Are there precise ratios of carbon : waste required? Or is It sufficient to put enough carbon to cover the dead animals, whatever that amount may be?

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Nothing precise, keep it simple!  Start with a 1:1 ratio of dry carbon material to wet waste, i.e. poultry.  If you start to see moisture building around the edges or bubbles appearing when the fan is on, increase the dry carbon material to 2 parts and 1 wet organic part.