monthly continuous (perpetual) inventory counts

Not sure what you are asking – inventory “for the month” doesn’t mean anything to me – you can talk about purchases for a month and sales for a month – but inventory is a balance at a point in time.

A “full physical inventory” at a reporting date will count all inventory with test checks “from physical-to-book” (existence)” and “from book-to-physical” (completeness).

If a client keeps perpetual records of inventory balances – imagine stock cards that show every goods movement in/out and a running total of the balance – it is not necessary to count all inventory at the reporting date as long as the records are shown to be accurate. To this end, there must be regular (e.g. monthly) counts throughout that year during which every product line must be counted at least once (e.g. on a rotational basis).