SP source relations: what can constrain the inputs left by the cap?
Immutable preparation inventory for C1132–C1431. No research actions are numbered or executed here. All material is from the supplied local sources; source hashes, exact excerpts, line locators, and exhaustive control-token search results are in sp_source_relations.json.
The full frozen Strategy was reread. Its principal demand is to explain local source quantities and their placement before pursuing another global endpoint. The candidate relations below are already present in the repository; none is presented as a newly discovered numeral or equation.
Best next source constraints
The equal 65 flanks and half-Adam relation are the strongest local continuation. File51a §8.3 explicitly states that Mahalalel and Enoch both beget at 65 and that 65 is half Adam's 130. Hold Adam130 and these named relations: Mahalalel65 and Enoch65 follow, and the already-established SP cap sum Jared+Enoch127 then fixes Jared62. The relation supplies the premise missing from the cap inverse; it is not generated by the cap itself. The source's MT162/Watcher interpretation remains distinct from the unchanged named age values that transfer to SP.
The first-five460 constraint adds one useful total, not five ages. File47 §8.4 prints 130+105+90+70+65=460. File20 §7A.10 supplies the Adam-to-Jared bookend and expressly relates it to SP data. With Adam130 and Mahalalel65 held, this reduces the unresolved early part to Seth+Enosh+Kenan265. No relation located in the relevant searched passages uniquely selects105/90/70 from that sum. Their actual named rows remain input unless a further source-controlled condition is found. The BJ appearance is not independent historical evidence for values the source says it transfers from SP.
Methuselah's ancestral969 has an explicit local path condition. File18 §4A.2 prints 187+182+600=969 and states that Methuselah dies at the Flood. Hold those three source intervals and the death-at-Flood alignment: the ancestral lifespan is determined, and comparison with SP720 then yields249. The alignment is an additional source premise relative to the cap. It is not proof that the discarded249 could be recovered from the clipped output alone.
Lamech offers two differently qualified routes. File18 §4A.4 explicitly relates MT777, main LXX753, and SP653 through24 and100, with the24 matched to the1656 Prophetic gain. The source calls these structural support, not historical production. File17 §§8.1 and12.1 also supplies 483+777=1260, explicitly marked inferred. If its two prophetic measures and its identification of the complement with Lamech are held,777 follows. That identification is a substantive extra premise; arbitrary decompositions of1260 do not select Lamech automatically.
Candidate roles at a glance
| Candidate | What it can constrain | What remains supplied |
|---|---|---|
| MT/SP identical early rows | Five SP values from corresponding MT rows | The five inherited magnitudes and names |
| LXX centenary repartition | SP/MT ages from LXX ages minus100 | LXX input ages and declared correspondence |
| First-five460 | One early-prefix sum | Its internal age distribution |
| Equal65 flanks, half Adam | Mahalalel and Enoch; then Jared from cap sum127 | Adam130 and the named source relation |
| Life minus remaining years | Complete first-five ages and Enoch65 | Both component ledgers; this is re-expression |
| Jared162+800 | Baseline962, then excess115 | Jared162 and remainder800 |
| Methuselah187+182+600 | Baseline969, then excess249 | The three ages and Flood-death alignment |
| MT Lamech182+595 | 777, then excess124 | The two biography components |
| Inferred483+777=1260 | 777 as the selected complement | 483,1260, and the source's Lamech bridge |
| Lamech24/100 relations | Conditional recovery of753;124=24+100 | Source-selected comparisons;753 remains attested input |
| First-five multiples of5 | Zero rounding residual | Exact magnitudes remain undetermined |
Given values versus derivation
The Actual MT remaining-year column in File51a §3.4 gives800,807,815,840,830 for the first five rows and300 for Enoch. Subtracting them from930,912,905,910,895 and365 recovers the begetting ages exactly. This is an alternative encoding of source biographies, not a reduction in the number of numerical premises. The same distinction applies to Jared162+800 and Lamech182+595. Rounded960 cannot uniquely recover exact962.
File18 treats753 as an attested LXX lifespan. Restored182 paired with it gives calculated571;571 is not an independently supplied number that can then explain753. The relations24 and100 are existing structural support for the selected lifespan. Their sum124 is dependent, and the cap cannot select MT777 versus LXX753 because both inputs collapse to SP653.
All cap states remain native G2. Lamech's53rd counted binding is52 completed units; the three cap lives remain847/720/653 inclusive with selected deaths at Flood start. Main LXX remains182/753,777 appendix-only, and188 nonoperative. File20's older exclusive frame and File53's optional Enoch364 compression are recorded as distinct material, not activated in this reconstruction. No primers, second decimal inversion, upstream Gear transport, source edits, or remote endpoint search were used.
Novelty and source audit
Project Procedures §18 was read. Entire current Restart Capsule, State Vocabulary Register, and Style Guide texts were mechanically searched for every candidate numeral, BC/AD form, and explicit relation string; the JSON retains all matching line numbers and source hashes. This inventory therefore classifies material as source-given, explicit existing relation, source-inferred relation, or derived consequence. It makes no repository-wide absence claim from a failed excerpt search.
The useful next comparison is between the cost of holding a number directly and the cost of holding the source relations that identify it. The present sources permit a sharper conditional model, especially through the local65 relation and Methuselah's Flood alignment. They do not yet supply a complete origin theory for the first-five age tuple or all discarded lifespan excesses.