Carry Signatures in a Binary Blockwise Reformulation of Goldbach's Conjecture

We present a binary blockwise reformulation of Goldbach’s conjecture. Fix a block size nand let B = 2ⁿ. Every even target integer E can be decomposed into base-B blocks, and thecondition E = P + Q can then be written as a finite system of local block equations coupled bycarry propagation. This yields, for each Goldbach decomposition, a carry signature recordingthe carry pattern across blocks. We also recall the standard XOR–AND carry-resolution mechanism for binary addition, whichprovides a lower-level logical interpretation of the same sum. The principal contribution ofthe paper is not a proof of Goldbach’s conjecture, but a precise reformulation together withstructural invariants naturally attached to Goldbach decompositions. Computational experimentsfor small even integers suggest that many distinct Goldbach pairs collapse into comparativelyfew carry classes, and that prefix-type carry signatures occur frequently. These observations areexploratory, but they indicate that Goldbach decompositions may admit a nontrivial internalcombinatorial organization beyond the classical additive statement.