### Pivots place the whole field

A duration ratio does not determine a date transformation until a held pivot is declared. Around pivot a, factor k sends coordinate x to a+k(x−a). Differences between points scale by k, while their absolute placement still depends on a. This separates the shape of a chronological field from where it sits.

For two successive Keys, integer placement has an exact test. Let their reduced ratios be p₁/q₁ and p₂/q₂, with integer pivots a and b. An integer input can have integer outputs at both stages only when gcd(p₁,q₂) divides b−a. If that condition holds, the admissible inputs form one residue class with spacing q₁q₂/gcd(p₁,q₂). A complete field belongs to that class when one point has the right placement and every internal difference is a multiple of the spacing.

For the fixed pivots 14006 and 4836, the diagnostic J→P composition has no such integer inputs: gcd(300,69)=3 does not divide their difference. This identifies a domain limit without inventing replacement pivots or authorizing new chronological routes.

Fractions themselves remain exact coordinates. The full File46 head pattern, with indices 92,89,46,43,0 on a ten-unit mesh, survives E expansion on mesh 250/23. Preserving that entire rational image explains the family more faithfully than discarding points to retain only integer endpoints.

*Sources:* File46 Harmonic Expansion Protocols §§2–3:14006 and4836 pivots; §6A.5: complete head field; 490d_Unification_Research_Strategy_v0_2_20260906.md §5C: transformations and domains; c482_c531/journal.json C506–C507: inherited placement witnesses.

*Research record:* C998, C999, C1000, C1001, C1002, C1003.