Finite variant configurations and their chronological representations

Prepared for C732–C831; no numbered research step has been executed here. Sources are local supplied files. The accompanying variant_field_literal_inputs.json contains 39 literal tables, 310 table rows, and 19 source extracts with line locations and source hashes. It contains no calculated row presented as a source datum.

Strongest new question

Can the same finite source choices be realized coherently in regular, cumulative, and Rounded chronology, once their different boundary domains are retained? The new work should explain why apparently identical variant names sometimes have different effects across modes, and when rounding genuinely commutes with those choices. Repeating the familiar 60|155|60 or 60|70|60 geometry is not needed.

Earlier work already established the local projection matrix (b,r) -> (b,b+r) (C487), Cainan's 130/460 vector and 330 bridge effect (C488), the native source differences (C483–491), aggregate rounding residuals (C492–494), and local Noah/Shem 60|70|60 geometry (C656). The proposed new contribution is the whole finite configuration, its boundary restriction maps, its mode-dependent source lifts, and its rounding interchange law.

1. The regular configuration is a nested support construction

Use source order and these three bits: s = minimum/full430 Sojourn position; t = baseline/+60 Terah birth position; c = second Cainan OFF/ON. The supported parameter values are only 0/1, not unrestricted repeated additions. Source: File18 §§1.3.1–1.3.2, 1.4, 1.6.

For a common retained birth row n, define masks:

Cainan's own row exists only when ON and has masks S=T=1, H=0. It has four positions, never eight. Kohath and Amram have no direct regular birth coordinates in this construction; do not manufacture a zero coordinate or call the fixed Moses row a zero-state object.

For each tradition's native minimum birth table, let c_native=0 for MT/SP and 1 for LXX. Then the complete retained birth coordinate is

R_tau(n;s,t,c) = base_tau(n) + 215*s*S(n) + 60*t*T(n) + 130*(c-c_native)*H(n).

This formula is a reconstruction proposal from the source's explicitly stated domains. It applies no new Gear operation. Native LXX is evaluated at c=1; c=0 is the explicit theoretical removal comparison. SP is native215 at s=0; its reports' equalized full430 frame is s=1. Standard MT is s=1 but MT minimum is s=0. Keep source-table Noah/Shem/Flood node bindings fixed.

This yields a family of regular birth configurations, with eight distinguishable placements upstream, four in the middle, two downstream, and one fixed terminal placement. The source's 0 direct pivots for Kohath/Amram means unassigned, not one fixed placement. The 12–8–4–2–0 narrative count ladder remains a separate source interpretation.

Possible exact tests: recover every literal MT Terah/Cainan example; recover native and equalized SP heads without mixing them; recover native LXX rather than inserting a second Cainan; check the entire retained-row difference vector is unchanged by matched common settings. Do not treat a remembered global '+1380' as applying to every LXX row: actual differences are row-dependent.

2. A finite action is not unbounded translation

On the binary configuration labels the three switches commute and square to identity. In numeric space switching s changes a coordinate by 215*(1-2s)*S, not by an unconditional +215. Analogous statements hold for t and c. The inverse transition removes the same admitted variant; it is not the deferred decimal inverse-of-inverse operation.

The nested masks give source-controlled restriction maps from the full configuration to a given genealogical block. Their kernels explain why a downstream interval can remain fixed while an upstream comparison changes. The common birth field is a finite configuration set; avoid calling it a finite field.

For any ordered same-side pair i,j, the span response is

delta(R_i-R_j)=215*ds*(S_i-S_j)+60*dt*(T_i-T_j)+130*dc*(H_i-H_j).

Consequently an internal interval in one block is invariant under every common active switch. A cross-boundary interval detects precisely the hinges separating its endpoints. This is a compact source-to-span incidence law, not a hunt through arbitrary boundaries. One wholefamily test should suffice.

3. The strongest cross-mode example: the Terah source triangle

File18 MT/LXX minimum gives Terah (begetting70, lifespan205), while MT/LXX +60 gives (begetting130, lifespan205), holding Abraham's selected birth fixed. Thus the compatible row anatomy is

MT/LXX baseline (b,r)=(70,135) -> alternative (130,75).

Its projected effects are (delta_regular,delta_cumulative)=(60,0): the raw cumulative lifespan is unchanged.

SP supplies a different local starting row: textual Terah (70,145), hence (70,75). File18 §3.2.2 says textual145 and architectural205 both target Abraham1951 and Terahdeath1876, while retaining the SP +60 possibility. Holding those two targets in the architectural205 branch implies birth2081, begetting130, hence (130,75). The exact SP ideal biography is a derived compatible construction from that statement, not a separately printed raw row. Independent review by transfer_review confirmed this reading.

The three source-supported row states therefore form this local triangle:

The two-step route (0,+60) then (+60,-60) equals direct (+60,0) in row coordinates. The regular/cumulative projected changes are correspondingly (0,60)+(60,0)=(60,60).

With regular Abraham1951 held, the three Terah heads/deaths are 2021/1876, 2021/1816, 2081/1876. With cumulative Abraham2435 held, the heads are 2580,2640,2640. The first two cumulative heads are literal File18 §§6A,6C values; the third is only the row-evaluation diagnostic. Official SP cumulative chronology retains145. A regular +60 effect therefore does not uniquely determine its cumulative lift: the source row and lifespan policy matter.

This triangle is more informative than another scalar variant shape: it realizes the Strategy's shared row grammar through actual MT/LXX/SP source differences and two chronological projections, while explaining an important obstruction to a universal mode conversion.

4. Keep the secondary Rounded cumulative +60 operation distinct

File51a §23.7 explicitly prints Rounded cumulative Creation blocks 14011–14006 and 14071–14066. These are source-admitted secondary macro comparison blocks. They are not generated by reprojecting the raw MT Terah70→130 lifespan-preserving change. The latter has cumulative effect0; the former explicitly translates its displayed block by60.

This is a positive distinction between two legitimate typed operations, not a reason to reject either table. It is an excellent limited test of whether the proposed common grammar retains operation provenance. Do not silently extend the secondary block translation to a full cumulative genealogy or to SP145.

5. Cainan insertion has one support but two weights

File18 §1.6 applies regular130 and cumulative460 at the same hinge: Adam through Arphaxad moves, Shelah and all downstream rows stay fixed. Cainan itself is inserted between Arphaxad and Shelah, not shifted by an extra own insertion.

For the common cumulative rows,

C_tau(n;c)=baseC_tau(n)+460*(c-c_native)*H(n).

The Sojourn position is a regular placement parameter; it is not a new cumulative lifespan. MT/LXX Terah's fixed205 repartition is also forgotten by the lifespan projection. Thus one should not force the entire regular binary cube into eight distinct cumulative source executions. The cumulative source realization of those raw choices retains only the admitted Cainan dimension. The SP ideal145→205 diagnostic is a separately tagged row change, not an automatically admitted cumulative switch.

The new test should compare the support profiles and the resulting restriction/forgetting maps, rather than repeating the already known single endpoint330 bridge effect.

6. Rounding commutes with this specific finite configuration

Let Q5 denote nearest-five rounding of an integer source duration. Then Q5(z+5m)=Q5(z)+5m, and the residual Q5(z)-z is invariant under an admitted multiple-of-five row increment. Every present hinge amount60,130,215,460 is divisible by5.

Important: File51a does not round naked BC labels. It rounds begetting ages and evaluates the anchored regular genealogy; separately it rounds lifespans and evaluates the cumulative chain; its theoretical regular deaths round b and r separately. State the interchange law at those source-data operations, then derive coordinate equivariance with the relevant anchor fixed.

For MT/LXX Terah (70,135)->(130,75), every component is already a multiple of5; the lifespan remains205 before and after rounding. For SP textual/ideal (70,75)->(130,75), every component is again fixed by rounding, but lifespan changes145→205. Cainan's inserted regular130 and cumulative460 are each fixed by Q5. A supported Sojourn placement changes the fixed-span component by215, also preserved.

Therefore the entire existing coordinate-residual vector, not merely its sum, is identical in all paired admitted configurations. File51a §3.1 provides23 regular rows and §16.2 provides26 cumulative rows for a complete MT test. At a Cainan insertion, add a row whose local duration has zero rounding residual and keep the old retained rows' coordinate residuals. The new Cainan coordinate inherits the downstream Shelah coordinate residual: regular2421+130 gives2551 on both paired rails, whereas cumulative4393+460=4853 and4396+460=4856 retain a +3 coordinate residual. A partial source-projected span may change by130/460 while its Actual–Rounded residual does not.

This proves local equivariance on these particular multiple-of-five moves; it does not make rounding globally additive, invertible, or an affine map between regular and cumulative dates. No new statistical claim follows. Do not extrapolate the source table's strict no+2 comparison into new Gear transport.

7. A particularly useful wholefamily cross-tradition test

The same packet suggests a strong concrete holdout: SP with regular Cainan restored and native LXX share the entire retained Noah-to-Terah birth suffix, although their death and cumulative structures still differ.

At native/minimum s=0,t=0, take the supplied SP Noah G2 primary3493, Shem Arphaxad-bound2991, Flood start2893, and Arphaxad2891. Adding the admitted upstream Cainan130 gives3623/3121/3023/3021, exactly the literal LXX values. Inserted Cainan is2886=2756+130 in both. Shelah2756,Eber2626,Peleg2492,Reu2362,Serug2230,Nahor2100,Terah2021 are already equal and remain fixed under insertion. Flood closes likewise2892+130=3022. Do not substitute SP G2 primary Shem2993 for the declared Arphaxad-bound2991; that would compare different existing bindings.

This is more than one head equality: the complete ordered suffix agrees because the birth-driving post-Flood begetting sequence agrees once Cainan is present. Matched s/t settings then preserve that equality over the whole admitted configuration by the shared masks.

The death differences reveal what birth equality forgets. LXX-minus-SP lifespan differences on this leveled suffix are Arphaxad27, Cainan0, Shelah27, Eber100, Peleg100, Reu100, Serug100, Nahor60, Terah60; Noah/Shem differences0. Their sum is574. These are derived differences of literal source rows, not additional source dates. Where births coincide, a longer LXX lifespan moves its death forward by exactly that amount, so the corresponding BC death-label difference is the negative of the lifespan difference. The SP textualTerah145 branch remains official here.

In the cumulative representation, those same life differences accumulate rather than disappearing. The upstream pre-Noah life differences Jared115,Methuselah249,Lamech100 total464;574+464=1038, equal to the nativeLXX versus restoredSP cumulative Creation difference14896−(13398+460). The aggregate endpoint1038 is inherited from the earlier source comparison; the new contribution would be the complete birth-agreement/death-divergence/suffix-accumulation explanation. Native un-restoredSP remains a legitimate separate state.

Suggested execution: freeze the literal ordered birth/lifespan rows; test the entire aligned suffix with every node displayed; reconstruct its death discrepancy vector and cumulative running sums; finally test matched finite configurations. This is one coherent dependent family, not a count of independent node matches.

Suggested useful sequence, to reassess after each completed step

1. Freeze source rows/domains and the native settings, rather than only a set of increments.
2. Reconstruct MT's literal four Terah states and four inserted-Cainan rows from the masked source rule.
3. Reconstruct the entire common-row birth configuration for MT/SP/nativeLXX, retaining absent Cainan and unassigned Kohath/Amram.
4. Express its finite switches and boundary restriction law; test all supported settings at once.
5. Recover the Terah row triangle and its regular/cumulative projections using independently supplied local anchors.
6. Identify why the same regular+60 has different possible cumulative lifts; preserve officialSP145.
7. Separate File51a's secondary cumulative+60 display block from rawTerah projection.
8. Show Cainan's shared support/two weights and the mode-specific forgetting of regular placement parameters.
9. Prove Q5 multiple-of-five equivariance, then regenerate the entire23-row regular and26-row cumulative residual vectors under the admitted states.
10. Integrate the result as a constructive explanation of how family variants fit together: common state choices, different source projections, and compatible rounding where the local arithmetic allows it.

The source tuple, date-role choices and phase convention remain inputs. This preparation does not open either deferred primer,1486, additional Gear transport, decimal inverse-of-inverse, or canonical revisions.