A common representation of the chronological families

Preparation for Stage G; no root numbered action is adopted here. The complete 9,147-word C1131 synthesis, its technical companion, the existing C831/C931 operator schema, the entire Strategy and the completed models through C1251 were read. The proposal reuses their proofs. It introduces one bounded interface verification and one presentation check, reported in the companion diagnostics: 70/70 checks pass.

Recommendation

The common presentation should be a short network of source-labelled structures, selected measurements and declared operations. Its central object is not a naked date. A record retains source edition and state, named rows or vertices, incidence/order, measure and count convention, coordinate chart, anchor, event role, and whether a value is supplied or generated. An operation record adds its exact input domain, permitted source route and information retained or lost.

This already joins the families at useful interfaces. Regular and Cumulative chronology measure the same named rows differently; a source insertion changes the row structure; the SP cap makes selected life measures depend on a birth path; row rounding records those measures more coarsely; an exact residual lift carries source changes through rounding; admitted component lists feed one-pass reversal and anchored Keys. Covenant and NT/count constructions reuse ordered measurement and explicitly supplied comparison relations.

No further general operator survey is needed. The Stage G block can chiefly compress the completed work. The desired advance is a readable dependency map, not another name for an inherited equality.

The verified architecture

This is a common format for different constructions, not a claim that every row can pass through every operation.

1. Row changes and the two genealogy measurements

For ordinary compatible rows, (L=b+r). The proved normal form uses

Every compatible change has the unique coefficients (d=\Delta b,e=\Delta L). The formulas commute on unrestricted compatible coordinates. A route through nonnegative rows also needs its own intermediate inequalities; source permission is a further restriction. The stored (1,1,2) diagnostic with (d=2,e=1) makes the distinction concrete: R-first leaves the domain while A-first does not.

For a fixed ordered path, write (U_{ij}=1_{j\ge i}). Its Regular and Cumulative realizations are (a\mathbf1+Ub) and (a\mathbf1+UL). Therefore complete differences are (U\Delta b) and (U\Delta L), with any anchor changes added explicitly. Adjacent differences invert this measurement when the terminal is retained. Shared topology does not mean equal weights. Cainan contributes 130 to one measurement and 460 to the other; the inserted named vertex remains part of the object.

The 19-shared-row MT/LXX five-amplitude model and its four-amplitude restriction are already verified applications. The restriction (e+n=u) is a fitted source relation; neither rank count derives the supports or source values. Native Cainan raises the matched regular/cumulative differences 1250/430 to 1380/890. These are inherited conclusions, now placed in the common diagram. Sources: File18 §§1.6,2–4,6; C1152–1160,C1182–1189.

2. The cap and the Rounded residual lift

The SP branch first measures the birth path, forms the nine capacities (c=Ub+601\mathbf1), and then sets (y_i=\min(L_i,c_i)). It supplies a dependence between Regular inputs and Cumulative outputs. It does not identify discarded excess life. The exact reductions 115/249/124 total 488; the post-Flood 120 and source-appointed frame/opening choices enter separately in the rectangle construction.

Monotone rounding satisfies (Q(\min(L,c))=\min(Q(L),Q(c))). This square is valid for resolved counts on both routes. It neither rounds date labels first nor converts inclusive SP cap counts into ordinary biographies. The branch-specific loss register reconstructs the baseline only by retaining the missing residual or excess information. Sources: File18 §3.1.3; C953–983,C1202–1212,C1223–1229.

For an ordinary integer measurement, retain (y=Q(x)) and (\rho=x-y\in[-2,2]). The proved translation lift is

It obeys (T_eT_d=T_{d+e}) because each route encodes the same exact value. Source admission of intermediate states remains separate. Only shifts in (5\mathbb Z) act on every rounded cell without residuals. The full 105-measurement test, the Lamech/Arphaxad/Shelah cancellation and its interior field already verify the chronological application. This lossless encoding preserves information rather than reducing the number of source inputs. Sources: File51a §§3.4,16–17; C1190–1201,C1213–1222.

3. Row-rounded paths need their own aggregate residuals

If (w=y+\rho) row by row, the exact path is reconstructed from

Here (U\rho) is an accumulated residual and need not lie in ([-2,2]). Normalizing it back to a nearest-five cell changes the displayed Rounded path. The same distinction holds for a source block-sum matrix (B): its published row-rounded measurement is (By), with residual (B\rho), while once-rounding the exact block gives (By+Q_{\mathbb Z}(B\rho)).

The bounded five-block check makes that distinction visible without inventing a partition:

The source path totals 2580 after row rounding, with exact residual 6; rounding its exact 2586 total once gives 2585. All 25 MT coordinates reconstruct with their located residual sums. This is a presentation safeguard applying inherited arithmetic, not a new rounding theorem. It also explains why the internal 129|801 → 130|800 change can preserve 930. Sources: File51a §§3.1,7A; C1233–1241; preparation IF02.

4. The precise new interface to the inverse family

The Moses model supplies Rounded Creation 4106 and Noah 3056. File52c separately supplies Conquest 1406, Nativity 6 and the selected Flood 2456 role. Reading the admitted paths from Nativity toward Creation produces exactly:

All three match the frozen inherited inverse inputs. This certifies that the new source-row reconstruction can feed the old admitted operation without changing its domain. Its one-pass outputs and 14720 completion are reused, not recounted as discoveries.

Two role checks are essential. Arphaxad birth and selected Flood share 2456 but retain their supplied roles. The transformed first Conquest leg lands at 4106, yet is not the original Creation node. The cumulative 14006/4836 source path remains a separate inherited measurement. No bridge from SP capped rows to the MT inverse domain is thereby admitted. Sources: File52c §§2.1,3.3–3.4; C1058–1068,C1234; preparation IF01/IF03.

5. Keys, Covenant and counts fit at explicit interfaces

Exact uniform Keys commute with summing a path. Different component Keys pass through a block-sum operation for all inputs precisely when their factors are constant within each block. The inherited partial/uniform 483 example reaches total 490 by two different interiors. A Key's anchored action (a+k(x-a)), calendar unit and any intermediate integer-grid condition must accompany it. Common day-volume (336E=360P=364J) is a calibration, not a universal conversion between chronological families. Sources: File12 active-Key clarification; File63 §1.3,§9.7; C984–1007,C1058–1068.

The Covenant construction supplies a further coupled source graph. Its first clutch join is forced by the definition (d=C_K-D); the next uses the shared Levi lifespan, and the next uses the retained 147 condition. The 42-coordinate array (X(i,j,k)=b_i+jp-kh) has nine unrestricted parameters. These are inherited dependencies of one field, not separate matches needing new operators. Sources: Files60–62; C1032–1057,C1070.

NT and list paths likewise retain distinct measures and labels. The NT fork fixes (w=69u); a separate 35-slot/2450 relation supplies (u=70). Esau's complete selected prefix basis preserves nine input counts, while Toledot multiplicities preserve occurrence measure without making reflection descend. The primary NT seven-edge grouping does preserve reflection. These establish a shared measurement method, not identical source objects. Sources: Files43,54,58,70; C1009–1029.

What the presentation costs, and where it stops

The architecture adds no fitted numerical parameter and no new chronological operation. Its new cost is explicit interface bookkeeping. It still retains source values and masks; count conventions; Cainan's incidence; cap baseline and threshold; selected row or block measure; anchors and cuts; digit register; units; and source/diagnostic status. Four fitted amplitudes do not erase these costs. Residuals and cap-loss data restore information only because they retain it. No numerical description-length optimum is asserted.

Composition is warranted where an executed operation's output meets the next declared input domain with these tags intact. The existing proofs give concrete coherent squares: accumulation of row changes; cap with consistently rounded bounds; source shifts through residual encoding; uniform conversion through block sums; and reflection through compatible grouping. They do not make every pair of operations commute. A common presentation can therefore proceed without declaring a universal equivalence, a universal naturality law, or a reversible transformation group.

The compact negative set should remain visible: the three-tradition scalar affine obstruction; R/A intermediate-domain failure; cap information loss; row-rounding versus block-rounding; whole-span 2700 reversal versus reversal of 1650|1050; equal 490 totals with different interiors; the fixed-pivot J→P integer-domain obstruction; and Toledot reflection failure. These are explanatory limits, not a new catalogue of searches.

Reader diagram design

Use a 1500×1200 static vector panel with three columns: source object, measure or operation, complete result and retained information. Six horizontal example lanes are enough:

1. MT/LXX named rows → (R_d,A_e), Regular/Cumulative measurement → full displacement fields; Cainan labelled separately.
2. SP birth path plus baseline → capacities, cap, count rounding → nine outputs and located excess/residual data.
3. MT path to Moses → row rounding and ordered sums → five blocks and full residual field.
4. Three File52c original regular component words → one reversal, declared whole-span E → inherited complete path images; proper prefixes remain rational.
5. Covenant local relationships → lifespan stack and defined clutch → first forced joins plus the additional 147 condition.
6. NT/list source labels and counts → measured accumulation and permitted grouping → complete slot/count fields with a reflection test.

A narrow labelled connector may run from lane 3 to lane 4: “matched source nodes and three original component words.” Do not draw a SP-cap-to-inverse connector. Put three short constraints in the footer: shared topology ≠ same weights; same total ≠ same interior; this is not a universal date conversion. Do not use decorative convergence lines between equal numbers. Keep exact details in the adjacent table rather than reducing label size.

The companion JSON gives the claim inventory, overlap audit, source costs and diagram schema. No additional cross-operator numerical campaign is recommended before synthesis.