The SolArc 12k is two 4kW 120vac inverters, so max continuous AC power is 8 kW or 4.8 kW continuous on either 120vac side when fed from batteries.

Battery AH size per kW of solar depends on battery type which is important factor in how much peak current the battery can take. LFP can take higher ripple current than lead-acid batteries.

For LFP battery with about 51vdc delivered to inverter input, for 8 kW AC power consumption, at 92% efficiency, at power factor = 1, will have average inverter DC input current of 170.6 amps and peak DC input current of 342.1 amps.

Assuming PV is matching AC output power there would be +/- 171 amps of ripple current on battery. It is your call how rough you want to treat your battery but I would say about 500 AH minimum if it was my battery. If you push to grid or AC load for extended time I would have an even larger AH battery for this amount of ripple current.

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Most 48v inverter manufacturers recommend 100 AH per kW of PV power for minimum battery sizing but this is more of a lead-acid battery number.