Three-Sign Cancellation Hypernumber Systems and Associator Curvature

We introduce and study a three–sign cancellation hypernumber system H which extends the real field by adjoining a third sign Λ. The underlying set is H = {0} ∪ {+, −, Λ} × ℝ_{>0}, with a single–valued multiplication and a hyperaddition ⊞ designed to encode cancellation phenomena between positive and negative reals. The classical real line embeds as a genuine subfield ℝ_cl ⊂ H, and all field operations agree with the usual ones on ℝ. The additive structure of H is almost associative but not a canonical hypergroup. We give an explicit description of where associativity fails and compute, for triples of the form (+, a), (−, b), (Λ, c), a closed formula for the associativity defect κ(a, b, c) = 2·min(a, b) = a + b − |a − b|, which coincides with the loss of absolute value when adding a and −b in ℝ. To explain this behaviour, we construct an ambient “cancellation monoid” (K, ⊕) on ℝ × ℝ_{≥0} which is strictly associative and records both real sums and accumulated cancellation mass. We prove that H cannot be recovered from K by any simple projection, and formulate an ambient reconstruction problem. In addition, scalar multiplication by real numbers (defined via the embedded copy of ℝ) distributes over ⊞, and the sign–layer admits a canonical hypergroup envelope governing the possible signs of hypersums. The results provide a controlled example of a nonassociative hyperaddition sitting over the real field and suggest several directions for multisign generalizations and connections with hyperfields and tropical geometry.This Zenodo record corresponds to the arXiv preprint:arXiv:2602.21207 [math.RA; math.GR], "Three-Sign Cancellation Hypernumber Systems and Associator Curvature".