This preprint gives a completed two-sign refinement of the 107-residual Erdős–Straus shell. It defines the completed tesserine sign algebra \ (S_= R s, s/ (s²-1, s²-1) \), proves the four primitive branch decomposition, and derives the exact sign-resolved target coefficient \ (^₁₀₇ (p_*) =2e^-_-o₃₄53\). The main finite calculation proves that the shell positivity is localized by the scalar Cayley–Dickson blade \ (1\). The tag-forgotten blade polynomial has a unique upper-circular survivor, and the tag-resolved target hits are exactly the coefficient of \ (53\) in the explicit finite product \ ( (34+72) (19+87) =253+15+91\). The paper also connects this calculation with the circular completion of the modulo-840 square fiber and proves the compressed sedenion zero-divisor formulas in the completed sign algebra. In that compressed form, the upper sign \ (s\) separates annihilation from survival, while the lower sign \ (s\) separates the two surviving diagonal transports. Files included in this record: PDF preprint, LaTeX source, and source bundle.
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