# Ranked integration candidates after the new domain tests

Preparation only; no numbered actions. Read C832–C858 and the NT, operator and Covenant/carrier packets. These candidates are intended after those packets' principal complete-family tests. If a domain step already answers a candidate in full, cite it rather than execute it again.

## 1. Recover both source endpoints from one span and one generated placement

**Question:** Can the new twelvefold macro family and the older weighted Creation bridge be explained as two coordinate choices on the same typed endpoint pair?

Write `D=C−R` and `Wλ=λC+(1−λ)R`. Then `R=Wλ−λD` and `C=Wλ+(1−λ)D`; the map is invertible for every λ. Use the complete six-source File63 packet at λ=12 and the established Actual/Rounded Creation comparison at λ=4/5. Verify the recovered source pairs, not merely their generated heads.

**New integration:** the macro coordinate and the12026 weighted coordinate have different source coefficients and different roles, but each combines with a span to retain the entire pair. Either coordinate alone loses one degree of source placement. This is a concrete shared representation with an exact inverse, not a fitted chronology conversion. It is not a second decimal inversion.

## 2. Distinguish whole-frame motion from changing only an anchor

**Question:** Which of the four new modules is covariant when the whole source object moves, and which apparent “anchor shifts” actually change its measured input?

Use the frozen32 paired inverse rows, the complete NT6/36 rail pair, the File63 six-pair macro family, and the anchored festival path. Whole-frame translation leaves durations, categories, indices and digit strings unchanged and therefore translates outputs uniformly. File52c's existing Nativity/Conquest comparison instead keeps the source date fixed while changing the anchor, changing the duration by1400 and sometimes changing digit length or borrowing state. It therefore produces the nonuniform paired gain field.

**New integration:** one precise distinction explains why the NT+30 field and File63 paired−30 changes are rigid translations while the inverse paired-anchor columns are not. C499/C608 already prove generic covariance; the new contribution is the complete contrast between these new source objects. Do not present a formal whole-frame diagnostic as a new admitted chronology.

## 3. Identify the minimal retained information for each whole-family reconstruction

**Question:** What information must survive projection if the original source object is to be reconstructed?

Build one compact, evidence-backed comparison: Tishri cell counts versus row/column margins; Genesis32 named rows versus purity/sex totals; NT names/row roles versus a homogeneous index grid; decimal digits/placeholders versus source-span value after a transformation; macro source pair versus expanded total; original biography versus retained post-expansion terminal unit.

Use one existing counterexample per lost feature, with one positive recovery formula where available. The new synthesis should say **why** the shared object needs categories, order, index, digit representation, endpoint roles and subdivision—not merely list these labels. Do not claim a globally minimal grammar; this is a finite necessity argument over the tested objects. In particular, a path projection need not preserve every relation of its original table or grid.

## 4. Derive the list-to-49/70 interface on the complete declared festival fields

**Question:** Which boundaries of the two fixed festival sequences lie on the common490 measure generated by the NT49/70 grid, and why?

The list operation turns a count c into duration7c. Such a duration lies in49Z iff7 divides c, in70Z iff10 divides c, and in the common490Z iff70 divides c. Apply this fixed criterion to **all** forward and reverse prefixes, not selected landings. Both prefix sets contain140, giving980=2×490; the complete overlap also includes59 and199, which are outside the490 lattice. The source total280 gives1960=4×490;189 and182 instead give27×49 and26×49.

**New integration:** the exact common-measure condition joins ritual count scaling to the NT/BJ coarsening while explaining their limits. The coordinates use different source anchors; equal duration lattice membership does not identify their events. This is a deterministic finite classification, not a numerical-hit or probability tally. If the NT/operator domain work already classifies this particular list family, omit the repeat.

## 5. Separate a textual name insertion from changing its measurement weight

**Question:** Does restoring the same Cainan node make the regular, cumulative, Luke70 and BJ57 representations agree, or only align their topology?

Use the complete new Luke-to-MT comparison through Jacob and the inherited MT/LXX/SP/BJ insertion results. On the common name graph, regular Cainan has130, cumulative has460, Luke's display has70 and the BJ secondary deleted birth interval has57. These are source-specific weights, not scalar conversions inferred from one shared name. When Cainan is absent, contract the surrounding NT two-edge block before comparing common nodes; when restored, keep the node and compare its edge weight explicitly.

**New integration:** the new NT object supplies another complete positive example of shared topology with different measures. Recover the entire change in the Luke/MT bridge from the one insertion support, including its downstream convergence, without introducing another Cainan state. Prior C795 already separates1/57/130/460 weights; only the full NT transfer is new.

## 6. Build a small operation-order table with one decisive source object per mechanism

**Question:** Which apparent order effects are caused by nonlinear measurement, changing anchors, intermediate grid requirements, or genuine reordering of source data?

Use four already frozen examples: linear scale with festival permutation (commutes, but permutation changes prefixes); decimal evaluation before/after anchor-induced digit change (piecewise gain); P/J equal final scalar product with different intermediate integer membership; File63 original versus retained terminal biography (same outer value, different inner node).

**New integration:** distinguish four mechanisms that a vague statement “order matters” would collapse. A final product equality is weaker than equality of the complete typed route. The table should state input object, what is held, what can change, and the preserved quantity. This is a finite comparison of completed constructions, not a search of new operation words.

## 7. Compare the two kinds of calendar-grid compatibility

**Question:** How do the NT49/70 common boundaries and the Key-preserved integer/quarter/five-year grids relate without becoming the same theorem?

The first is common coarsening of two integer measures, `70k=49j`, with period490. The second is preservation of a chosen grid by a reduced ratio p/q, requiring divisibility of the input index by q. Present them as two exact divisibility conditions on a declared measurement index. Show their different outputs on one complete frozen field each: the12 NT common boundaries and the five quarter-carrier rows.

**New integration:** both fit modular measurement rules, but coarsening compares two representations whereas Key-grid preservation tests an operator's domain. Neither derives source anchors or authorizes a new historical route. If the operator synthesis already makes this exact distinction with these complete objects, this is editorial integration only.

## 8. Turn the macro terminal-biography distinction into a general subdivision criterion

**Question:** When can an original named terminal subinterval be retained after scaling the outer span without claiming it was uniformly transported?

Let the original outer span be S and its terminal biography l. Uniform k-scaling gives terminal kl. The source's retained-biography construction instead places a new outer head at kS while holding terminal l, leaving prefix kS−l. Their internal cuts differ by `(k−1)l` although their outer endpoints agree. Apply this to the full File63 P branch with l147 and compare the inherited File52c component-evaluation and Actual/Rounded retained-path conclusions.

**New integration:** the final biography is an explicit additional retention choice, rather than the image of the old internal cut. This gives a clear general rule for one more way source families can share endpoints while disagreeing internally. The scalar formula is simple; use it once with the exact complete new source path and stop.

## 9. Distinguish forced identities from additional source constraints across the four domains

**Question:** Which conclusions would hold for any inputs of the declared type, and which depend on the particular source values or roles?

Use a small ledger with concrete contrasts: equal Tishri grand totals are forced by the common matrix; equal Genesis32 purity/sex marginals require the extra50=50 condition; homogeneous NT grid reflection is automatic but BJ placement of its named internal boundaries is extra; File52c paired-field formulas are forced by fixed decimal rules but source dates select the branches; File63 paired-translation responses are algebraic but the supplied12 multiplier and retained147 biography are premises.

**New integration:** this is an explanatory constraint budget for the current new domains. It identifies the residual research burden without probability claims or declaring every correctly regenerated coordinate an independent witness. C716 had an earlier dependency map; use the new complete objects and their exact nonredundant constraints rather than restating the old map.

## 10. Test whether the updated grammar can regenerate a source family from a compact packet

**Question:** Can the final explanation be made constructive enough that a reader can regenerate all relevant outputs without consulting a table of generated dates?

Choose one new complete family, preferably the festival forward/reverse pair or the six macro heads, and retain only its smallest *source-supported* input packet, row/role definitions, operation and anchor. Generate the complete output table, then compare it with the frozen literal table. The numbered action must focus on genuinely removing redundant generated inputs from the explanation; it should not merely rerun the same formula or claim source-independent prediction.

**New integration:** a worked compiler-like example can make the Strategy's larger answer usable. Pair it with a clear statement of the premises it does not explain. One demonstration is enough; additional code tests should address a specific unresolved risk.

## Suggested final emphasis

The strongest new statement is that these source families share a compact **measurement-and-construction grammar** whose inputs retain their own categories, order, index, notation, roles and subdivisions. The grammar explains complete output fields and their differences. It is not a universal numerical map, and representation fidelity is not historical causation. Lead with the whole-family examples and the positive reconstruction before discussing the remaining source choices.