What do you get for winning a Seize Imperial Duchy war in CK3?

The attacker gains the contested Title (patch 1.20.0.3).

What does the Seize Imperial Duchy casus belli cost?

piety and prestige; the amount depends on the titles, claims and distance, worked out when you declare.

Casus belli · Conquest

Seize Imperial Duchy

Seize Imperial Duchy is a casus belli in Crusader Kings III (conquest). If the attacker wins: The attacker gains the contested Title. If the defender wins: The defender keeps the contested Title. Declaring it costs piety and prestige. The war goal is worth 80% war score. Patch 1.20.0.3, from game/common/casus_belli_types/07_ep3_wars.txt.

Quick answers (3)

What do you get for winning a Seize Imperial Duchy war in CK3?: The attacker gains the contested Title (patch 1.20.0.3).
What happens on a white peace?: The defender keeps the contested Title.
What does the Seize Imperial Duchy casus belli cost?: piety and prestige; the amount depends on the titles, claims and distance, worked out when you declare.

How the war ends

Attacker wins

The attacker gains the contested Title.

White peace

The defender keeps the contested Title.

Defender wins

The defender keeps the contested Title.

If the war becomes invalid

The … has ended: the defender no longer … any land in the target region.

Who can declare it

The game shows no text of its own for these conditions; the script is below.

Every conquest war (the group's allowed_for_character) (7 lines of game script)

herders_and_tributary_constraints = yes
pam_personal_pacifism_offensive_war_ban_trigger = yes
tgp_dynastic_cycle_offensive_wars_ban_trigger = yes
tgp_war_ban_all_ministers_trigger = yes
NOT = {
	government_has_flag = government_is_theocracy
}

As written in game/common/casus_belli_groups/00_casus_belli_groups.txt.

Attacker (allowed_for_character) (31 lines of game script)

exists = global_var:byz_collapse_counter
OR = {
	NOT = {
		exists = title:e_byzantium.holder
	}
	top_liege = {
		this = title:e_latin_empire.holder
	}
	AND = {
		top_liege = title:e_byzantium.holder
		title:e_latin_empire = {
			any_in_de_jure_hierarchy = {
				tier <= tier_kingdom
			}
		}
	}
}
any_held_title = {
	tier <= tier_empire
	OR = {
		this = title:e_byzantium
		this = title:e_latin_empire
		empire = title:e_byzantium
		empire = title:e_latin_empire
	}
}
title:e_byzantium = {
	any_in_de_jure_hierarchy = {
		tier <= tier_kingdom
	}
}

As written in game/common/casus_belli_types/07_ep3_wars.txt.

Defender (allowed_against_character) (24 lines of game script)

scope:attacker = {
	NOR = {
		top_liege = scope:defender.top_liege
		liege = scope:defender
	}
}
scope:defender = {
	any_realm_county = {
		OR = {
			empire = title:e_byzantium
			empire = title:e_latin_empire
		}
	}
}
trigger_if = {
	limit = {
		top_liege = title:e_byzantium.holder
	}
	scope:defender = {
		any_realm_county = {
			empire = title:e_latin_empire
		}
	}
}

As written in game/common/casus_belli_types/07_ep3_wars.txt.

When it can start (valid_to_start) (40 lines of game script)

scope:target = {
	OR = {
		empire = title:e_byzantium
		empire = title:e_latin_empire
	}
}
scope:target = {
	any_in_de_jure_hierarchy = {
		tier = tier_county
		OR = {
			scope:attacker = {
				is_ai = no
			}
			any_neighboring_county = {
				holder = {
					target_is_same_character_or_above = scope:attacker
				}
			}
			scope:attacker.top_liege = {
				primary_title = {
					OR = {
						this = title:e_byzantium
						this = title:e_latin_empire
					}
				}
			}
			scope:attacker = {
				government_has_flag = government_is_republic
			}
			scope:attacker = {
				government_has_flag = government_is_administrative
				vassal_contract_has_flag = admin_theme_naval
			}
			scope:attacker = {
				ai_greed >= high_positive_ai_value
				ai_boldness >= high_positive_ai_value
			}
		}
	}
}

As written in game/common/casus_belli_types/07_ep3_wars.txt.

Cost to declare

Costs piety and prestige. The amount is worked out when you declare (by the titles targeted, claims, distance and more), so the script is shown as written:

Cost (119 lines of game script)

piety = {
	add = common_cb_impious_piety_cost
	if = {
		limit = {
			scope:defender.faith = scope:attacker.faith
		}
		add = {
			value = 100
			desc = CB_SAME_FAITH_COST
		}
		add = {
			value = 10
			multiply = scope:attacker.realm_size
			desc = CB_REALM_SIZE_COST
			max = 800
		}
	}
	multiply = common_cb_piety_cost_multiplier
	if = {
		limit = {
			scope:attacker = {
				NOR = {
					top_liege.primary_title = {
						OR = {
							this = title:e_latin_empire
							this = title:e_byzantium
						}
					}
					culture = culture:greek
					culture = {
						any_parent_culture_or_above = {
							this = culture:greek
						}
					}
					has_trait = despoiler_of_byzantium
				}
			}
		}
		multiply = 2
	}
}
prestige = {
	value = 0
	if = {
		limit = {
			scope:defender.faith != scope:attacker.faith
		}
		add = {
			value = 150
			desc = CB_COST_DIFFERENT_FAITH
		}
		add = {
			value = 10
			multiply = scope:attacker.realm_size
			desc = CB_REALM_SIZE_COST
			max = 800
		}
	}
	multiply = common_cb_prestige_cost_multiplier
	if = {
		limit = {
			exists = scope:attacker.house
			scope:attacker.house = {
				OR = {
					has_house_modifier = fp2_struggle_hostility_house_culture_modifier
					has_house_modifier = fp2_struggle_hostility_house_combined_modifier
				}
			}
			any_in_global_list = {
				variable = fp2_struggle_ending_culture_list
				this = scope:attacker.culture
			}
			any_in_global_list = {
				variable = fp2_struggle_ending_culture_list
				this = scope:defender.culture
			}
		}
		divide = {
			value = 2
			desc = CB_COST_FP2_STRUGGLE_HOSTILITY_CULTURE_DISCOUNT
		}
	}
	if = {
		limit = {
			has_global_variable = fp2_struggle_compromise_ending
			any_in_global_list = {
				variable = fp2_struggle_ending_culture_list
				this = scope:attacker.culture
			}
			any_in_global_list = {
				variable = fp2_struggle_ending_culture_list
				this = scope:defender.culture
			}
		}
		multiply = {
			value = 2
			desc = CB_COST_FP2_STRUGGLE_COMPROMISE_MULTIPLIER
		}
	}
	if = {
		limit = {
			scope:attacker = {
				NOR = {
					top_liege.primary_title = {
						OR = {
							this = title:e_latin_empire
							this = title:e_byzantium
						}
					}
					ep3_orthodox_faith_trigger = yes
					faith = faith:hellenic_pagan
					faith = faith:catholic
					has_trait = despoiler_of_byzantium
				}
			}
		}
		multiply = 2
	}
}

As written in game/common/casus_belli_types/07_ep3_wars.txt.

Source: game/common/casus_belli_types/07_ep3_wars.txt and the English localisation, Crusader Kings III 1.20.0.3. “The attacker”, “the defender” stand where the game names the character; “…” marks other words it fills in during play.