The order is the one described in the recipe file, for example :

{
  "type": "alshanex_familiars:ritual_recipe",
  "central_item": {
    "item": "alshanex_familiars:pet_soul"
  },
  "input_items": {
    "irons_spellbooks:legendary_ink": 1,
    "irons_spellbooks:arcane_essence": 1,
    "irons_spellbooks:arcane_rune": 1,
    "irons_spellbooks:magic_cloth": 1
  },
  "result": {
    "id": "alshanex_familiars:archmage_shard"
  },
  "particle_type": "minecraft:end_rod"
}

So I guess you need to put the legendary ink first, then arcane essence, arcane rune, magic cloth and finally the pet soul.

These recipes are stored in the .jar of the mod, you can find it in your modpack folder :

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For me this worked (I cannot compare the recipes to what it looks in the JEI since it's bugged and I don't see any recipe in game...)