Supersignum Completion of the Modulo-840 Residual Star: Full signed support, branch projection, and the R=107 sedenion shell

This note gives the supersignum refinement of the modulo-840 residual completion note. The scalar algebra is 𝔸_ss = ℝ[s, g] / (s² − 1, g² − s). The idempotent decomposition separates the circular branch s = −1 from the hyperbolic branch s = 1. The hyperbolic branch recovers the positive divisor-counting shell coefficient. The circular branch recovers the ordered signed Cayley–Dickson coefficient. The unbranched supersignum coefficient retains both signs before cancellation and therefore refines the split-zero support formalism. The circular endpoint completion S₈₄₀ → S_A₈, the full edge branch, the target star, the 744 ↦ 750 endpoint flip, the R = 107 champion coefficient, and the sedenion associator residue are all written in supersignum form.