A cryogenic purification system configured for purifying a hydrogen, nitrogen, methane and argon containing feed stream, the system comprising: a synthesis gas rectification column configured to receive the feed stream and produce a hydrogen and nitrogen enriched overhead vapor stream and a methane-rich condensed phase stream at the bottom of the synthesis gas rectification column; a hydrogen stripping column; a subcooler configured to subcool a portion of the methane-rich condensed phase stream via indirect heat exchange with a first portion of a hydrogen free methane bottom stream from the hydrogen stripping column to produce a warm first portion of the hydrogen free methane bottom stream; a first expansion valve disposed downstream of the subcooler and upstream of the hydrogen stripping column and configured to expand the subcooled methane-rich stream; the hydrogen stripping column configured to receive the subcooled and expanded methane-rich condensed phase stream proximate the top of the column and the warmed first portion of the hydrogen free methane bottom stream, strip hydrogen from the subcooled methane-rich condensed phase stream and produce the hydrogen free methane bottom stream and a hydrogen enriched gaseous overhead; a condenser-reboiler disposed within the synthesis gas rectification column configured to receive a second portion of the hydrogen free methane bottom stream and to vaporize the second portion of the hydrogen free methane stream against the hydrogen and nitrogen enriched overhead vapor to produce a vaporized or partially vaporized hydrogen free methane-rich stream; a second expansion valve disposed downstream of the hydrogen stripping column and upstream of the condenser-reboiler and configured to expand the second portion of the hydrogen free methane bottom stream; a heat exchanger configured to (a) warm the hydrogen and nitrogen enriched overhead vapor stream via indirect heat exchange with the feed stream to produce a hydrogen and nitrogen containing synthesis gas; and (b) to warm the vaporized or partially vaporized hydrogen free methane-rich stream via indirect heat exchange with the feed stream to produce a methane fuel gas; and a recycle circuit configured to recycle the hydrogen rich gaseous overhead from the hydrogen stripping column back to the hydrogen, nitrogen, methane and argon containing feed stream. a synthesis gas rectification column configured to receive the feed stream and produce a hydrogen and nitrogen enriched overhead vapor stream and a methane-rich condensed phase stream at the bottom of the synthesis gas rectification column; a hydrogen stripping column; a subcooler configured to subcool a portion of the methane-rich condensed phase stream via indirect heat exchange with a first portion of a hydrogen free methane bottom stream from the hydrogen stripping column to produce a warm first portion of the hydrogen free methane bottom stream; a first expansion valve disposed downstream of the subcooler and upstream of the hydrogen stripping column and configured to expand the subcooled methane-rich stream; the hydrogen stripping column configured to receive the subcooled and expanded methane-rich condensed phase stream proximate the top of the column and the warmed first portion of the hydrogen free methane bottom stream, strip hydrogen from the subcooled methane-rich condensed phase stream and produce the hydrogen free methane bottom stream and a hydrogen enriched gaseous overhead; a condenser-reboiler disposed within the synthesis gas rectification column configured to receive a second portion of the hydrogen free methane bottom stream and to vaporize the second portion of the hydrogen free methane stream against the hydrogen and nitrogen enriched overhead vapor to produce a vaporized or partially vaporized hydrogen free methane-rich stream; a second expansion valve disposed downstream of the hydrogen stripping column and upstream of the condenser-reboiler and configured to expand the second portion of the hydrogen free methane bottom stream; a heat exchanger configured to (a) warm the hydrogen and nitrogen enriched overhead vapor stream via indirect heat exchange with the feed stream to produce a hydrogen and nitrogen containing synthesis gas; and (b) to warm the vaporized or partially vaporized hydrogen free methane-rich stream via indirect heat exchange with the feed stream to produce a methane fuel gas; and a recycle circuit configured to recycle the hydrogen rich gaseous overhead from the hydrogen stripping column back to the hydrogen, nitrogen, methane and argon containing feed stream.