Inverse–Key interfaces: thirteen worthwhile questions

Preparation only. I found thirteen useful candidate questions, of which several should be combined as supporting context. Twenty to twenty-five would repeat previous results or fragment one theorem into small checks. The full list was frozen in questions_before_calculation.json before calculation. The strongest new object is the complete four-path File52c family under its already declared whole-span Priestly completion.

Files: inputs.json contains the literal path and endpoint inputs, exact locators and source identities; calculations.json holds exact Fraction results; prepare.py reproduces them. No numbered root action or source edit was performed.

1. An integral outer completion need not preserve integer internal coordinates

File52c §3.4 supplies four complete paths, read from Nativity toward Creation. Apply digit reversal only once to each original component. The resulting component paths are:

All total14720 and therefore share the already established E·14720=16000. But every proper prefix in every path has denominator23 after E=25/23. With the source anchor6 held, the final endpoint is the existing16006BC; no internal image acquires an integer BC coordinate.

This is a new complete-family interface test, not a new endpoint discovery. Uniform E remains exactly additive: transforming the pieces and then summing gives the same rational answer as transforming their sum. What fails is the stronger demand that every intermediate coordinate remain integral. The componentwise rational display is a mathematical extension of the source's whole-span conversion; it is not a newly adopted chronology.

For a general reduced factor p/q, a merged block has integral transformed length exactly when the block sum is divisible by q. Consecutive cuts from an integral held anchor must therefore occur at original prefix sums congruent to zero modulo q. Applying this rule to all four paths permits only the final cut. Thus the finest numerical coarsening with integral E block lengths is the whole path in every case. The source boundaries and their roles remain in the retained path record; they are not deleted from chronology.

This complements the Key packet's operator identity C D = Dbar C: a uniform factor commutes with coarsening over rational values, while preservation of a chosen integer grid imposes additional congruences. The two assertions have different domains.

Recommended questions: IQ01–IQ03, with IQ05's held-anchor reminder included in their exposition rather than treated as another discovery.

2. The two apparent branch checks are equivalent on the fixed register domain

C940's cumulative total/register theorem permits one-pass component sums10620 and11610. Append the source's original1400 tail, whose single-pass value is4100:

Within this exact two-branch domain, the following select the same branch:

• agreement with the regular primary inverse total;
• integral Priestly completion after the same source tail.

The equivalence follows because the branch displacement990 is congruent to1 modulo23. It does not assert that decimal reversal generally preserves the23 lattice. Nor should these selectors be counted as independent evidence after the two-branch source domain is fixed. The original tail and its anchor provenance remain explicit, preserving C952's correction.

Recommended question: IQ04. This is a genuinely new dependency relation between the completed register theorem and the Key domain.

3. Calibration transfers; a source-appointed allocation need not

The same14720 scalar has exact calibrated comparisons:

The P and J rows are conditional scalar/calendrical comparisons. File52c's source construction explicitly licenses the E completion; these additional rows do not create new literal chronological paths. They show that common calibrated volume is coherent even when integer year-count closure differs.

Now test transfer of the previously established allocation identities. To imitate whole P by applying E to only part u of S, the selected amount must be u=S/6. To imitate J it must be u=S/26. At S=14720 these are 7360/3 and 7360/13.

Both are noninteger. Every supplied component and every sum of supplied whole components is an integer. Therefore neither allocation can be realized by selecting whole components from any of the four admitted completed inverse paths. A rational subdivision would require a further source-appointed cut; the algebra does not supply its location or narrative role.

The result distinguishes a valid factor identity from an available chronological realization. It does not refute the source1/6 and1/26 allocations in Files63's own paths, and it does not undo calendar calibration.

Recommended questions: IQ06–IQ08, preferably as two substantive steps: exact scalar transfer, then the source-support obstruction to allocation transfer.

4. The paired-anchor digit branches constrain common Key placement

For a reduced Key p/q and integer coordinates x,y,a, both affine expansions about one common integer pivot a are integral only if q | (x−y). If the difference is divisible by q, a correct pivot residue remains necessary; difference divisibility alone does not license a source pivot.

The established long-core branch differences are990 and1980. They are congruent to1 and2 modulo23. Since all three Key denominators contain23, neither such endpoint pair can have both members expand integrally about any one common integer Key pivot. This is independent of which common pivot is considered; no pivot or target search is performed. It concerns the already supplied pair relation, not a fresh count of successful targets.

The remaining branch law can be expressed symbolically without enumerating outcomes:

Here a,b,c retain the source-core digit roles and bounds from C850–858. The reduced divisibility conditions account for gcd(90,69)=gcd(990,69)=3; no target columns are used.

The positive companion is an exact comparison mesh. The differences among all64 existing endpoint literals generate5Z. Under a common E comparison their differences generate (125/23)Z. File52c's already supplied nine-member pillar field instead has difference lattice115Z and hence E-image lattice125Z. Using the existing Mahalalel pair1731→601 with14726 held as the reference, these are:

full comparison field ⊂ 601+(125/23)Z;

existing nine-member field ⊂ 601+125Z.

The full exact rational field is coherent; it need not be repaired by rounding dates or by broadening the selected integer pillar family. The nine-member integer normal form is inherited from C513; the new information is its relation to the full paired field's finer rational mesh. No new target outcomes, frequencies or member admissions were computed.

Recommended questions: IQ09–IQ11. Merge IQ10 with IQ09 unless the symbolic branch table materially helps the synthesis.

5. Added uniform Key outputs do not restore lost source information

On the aggregate digit vector, the raw and reversed measurement rows are (100,10,1) and (1,10,100), apart from the common outside factor10. Uniform E/P/J applied to the reversed total append scalar multiples of the second row. The enlarged measurement matrix still has rank2.

Consequently, after the original and reversed totals are known, the three uniform Key measurements add no new information about the aggregate columns. Still less can they recover how digits were allocated between source components or what event occupied the internal boundary. This is an explicit interface dependency, not a claim that the original two measurements are chronologically sufficient.

Recommended question: IQ12, followed by IQ13's short classification ledger:

Novelty and scope

The already known14720→16000 value, general calibration, allocation fractions, affine integrality rule, nine-member125 grid, and digit-branch values are premises. The new work is their complete-object interaction: prefix closure, equivalent finite-domain branch selectors, source-supported allocation failure, and the full paired-field mesh. Relevant prior checks include C497–499,C505–520,C850–858,C881 and the present C939–952, together with the prepared Key packet.

Every decimal call in the preparation receives an original File52c component or the original1400 tail. Key outputs are never passed to reversal. Calendar comparisons retain their conditional role. No primer, new Gear, source edit, statistical tally or historical-causation claim is opened.