# What the common grammar retains

The compact account reuses rules across many outputs. Its input cost includes source choices and placement, not only the numerals in a formula.

| Family | Retained source information | Generated together |
|---|---|---|
| Ordinary genealogies | Selected rows, order, named change supports, terminal and count convention | Complete Regular/Cumulative differences and interior ladders |
| Regular variant field | Terah, Cainan and Sojourn choices; applicable support | Eight offsets and their consecutive gaps |
| SP cap | Native births, baseline lives, inclusive capacity rule | Nine capped outputs and located reduction 488 |
| SP rectangle | The source fields, standard frames, −215 state, week/decade openings and endpoints | Two equal Key gains, retained 9200 and Priestly 10000 |
| Moses family | Selected MT path, late event edges, Moses 1526 and named cuts | Twenty-five coordinates, five blocks and related cycle spans |
| Inverse/Keys | Original component words, retained-zero policy, appointed factor and pivot | Whole transformed paths, shared totals and exact rational prefixes |
| Covenant | Local ages and offsets, lifespans, terminal, state and phase | Regular field, stack, clutch columns and forced/conditional joins |
| Counts/NT | Category labels, source order, templates, named slots, scale and grouping | Recoverable registers, comparisons and permitted symmetries |

The five-amplitude row model describes differences rather than replacing all source rows. The century amplitude is derived from those paired rows. Similarly, measured Covenant gaps and cumulative prefixes are consequences once their input path is given.

Residuals and clipped excesses have a different role: they retain information discarded by a chosen representation. Keeping them permits exact recovery, but does not count as generating that information from the rounded or capped output.

No numerical description-cost weights were fixed for this study. The result is therefore a transparent structural compression, not a globally minimal model or a statistical ranking of possible source histories.

## Claim, source and status table

The results distinguish retained source premises, verified calculations, conditional mechanisms and interpretive comparisons. A conditional mechanism explains the complete selected family from its named inputs; it does not establish how those inputs originated.

| Family | What is established | Status | Principal source |
|---|---|---|---|
| SP ages and cap | Source age relations and located capacity losses explain the full nine-row cap and its 115/249/124 reductions. | Conditional mechanism | File 18 §§2.1.3,3.1–3.1.3; File 51a §8.3 |
| Regular/Cumulative rows | Different preserved row components generate complete fields; native Cainan is measured as 130 or 460. | Exact source reconstruction | File 18 §§1.6,2–4,4A |
| Rounded residuals | Located residuals explain exact source changes and recover the full paths when retained. | Operation relation and source application | File 51a §§3.1,3.4,16; File 18 §3.1 |
| SP rectangle | The row/cap mechanisms feed the established 2760/11960 equal-gain construction after the declared openings. | Provenance of an existing comparison | Strategy §4.3; File 18 §§3,6C |
| Moses path | Five named blocks explain the connected 430/490 spans and the composite 130|800 comparison. | Complete calculation; typology separate | File 51a §§3.1,7A |
| Regular kernel | The 1250/650 row field and 405 variant field meet through the stated two-condition calibration. | Source construction plus observed calibration | Strategy §4.1; File 18 §§1–4; File 07 §2.11.1a |
| NT interface | A complete 22-name birth comparison exposes what shared births preserve and differing lives leave unseen. | Integration of supplied fields | File 43 §3.4; File 18 §§2–4; File 54 §§1–2 |
| Inverse interface | Three original words reconstructed from the genealogy enter the inherited single-reversal paths. | Verified interface; completion inherited | Latest File 52c §§2.1,3.3–3.4 |

Keep the source choices alongside the relevant claim: SP count bindings and baseline lives; named row supports and native Cainan; the quantity rounded; the rectangle’s frame and openings; Moses’ selected event path; the kernel’s two calibration conditions; NT names and metric; and File 52c’s original cuts and anchors.

The main interpretive example is Arphaxad as a “second Adam.” The calculation establishes a 130|800 composite path equal to Adam’s two biography components. The typological identification remains the source’s reading. Likewise, the midpoint 950 may be compared with Noah’s lifespan without becoming an independent derivation of the 1250/650 pair.

The new source mechanism or interface is stated separately from inherited results. Re-expressing a known endpoint, constructing an operation register or reproducing a dependent total does not create an additional witness. The JSON companion supplies precise retained premises, chronological scope, step references and source hashes for editorial integration.