Integrated Supersignum, Circular Completion, and Blade-One Positivity for the 107-Residual Erdős–Straus Shell

We give a complete finite derivation of the four-way completed supersignum/tesserine refinement of the residual Erdős–Straus shell p∗ = 8,803,369, R = 107, where a∗ = (p∗ + 107)/4 = 3² · 11² · 43 · 47. The construction starts from the elementary fixed-shell divisor identity, passes through the projectively normalized two-target star {−1, −p⁻¹}, and then refines the divisor exponent box by two independent sign coordinates. The first sign records the ordered Cayley–Dickson power sign; the second records the quadratic-character parity of the residual logarithm. These two signs generate a completed tesserine sign algebra 𝕊_□ = ℝ[ŝ, š] / (ŝ² − 1, š² − 1), with four primitive branches e^±±. In this refinement the R = 107 target coefficient is not merely a signed scalar; it is the idempotent-sector statement Θ^□₁₀₇(p∗) = 2e^-- o₃₄[53]. The main calculation proves that this positivity is localized by the scalar Cayley–Dickson blade 1. The tag-forgotten blade polynomial has only one upper-circular survivor, namely the blade 1 coefficient 5 + 4ŝ. Tag-resolved, the upper-minus scalar blade is B₁⁻ = [34] + [44] + [62] + [72], and the two target hits are exactly the coefficient of [53] in ([34] + [72])([19] + [87]) = 2[53] + [15] + [91]. Equivalently, Ω₁₀₇(p∗) = [53](([34] + [72])([19] + [87])) = 2. We also connect this blade-one localization with the circular modulo-840 completion: the hard square fiber S₈₄₀ = {1, 121, 169, 289, 361, 529} is a cyclic C₆-support, the target edge for p∗ ≡ 169 (mod 840) is the circular edge E₃, and the congruence 744 ↦ 750 is the endpoint flip −3 ↦ +3 in the A₈-discriminant coordinate. All computations are explicit and finite.