XI Sum--Gap Coordinates and Finite Local Factors for Prime-Pair Fibres

This article develops sum--gap coordinates for prime-pair geometry in the integer addition table. Constant-sum and constant-difference fibres are organized as transverse slices of a common finite CRT survivor torus in center--half-gap coordinates. Exact torus and slice counts, parity boundary laws, normalized finite local factors, and arithmetic-progression decompositions are derived. The local factors are classical Hardy--Littlewood prime-pair factors; the contribution is their joint geometric organization. The article does not prove Goldbach, twin-prime, or general Hardy--Littlewood prime-pair conjectures.