diff --git a/Library/Homebrew/os/mac/mach.rb b/Library/Homebrew/os/mac/mach.rb
index 44ddf94636..4f29aedcad 100644
--- a/Library/Homebrew/os/mac/mach.rb
+++ b/Library/Homebrew/os/mac/mach.rb
@@ -1,4 +1,4 @@
-# typed: true # rubocop:todo Sorbet/StrictSigil
+# typed: strict
 # frozen_string_literal: true
 
 require "macho"
@@ -12,26 +12,28 @@ module MachOShim
 
   delegate [:dylib_id] => :macho
 
+  sig { params(args: T.untyped).void }
   def initialize(*args)
-    @macho = T.let(nil, T.nilable(MachO::MachOFile))
-    @mach_data = T.let(nil, T.nilable(T::Array[T::Hash[Symbol, T.untyped]]))
+    @macho = T.let(nil, T.nilable(T.any(MachO::MachOFile, MachO::FatFile)))
+    @mach_data = T.let(nil, T.nilable(T::Array[T::Hash[Symbol, Symbol]]))
 
     super
   end
 
+  sig { returns(T.any(MachO::MachOFile, MachO::FatFile)) }
   def macho
     @macho ||= MachO.open(to_s)
   end
   private :macho
 
-  sig { returns(T::Array[T::Hash[Symbol, T.untyped]]) }
+  sig { returns(T::Array[T::Hash[Symbol, Symbol]]) }
   def mach_data
     @mach_data ||= begin
       machos = []
       mach_data = []
 
-      if MachO::Utils.fat_magic?(macho.magic)
-        machos = macho.machos
+      if macho.is_a?(MachO::FatFile)
+        machos = T.cast(macho, MachO::FatFile).machos
       else
         machos << macho
       end
@@ -68,6 +70,7 @@ module MachOShim
 
   # TODO: See if the `#write!` call can be delayed until
   #       we know we're not making any changes to the rpaths.
+  sig { params(rpath: String, options: T::Boolean).void }
   def delete_rpath(rpath, **options)
     candidates = rpaths(resolve_variable_references: false).select do |r|
       resolve_variable_name(r) == resolve_variable_name(rpath)
@@ -81,21 +84,25 @@ module MachOShim
     macho.write!
   end
 
+  sig { params(old: String, new: String, options: T::Boolean).void }
   def change_rpath(old, new, **options)
     macho.change_rpath(old, new, options)
     macho.write!
   end
 
+  sig { params(id: String, options: T::Boolean).void }
   def change_dylib_id(id, **options)
     macho.change_dylib_id(id, options)
     macho.write!
   end
 
+  sig { params(old: String, new: String, options: T::Boolean).void }
   def change_install_name(old, new, **options)
     macho.change_install_name(old, new, options)
     macho.write!
   end
 
+  sig { params(except: Symbol, resolve_variable_references: T::Boolean).returns(T::Array[String]) }
   def dynamically_linked_libraries(except: :none, resolve_variable_references: true)
     lcs = macho.dylib_load_commands
     lcs.reject! { |lc| lc.flag?(except) } if except != :none
@@ -105,6 +112,7 @@ module MachOShim
     names
   end
 
+  sig { params(resolve_variable_references: T::Boolean).returns(T::Array[String]) }
   def rpaths(resolve_variable_references: true)
     names = macho.rpaths
     # Don't recursively resolve rpaths to avoid infinite loops.
@@ -113,11 +121,12 @@ module MachOShim
     names
   end
 
+  sig { params(name: String, resolve_rpaths: T::Boolean).returns(String) }
   def resolve_variable_name(name, resolve_rpaths: true)
     if name.start_with? "@loader_path"
-      Pathname(name.sub("@loader_path", dirname)).cleanpath.to_s
+      Pathname(name.sub("@loader_path", dirname.to_s)).cleanpath.to_s
     elsif name.start_with?("@executable_path") && binary_executable?
-      Pathname(name.sub("@executable_path", dirname)).cleanpath.to_s
+      Pathname(name.sub("@executable_path", dirname.to_s)).cleanpath.to_s
     elsif resolve_rpaths && name.start_with?("@rpath") && (target = resolve_rpath(name)).present?
       target
     else
@@ -125,6 +134,7 @@ module MachOShim
     end
   end
 
+  sig { params(name: String).returns(T.nilable(String)) }
   def resolve_rpath(name)
     target = T.let(nil, T.nilable(String))
     return unless rpaths(resolve_variable_references: true).find do |rpath|
@@ -134,48 +144,58 @@ module MachOShim
     target
   end
 
+  sig { returns(T::Array[Symbol]) }
   def archs
     mach_data.map { |m| m.fetch :arch }
   end
 
+  sig { returns(Symbol) }
   def arch
     case archs.length
     when 0 then :dunno
-    when 1 then archs.first
+    when 1 then archs.fetch(0)
     else :universal
     end
   end
 
+  sig { returns(T::Boolean) }
   def universal?
     arch == :universal
   end
 
+  sig { returns(T::Boolean) }
   def i386?
     arch == :i386
   end
 
+  sig { returns(T::Boolean) }
   def x86_64?
     arch == :x86_64
   end
 
+  sig { returns(T::Boolean) }
   def ppc7400?
     arch == :ppc7400
   end
 
+  sig { returns(T::Boolean) }
   def ppc64?
     arch == :ppc64
   end
 
+  sig { returns(T::Boolean) }
   def dylib?
     mach_data.any? { |m| m.fetch(:type) == :dylib }
   end
 
+  sig { returns(T::Boolean) }
   def mach_o_executable?
     mach_data.any? { |m| m.fetch(:type) == :executable }
   end
 
   alias binary_executable? mach_o_executable?
 
+  sig { returns(T::Boolean) }
   def mach_o_bundle?
     mach_data.any? { |m| m.fetch(:type) == :bundle }
   end