5516\\to5286\\to4136=230+1150=1380.}\n\\]\n\nF54 already contains this numerical `5516/5286/4136` decomposition in its Luke/Rounded discussion. The new comparison supplies a different node-class for `5286`: the lower rounded LXX-off cumulative Flood. Shared coordinates are corroborative intersections, not event identity.\n\nThe Adam–Christ age-30 analogy is a typological reason for testing the localized Apparent-Age state; it is not a textual assertion that Genesis supplies Adam's age as `30`. Luke 3:23 supplies Jesus' “about thirty” ministry-age comparison. [F54; F22; DISC]\n\n## 36. The `920/9200` decimal correspondence\n\nMT cumulative Apparent Adam is:\n\n\\[\n14036=14006+30,\n\\]\n\nand:\n\n\\[\n14036-4836=9200=20\\times460=23\\times400.\n\\]\n\nThe upper rail works identically:\n\n\\[\n14041-4841=9200.\n\\]\n\nAdding the MT Flood-to-Christ span gives:\n\n\\[\n9200+4830=14030=122u.\n\\]\n\nThus:\n\n\\[\n14041/14036\\to4841/4836\\to11/6\\text{ BC}\n=80u+42u.\n\\]\n\nNative LXX Apparent Adam gives:\n\n\\[\n14926-14006=920=430+460+30=2\\times460.\n\\]\n\nTherefore:\n\n\\[\n\\boxed{920\\to1000,\\qquad9200\\to10000}\n\\]\n\nunder `25/23`. The same `920→1000` also appears as the transformed gains of the LXX Flood Mirror `10580→11500→12500`. This is the principal finite decimal correspondence retained from this part of the discussion. [F51a §16.2; F22; DISC]\n\n## 37. The common `1500` completion\n\nWithin the fixed regular four-state field, native LXX against MT without Cainan gives:\n\n\\[\n1380=12u\\xrightarrow{25/23}1500.\n\\]\n\nWithin the rounded cumulative field, native LXX against SP without Cainan gives:\n\n\\[\n1495=13u=5\\times299\\xrightarrow{300/299}1500.\n\\]\n\nThe gains recover the warning/phase cell:\n\n\\[\n\\boxed{120=115+5.}\n\\]\n\nThe two inputs are in ratio `12:13`; the operators are different; the target is the same. This is one of the most compact regular/cumulative correspondences reached in the discussion.\n\n## 38. `1495` measures the LXX/SP Creation-to-Exodus field\n\nUsing the upper rail:\n\n\\[\n14901-1446=13455=9\\times1495=117u=45\\times299,\n\\]\n\n\\[\n13406-1446=11960=8\\times1495=104u=40\\times299.\n\\]\n\nThe lower members give identical spans to `1441 BC`.\n\nThus the manuscript difference is one block, SP is eight blocks from the target, and LXX is nine. The `300/299` completion acts coherently:\n\n\\[\n1495\\to1500,\\quad11960\\to12000,\\quad13455\\to13500.\n\\]\n\nA separate Prophetic completion of the LXX span is:\n\n\\[\n13455=69\\times195\\xrightarrow{70/69}13650=37.5\\times364,\n\\]\n\nwith gain `195`, half of `390`. The `13` in `13455=13×1035` was compared with the repository's Cainan/curse symbolism; that remains theological interpretation.\n\n## 39. Apparent Adam, Joshua, and symmetric outer-flank expansion\n\nThe unshifted chain is:\n\n\\[\n14901/14896\\xrightarrow{1495}13406/13401\n\\xrightarrow{11960}1446/1441\n\\xrightarrow{1495}\\mathrm{AD}\\ 50/55.\n\\]\n\nIts total is:\n\n\\[\n1495+8(1495)+1495=14950.\n\\]\n\nThe Apparent-Adam comparison shifts the BC heads by `+30` and the AD targets by `−30`:\n\n\\[\n14931/14926\\xrightarrow{1495}13436/13431\n\\xrightarrow{11960}1476/1471\n\\xrightarrow{1495}\\mathrm{AD}\\ 20/25.\n\\]\n\n`1476 BC` is the repository Joshua-birth anchor; `1471` is the generated five-year companion, not an independently attested birth date.\n\nExpanding only the two outer flanks `1495→1500`, while keeping the central `11960` fixed, gives:\n\n\\[\n14936/14931\\xrightarrow{1500}13436/13431\n\\xrightarrow{11960}1476/1471\n\\xrightarrow{1500}\\mathrm{AD}\\ 25/30.\n\\]\n\nThe ordinary counterpart is:\n\n\\[\n14906/14901\\xrightarrow{1500}13406/13401\n\\xrightarrow{11960}1446/1441\n\\xrightarrow{1500}\\mathrm{AD}\\ 55/60.\n\\]\n\nBoth totals become `14960`. This is **not** a whole-span `300/299` expansion of `14950`; it is a stated symmetric expansion of the two flanks only. The Christological reading of the AD endpoints remains a schematic comparison, not a replacement for the repository's historical Passion-date distinctions.\n\n\n---\n\n# Part VII — The overlapping comparison aggregates and `1150/1152`\n\n## 40. The preserved cumulative aggregate\n\nThe author specifically requested that this observation be retained, and then reordered it to place `30` at the center:\n\n\\[\n\\boxed{\n(460+575+460)+(430+30+430)+(460+145+460)=3450.\n}\n\\]\n\nThe cell widths are:\n\n\\[\n1495+890+1065=3450,\n\\]\n\nso:\n\n\\[\n\\boxed{3450/3=1150=10u=23\\times50\\xrightarrow{25/23}1250.}\n\\]\n\nFlattening the written expression gives nine terms with `30` in the fifth position. Its center-column anatomy is:\n\n\\[\n575\\mid30\\mid145,\n\\]\n\nwhere:\n\n\\[\n575+145=720,\\quad575-145=430,\\quad575+30+145=750.\n\\]\n\nThe six flanking terms give:\n\n\\[\n4(460)+2(430)=2700,\n\\]\n\nso:\n\n\\[\n3450=2700+750.\n\\]\n\nThe author regards this as an elegant, potentially central relation whose broader architectural purpose remains to be understood. That priority is retained. These are **overlapping** comparison cells, however, not three disjoint temporal intervals.\n\n## 41. Actual cumulative raises the aggregate to `3456`\n\nSP actual Creation is three years later than its rounded Nisan head. Consequently the actual SP-related widths are three years larger:\n\n\\[\n1495\\to1498,\\qquad1065\\to1068,\n\\]\n\nwhile `890` is unchanged.\n\nThe actual cell decomposition is:\n\n\\[\n\\boxed{\n(460+578+460)+(430+30+430)+(460+148+460)=3456.\n}\n\\]\n\nThus:\n\n\\[\n\\boxed{3456/3=1152=16\\times72.}\n\\]\n\nA single SP displacement is counted in two of the overlapping cells, so its effect on the mean is:\n\n\\[\n\\frac{2\\times3}{3}=2.\n\\]\n\nThis explains how a three-year source difference produces the two-year mean-pair:\n\n\\[\n\\boxed{1150/1152.}\n\\]\n\n## 42. Audit-level algebraic reduction of the aggregate\n\nThe following is an explicit compression of the already discussed sums, added during report preparation; it is not a new empirical discovery or a replacement for the author's interpretation.\n\nLet the leveled cumulative Creation widths be:\n\n\\[\na=L-M=430,\\qquad b=M-S=605\\ \\text{or}\\ 608,\n\\qquad c=460.\n\\]\n\nThe three native/optional-Cainan cell totals are:\n\n\\[\nL-S+c=a+b+c,\n\\quad L-M+c=a+c,\n\\quad M-S+c=b+c.\n\\]\n\nTheir sum is therefore:\n\n\\[\n\\boxed{\\Sigma_c=2(a+b)+3c,\\qquad\\overline{\\Sigma}_c=c+\\frac{2}{3}(a+b).}\n\\]\n\nRounded:\n\n\\[\n460+\\frac{2}{3}(1035)=460+690=1150.\n\\]\n\nActual:\n\n\\[\n460+\\frac{2}{3}(1038)=460+692=1152.\n\\]\n\nThis identifies the exact overlap accounting. It also places the rounded mean directly in the already discussed `460+690=1150` family. What remains interpretively interesting is that the independently supplied chronology makes these widths land on the familiar Key and Shem/Conquest values; the arithmetic averaging itself is no longer unexplained.\n\n## 43. The regular aggregate and the overlapping `2500`s\n\nThe rounded regular cells are:\n\n\\[\n130+815+130=1075,\n\\]\n\n\\[\n130+1120+130=1380,\n\\]\n\n\\[\n130+175+130=435.\n\\]\n\nTheir aggregate is:\n\n\\[\n1075+1380+435=2890=390+2110+390.\n\\]\n\nEach flank plus the shared center gives:\n\n\\[\n\\boxed{390+2110=2110+390=2500=50^2.}\n\\]\n\nThese are two **overlapping** `2500` windows; they do not sum to a disjoint `5000` span.\n\nThe actual regular middle terms are `820`, `1120`, and `170`. Thus:\n\n\\[\n820+1120+170=815+1120+175=2110,\n\\]\n\nand the same aggregate survives. Rounding transfers `5` from the first cell to the third:\n\n\\[\n1080/1380/430\\to1075/1380/435.\n\\]\n\nThe actual totals expose:\n\n\\[\n1080=3\\times360=15\\times72,\n\\]\n\nThe rounded companion is `1075=5×215=2.5×430`. F51a already identifies it with fifteen alternate precessional-day units of `215/3`, against fifteen standard `72` units in `1080`; this is source context rather than an additional discovery of the present discussion. [F51a §§7A.5, 17.4]\n\nand their pairwise differences are:\n\n\\[\n1080-430=650,\\quad1380-1080=300,\\quad1380-430=950.\n\\]\n\nThe rounded analogues are `640`, `305`, and `945`. Seth's `130` was proposed as a further interpretive link for the `3×130` flanks; this was not separately developed.\n\nFor overlap accounting, write regular `L−M=a=1250`, `S−M=b=300/305`, and `c=130`. Then:\n\n\\[\n\\boxed{\\Sigma_r=(a-b+c)+(a+c)+(b+c)=2a+3c=2890.}\n\\]\n\nThe SP location cancels algebraically. This is why the regular aggregate remains fixed while the SP-related cells exchange five years.\n\n## 44. The `1152` precessional connection\n\nThe newly obtained actual cumulative mean belongs to the repository's `72`-year schematic precessional-day vocabulary:\n\n\\[\n1152=16\\times72=2^4\\times72.\n\\]\n\nOther discussed members are:\n\n\\[\n720=10\\times72,\\quad1080=15\\times72,\n\\quad1656=23\\times72,\\quad2880=40\\times72.\n\\]\n\nMost importantly, the regular MT Shem pair independently gives:\n\n\\[\n2558-1406=1152,\n\\qquad2556-1406=1150.\n\\]\n\nThus the aggregate mean-pair reappears point-to-point in the source-controlled Shem/Conquest comparison. This is the stronger observation; the factorization of every coefficient is not assigned independent significance. The `72` here is the repository's idealized precessional unit, not a newly researched modern astronomical rate. [R1–R3; F17; DISC]\n\n## 45. Five-year and seven-year reconciliation\n\nThe difference is:\n\n\\[\n1152-1150=2,\n\\]\n\nwhich equals the width difference:\n\n\\[\n7-5=2.\n\\]\n\nThe discussion proposed that the mean-pair can therefore mediate between compressed five-year rails and seven-year actual envelopes. This is a useful resolution hypothesis, but actual endpoint alignment must still be checked individually. In particular, the `+5` generated by `123→125` at Aaron should not be confused with simply deleting one term from every `2+3+2` actual cell.\n\n## 46. MT and LXX use the mean-pair in opposite directions\n\nMT:\n\n\\[\n14004+1152=14006+1150=15156.\n\\]\n\nThe generated common head then gives:\n\n\\[\n15156-4116=11040=23\\times480\\xrightarrow{25/23}12000.\n\\]\n\nLXX:\n\n\\[\n14896-1150=13746,\n\\qquad14896-1152=13744.\n\\]\n\nAdding the localized Apparent-Age `30` to these generated heads gives `13776/13774`. Against the actual regular LXX Year-6/Year-7 pair:\n\n\\[\n13776-5496=13774-5494=8280=23\\times360\n\\xrightarrow{25/23}9000.\n\\]\n\nTherefore:\n\n\\[\n11040-8280=2760=23\\times120,\n\\]\n\nand after expansion:\n\n\\[\n12000-9000=3000=25\\times120.\n\\]\n\nEquivalently:\n\n\\[\n8280=18\\times460,\\qquad11040=24\\times460.\n\\]\n\nThe averages are here used as **substitute comparison carriers** by explicit author proposal. They are not newly transmitted chronology totals.\n\n## 47. `8280` fuses the `72` and `115` descriptions\n\n\\[\n\\boxed{8280=72\\times115=23\\times360.}\n\\]\n\nIts Key completion is:\n\n\\[\n72\\times115\\to72\\times125=9000.\n\\]\n\nThe MT counterpart is:\n\n\\[\n11040=96\\times115\\to96\\times125=12000.\n\\]\n\nThe pair therefore compresses to:\n\n\\[\n\\boxed{24\\times(3\\mid4)\\times(115\\to125).}\n\\]\n\nThe vertical separation is `24×115→24×125`, or `2760→3000`. This restates the same exact rectangle in a more economical factorization; it is not an additional independent witness.\n\n## 48. The provisional SP `441×23` probe\n\nBefore the native SP resolution emerged, the following was tested:\n\n\\[\n4411-1=4410=9\\times490,\n\\]\n\n\\[\n10143=441\\times23=21\\times483.\n\\]\n\nProjecting backward from `4411 BC` gives:\n\n\\[\n4411+10143=14554.\n\\]\n\nThe Prophetic completion gives:\n\n\\[\n10143\\times\\frac{70}{69}=10290=21\\times490,\n\\]\n\nwith gain `147`, and the expanded head is `14701 BC`. Therefore:\n\n\\[\n14701-1=14700=(21+9)\\times490.\n\\]\n\nThat scalar chain is exact. The proposed mean-pair tie was less settled:\n\n\\[\n14554-1150=13404,\\quad14554-1152=13402,\n\\]\n\nwhose midpoint is actual SP envelope head `13403`; equivalently `14554−13403=1151`.\n\nA generated `+2` companion gives:\n\n\\[\n14556-1150=13406,\n\\qquad14556-1152=13404.\n\\]\n\nThese do not recover the complete rounded `13406/13401` pair. The earlier suggestion that SP might preferentially use the midpoint `1151` remains exploratory and unadopted. It should not be elevated above the later, source-supported SP `−215` / `9200` bridge.\n\n---\n\n# Part VIII — The SP's native-state resolution\n\n## 49. Why SP's phase behavior is distinctive\n\nActual and rounded SP Creation envelopes are:\n\n| Mode | Actual | Rounded |\n|---|---:|---:|\n| Cumulative | `13403/13396` | `13406/13401` |\n| Regular on the leveled `+215` plane | `4421/4414` | `4416/4411` |\n\nCumulative rounded shifts earlier by `3/5`; regular rounded shifts later by `5/3`. Corresponding actual cumulative-to-regular spans are:\n\n\\[\n13403-4421=13396-4414=8982,\n\\]\n\nwhereas corresponding rounded spans are:\n\n\\[\n13406-4416=13401-4411=8990.\n\\]\n\nThus `8990−8982=8=3+5`.\n\nThe author redirected attention from this `8` to the larger union of evenly spaced comparison boundaries:\n\n\\[\n4421/4416/4411,\n\\qquad13406/13401/13396.\n\\]\n\nEach spans `5+5=10`. They are mixed actual/rounded comparison spines, not single transmitted envelopes.\n\n## 50. Applying the SP-favored `−215` state\n\nApply the SP Sojourn translation to the regular comparison spine:\n\n\\[\n4421-215=4206,\\quad4416-215=4201,\\quad4411-215=4196.\n\\]\n\nNow:\n\n\\[\n13406-4206=13401-4201=13396-4196=9200.\n\\]\n\nTherefore:\n\n\\[\n\\boxed{\n13406/13401/13396\\xrightarrow{-9200}4206/4201/4196.\n}\n\\]\n\nThe established source theorem already gives:\n\n\\[\n13398-4198=9200=23\\times400\\xrightarrow{25/23}10000.\n\\]\n\nThe newly assembled triplet expands the phase presentation around that source-controlled SP bridge. F18 §6C.1 also already records `13406/13401/13396` as an SP Day-4 phase-anchor display with different node labels; that is a shared-coordinate corroboration, not proof that the mixed-rounding and Day-4 constructions are identical. [F11 §9; F18 §6C.1]\n\n## 51. Leveled comparison and native-state execution are both required\n\nThe regular LXX naturally includes Cainan `130`, and its cumulative chain includes `460`. The SP naturally favors the Canaan/Egypt Sojourn execution used here as `−215` from the leveled plane.\n\nThe SP Exodus 12:40 reading supplies the Canaan/Egypt scope. The explicit `215+215=430` division is the repository's chronological execution of that scope, not a claim that the verse itself prints two separate `215` figures. F04/F05 also preserve the distinct LXX `33+397` family-unit interpretation rather than forcing all LXX applications into one SP-like arrangement.\n\nThe method is consequently:\n\n> **First compare equivalent variant states to expose common structure; then allow each manuscript's native or favored configuration to disclose its particular internal relationships.**\n\nThe SP `−215` state does not invalidate the leveled discoveries. It explains why remaining solely on the leveled plane made the SP cumulative/regular connection unnecessarily difficult. [F01/F04/F05; F18; DISC]\n\n## 52. Both SP modes generate the wilderness `40`\n\nRegular SP `−215`:\n\n\\[\n4206-1446=2760=69\\times40,\n\\]\n\n\\[\n2760\\times\\frac{70}{69}=2800=4206-1406.\n\\]\n\nCumulative rounded SP:\n\n\\[\n13406-1446=11960=299\\times40,\n\\]\n\n\\[\n11960\\times\\frac{300}{299}=12000=13406-1406.\n\\]\n\nThe cumulative Conquest span is also Mosaic at hundredfold scale:\n\n\\[\n13406-1526=11880=99\\times120=165\\times72,\n\\]\n\n\\[\n1526-1406=120,\\qquad11880+120=12000=100\\times120.\n\\]\n\nBoth rounded members preserve the same `12000` to their corresponding Conquest companions: `13406−1406=13401−1401=12000`.\n\nThe two different operators add the same `40`. Their difference remains invariant:\n\n\\[\n11960-2760=12000-2800=9200.\n\\]\n\nThus:\n\n\\[\n\\boxed{\n\\begin{array}{ccc}\n299\\times40&\\xrightarrow{300/299}&300\\times40\\\\\n\\downarrow9200&&\\downarrow9200\\\\\n69\\times40&\\xrightarrow{70/69}&70\\times40.\n\\end{array}\n}\n\\]\n\nThe algebraic difference is:\n\n\\[\n40(299-69)=40(300-70)=40\\times230=9200.\n\\]\n\nThis is the strongest SP-specific compression reached by the end of the discussion. It joins its cumulative/regular bridge to the same Exodus–Conquest `40` on both rows. [R1; F11; F63; DISC]\n\n## 53. Restored SP cumulative Cainan gives the MT `12600` on the Exodus frame\n\nRestore cumulative Cainan to the SP Exodus-directed upper-rail span:\n\n\\[\n11960+460=12420=69\\times180=27\\times460.\n\\]\n\nThen:\n\n\\[\n12420\\times\\frac{70}{69}=12600.\n\\]\n\nHolding Exodus `1446 BC` fixed generates:\n\n\\[\n1446+12600=14046\\text{ BC}.\n\\]\n\nThe MT source relation is:\n\n\\[\n14006-1406=12600.\n\\]\n\nTherefore:\n\n\\[\n\\boxed{\n14046\\to1446=14006\\to1406=12600,\n}\n\\]\n\nwith the whole frame translated by `40`. The generated SP head is not a replacement for native SP Creation.\n\n## 54. The lower SP rail reproduces the Revelation-11 comparison\n\nThe lower companion uses the target `1441 BC`:\n\n\\[\n13401-1441=11960,\n\\]\n\nand the same restored-Cainan/Prophetic expansion generates `14041 BC`. Hence:\n\n\\[\n14046/14041\\to1446/1441=12600.\n\\]\n\nTo the fixed Conquest `1406 BC`:\n\n\\[\n14046-1406=12640=12600+40,\n\\]\n\n\\[\n14041-1406=12635=12600+35=10(1260+3.5).\n\\]\n\nF22 §9A.2 already controls the second arithmetic and its Revelation-11 typological reading, with `1441` as the inferred forward-order joint. The new SP route regenerates the label `14041` by a different operation. It must remain distinct from MT cumulative Apparent Adam at the same label.\n\nThe five-year comparison therefore displays both the historical-model wilderness `40` and the prophetic-scale final `35`, without claiming that Revelation's `1260` literally means forty years. [F22 §9A; DISC]\n\n## 55. SP cumulative Shem/Flood to Abraham\n\nThe actual SP cumulative Nisan spine gives:\n\n\\[\n5311-1951=3360=10\\times336,\n\\]\n\n\\[\n4711-1951=2760=23\\times120.\n\\]\n\n`1951 BC` is the SP-favored Abraham birth. The difference of the two spans is Shem's `600`.\n\nThe rounded Shem lower member selects the Call/Sojourn framework:\n\n\\[\n5316-1876=3440=8\\times430,\n\\]\n\n\\[\n5316-1446=3870=9\\times430.\n\\]\n\nTherefore:\n\n\\[\n5316\\to1876\\to1446=8(430)+430.\n\\]\n\nThis retains both aspects of the SP: the `−215` position for Abraham's birth/Call and the full `430` from Call to Exodus. The selected actual and rounded cumulative members highlight different Abrahamic nodes without changing the source biography `1951/1876/1776`. [F01; F05; F18; DISC]\n\n## 56. SP actual/rounded Flood bridges: `1610/1820/1825`\n\nUsing the SP regular `−215` post-Flood comparison nodes:\n\n\\[\n\\text{Shem}=2991,\\qquad\\text{Arphaxad/Flood-close seam}=2891.\n\\]\n\nActual cumulative Flood/Arphaxad `4711` gives:\n\n\\[\n4711-2891=1820=5\\times364,\n\\]\n\n\\[\n4711-2991=1720=4\\times430.\n\\]\n\nThe rounded cumulative lower member `4716` gives:\n\n\\[\n4716-2891=1825=5\\times365,\n\\]\n\n\\[\n4716-2991=1725=15u.\n\\]\n\nThe leveled regular rounded Flood instead gives:\n\n\\[\n4716-3106=1610=7\\times230=14u.\n\\]\n\nThus:\n\n\\[\n\\boxed{1610\\xrightarrow{+215}1825\\xrightarrow{-5}1820,}\n\\]\n\nand:\n\n\\[\n\\boxed{1820-1610=210=215-5.}\n\\]\n\nThese are actual-versus-rounded cumulative comparison members, not the two members of the single rounded Aaron/Moses pair: the full rounded SP Flood pair is `4721/4716`. The label `4716` also appears as an upper member of the actual envelope; shared label does not erase this distinction.\n\n**Flood-boundary guard:** F18's raw SP Flood year is `2892–2891`; `2892` is the initial Flood/death cluster, while `2891` is its terminal/post-Flood Arphaxad comparison boundary. The `1820` relation selects `2891`. It is not the interval from `4711` to the raw Flood head `2892`, which would be `1819`. In the rounded `−215` biography, Flood is `2891` and Noah death is `2541`, preserving `350`; the raw actual Noah death is `2542`.\n\nThe cross-manuscript alternative:\n\n\\[\n4716-2166=2550=1260+1290\n\\]\n\nalso remains exact, but the author favors the internally SP-specific alignments above when explaining the SP on its own terms. [F18 §§3, 6C; DISC]\n\n## 57. SP and the Book of Jubilees: the `350/343`, `30`, and `8050` bridge\n\n### 57.1. Why the comparison belongs in the SP native-state section\n\nWithin the adopted reconstruction, the Book of Jubilees (`BJ`) is especially close to the SP from Creation to the Flood. Its later chronology also remains close to the SP `−215` execution from Abraham to the Exodus. The Abrahamic birth coordinates are:\n\n\\[\n1981_{\\mathrm{BJ}}-1951_{\\mathrm{SP}}=30.\n\\]\n\nThe `30` is arithmetically exact and structurally suggestive because Adam and Abram each open a people-field: Adam as father of humanity and Abram/Abraham as covenantal father of a people and many nations. This analogy gives the `30` interpretive relevance in a report already concerned with Apparent Adam's `30`, but it does not establish that the two occurrences have the same historical cause. [F01; BJ/DISC]\n\nThe author's working explanation is that BJ relies on an SP-like chronological foundation, particularly in the virtually identical antediluvian sequence. Chronological affinity is distinguished here from proof of direct literary dependence on the extant SP. Corresponding Noah/Shem/Flood comparisons should retain the shared sequence while tracking the Creation/Fall origin and one-year Flood phase; the Creation offset must not be propagated indiscriminately between differently labeled event boundaries.\n\n### 57.2. The uniform `350` Creation/Fall translation\n\nThe actual SP regular `−215` Creation week is:\n\n\\[\n4206\\text{–}4199\\ \\mathrm{BC}.\n\\]\n\nThe BJ Eve/Fall week is:\n\n\\[\n3856\\text{–}3849\\ \\mathrm{BC}.\n\\]\n\nMatching the corresponding endpoints gives one uniform translation:\n\n\\[\n4206-3856=350,\n\\]\n\n\\[\n4199-3849=350.\n\\]\n\nTherefore:\n\n\\[\n\\boxed{4206/4199\\xrightarrow{-350}3856/3849,\\qquad350=5\\times70.}\n\\]\n\nThis correspondence is particularly apt to BJ's Jubilee organization: the SP seven-year Creation field stands five `70`-year generations before the BJ seven-year Eve/Fall field. It is not itself seven Jubilees; that more specifically Jubilee-shaped relation appears in the next subsection.\n\n### 57.3. The uniform `343` Jubilee translation and the separate Flood phase\n\nWhen BJ's preceding seven years of Creation are included, its initial Creation week is:\n\n\\[\n3863\\text{–}3856\\ \\mathrm{BC}.\n\\]\n\nThe exact `343` applies to both corresponding Creation-week boundaries:\n\n\\[\n\\boxed{4206-3863=4199-3856=343=7\\times49=7^3.}\n\\]\n\nThis is the desired seven-Jubilee fit: a seven-day/year Creation field is separated from its BJ counterpart by seven jubilees of `49` years.\n\nThe same SP terminal Creation boundary also supplies the Conquest relation:\n\n\\[\n\\boxed{4199-1406=2793=57\\times49.}\n\\]\n\nThus the selected SP boundary participates both in the seven-Jubilee BJ bridge and in a fifty-seven-Jubilee span to Conquest.\n\nThe earlier `3862` was an addition error, corrected by the author to `3863`. Both Creation weeks span seven elapsed years. The earlier explanation invoking seven inclusively named BJ years is withdrawn; no one-year Creation-boundary discrepancy remains.\n\nA separate `343/344` distinction remains at the Flood. From the BJ `3856` coordinate:\n\n\\[\n3856-(7+1300)=2549\\ \\mathrm{BC}\n\\]\n\nfor the Flood, while BJ assigns the preceding full Ark-building year to `2550 BC`. Against the SP Gear-2 normal Flood comparator `2893 BC`:\n\n\\[\n2893-2549=344,\n\\]\n\n\\[\n\\boxed{2893-2550=343=7\\times49.}\n\\]\n\nThus the one-year Ark lead converts the Flood-to-Flood `344` into a Jubilee-exact `343`, matching the uniform seven-Jubilee Creation translation. This correspondence supports the author's inclusive reading of the SP Flood as completing Noah's six-hundredth year rather than treating every adjacent Flood label as interchangeable. It does not erase the other SP controls: `2893` is the Gear-2 normal comparator, while §56 retains the raw `2892–2891` Flood-year/seam and the rounded `−215` `2891` state. [R2; F18; BJ/DISC]\n\nWhether BJ's Ark year was intended to compensate for this difference is the author's proposed explanation, not a demonstrated authorial purpose. The observed fit is compatible with inclusive reckoning but does not independently decide between all possible phase conventions.\n\n### 57.4. Rounded `345→350` and the cumulative `10350`\n\nThe rounded SP `−215` Creation pair and BJ rounded Fall pair give:\n\n\\[\n4201/4196\\quad\\text{and}\\quad3856/3851.\n\\]\n\nOn the same side:\n\n\\[\n4201-3856=4196-3851=345=5\\times69,\n\\]\n\nand therefore:\n\n\\[\n\\boxed{345\\times\\frac{70}{69}=350.}\n\\]\n\nThe rounding contraction `350→345` and Prophetic restoration `345→350` are the same relation already encountered in the SP Noah-death field (§11), now attached to the SP–BJ Creation/Fall translation.\n\nThis also adjoins the SP cumulative/regular bridge:\n\n\\[\n13406/13401/13396-4206/4201/4196=9200,\n\\]\n\n\\[\n9200\\times\\frac{25}{23}=10000.\n\\]\n\nAdding the independently recovered `350` gives:\n\n\\[\n\\boxed{10000+350=10350=23\\times450.}\n\\]\n\nEquivalently, the rounded input may be displayed as `10000+(345→350)`. The result is an exact compound construction, not one uninterrupted transmitted chronological span. Its possible significance remains open.\n\n### 57.5. The rounded Mirror cross-pair gives `8050=23×350`\n\nUnder Protocol-1 civil Mirror conversion:\n\n\\[\n3856\\ \\mathrm{BC}\\leftrightarrow\\mathrm{AD}\\ 3855,\n\\qquad\n3851\\ \\mathrm{BC}\\leftrightarrow\\mathrm{AD}\\ 3850.\n\\]\n\nCross-pairing the two rounded endpoints preserves one span:\n\n\\[\nS(4201\\ \\mathrm{BC},\\mathrm{AD}\\ 3850)=4201+3850-1=8050,\n\\]\n\n\\[\nS(4196\\ \\mathrm{BC},\\mathrm{AD}\\ 3855)=4196+3855-1=8050.\n\\]\n\nHence:\n\n\\[\n\\boxed{8050=23\\times350.}\n\\]\n\nThe construction is noteworthy because the Mirror span contains `23` copies of the same `350` that separates actual SP Creation from the BJ Eve/Fall week, while its same-side rounded precursor is `345→350`. It remains a generated Mirror relation, not a claim that BJ itself dates an event to the AD counterpart. [F17; CTRL; BJ/DISC]\n\n\n---\n\n# Part IX — Compression, retained uncertainties, and continuation\n\n## 58. Three operations that should not be confused\n\nThe discussion repeatedly used “transfer,” “reversal,” and “Mirror.” Their common arithmetic is useful, but the operations are not identical.\n\n**Partition redistribution:**\n\n\\[\n(a,b)\\mapsto(a+\\delta,b-\\delta),\\qquad a+b\\text{ unchanged}.\n\\]\n\nThe principal instance is:\n\n\\[\n(430,605)\\mapsto(460,575),\\qquad\\delta=30.\n\\]\n\n**Common chronological translation across the BC/AD axis:**\n\n\\[\nS=B+A-1,\n\\]\n\n\\[\n(B,A)\\mapsto(B+\\delta,A-\\delta),\\qquad S\\text{ unchanged}.\n\\]\n\nThe numerical signs are opposite because `B` and `A` are civil-era magnitudes. This can describe both endpoints moving earlier together; it is not necessarily opposite movement in chronological time.\n\n**Bilateral outward extension:**\n\n\\[\n(B,A)\\mapsto(B+\\delta,A+\\delta),\\qquad S\\mapsto S+2\\delta.\n\\]\n\nThe first LXX native-Cainan Mirror extension uses `δ=460`, hence gain `920`. Its second symmetric completion uses `δ=500`, hence gain `1000`.\n\nKeeping these distinct preserves the genuine relations without turning every matching sum into the same geometric operation.\n\n## 59. The smallest useful operator vocabulary reached so far\n\nA compact working vocabulary is:\n\n\\[\n\\boxed{5,\\ 23,\\ 30,\\ 40,\\ 120,\\ 130,\\ 215,\\ 430,\\ 460}\n\\]\n\nwith the declared derived values:\n\n\\[\n115=5\\times23,\\quad125=5\\times25,\n\\quad345=3\\times115,\\quad575=5\\times115,\n\\]\n\n\\[\n690=6\\times115,\\quad1035=9\\times115,\n\\quad1150=10\\times115,\n\\]\n\n\\[\n1380=12\\times115,\\quad1495=13\\times115.\n\\]\n\nThe principal scalar operations remain:\n\n\\[\n\\boxed{K_P(x)=\\frac{25}{23}x,\n\\quad K_H(x)=\\frac{70}{69}x,\n\\quad K_E(x)=\\frac{300}{299}x.}\n\\]\n\nTheir gain laws are:\n\n\\[\nK_P(23m)-23m=2m,\n\\]\n\n\\[\nK_H(69m)-69m=m,\n\\]\n\n\\[\nK_E(299m)-299m=m.\n\\]\n\nThese laws explain why `5`, `30`, `40`, `100`, `120`, `460`, `920`, and `1000` recur as gains once the relevant source spans are specified. They are a compression of the observed architecture, not yet a demonstrated unique minimal generating set.\n\n## 60. Candidate minimal equations across Reports I–IV\n\nThe most useful current shortlist is:\n\n### A. The inherited regular macro-engine\n\n\\[\n\\boxed{1250=950+300,\\qquad650=950-300.}\n\\]\n\n### B. The rounded warning/terminal hinge\n\n\\[\n\\boxed{115+5=120,\\qquad120+5=125,\\qquad115\\frac{25}{23}=125.}\n\\]\n\n### C. The Apparent-Age/Cainan transfer\n\n\\[\n\\boxed{430+30=460,\\quad605-30=575,\n\\quad430+605=460+575=1035.}\n\\]\n\n### D. The two Creation-cell presentations\n\n\\[\n\\boxed{430\\mid30\\mid430,\n\\qquad460\\mid575\\mid460.}\n\\]\n\nThe MT–SP companion is `460|145|460`, with `145=115+30`.\n\n### E. The regular/cumulative common completion\n\n\\[\n\\boxed{12(115)\\frac{25}{23}=13(115)\\frac{300}{299}=1500.}\n\\]\n\n### F. The dated Noah–Moses `690` invariant\n\n\\[\n\\boxed{570+120=650+40=690.}\n\\]\n\nEach addend retains the dated construction in §18.\n\n### G. The cumulative aggregate and its actual correction\n\n\\[\n\\boxed{\\overline{\\Sigma}_c=460+\\frac23(L-S)=1150/1152.}\n\\]\n\nHere `L−S=1035/1038`, and these are alternate state means, not the arithmetic mean of `1150` and `1152`.\n\n### H. The native/generated Cainan Mirror junction\n\n\\[\n\\boxed{23(460)\\to25(460)=23(500)\\to25(500).}\n\\]\n\n### I. The decimal bridge\n\n\\[\n\\boxed{920/9200\\xrightarrow{25/23}1000/10000.}\n\\]\n\n### J. The SP favored-state wilderness rectangle\n\n\\[\n\\boxed{40(299-69)=40(300-70)=9200.}\n\\]\n\nIts coordinate execution is `13406/4206 → 1446/1406` with the two different exact Keys, as in §52.\n\n### K. The SP–BJ Jubilee and Mirror companion\n\n\\[\n\\boxed{350-7=343=7\\times49,\\qquad345\\frac{70}{69}=350,\\qquad8050=23\\times350.}\n\\]\n\nIts selected dated executions are in §57, including `4199−1406=57×49`. The compound `9200→10000`, followed by `+350=10350=23×450`, remains a lower-weight observation rather than a new core theorem.\n\nThese equations are not substitutes for the six tables. They are the current index into those tables and their permitted operations.\n\n## 61. What is most firmly established within the declared model\n\nThe rounded tables and their source-qualified actual comparators generate the stated offsets exactly. The cumulative Aaron/Moses branch supplies a genuine five-year paired state. The regular death method and cumulative lifespan method remain distinct. The `115` unit is the exact mod-5/23 least common multiple, and its `115→125` Key action explains the repeated gains.\n\nThe Cainan-OFF/native LXX Mirror executions converge exactly at `11500`. The `1380` and `1495` widths complete to `1500` by different operators. The cumulative overlapping comparison-cell means are exactly `1150/1152`; their overlap accounting explains the `+2` change. The SP `−215` configuration generates a three-point `9200` bridge consistent with an existing source theorem, while its regular and cumulative Exodus-directed spans independently expand by the same `40` to Conquest.\n\nThese are the principal arithmetic results. Their theological meaning and their status as intentional or providential architecture are interpretive conclusions built upon them, not part of the arithmetic verification itself.\n\n## 62. What should remain open\n\nThe report does not settle a unique smallest equation system for the entire repository. It also does not show that every repeated number is an independent witness, that every Key may be applied iteratively, or that generated Christ-side dates are all literal historical dates.\n\nThe `625` half-hub has additional finite-scale support, but no transmitted midpoint event has been supplied. The exact Eber `464/465` contraction comparison remains worth further investigation. The regular `390|2110|390` and cumulative `3450/3456` aggregates remain high-priority synthesis observations, with their overlap algebra now explicit. Seth's role in the regular flanks and the larger meaning of the central `30` deserve further study without assuming a conclusion.\n\nThe `441×23` SP construction is arithmetically sound as a generated `21×483→21×490` chain, but its complete source-state relation to `1150/1152` has not been established. The later `−215` bridge is the preferred explanation actually reached in this discussion; it does not prove the earlier lead false.\n\nThe actual Christ-envelope `5+1+1+5+1=13` field was referenced, but not fully rebuilt here. The broader File_51b/52 inverse-number and Mirror architecture remains outside this report's proof scope. No statistical significance test was performed for the new comparative patterns.\n\n## 63. Corrections and safeguards that must survive into the next chat\n\n| Issue from the discussion | Retained handling |\n|---|---|\n| Initial “unresolved rounded envelope” statement | The ordinary cumulative rounded `+5/base` branch is resolved by Aaron `125` versus Moses `120`. Full exact-phase actual mappings remain separately controlled. |\n| “MT/LXX regular Creation dates stay unchanged” | Their inter-manuscript gap stays unchanged. Standard actual Year-7 endpoints are `+8` relative to lower rounded regular members; cumulative MT/LXX spine heads truly remain stationary. |\n| `9100` called `35×364` | Correct is `25×364`; `35×364=12740`. |\n| `123→125` assigned to Levi | It belongs to Aaron. Levi is `137→135`. |\n| `13406` called the lower rounded SP Creation member | It is the upper member of `13406/13401`. |\n| `570+120` treated as a single contained Flood-row span | Use the explicitly dated warning/Flood diagonal comparison in §18. |\n| `4836−2576=2500` | Correct is `2260`. |\n| `4716−3226=1610` | Correct is `1490`; `4836−3226=4716−3106=1610`. |\n| `13455×70/69=195` | The output is `13650`; `195` is the gain. |\n| `1150/1152` called their own mean | They are two state-specific means. Their arithmetic midpoint is `1151`. |\n| `1152=72×2×2×2` | Four factors of two are needed: `1152=72×2⁴`. |\n| `+30` treated as a propagated change to all patriarchs | Apparent Age is local to the Adam/Creation overlay; uniform translations of comparison constructions must be declared separately. |\n| `430+30+430` and `460+575+460` treated as disjoint additive chronology | They are overlapping four-state comparison cells. |\n| One flank expansion treated as whole-span expansion | State which endpoint is fixed and whether one flank, two flanks, or the complete span is transformed. |\n| `10580→11500→12500` treated as merely an unsupported second iteration | Preserve both provenances of `11500`, and the `23²` structure; do not generalize this license to arbitrary inputs. |\n| SP `2891` called simply the actual Flood date | Retain the raw `2892–2891` Flood-year distinction, post-Flood Arphaxad seam, and rounded `−215` Flood state. |\n| `4711/4716` called one rounded Moses/Aaron pair | These are actual/rounded comparison members. The rounded SP Flood pair is `4721/4716`. |\n| Literal `120` countdown or literal Joshua `1471` assumed | The countdown is the adopted Genesis 6:3 reading; `1471` is a generated rounded companion. |\n| Mirror “inverse” confused with number reversal | Civil Mirror, same-side complements, span expansion, and decimal digit reversal remain different operations. |\n| “Maximum” taken globally across every variant | The `1380/1495` maxima are local to the stated Creation comparison fields, with other variants held fixed. |\n\n## 64. Restart instructions for the next Project chat\n\n**Research objective:** Use Reports I–IV as the concentrated evidence ledger for reducing the Unified Chronology's apparent complexity to its simplest informative equations. The wider repository remains the source of exact data and operator controls, not material to be replaced by the reports.\n\n**Current working position:** The rounded `115`/five-year structure, the Apparent-Age/Cainan transfer, the native/generated LXX Mirror, and the SP `−215` / `9200` dual-expansion rectangle are now established as the main directions of inquiry. The SP is not to be forced back onto the leveled plane for every internal calculation.\n\n**Discussion method:** Proceed a little at a time with medium-length responses. Check arithmetic, identify states and endpoints, distinguish a gain from an output, and retain worthwhile unresolved observations without repeatedly expanding the entire repository. Do not run ahead into a new full synthesis until the author opens it.\n\n**September 5 addition, corrected:** Preserve §57's SP–BJ `30`, `350/343/344`, `345→350`, `57×49`, and Mirror `8050=23×350` relations. BJ initial Creation is `3863–3856 BC`, giving a uniform `343` translation from SP `4206–4199 BC`. The `344/343` distinction remains separately at Flood/Ark. Direct SP dependence and deliberate Ark-year compensation remain interpretive proposals; `10350=23×450` remains exploratory. This addition does not change the six tables or §56's distinct Flood states.\n\n**Author's standing clarification preference:** If a supplied figure or statement looks inconsistent with the logically expected result or apparent intent, flag the discrepancy and ask what was intended before building an explanation around it. Do not assume an unusual counting convention when an arithmetic or transcription error may explain the discrepancy.\n\n**Source hierarchy:** Consult the six tables and state register first. Use R1–R3 for the inherited regular and actual-cumulative arguments. Consult F18/F51a for data and rounding, F09/F11/F22 for cumulative/phase bridges, and the appropriate related files for broader interpretation. Do not modify canonical files or controls merely because this discussion report contains a new observation.\n\n**Last reached SP construction:**\n\n\\[\n4711\\to2891=1820=5\\times364,\n\\quad4716\\to2891=1825=5\\times365,\n\\]\n\n\\[\n4711\\to2991=1720=4\\times430,\n\\quad4716\\to2991=1725=15\\times115,\n\\]\n\n\\[\n4716\\to3106=1610,\n\\quad1610+215-5=1820.\n\\]\n\nThis sits beside the stronger Creation-side invariant:\n\n\\[\n\\boxed{13406-4206=11960-2760=12000-2800=9200.}\n\\]\n\n**The central methodological conclusion:**\n\n> The leveled chronologies reveal common structure; native/favored configurations reveal how each manuscript participates in it. Small phase differences are to be tracked, not erased. The aim is to discover which few operations genuinely generate the repeated structure, rather than merely to collect more matching numbers.\n\n\n---\n\n# Appendix A — The six rounded Creation–Flood tables\n\nThese tables preserve the opening six-table exercise. The regular tables use rounded begetting ages and the separate remaining-years method for the theoretical death layer. The cumulative tables use whole-lifespan rounding and display the lower Nisan spine; add the declared Aaron `+5` branch to obtain the upper cumulative member. Dates are BC. The full table source states are in §§2–8.\n\n\n## A.1. MT regular rounded\n\nCreation boundary block: **`4111/4106 BC`**. \n\n| Patriarch | Begetting age, raw → rounded | Rounded birth | Rounded lifespan, remaining-years method | Rounded terminal |\n|---|---:|---:|---:|---:|\n| Adam | `130→130` | `4106` | `930` | `3176` |\n| Seth | `105→105` | `3976` | `910` | `3066` |\n| Enosh | `90→90` | `3871` | `905` | `2966` |\n| Kenan | `70→70` | `3781` | `910` | `2871` |\n| Mahalalel | `65→65` | `3711` | `895` | `2816` |\n| Jared | `162→160` | `3646` | `960` | `2686` |\n| Enoch | `65→65` | `3486` | `365` | `3121 (translation/close)` |\n| Methuselah | `187→185` | `3421` | `965` | `2456` |\n| Lamech | `182→180` | `3236` | `775` | `2461` |\n| Noah | `500→500` | `3056` | `950` | `2106` |\n| Shem | `100→100` | `2556` | `600` | `1956` |\n\nFlood / Arphaxad comparison seam: **`2456 BC`**. Noah birth to Flood is `600`; Flood to Noah death is `350`.\n\n\n## A.2. SP regular rounded\n\nCreation boundary block: **`4416/4411 BC`**. \n\n| Patriarch | Begetting age, raw → rounded | Rounded birth | Rounded lifespan, remaining-years method | Rounded terminal |\n|---|---:|---:|---:|---:|\n| Adam | `130→130` | `4411` | `930` | `3481` |\n| Seth | `105→105` | `4281` | `910` | `3371` |\n| Enosh | `90→90` | `4176` | `905` | `3271` |\n| Kenan | `70→70` | `4086` | `910` | `3176` |\n| Mahalalel | `65→65` | `4016` | `895` | `3121` |\n| Jared | `62→60` | `3951` | `845` | `3106` |\n| Enoch | `65→65` | `3891` | `365` | `3526 (translation/close)` |\n| Methuselah | `67→65` | `3826` | `720` | `3106` |\n| Lamech | `53→55` | `3761` | `655` | `3106` |\n| Noah | `502→500` | `3706` | `950` | `2756` |\n| Shem | `100→100` | `3206` | `600` | `2606` |\n\nFlood / Arphaxad comparison seam: **`3106 BC`**. Noah birth to Flood is `600`; Flood to Noah death is `350`.\n\nJared, Methuselah, and Lamech all terminate at rounded Flood `3106 BC`. The optional local `4406` Creation alignment in F51a is not the adopted transmitted `4411` state and is not used to regenerate this table.\n\n\n## A.3. LXX regular rounded\n\nCreation boundary block: **`5491/5486 BC`**. Native second Cainan is retained.\n\n| Patriarch | Begetting age, raw → rounded | Rounded birth | Rounded lifespan, remaining-years method | Rounded terminal |\n|---|---:|---:|---:|---:|\n| Adam | `230→230` | `5486` | `930` | `4556` |\n| Seth | `205→205` | `5256` | `910` | `4346` |\n| Enosh | `190→190` | `5051` | `905` | `4146` |\n| Kenan | `170→170` | `4861` | `910` | `3951` |\n| Mahalalel | `165→165` | `4691` | `895` | `3796` |\n| Jared | `162→160` | `4526` | `960` | `3566` |\n| Enoch | `165→165` | `4366` | `365` | `4001 (translation/close)` |\n| Methuselah | `187→185` | `4201` | `965` | `3236` |\n| Lamech | `182→180` | `4016` | `775` | `3241` |\n| Noah | `500→500` | `3836` | `950` | `2886` |\n| Shem | `100→100` | `3336` | `600` | `2736` |\n\nFlood / Arphaxad comparison seam: **`3236 BC`**. Noah birth to Flood is `600`; Flood to Noah death is `350`.\n\nSubtract `130` from the Creation-through-Arphaxad coordinates to obtain the common Cainan-OFF regular comparison used in the main report. Regular Lamech here uses `182/777`, not the cumulative `753` chain input.\n\n\n## A.4. MT cumulative rounded\n\n| Cumulative node | Whole lifespan, raw → rounded | Actual Nisan | Rounded Nisan | Rounded − actual |\n|---|---:|---:|---:|---:|\n| Adam / Creation | `930→930` | `14006` | `14006` | `0` |\n| Seth | `912→910` | `13076` | `13076` | `0` |\n| Enosh | `905→905` | `12164` | `12166` | `+2` |\n| Kenan | `910→910` | `11259` | `11261` | `+2` |\n| Mahalalel | `895→895` | `10349` | `10351` | `+2` |\n| Jared | `962→960` | `9454` | `9456` | `+2` |\n| Enoch | `365→365` | `8492` | `8496` | `+4` |\n| Methuselah | `969→970` | `8127` | `8131` | `+4` |\n| Lamech | `777→775` | `7158` | `7161` | `+3` |\n| Noah | `950→950` | `6381` | `6386` | `+5` |\n| Shem | `600→600` | `5431` | `5436` | `+5` |\n| Arphaxad / Flood seam | `438→440` | `4831` | `4836` | `+5` |\n\nTerminal anchor: Moses death `1406 BC`. Total lifespan sum stays `12600`; cumulative Creation is `14006` in both actual and rounded. The regular Methuselah `965` must not replace cumulative `970`.\n\n\n## A.5. SP cumulative rounded\n\n| Cumulative node | Whole lifespan, raw → rounded | Actual Nisan | Rounded Nisan | Rounded − actual |\n|---|---:|---:|---:|---:|\n| Adam / Creation | `930→930` | `13398` | `13401` | `+3` |\n| Seth | `912→910` | `12468` | `12471` | `+3` |\n| Enosh | `905→905` | `11556` | `11561` | `+5` |\n| Kenan | `910→910` | `10651` | `10656` | `+5` |\n| Mahalalel | `895→895` | `9741` | `9746` | `+5` |\n| Jared | `847→845` | `8846` | `8851` | `+5` |\n| Enoch | `365→365` | `7999` | `8006` | `+7` |\n| Methuselah | `720→720` | `7634` | `7641` | `+7` |\n| Lamech | `653→655` | `6914` | `6921` | `+7` |\n| Noah | `950→950` | `6261` | `6266` | `+5` |\n| Shem | `600→600` | `5311` | `5316` | `+5` |\n| Arphaxad / Flood seam | `438→440` | `4711` | `4716` | `+5` |\n\nTerminal anchor: Joshua death `1296 BC`, after including Joshua `110`. Total lifespan sum changes from `12102` to `12105`, giving Creation `13398→13401`.\n\n\n## A.6. LXX cumulative rounded\n\n| Cumulative node | Whole lifespan, raw → rounded | Actual Nisan | Rounded Nisan | Rounded − actual |\n|---|---:|---:|---:|---:|\n| Adam / Creation | `930→930` | `14896` | `14896` | `0` |\n| Seth | `912→910` | `13966` | `13966` | `0` |\n| Enosh | `905→905` | `13054` | `13056` | `+2` |\n| Kenan | `910→910` | `12149` | `12151` | `+2` |\n| Mahalalel | `895→895` | `11239` | `11241` | `+2` |\n| Jared | `962→960` | `10344` | `10346` | `+2` |\n| Enoch | `365→365` | `9382` | `9386` | `+4` |\n| Methuselah | `969→970` | `9017` | `9021` | `+4` |\n| Lamech | `753→755` | `8048` | `8051` | `+3` |\n| Noah | `950→950` | `7295` | `7296` | `+1` |\n| Shem | `600→600` | `6345` | `6346` | `+1` |\n| Arphaxad / Flood seam | `465→465` | `5745` | `5746` | `+1` |\n\nTerminal anchor: Moses death `1406 BC`. Native Cainan `460` is retained in the full chain. Total lifespan sum stays `13490`, giving Creation `14896` in both actual and rounded. Subtract `460` from Creation through Arphaxad to level Cainan OFF.\n\n\n## A.7. Full cumulative input-chain register\n\nThis compact ledger makes the reconstructed SP/LXX tables reproducible. Round each **whole lifespan** to the nearest multiple of five, sum suffixes backward from the stated terminal anchor, and then apply a separately declared Aaron `+5` rail. `—` means that the name is absent from that native chain, not a lifespan of zero.\n\n| Patriarch | MT actual lifespan | SP actual lifespan | LXX actual lifespan |\n|---|---:|---:|---:|\n| Adam | `930` | `930` | `930` |\n| Seth | `912` | `912` | `912` |\n| Enosh | `905` | `905` | `905` |\n| Kenan | `910` | `910` | `910` |\n| Mahalalel | `895` | `895` | `895` |\n| Jared | `962` | `847` | `962` |\n| Enoch | `365` | `365` | `365` |\n| Methuselah | `969` | `720` | `969` |\n| Lamech | `777` | `653` | `753` |\n| Noah | `950` | `950` | `950` |\n| Shem | `600` | `600` | `600` |\n| Arphaxad | `438` | `438` | `465` |\n| Cainan II | — | — | `460` |\n| Shelah | `433` | `433` | `460` |\n| Eber | `464` | `404` | `504` |\n| Peleg | `239` | `239` | `339` |\n| Reu | `239` | `239` | `339` |\n| Serug | `230` | `230` | `330` |\n| Nahor | `148` | `148` | `208` |\n| Terah | `205` | `145` | `205` |\n| Abraham | `175` | `175` | `175` |\n| Isaac | `180` | `180` | `180` |\n| Jacob | `147` | `147` | `147` |\n| Levi | `137` | `137` | `137` |\n| Kohath | `133` | `133` | `133` |\n| Amram | `137` | `137` | `137` |\n| Moses | `120` | `120` | `120` |\n| Joshua | — | `110` | — |\n\nLXX cumulative Lamech `753` is the explicit repository chain input; the regular `777` and cumulative row-local `777` overlay are not propagated into this register. The input ledger and all six calculated tables were re-executed during preparation of this report. This is a bounded arithmetic check, not a new pressure test of the repository sources.\n\n---\n\n# Appendix B — Preparation and verification scope\n\nThe report was assembled from the present discussion, the three attached parent reports, and the mounted repository sources identified in §1. It preserves the author's sequence of discoveries while grouping related equations, marking exploratory leads, and recording the corrections made during the discussion. The explicit overlap reductions in §§42–43 and the endpoint realization in §30 are labeled audit-level clarifications of existing calculations, not new source facts.\n\nThe six rounded tables were reconstructed from the declared begetting-age and lifespan inputs. The numerical check covered **293 executable adjacent-equality comparisons** extracted from the report and **81 separately specified endpoint, ratio, and paired-chain checks**. All passed in exact rational arithmetic, with an additional SymPy re-execution of those same checks. The table-frame check found all **20 Markdown tables** structurally consistent; display-math delimiters are balanced and section headings are unique.\n\nThese are bounded preparation checks. They do not establish the independence, historical intentionality, statistical rarity, or theological interpretation of the patterns, and do not constitute a new full pressure test of any repository file. No canonical source or control file was revised.\n\n**September 5 revision scope:** Added the author-supplied SP–BJ comparison in §57 and updated the heading, source register, equation shortlist, and restart instructions. The preceding preparation-check counts describe the September 4 baseline, not a fresh execution of the full original audit. The new calculations were checked separately, with explicit endpoint, Mirror-crossing, and inference guards. No canonical repository source or control file was revised.\n\n**End of Report IV.**\n"}