Integrated Completed Tesserine Sign Algebra, Circular Completion, and Blade-One Positivity for the 107-Residual Erdős–Straus Shell

This preprint gives a completed two-sign refinement of the 107-residual Erdős–Straus shell. It defines the completed tesserine sign algebra \(\mathbb S_{\square}=\mathbb R[\widehat s,\check s]/(\widehat s^2-1,\check s^2-1)\), proves the four primitive branch decomposition, and derives the exact sign-resolved target coefficient \(\Theta^{\square}_{107}(p_*)=2e^-_-o_{34}[53]\). 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 \(([34]+[72])([19]+[87])=2[53]+[15]+[91]\). 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 \(\widehat s\) separates annihilation from survival, while the lower sign \(\check s\) separates the two surviving diagonal transports. Files included in this record: PDF preprint, LaTeX source, and source bundle.