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The ghost conjecture is first provided by Bergdall and Pollack in [BP-1,BP-2]to study the Up-slopes of spaces of modular forms, which, so far, has alreadybrought plenty of important results. The local version of this conjecture undergenericity condition has been solved by Liu-Truong-Xiao-Zhao in [LTXZ-1,LTXZ-2].  In the current paper, we prove a necessary and sufficient condition for asequence of local ghost series to satisfy that their product has the sameNewton polygon to the ghost series build from the direct sum of theirassociated modules. That answers a common question asked in both [BP2,LTXZ-1].

The slope-invariant of local ghost series under direct sum