diff --git a/Library/.rubocop.yml b/Library/.rubocop.yml
index 7fca2f268c..7cb07e6f70 100644
--- a/Library/.rubocop.yml
+++ b/Library/.rubocop.yml
@@ -280,9 +280,14 @@ RSpec/Focus:
 Sorbet/ConstantsFromStrings:
   Enabled: false
 
+# This is already the default
 Sorbet/FalseSigil:
   Enabled: false
 
+# T::Sig is monkey-patched into Module
+Sorbet/RedundantExtendTSig:
+  Enabled: true
+
 Sorbet/StrictSigil:
   inherit_mode:
     override:
diff --git a/Library/Homebrew/PATH.rb b/Library/Homebrew/PATH.rb
index 0c949555f9..5730b2a11f 100644
--- a/Library/Homebrew/PATH.rb
+++ b/Library/Homebrew/PATH.rb
@@ -5,8 +5,6 @@
 #
 # @api private
 class PATH
-  extend T::Sig
-
   include Enumerable
   extend Forwardable
 
diff --git a/Library/Homebrew/api.rb b/Library/Homebrew/api.rb
index 4b154f4432..ebdb3d77df 100644
--- a/Library/Homebrew/api.rb
+++ b/Library/Homebrew/api.rb
@@ -11,8 +11,6 @@ module Homebrew
   #
   # @api private
   module API
-    extend T::Sig
-
     extend Cachable
 
     HOMEBREW_CACHE_API = (HOMEBREW_CACHE/"api").freeze
diff --git a/Library/Homebrew/api/analytics.rb b/Library/Homebrew/api/analytics.rb
index 9609eb37c1..fb5ddd7b44 100644
--- a/Library/Homebrew/api/analytics.rb
+++ b/Library/Homebrew/api/analytics.rb
@@ -8,8 +8,6 @@ module Homebrew
     # @api private
     module Analytics
       class << self
-        extend T::Sig
-
         sig { returns(String) }
         def analytics_api_path
           "analytics"
diff --git a/Library/Homebrew/api/cask.rb b/Library/Homebrew/api/cask.rb
index 32be78af09..50c22b981d 100644
--- a/Library/Homebrew/api/cask.rb
+++ b/Library/Homebrew/api/cask.rb
@@ -11,7 +11,6 @@ module Homebrew
     module Cask
       class << self
         include Cachable
-        extend T::Sig
 
         private :cache
 
diff --git a/Library/Homebrew/api/formula.rb b/Library/Homebrew/api/formula.rb
index a93fa4043d..adf2226ca9 100644
--- a/Library/Homebrew/api/formula.rb
+++ b/Library/Homebrew/api/formula.rb
@@ -11,7 +11,6 @@ module Homebrew
     module Formula
       class << self
         include Cachable
-        extend T::Sig
 
         private :cache
 
diff --git a/Library/Homebrew/build_environment.rb b/Library/Homebrew/build_environment.rb
index d1b7f8b861..e8427cb5c0 100644
--- a/Library/Homebrew/build_environment.rb
+++ b/Library/Homebrew/build_environment.rb
@@ -5,8 +5,6 @@
 #
 # @api private
 class BuildEnvironment
-  extend T::Sig
-
   sig { params(settings: Symbol).void }
   def initialize(*settings)
     @settings = Set.new(settings)
@@ -31,8 +29,6 @@ class BuildEnvironment
 
   # DSL for specifying build environment settings.
   module DSL
-    extend T::Sig
-
     # Initialise @env for each class which may use this DSL (e.g. each formula subclass).
     # `env` may never be called, and it needs to be initialised before the class is frozen.
     def inherited(child)
diff --git a/Library/Homebrew/bundle_version.rb b/Library/Homebrew/bundle_version.rb
index 92c522659a..b13cbf28ea 100644
--- a/Library/Homebrew/bundle_version.rb
+++ b/Library/Homebrew/bundle_version.rb
@@ -8,8 +8,6 @@ module Homebrew
   #
   # @api private
   class BundleVersion
-    extend T::Sig
-
     include Comparable
 
     extend SystemCommand::Mixin
diff --git a/Library/Homebrew/cask/artifact/abstract_artifact.rb b/Library/Homebrew/cask/artifact/abstract_artifact.rb
index 752dabfa57..b5f120cc47 100644
--- a/Library/Homebrew/cask/artifact/abstract_artifact.rb
+++ b/Library/Homebrew/cask/artifact/abstract_artifact.rb
@@ -9,7 +9,6 @@ module Cask
     #
     # @api private
     class AbstractArtifact
-      extend T::Sig
       extend T::Helpers
       abstract!
 
diff --git a/Library/Homebrew/cask/artifact/abstract_uninstall.rb b/Library/Homebrew/cask/artifact/abstract_uninstall.rb
index 7b4d522a7a..423b4c11d4 100644
--- a/Library/Homebrew/cask/artifact/abstract_uninstall.rb
+++ b/Library/Homebrew/cask/artifact/abstract_uninstall.rb
@@ -13,8 +13,6 @@ module Cask
     #
     # @api private
     class AbstractUninstall < AbstractArtifact
-      extend T::Sig
-
       ORDERED_DIRECTIVES = [
         :early_script,
         :launchctl,
diff --git a/Library/Homebrew/cask/artifact/artifact.rb b/Library/Homebrew/cask/artifact/artifact.rb
index a3b25a5972..4ab625dfdf 100644
--- a/Library/Homebrew/cask/artifact/artifact.rb
+++ b/Library/Homebrew/cask/artifact/artifact.rb
@@ -9,8 +9,6 @@ module Cask
     #
     # @api private
     class Artifact < Moved
-      extend T::Sig
-
       sig { returns(String) }
       def self.english_name
         "Generic Artifact"
diff --git a/Library/Homebrew/cask/artifact/keyboard_layout.rb b/Library/Homebrew/cask/artifact/keyboard_layout.rb
index cd853e921f..c7d549be00 100644
--- a/Library/Homebrew/cask/artifact/keyboard_layout.rb
+++ b/Library/Homebrew/cask/artifact/keyboard_layout.rb
@@ -9,8 +9,6 @@ module Cask
     #
     # @api private
     class KeyboardLayout < Moved
-      extend T::Sig
-
       def install_phase(**options)
         super(**options)
         delete_keyboard_layout_cache(**options)
diff --git a/Library/Homebrew/cask/artifact/mdimporter.rb b/Library/Homebrew/cask/artifact/mdimporter.rb
index d85cf27438..27419fa7c2 100644
--- a/Library/Homebrew/cask/artifact/mdimporter.rb
+++ b/Library/Homebrew/cask/artifact/mdimporter.rb
@@ -9,8 +9,6 @@ module Cask
     #
     # @api private
     class Mdimporter < Moved
-      extend T::Sig
-
       sig { returns(String) }
       def self.english_name
         "Spotlight metadata importer"
diff --git a/Library/Homebrew/cask/artifact/moved.rb b/Library/Homebrew/cask/artifact/moved.rb
index 477cd0470a..0c117d2611 100644
--- a/Library/Homebrew/cask/artifact/moved.rb
+++ b/Library/Homebrew/cask/artifact/moved.rb
@@ -9,8 +9,6 @@ module Cask
     #
     # @api private
     class Moved < Relocated
-      extend T::Sig
-
       sig { returns(String) }
       def self.english_description
         "#{english_name}s"
diff --git a/Library/Homebrew/cask/artifact/prefpane.rb b/Library/Homebrew/cask/artifact/prefpane.rb
index a2a1057b34..04483f3170 100644
--- a/Library/Homebrew/cask/artifact/prefpane.rb
+++ b/Library/Homebrew/cask/artifact/prefpane.rb
@@ -9,8 +9,6 @@ module Cask
     #
     # @api private
     class Prefpane < Moved
-      extend T::Sig
-
       sig { returns(String) }
       def self.english_name
         "Preference Pane"
diff --git a/Library/Homebrew/cask/artifact/qlplugin.rb b/Library/Homebrew/cask/artifact/qlplugin.rb
index e5766f4ee3..ed3994aa28 100644
--- a/Library/Homebrew/cask/artifact/qlplugin.rb
+++ b/Library/Homebrew/cask/artifact/qlplugin.rb
@@ -9,8 +9,6 @@ module Cask
     #
     # @api private
     class Qlplugin < Moved
-      extend T::Sig
-
       sig { returns(String) }
       def self.english_name
         "QuickLook Plugin"
diff --git a/Library/Homebrew/cask/artifact/relocated.rb b/Library/Homebrew/cask/artifact/relocated.rb
index 80d2e727b1..0642db0e2b 100644
--- a/Library/Homebrew/cask/artifact/relocated.rb
+++ b/Library/Homebrew/cask/artifact/relocated.rb
@@ -9,8 +9,6 @@ module Cask
     #
     # @api private
     class Relocated < AbstractArtifact
-      extend T::Sig
-
       def self.from_args(cask, *args)
         source_string, target_hash = args
 
diff --git a/Library/Homebrew/cask/artifact/stage_only.rb b/Library/Homebrew/cask/artifact/stage_only.rb
index ced0f5eed8..41f23e0c11 100644
--- a/Library/Homebrew/cask/artifact/stage_only.rb
+++ b/Library/Homebrew/cask/artifact/stage_only.rb
@@ -9,8 +9,6 @@ module Cask
     #
     # @api private
     class StageOnly < AbstractArtifact
-      extend T::Sig
-
       def self.from_args(cask, *args, **kwargs)
         if (args != [true] && args != ["true"]) || kwargs.present?
           raise CaskInvalidError.new(cask.token, "'stage_only' takes only a single argument: true")
diff --git a/Library/Homebrew/cask/artifact/suite.rb b/Library/Homebrew/cask/artifact/suite.rb
index 5eb8462640..6d7a2e3ccf 100644
--- a/Library/Homebrew/cask/artifact/suite.rb
+++ b/Library/Homebrew/cask/artifact/suite.rb
@@ -9,8 +9,6 @@ module Cask
     #
     # @api private
     class Suite < Moved
-      extend T::Sig
-
       sig { returns(String) }
       def self.english_name
         "App Suite"
diff --git a/Library/Homebrew/cask/artifact/symlinked.rb b/Library/Homebrew/cask/artifact/symlinked.rb
index 3cef93a179..3d920c1ede 100644
--- a/Library/Homebrew/cask/artifact/symlinked.rb
+++ b/Library/Homebrew/cask/artifact/symlinked.rb
@@ -9,8 +9,6 @@ module Cask
     #
     # @api private
     class Symlinked < Relocated
-      extend T::Sig
-
       sig { returns(String) }
       def self.link_type_english_name
         "Symlink"
diff --git a/Library/Homebrew/cask/artifact/vst3_plugin.rb b/Library/Homebrew/cask/artifact/vst3_plugin.rb
index 3ffe75f82f..313c09cb95 100644
--- a/Library/Homebrew/cask/artifact/vst3_plugin.rb
+++ b/Library/Homebrew/cask/artifact/vst3_plugin.rb
@@ -9,8 +9,6 @@ module Cask
     #
     # @api private
     class Vst3Plugin < Moved
-      extend T::Sig
-
       sig { returns(String) }
       def self.english_name
         "VST3 Plugin"
diff --git a/Library/Homebrew/cask/artifact/vst_plugin.rb b/Library/Homebrew/cask/artifact/vst_plugin.rb
index 9c3b5f5de0..d2cdca6073 100644
--- a/Library/Homebrew/cask/artifact/vst_plugin.rb
+++ b/Library/Homebrew/cask/artifact/vst_plugin.rb
@@ -9,8 +9,6 @@ module Cask
     #
     # @api private
     class VstPlugin < Moved
-      extend T::Sig
-
       sig { returns(String) }
       def self.english_name
         "VST Plugin"
diff --git a/Library/Homebrew/cask/audit.rb b/Library/Homebrew/cask/audit.rb
index 2798a41779..a99698862c 100644
--- a/Library/Homebrew/cask/audit.rb
+++ b/Library/Homebrew/cask/audit.rb
@@ -14,8 +14,6 @@ module Cask
   #
   # @api private
   class Audit
-    extend T::Sig
-
     extend Predicable
 
     attr_reader :cask, :download
diff --git a/Library/Homebrew/cask/cache.rb b/Library/Homebrew/cask/cache.rb
index 911a4eb833..62e7662e7f 100644
--- a/Library/Homebrew/cask/cache.rb
+++ b/Library/Homebrew/cask/cache.rb
@@ -6,8 +6,6 @@ module Cask
   #
   # @api private
   module Cache
-    extend T::Sig
-
     sig { returns(Pathname) }
     def self.path
       @path ||= HOMEBREW_CACHE/"Cask"
diff --git a/Library/Homebrew/cask/cask.rb b/Library/Homebrew/cask/cask.rb
index 4a161ea23e..8932110cf9 100644
--- a/Library/Homebrew/cask/cask.rb
+++ b/Library/Homebrew/cask/cask.rb
@@ -13,8 +13,6 @@ module Cask
   #
   # @api private
   class Cask
-    extend T::Sig
-
     extend Forwardable
     extend Predicable
     extend APIHashable
diff --git a/Library/Homebrew/cask/cask_loader.rb b/Library/Homebrew/cask/cask_loader.rb
index 7d20ee0af0..8e5fa074b9 100644
--- a/Library/Homebrew/cask/cask_loader.rb
+++ b/Library/Homebrew/cask/cask_loader.rb
@@ -13,7 +13,6 @@ module Cask
     extend Context
 
     module ILoader
-      extend T::Sig
       extend T::Helpers
       interface!
 
@@ -110,8 +109,6 @@ module Cask
 
     # Loads a cask from a URI.
     class FromURILoader < FromPathLoader
-      extend T::Sig
-
       def self.can_load?(ref)
         # Cache compiled regex
         @uri_regex ||= begin
@@ -362,8 +359,6 @@ module Cask
 
     # Pseudo-loader which raises an error when trying to load the corresponding cask.
     class NullLoader < FromPathLoader
-      extend T::Sig
-
       def self.can_load?(*)
         true
       end
diff --git a/Library/Homebrew/cask/caskroom.rb b/Library/Homebrew/cask/caskroom.rb
index 1d35d466d7..a478f3c3b6 100644
--- a/Library/Homebrew/cask/caskroom.rb
+++ b/Library/Homebrew/cask/caskroom.rb
@@ -8,8 +8,6 @@ module Cask
   #
   # @api private
   module Caskroom
-    extend T::Sig
-
     sig { returns(Pathname) }
     def self.path
       @path ||= HOMEBREW_PREFIX/"Caskroom"
diff --git a/Library/Homebrew/cask/config.rb b/Library/Homebrew/cask/config.rb
index 164f98c5d4..706ab32300 100644
--- a/Library/Homebrew/cask/config.rb
+++ b/Library/Homebrew/cask/config.rb
@@ -11,8 +11,6 @@ module Cask
   #
   # @api private
   class Config
-    extend T::Sig
-
     DEFAULT_DIRS = {
       appdir:               "/Applications",
       keyboard_layoutdir:   "/Library/Keyboard Layouts",
diff --git a/Library/Homebrew/cask/denylist.rb b/Library/Homebrew/cask/denylist.rb
index 6b3a0ca589..7afbcb46b1 100644
--- a/Library/Homebrew/cask/denylist.rb
+++ b/Library/Homebrew/cask/denylist.rb
@@ -6,8 +6,6 @@ module Cask
   #
   # @api private
   module Denylist
-    extend T::Sig
-
     sig { params(name: String).returns(T.nilable(String)) }
     def self.reason(name)
       case name
diff --git a/Library/Homebrew/cask/dsl/depends_on.rb b/Library/Homebrew/cask/dsl/depends_on.rb
index 9522f0553e..7af6ff40e0 100644
--- a/Library/Homebrew/cask/dsl/depends_on.rb
+++ b/Library/Homebrew/cask/dsl/depends_on.rb
@@ -11,7 +11,6 @@ module Cask
     #
     # @api private
     class DependsOn < SimpleDelegator
-      extend T::Sig
       VALID_KEYS = Set.new([
         :formula,
         :cask,
diff --git a/Library/Homebrew/cask/dsl/version.rb b/Library/Homebrew/cask/dsl/version.rb
index 2b5170406b..20d26a7195 100644
--- a/Library/Homebrew/cask/dsl/version.rb
+++ b/Library/Homebrew/cask/dsl/version.rb
@@ -7,8 +7,6 @@ module Cask
     #
     # @api private
     class Version < ::String
-      extend T::Sig
-
       DIVIDERS = {
         "." => :dots,
         "-" => :hyphens,
diff --git a/Library/Homebrew/cask/exceptions.rb b/Library/Homebrew/cask/exceptions.rb
index dcf76029fb..e37817b6c3 100644
--- a/Library/Homebrew/cask/exceptions.rb
+++ b/Library/Homebrew/cask/exceptions.rb
@@ -11,8 +11,6 @@ module Cask
   #
   # @api private
   class MultipleCaskErrors < CaskError
-    extend T::Sig
-
     def initialize(errors)
       super()
 
@@ -32,8 +30,6 @@ module Cask
   #
   # @api private
   class AbstractCaskErrorWithToken < CaskError
-    extend T::Sig
-
     sig { returns(String) }
     attr_reader :token
 
@@ -52,8 +48,6 @@ module Cask
   #
   # @api private
   class CaskNotInstalledError < AbstractCaskErrorWithToken
-    extend T::Sig
-
     sig { returns(String) }
     def to_s
       "Cask '#{token}' is not installed."
@@ -64,8 +58,6 @@ module Cask
   #
   # @api private
   class CaskConflictError < AbstractCaskErrorWithToken
-    extend T::Sig
-
     attr_reader :conflicting_cask
 
     def initialize(token, conflicting_cask)
@@ -83,8 +75,6 @@ module Cask
   #
   # @api private
   class CaskUnavailableError < AbstractCaskErrorWithToken
-    extend T::Sig
-
     sig { returns(String) }
     def to_s
       "Cask '#{token}' is unavailable#{reason.empty? ? "." : ": #{reason}"}"
@@ -95,8 +85,6 @@ module Cask
   #
   # @api private
   class CaskUnreadableError < CaskUnavailableError
-    extend T::Sig
-
     sig { returns(String) }
     def to_s
       "Cask '#{token}' is unreadable#{reason.empty? ? "." : ": #{reason}"}"
@@ -107,8 +95,6 @@ module Cask
   #
   # @api private
   class TapCaskUnavailableError < CaskUnavailableError
-    extend T::Sig
-
     attr_reader :tap
 
     def initialize(tap, token)
@@ -128,8 +114,6 @@ module Cask
   #
   # @api private
   class TapCaskAmbiguityError < CaskError
-    extend T::Sig
-
     def initialize(ref, loaders)
       super <<~EOS
         Cask #{ref} exists in multiple taps:
@@ -142,8 +126,6 @@ module Cask
   #
   # @api private
   class CaskAlreadyCreatedError < AbstractCaskErrorWithToken
-    extend T::Sig
-
     sig { returns(String) }
     def to_s
       %Q(Cask '#{token}' already exists. Run #{Formatter.identifier("brew edit --cask #{token}")} to edit it.)
@@ -154,8 +136,6 @@ module Cask
   #
   # @api private
   class CaskAlreadyInstalledError < AbstractCaskErrorWithToken
-    extend T::Sig
-
     sig { returns(String) }
     def to_s
       <<~EOS
@@ -171,8 +151,6 @@ module Cask
   #
   # @api private
   class CaskCyclicDependencyError < AbstractCaskErrorWithToken
-    extend T::Sig
-
     sig { returns(String) }
     def to_s
       "Cask '#{token}' includes cyclic dependencies on other Casks#{reason.empty? ? "." : ": #{reason}"}"
@@ -183,8 +161,6 @@ module Cask
   #
   # @api private
   class CaskSelfReferencingDependencyError < CaskCyclicDependencyError
-    extend T::Sig
-
     sig { returns(String) }
     def to_s
       "Cask '#{token}' depends on itself."
@@ -195,8 +171,6 @@ module Cask
   #
   # @api private
   class CaskUnspecifiedError < CaskError
-    extend T::Sig
-
     sig { returns(String) }
     def to_s
       "This command requires a Cask token."
@@ -207,8 +181,6 @@ module Cask
   #
   # @api private
   class CaskInvalidError < AbstractCaskErrorWithToken
-    extend T::Sig
-
     sig { returns(String) }
     def to_s
       "Cask '#{token}' definition is invalid#{reason.empty? ? "." : ": #{reason}"}"
@@ -228,8 +200,6 @@ module Cask
   #
   # @api private
   class CaskQuarantineError < CaskError
-    extend T::Sig
-
     attr_reader :path, :reason
 
     def initialize(path, reason)
@@ -257,8 +227,6 @@ module Cask
   #
   # @api private
   class CaskQuarantinePropagationError < CaskQuarantineError
-    extend T::Sig
-
     sig { returns(String) }
     def to_s
       s = +"Failed to quarantine one or more files within #{path}."
@@ -277,8 +245,6 @@ module Cask
   #
   # @api private
   class CaskQuarantineReleaseError < CaskQuarantineError
-    extend T::Sig
-
     sig { returns(String) }
     def to_s
       s = +"Failed to release #{path} from quarantine."
diff --git a/Library/Homebrew/cask/installer.rb b/Library/Homebrew/cask/installer.rb
index 1321e8b32e..0ae2d8e935 100644
--- a/Library/Homebrew/cask/installer.rb
+++ b/Library/Homebrew/cask/installer.rb
@@ -16,8 +16,6 @@ module Cask
   #
   # @api private
   class Installer
-    extend T::Sig
-
     extend Predicable
 
     def initialize(cask, command: SystemCommand, force: false, adopt: false,
diff --git a/Library/Homebrew/cask/pkg.rb b/Library/Homebrew/cask/pkg.rb
index 83447c4b6e..cc3d4d06cb 100644
--- a/Library/Homebrew/cask/pkg.rb
+++ b/Library/Homebrew/cask/pkg.rb
@@ -8,8 +8,6 @@ module Cask
   #
   # @api private
   class Pkg
-    extend T::Sig
-
     sig { params(regexp: String, command: T.class_of(SystemCommand)).returns(T::Array[Pkg]) }
     def self.all_matching(regexp, command)
       command.run("/usr/sbin/pkgutil", args: ["--pkgs=#{regexp}"]).stdout.split("\n").map do |package_id|
diff --git a/Library/Homebrew/cask/quarantine.rb b/Library/Homebrew/cask/quarantine.rb
index 7333b4e231..d006d096ff 100644
--- a/Library/Homebrew/cask/quarantine.rb
+++ b/Library/Homebrew/cask/quarantine.rb
@@ -9,8 +9,6 @@ module Cask
   #
   # @api private
   module Quarantine
-    extend T::Sig
-
     QUARANTINE_ATTRIBUTE = "com.apple.quarantine"
 
     QUARANTINE_SCRIPT = (HOMEBREW_LIBRARY_PATH/"cask/utils/quarantine.swift").freeze
diff --git a/Library/Homebrew/cask/staged.rb b/Library/Homebrew/cask/staged.rb
index bc05bb0b79..bd7b2f2a53 100644
--- a/Library/Homebrew/cask/staged.rb
+++ b/Library/Homebrew/cask/staged.rb
@@ -8,8 +8,6 @@ module Cask
   #
   # @api private
   module Staged
-    extend T::Sig
-
     # FIXME: Enable cop again when https://github.com/sorbet/sorbet/issues/3532 is fixed.
     # rubocop:disable Style/MutableConstant
     Paths = T.type_alias { T.any(String, Pathname, T::Array[T.any(String, Pathname)]) }
diff --git a/Library/Homebrew/cask/upgrade.rb b/Library/Homebrew/cask/upgrade.rb
index 41c52954f9..ac7e53c9f4 100644
--- a/Library/Homebrew/cask/upgrade.rb
+++ b/Library/Homebrew/cask/upgrade.rb
@@ -7,8 +7,6 @@ require "cask/config"
 module Cask
   # @api private
   class Upgrade
-    extend T::Sig
-
     sig {
       params(
         casks:               Cask,
diff --git a/Library/Homebrew/cask/url.rb b/Library/Homebrew/cask/url.rb
index 09bc6ad9c7..f7ef0197d3 100644
--- a/Library/Homebrew/cask/url.rb
+++ b/Library/Homebrew/cask/url.rb
@@ -5,12 +5,8 @@
 #
 # @api private
 class URL < Delegator
-  extend T::Sig
-
   # @api private
   class DSL
-    extend T::Sig
-
     attr_reader :uri, :specs,
                 :verified, :using,
                 :tag, :branch, :revisions, :revision,
@@ -79,12 +75,8 @@ class URL < Delegator
 
   # @api private
   class BlockDSL
-    extend T::Sig
-
     # To access URL associated with page contents.
     module PageWithURL
-      extend T::Sig
-
       # @api public
       sig { returns(URI::Generic) }
       attr_accessor :url
diff --git a/Library/Homebrew/cask/utils.rb b/Library/Homebrew/cask/utils.rb
index c512e898d6..dd5ba92ca0 100644
--- a/Library/Homebrew/cask/utils.rb
+++ b/Library/Homebrew/cask/utils.rb
@@ -11,8 +11,6 @@ module Cask
   #
   # @api private
   module Utils
-    extend T::Sig
-
     def self.gain_permissions_remove(path, command: SystemCommand)
       if path.respond_to?(:rmtree) && path.exist?
         gain_permissions(path, ["-R"], command) do |p|
diff --git a/Library/Homebrew/cleanup.rb b/Library/Homebrew/cleanup.rb
index cc690902b6..2439afcb3e 100644
--- a/Library/Homebrew/cleanup.rb
+++ b/Library/Homebrew/cleanup.rb
@@ -16,8 +16,6 @@ module Homebrew
     private_constant :CLEANUP_DEFAULT_DAYS
 
     class << self
-      extend T::Sig
-
       sig { params(pathname: Pathname).returns(T::Boolean) }
       def incomplete?(pathname)
         pathname.extname.end_with?(".incomplete")
diff --git a/Library/Homebrew/cli/args.rb b/Library/Homebrew/cli/args.rb
index 414448fcb7..cf49a4dcc2 100644
--- a/Library/Homebrew/cli/args.rb
+++ b/Library/Homebrew/cli/args.rb
@@ -6,8 +6,6 @@ require "ostruct"
 module Homebrew
   module CLI
     class Args < OpenStruct
-      extend T::Sig
-
       attr_reader :options_only, :flags_only
 
       # undefine tap to allow --tap argument
diff --git a/Library/Homebrew/cli/named_args.rb b/Library/Homebrew/cli/named_args.rb
index d87ec0b655..97646e4d67 100644
--- a/Library/Homebrew/cli/named_args.rb
+++ b/Library/Homebrew/cli/named_args.rb
@@ -11,8 +11,6 @@ module Homebrew
     #
     # @api private
     class NamedArgs < Array
-      extend T::Sig
-
       def initialize(*args, parent: Args.new, override_spec: nil, force_bottle: false, flags: [], cask_options: false)
         require "cask/cask"
         require "cask/cask_loader"
diff --git a/Library/Homebrew/cli/parser.rb b/Library/Homebrew/cli/parser.rb
index 0fc1cf2f09..76e7dcc8b0 100644
--- a/Library/Homebrew/cli/parser.rb
+++ b/Library/Homebrew/cli/parser.rb
@@ -15,8 +15,6 @@ HIDDEN_DESC_PLACEHOLDER = "@@HIDDEN@@"
 module Homebrew
   module CLI
     class Parser
-      extend T::Sig
-
       attr_reader :processed_options, :hide_from_man_page, :named_args_type
 
       def self.from_cmd_path(cmd_path)
@@ -680,8 +678,6 @@ module Homebrew
     end
 
     class MaxNamedArgumentsError < UsageError
-      extend T::Sig
-
       sig { params(maximum: Integer, types: T::Array[Symbol]).void }
       def initialize(maximum, types: [])
         super case maximum
@@ -698,8 +694,6 @@ module Homebrew
     end
 
     class MinNamedArgumentsError < UsageError
-      extend T::Sig
-
       sig { params(minimum: Integer, types: T::Array[Symbol]).void }
       def initialize(minimum, types: [])
         types << :named if types.empty?
@@ -711,8 +705,6 @@ module Homebrew
     end
 
     class NumberOfNamedArgumentsError < UsageError
-      extend T::Sig
-
       sig { params(minimum: Integer, types: T::Array[Symbol]).void }
       def initialize(minimum, types: [])
         types << :named if types.empty?
diff --git a/Library/Homebrew/cmd/--cache.rb b/Library/Homebrew/cmd/--cache.rb
index 00d0847748..26cb74200d 100644
--- a/Library/Homebrew/cmd/--cache.rb
+++ b/Library/Homebrew/cmd/--cache.rb
@@ -6,8 +6,6 @@ require "cli/parser"
 require "cask/download"
 
 module Homebrew
-  extend T::Sig
-
   extend Fetch
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/--caskroom.rb b/Library/Homebrew/cmd/--caskroom.rb
index 681cdf0331..e3d64ec694 100644
--- a/Library/Homebrew/cmd/--caskroom.rb
+++ b/Library/Homebrew/cmd/--caskroom.rb
@@ -2,8 +2,6 @@
 # frozen_string_literal: true
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/--env.rb b/Library/Homebrew/cmd/--env.rb
index f9814ccd94..84a4e46f7e 100644
--- a/Library/Homebrew/cmd/--env.rb
+++ b/Library/Homebrew/cmd/--env.rb
@@ -7,8 +7,6 @@ require "utils/shell"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/--prefix.rb b/Library/Homebrew/cmd/--prefix.rb
index 7204cfade7..95e21d38d6 100644
--- a/Library/Homebrew/cmd/--prefix.rb
+++ b/Library/Homebrew/cmd/--prefix.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.__prefix_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/cmd/--repository.rb b/Library/Homebrew/cmd/--repository.rb
index c72c0510fe..0d15e532ec 100644
--- a/Library/Homebrew/cmd/--repository.rb
+++ b/Library/Homebrew/cmd/--repository.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/analytics.rb b/Library/Homebrew/cmd/analytics.rb
index 984ad24db1..b6d622ab71 100644
--- a/Library/Homebrew/cmd/analytics.rb
+++ b/Library/Homebrew/cmd/analytics.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/cleanup.rb b/Library/Homebrew/cmd/cleanup.rb
index 897cd07df5..885a7bd524 100644
--- a/Library/Homebrew/cmd/cleanup.rb
+++ b/Library/Homebrew/cmd/cleanup.rb
@@ -5,8 +5,6 @@ require "cleanup"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/commands.rb b/Library/Homebrew/cmd/commands.rb
index f7ea8186b4..5f87e44284 100644
--- a/Library/Homebrew/cmd/commands.rb
+++ b/Library/Homebrew/cmd/commands.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/completions.rb b/Library/Homebrew/cmd/completions.rb
index 4003dd0de0..c9655658e0 100644
--- a/Library/Homebrew/cmd/completions.rb
+++ b/Library/Homebrew/cmd/completions.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "completions"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/config.rb b/Library/Homebrew/cmd/config.rb
index 7271881e56..b71022140c 100644
--- a/Library/Homebrew/cmd/config.rb
+++ b/Library/Homebrew/cmd/config.rb
@@ -5,8 +5,6 @@ require "system_config"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/deps.rb b/Library/Homebrew/cmd/deps.rb
index 444da12f15..56a0db6dca 100644
--- a/Library/Homebrew/cmd/deps.rb
+++ b/Library/Homebrew/cmd/deps.rb
@@ -7,8 +7,6 @@ require "cask/caskroom"
 require "dependencies_helpers"
 
 module Homebrew
-  extend T::Sig
-
   extend DependenciesHelpers
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/desc.rb b/Library/Homebrew/cmd/desc.rb
index 8abbf08588..c415d7526e 100644
--- a/Library/Homebrew/cmd/desc.rb
+++ b/Library/Homebrew/cmd/desc.rb
@@ -7,8 +7,6 @@ require "description_cache_store"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/developer.rb b/Library/Homebrew/cmd/developer.rb
index 131f18cef5..9f3ebc30ea 100755
--- a/Library/Homebrew/cmd/developer.rb
+++ b/Library/Homebrew/cmd/developer.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/docs.rb b/Library/Homebrew/cmd/docs.rb
index 31aa410df3..f3d7d713d1 100644
--- a/Library/Homebrew/cmd/docs.rb
+++ b/Library/Homebrew/cmd/docs.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/doctor.rb b/Library/Homebrew/cmd/doctor.rb
index d5956b2242..cfe47fae8d 100644
--- a/Library/Homebrew/cmd/doctor.rb
+++ b/Library/Homebrew/cmd/doctor.rb
@@ -6,8 +6,6 @@ require "cli/parser"
 require "cask/caskroom"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.doctor_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/cmd/fetch.rb b/Library/Homebrew/cmd/fetch.rb
index af75f9995e..744e317b29 100644
--- a/Library/Homebrew/cmd/fetch.rb
+++ b/Library/Homebrew/cmd/fetch.rb
@@ -7,8 +7,6 @@ require "cli/parser"
 require "cask/download"
 
 module Homebrew
-  extend T::Sig
-
   extend Fetch
 
   FETCH_MAX_TRIES = 5
diff --git a/Library/Homebrew/cmd/gist-logs.rb b/Library/Homebrew/cmd/gist-logs.rb
index 9bf01bc759..dba4217ee2 100644
--- a/Library/Homebrew/cmd/gist-logs.rb
+++ b/Library/Homebrew/cmd/gist-logs.rb
@@ -9,8 +9,6 @@ require "socket"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   extend Install
 
   module_function
diff --git a/Library/Homebrew/cmd/home.rb b/Library/Homebrew/cmd/home.rb
index 649c3d09da..8fe5bcd9a6 100644
--- a/Library/Homebrew/cmd/home.rb
+++ b/Library/Homebrew/cmd/home.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "formula"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/info.rb b/Library/Homebrew/cmd/info.rb
index 0d0580471b..266cf96b82 100644
--- a/Library/Homebrew/cmd/info.rb
+++ b/Library/Homebrew/cmd/info.rb
@@ -14,8 +14,6 @@ require "deprecate_disable"
 require "api"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   VALID_DAYS = %w[30 90 365].freeze
diff --git a/Library/Homebrew/cmd/install.rb b/Library/Homebrew/cmd/install.rb
index 36bf3e2160..3741bd6b2a 100644
--- a/Library/Homebrew/cmd/install.rb
+++ b/Library/Homebrew/cmd/install.rb
@@ -13,8 +13,6 @@ require "cli/parser"
 require "upgrade"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.install_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/cmd/leaves.rb b/Library/Homebrew/cmd/leaves.rb
index 621d64ae84..76fb8784ca 100644
--- a/Library/Homebrew/cmd/leaves.rb
+++ b/Library/Homebrew/cmd/leaves.rb
@@ -5,8 +5,6 @@ require "formula"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/link.rb b/Library/Homebrew/cmd/link.rb
index 9bacec52f3..3b6520e91c 100644
--- a/Library/Homebrew/cmd/link.rb
+++ b/Library/Homebrew/cmd/link.rb
@@ -6,8 +6,6 @@ require "cli/parser"
 require "unlink"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/list.rb b/Library/Homebrew/cmd/list.rb
index 2a5ee074cb..624be479f3 100644
--- a/Library/Homebrew/cmd/list.rb
+++ b/Library/Homebrew/cmd/list.rb
@@ -7,8 +7,6 @@ require "cli/parser"
 require "cask/list"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/log.rb b/Library/Homebrew/cmd/log.rb
index b598f00a3c..a0f56165b4 100644
--- a/Library/Homebrew/cmd/log.rb
+++ b/Library/Homebrew/cmd/log.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.log_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/cmd/migrate.rb b/Library/Homebrew/cmd/migrate.rb
index 68cae13b40..1a50c1eddb 100644
--- a/Library/Homebrew/cmd/migrate.rb
+++ b/Library/Homebrew/cmd/migrate.rb
@@ -5,8 +5,6 @@ require "migrator"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/missing.rb b/Library/Homebrew/cmd/missing.rb
index d4784c683d..5cd689b13b 100644
--- a/Library/Homebrew/cmd/missing.rb
+++ b/Library/Homebrew/cmd/missing.rb
@@ -7,8 +7,6 @@ require "diagnostic"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/options.rb b/Library/Homebrew/cmd/options.rb
index 8263c1cac3..6d3af41063 100644
--- a/Library/Homebrew/cmd/options.rb
+++ b/Library/Homebrew/cmd/options.rb
@@ -5,8 +5,6 @@ require "formula"
 require "options"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/outdated.rb b/Library/Homebrew/cmd/outdated.rb
index 677d2d6878..6df2c6e2d2 100644
--- a/Library/Homebrew/cmd/outdated.rb
+++ b/Library/Homebrew/cmd/outdated.rb
@@ -8,8 +8,6 @@ require "cask/caskroom"
 require "api"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.outdated_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/cmd/pin.rb b/Library/Homebrew/cmd/pin.rb
index 304105ffe9..cd5d91edbf 100644
--- a/Library/Homebrew/cmd/pin.rb
+++ b/Library/Homebrew/cmd/pin.rb
@@ -5,8 +5,6 @@ require "formula"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/postinstall.rb b/Library/Homebrew/cmd/postinstall.rb
index 821195f6dd..93ed03f059 100644
--- a/Library/Homebrew/cmd/postinstall.rb
+++ b/Library/Homebrew/cmd/postinstall.rb
@@ -6,8 +6,6 @@ require "formula_installer"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/readall.rb b/Library/Homebrew/cmd/readall.rb
index 98efb088a9..5659889cec 100644
--- a/Library/Homebrew/cmd/readall.rb
+++ b/Library/Homebrew/cmd/readall.rb
@@ -6,8 +6,6 @@ require "cli/parser"
 require "env_config"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/reinstall.rb b/Library/Homebrew/cmd/reinstall.rb
index 1aa5c77e51..6a88b3eb0e 100644
--- a/Library/Homebrew/cmd/reinstall.rb
+++ b/Library/Homebrew/cmd/reinstall.rb
@@ -15,8 +15,6 @@ require "upgrade"
 require "api"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.reinstall_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/cmd/search.rb b/Library/Homebrew/cmd/search.rb
index 064b10911b..43733bb427 100644
--- a/Library/Homebrew/cmd/search.rb
+++ b/Library/Homebrew/cmd/search.rb
@@ -8,8 +8,6 @@ require "cli/parser"
 require "search"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   PACKAGE_MANAGERS = {
diff --git a/Library/Homebrew/cmd/tap-info.rb b/Library/Homebrew/cmd/tap-info.rb
index cea6eb2528..281db2a6c1 100644
--- a/Library/Homebrew/cmd/tap-info.rb
+++ b/Library/Homebrew/cmd/tap-info.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/tap.rb b/Library/Homebrew/cmd/tap.rb
index 2361e6b0e9..7b1a9c1259 100644
--- a/Library/Homebrew/cmd/tap.rb
+++ b/Library/Homebrew/cmd/tap.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/uninstall.rb b/Library/Homebrew/cmd/uninstall.rb
index e0db1db786..65642fb159 100644
--- a/Library/Homebrew/cmd/uninstall.rb
+++ b/Library/Homebrew/cmd/uninstall.rb
@@ -13,8 +13,6 @@ require "cask/uninstall"
 require "uninstall"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/unlink.rb b/Library/Homebrew/cmd/unlink.rb
index 5034b6a2a7..f7572d15f6 100644
--- a/Library/Homebrew/cmd/unlink.rb
+++ b/Library/Homebrew/cmd/unlink.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "unlink"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/unpin.rb b/Library/Homebrew/cmd/unpin.rb
index 7e287d5956..9b26e4eda5 100644
--- a/Library/Homebrew/cmd/unpin.rb
+++ b/Library/Homebrew/cmd/unpin.rb
@@ -5,8 +5,6 @@ require "formula"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/untap.rb b/Library/Homebrew/cmd/untap.rb
index 89b8421536..bf12b54b8d 100644
--- a/Library/Homebrew/cmd/untap.rb
+++ b/Library/Homebrew/cmd/untap.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/update-report.rb b/Library/Homebrew/cmd/update-report.rb
index 41e67f8c9f..0292c9f8eb 100644
--- a/Library/Homebrew/cmd/update-report.rb
+++ b/Library/Homebrew/cmd/update-report.rb
@@ -11,8 +11,6 @@ require "settings"
 require "linuxbrew-core-migration"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   def auto_update_header(args:)
@@ -607,8 +605,6 @@ class Reporter
 end
 
 class ReporterHub
-  extend T::Sig
-
   extend Forwardable
 
   attr_reader :reporters
diff --git a/Library/Homebrew/cmd/upgrade.rb b/Library/Homebrew/cmd/upgrade.rb
index c30792e1c3..5480a2ec5a 100644
--- a/Library/Homebrew/cmd/upgrade.rb
+++ b/Library/Homebrew/cmd/upgrade.rb
@@ -11,8 +11,6 @@ require "cask/macos"
 require "api"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/cmd/uses.rb b/Library/Homebrew/cmd/uses.rb
index 9b77b71723..451eb6d447 100644
--- a/Library/Homebrew/cmd/uses.rb
+++ b/Library/Homebrew/cmd/uses.rb
@@ -12,8 +12,6 @@ require "dependencies_helpers"
 require "ostruct"
 
 module Homebrew
-  extend T::Sig
-
   extend DependenciesHelpers
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/compilers.rb b/Library/Homebrew/compilers.rb
index 225bf27a8d..12ac6cdb22 100644
--- a/Library/Homebrew/compilers.rb
+++ b/Library/Homebrew/compilers.rb
@@ -100,7 +100,6 @@ end
 #
 # @api private
 class CompilerSelector
-  extend T::Sig
   include CompilerConstants
 
   Compiler = Struct.new(:type, :name, :version)
diff --git a/Library/Homebrew/completions.rb b/Library/Homebrew/completions.rb
index e928a5b837..b61499a6e8 100644
--- a/Library/Homebrew/completions.rb
+++ b/Library/Homebrew/completions.rb
@@ -10,8 +10,6 @@ module Homebrew
   #
   # @api private
   module Completions
-    extend T::Sig
-
     Variables = Struct.new(
       :aliases,
       :builtin_command_descriptions,
diff --git a/Library/Homebrew/cxxstdlib.rb b/Library/Homebrew/cxxstdlib.rb
index ed43d0c387..a87523c472 100644
--- a/Library/Homebrew/cxxstdlib.rb
+++ b/Library/Homebrew/cxxstdlib.rb
@@ -5,8 +5,6 @@ require "compilers"
 
 # Combination of C++ standard library and compiler.
 class CxxStdlib
-  extend T::Sig
-
   def self.create(type, compiler)
     raise ArgumentError, "Invalid C++ stdlib type: #{type}" if type && [:libstdcxx, :libcxx].exclude?(type)
 
diff --git a/Library/Homebrew/debrew.rb b/Library/Homebrew/debrew.rb
index 05acc2da8b..b64ca304ff 100644
--- a/Library/Homebrew/debrew.rb
+++ b/Library/Homebrew/debrew.rb
@@ -28,8 +28,6 @@ module Debrew
 
   # Module for displaying a debugging menu.
   class Menu
-    extend T::Sig
-
     Entry = Struct.new(:name, :action)
 
     attr_accessor :prompt, :entries
diff --git a/Library/Homebrew/dependencies.rb b/Library/Homebrew/dependencies.rb
index fedb68cb72..018c9f438e 100644
--- a/Library/Homebrew/dependencies.rb
+++ b/Library/Homebrew/dependencies.rb
@@ -7,8 +7,6 @@ require "delegate"
 #
 # @api private
 class Dependencies < SimpleDelegator
-  extend T::Sig
-
   def initialize(*args)
     super(args)
   end
@@ -45,8 +43,6 @@ end
 #
 # @api private
 class Requirements < SimpleDelegator
-  extend T::Sig
-
   def initialize(*args)
     super(Set.new(args))
   end
diff --git a/Library/Homebrew/dependency.rb b/Library/Homebrew/dependency.rb
index 15ea12ecbd..0879cf268e 100644
--- a/Library/Homebrew/dependency.rb
+++ b/Library/Homebrew/dependency.rb
@@ -7,8 +7,6 @@ require "dependable"
 #
 # @api private
 class Dependency
-  extend T::Sig
-
   extend Forwardable
   include Dependable
   extend Cachable
@@ -82,8 +80,6 @@ class Dependency
   end
 
   class << self
-    extend T::Sig
-
     # Expand the dependencies of each dependent recursively, optionally yielding
     # `[dependent, dep]` pairs to allow callers to apply arbitrary filters to
     # the list.
diff --git a/Library/Homebrew/dependency_collector.rb b/Library/Homebrew/dependency_collector.rb
index 28e028b5ca..abaead8394 100644
--- a/Library/Homebrew/dependency_collector.rb
+++ b/Library/Homebrew/dependency_collector.rb
@@ -18,8 +18,6 @@ require "extend/cachable"
 # This class is used by `depends_on` in the formula DSL to turn dependency
 # specifications into the proper kinds of dependencies and requirements.
 class DependencyCollector
-  extend T::Sig
-
   extend Cachable
 
   attr_reader :deps, :requirements
diff --git a/Library/Homebrew/dev-cmd/audit.rb b/Library/Homebrew/dev-cmd/audit.rb
index 702b34a926..e8575cbf3b 100644
--- a/Library/Homebrew/dev-cmd/audit.rb
+++ b/Library/Homebrew/dev-cmd/audit.rb
@@ -20,8 +20,6 @@ require "formula_auditor"
 require "tap_auditor"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.audit_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/dev-cmd/bottle.rb b/Library/Homebrew/dev-cmd/bottle.rb
index 48bd0f9e9e..610c3e4658 100644
--- a/Library/Homebrew/dev-cmd/bottle.rb
+++ b/Library/Homebrew/dev-cmd/bottle.rb
@@ -36,8 +36,6 @@ ALLOWABLE_HOMEBREW_REPOSITORY_LINKS = [
 ].freeze
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.bottle_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/dev-cmd/bump-cask-pr.rb b/Library/Homebrew/dev-cmd/bump-cask-pr.rb
index 6d9af45ed9..c3a0cf1e63 100644
--- a/Library/Homebrew/dev-cmd/bump-cask-pr.rb
+++ b/Library/Homebrew/dev-cmd/bump-cask-pr.rb
@@ -7,8 +7,6 @@ require "cli/parser"
 require "utils/tar"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/bump-formula-pr.rb b/Library/Homebrew/dev-cmd/bump-formula-pr.rb
index abb1465d88..c963291903 100644
--- a/Library/Homebrew/dev-cmd/bump-formula-pr.rb
+++ b/Library/Homebrew/dev-cmd/bump-formula-pr.rb
@@ -7,8 +7,6 @@ require "utils/pypi"
 require "utils/tar"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/bump-revision.rb b/Library/Homebrew/dev-cmd/bump-revision.rb
index 897d8e67cb..78aa44c8f7 100644
--- a/Library/Homebrew/dev-cmd/bump-revision.rb
+++ b/Library/Homebrew/dev-cmd/bump-revision.rb
@@ -5,8 +5,6 @@ require "formula"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/bump-unversioned-casks.rb b/Library/Homebrew/dev-cmd/bump-unversioned-casks.rb
index ed363ed391..f6eafe1567 100644
--- a/Library/Homebrew/dev-cmd/bump-unversioned-casks.rb
+++ b/Library/Homebrew/dev-cmd/bump-unversioned-casks.rb
@@ -10,8 +10,6 @@ require "tap"
 require "unversioned_cask_checker"
 
 module Homebrew
-  extend T::Sig
-
   extend SystemCommand::Mixin
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/bump.rb b/Library/Homebrew/dev-cmd/bump.rb
index 50916f9a2f..9dd4d193f3 100644
--- a/Library/Homebrew/dev-cmd/bump.rb
+++ b/Library/Homebrew/dev-cmd/bump.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "livecheck/livecheck"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/cat.rb b/Library/Homebrew/dev-cmd/cat.rb
index 7b9faa1c25..6a1cb11791 100644
--- a/Library/Homebrew/dev-cmd/cat.rb
+++ b/Library/Homebrew/dev-cmd/cat.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.cat_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/dev-cmd/command.rb b/Library/Homebrew/dev-cmd/command.rb
index 0a9aa85bfa..18afa2316b 100644
--- a/Library/Homebrew/dev-cmd/command.rb
+++ b/Library/Homebrew/dev-cmd/command.rb
@@ -5,8 +5,6 @@ require "commands"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/contributions.rb b/Library/Homebrew/dev-cmd/contributions.rb
index 3abc22f96e..ca60d25a4b 100755
--- a/Library/Homebrew/dev-cmd/contributions.rb
+++ b/Library/Homebrew/dev-cmd/contributions.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "csv"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   PRIMARY_REPOS = %w[brew core cask].freeze
diff --git a/Library/Homebrew/dev-cmd/create.rb b/Library/Homebrew/dev-cmd/create.rb
index 678cd5409b..5cbba2bd73 100644
--- a/Library/Homebrew/dev-cmd/create.rb
+++ b/Library/Homebrew/dev-cmd/create.rb
@@ -9,8 +9,6 @@ require "utils/pypi"
 require "cask/cask_loader"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/determine-test-runners.rb b/Library/Homebrew/dev-cmd/determine-test-runners.rb
index c3903988fe..d85319e99e 100755
--- a/Library/Homebrew/dev-cmd/determine-test-runners.rb
+++ b/Library/Homebrew/dev-cmd/determine-test-runners.rb
@@ -6,8 +6,6 @@ require "test_runner_formula"
 require "github_runner_matrix"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(Homebrew::CLI::Parser) }
   def self.determine_test_runners_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/dev-cmd/dispatch-build-bottle.rb b/Library/Homebrew/dev-cmd/dispatch-build-bottle.rb
index 7ed05e1abf..dfd525d5bc 100644
--- a/Library/Homebrew/dev-cmd/dispatch-build-bottle.rb
+++ b/Library/Homebrew/dev-cmd/dispatch-build-bottle.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "utils/github"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/edit.rb b/Library/Homebrew/dev-cmd/edit.rb
index 04b1bb39f6..af1a5eb2b2 100644
--- a/Library/Homebrew/dev-cmd/edit.rb
+++ b/Library/Homebrew/dev-cmd/edit.rb
@@ -5,8 +5,6 @@ require "formula"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/extract.rb b/Library/Homebrew/dev-cmd/extract.rb
index ac002199cc..8c6bab1acc 100644
--- a/Library/Homebrew/dev-cmd/extract.rb
+++ b/Library/Homebrew/dev-cmd/extract.rb
@@ -69,8 +69,6 @@ ensure
 end
 
 module Homebrew
-  extend T::Sig
-
   BOTTLE_BLOCK_REGEX = /  bottle (?:do.+?end|:[a-z]+)\n\n/m.freeze
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/formula.rb b/Library/Homebrew/dev-cmd/formula.rb
index ec5dca7997..66afd0b7e7 100644
--- a/Library/Homebrew/dev-cmd/formula.rb
+++ b/Library/Homebrew/dev-cmd/formula.rb
@@ -5,8 +5,6 @@ require "formula"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/generate-cask-api.rb b/Library/Homebrew/dev-cmd/generate-cask-api.rb
index 614cf5a7ff..6d001a287d 100644
--- a/Library/Homebrew/dev-cmd/generate-cask-api.rb
+++ b/Library/Homebrew/dev-cmd/generate-cask-api.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "cask/cask"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/generate-formula-api.rb b/Library/Homebrew/dev-cmd/generate-formula-api.rb
index fb05cb187c..b5c2aded77 100644
--- a/Library/Homebrew/dev-cmd/generate-formula-api.rb
+++ b/Library/Homebrew/dev-cmd/generate-formula-api.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "formula"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/generate-man-completions.rb b/Library/Homebrew/dev-cmd/generate-man-completions.rb
index 8c48755473..25dd019278 100644
--- a/Library/Homebrew/dev-cmd/generate-man-completions.rb
+++ b/Library/Homebrew/dev-cmd/generate-man-completions.rb
@@ -6,8 +6,6 @@ require "completions"
 require "manpages"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/install-bundler-gems.rb b/Library/Homebrew/dev-cmd/install-bundler-gems.rb
index 742ed9ef6d..45cf04e19c 100644
--- a/Library/Homebrew/dev-cmd/install-bundler-gems.rb
+++ b/Library/Homebrew/dev-cmd/install-bundler-gems.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/irb.rb b/Library/Homebrew/dev-cmd/irb.rb
index 07e42818f3..8b80ed8bb3 100644
--- a/Library/Homebrew/dev-cmd/irb.rb
+++ b/Library/Homebrew/dev-cmd/irb.rb
@@ -26,8 +26,6 @@ class Symbol
 end
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/linkage.rb b/Library/Homebrew/dev-cmd/linkage.rb
index 55431777a2..1eee527d66 100644
--- a/Library/Homebrew/dev-cmd/linkage.rb
+++ b/Library/Homebrew/dev-cmd/linkage.rb
@@ -6,8 +6,6 @@ require "linkage_checker"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/livecheck.rb b/Library/Homebrew/dev-cmd/livecheck.rb
index 0fa2910d3e..a63359cf6e 100644
--- a/Library/Homebrew/dev-cmd/livecheck.rb
+++ b/Library/Homebrew/dev-cmd/livecheck.rb
@@ -7,8 +7,6 @@ require "livecheck/livecheck"
 require "livecheck/strategy"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   WATCHLIST_PATH = File.expand_path(Homebrew::EnvConfig.livecheck_watchlist).freeze
diff --git a/Library/Homebrew/dev-cmd/pr-automerge.rb b/Library/Homebrew/dev-cmd/pr-automerge.rb
index 2ab5e631ba..e3d698f78f 100644
--- a/Library/Homebrew/dev-cmd/pr-automerge.rb
+++ b/Library/Homebrew/dev-cmd/pr-automerge.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "utils/github"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/pr-publish.rb b/Library/Homebrew/dev-cmd/pr-publish.rb
index c6c64dd1aa..da152254ce 100644
--- a/Library/Homebrew/dev-cmd/pr-publish.rb
+++ b/Library/Homebrew/dev-cmd/pr-publish.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "utils/github"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/pr-pull.rb b/Library/Homebrew/dev-cmd/pr-pull.rb
index 258c2e8c26..1d1384df57 100644
--- a/Library/Homebrew/dev-cmd/pr-pull.rb
+++ b/Library/Homebrew/dev-cmd/pr-pull.rb
@@ -8,8 +8,6 @@ require "tmpdir"
 require "formula"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.pr_pull_args
     Homebrew::CLI::Parser.new do
@@ -530,8 +528,6 @@ module Homebrew
 end
 
 class GitHubArtifactDownloadStrategy < AbstractFileDownloadStrategy
-  extend T::Sig
-
   def fetch(timeout: nil)
     ohai "Downloading #{url}"
     if cached_location.exist?
diff --git a/Library/Homebrew/dev-cmd/pr-upload.rb b/Library/Homebrew/dev-cmd/pr-upload.rb
index c966dcde1c..8b7f3128b0 100644
--- a/Library/Homebrew/dev-cmd/pr-upload.rb
+++ b/Library/Homebrew/dev-cmd/pr-upload.rb
@@ -6,8 +6,6 @@ require "github_packages"
 require "github_releases"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/prof.rb b/Library/Homebrew/dev-cmd/prof.rb
index 04bbc62079..6457c08f60 100644
--- a/Library/Homebrew/dev-cmd/prof.rb
+++ b/Library/Homebrew/dev-cmd/prof.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/release.rb b/Library/Homebrew/dev-cmd/release.rb
index e7e87a4486..bd981a221b 100755
--- a/Library/Homebrew/dev-cmd/release.rb
+++ b/Library/Homebrew/dev-cmd/release.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/ruby.rb b/Library/Homebrew/dev-cmd/ruby.rb
index 5db9c72641..9cdfe64d1e 100644
--- a/Library/Homebrew/dev-cmd/ruby.rb
+++ b/Library/Homebrew/dev-cmd/ruby.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/sh.rb b/Library/Homebrew/dev-cmd/sh.rb
index 4fbc9e2e78..7ac99cb5fa 100644
--- a/Library/Homebrew/dev-cmd/sh.rb
+++ b/Library/Homebrew/dev-cmd/sh.rb
@@ -6,8 +6,6 @@ require "formula"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/style.rb b/Library/Homebrew/dev-cmd/style.rb
index 01744ac29b..f186aada1e 100644
--- a/Library/Homebrew/dev-cmd/style.rb
+++ b/Library/Homebrew/dev-cmd/style.rb
@@ -7,8 +7,6 @@ require "style"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/tap-new.rb b/Library/Homebrew/dev-cmd/tap-new.rb
index 510b1e3030..263922b2ff 100644
--- a/Library/Homebrew/dev-cmd/tap-new.rb
+++ b/Library/Homebrew/dev-cmd/tap-new.rb
@@ -5,8 +5,6 @@ require "tap"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.tap_new_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/dev-cmd/test.rb b/Library/Homebrew/dev-cmd/test.rb
index 0641fe8ad9..7187f77337 100644
--- a/Library/Homebrew/dev-cmd/test.rb
+++ b/Library/Homebrew/dev-cmd/test.rb
@@ -7,8 +7,6 @@ require "timeout"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/tests.rb b/Library/Homebrew/dev-cmd/tests.rb
index fb1e8f3c19..41bb42c048 100644
--- a/Library/Homebrew/dev-cmd/tests.rb
+++ b/Library/Homebrew/dev-cmd/tests.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "fileutils"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/typecheck.rb b/Library/Homebrew/dev-cmd/typecheck.rb
index 37805c83ba..beb5878cba 100644
--- a/Library/Homebrew/dev-cmd/typecheck.rb
+++ b/Library/Homebrew/dev-cmd/typecheck.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.typecheck_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/dev-cmd/unbottled.rb b/Library/Homebrew/dev-cmd/unbottled.rb
index 8cbe3ad802..f708d167ea 100644
--- a/Library/Homebrew/dev-cmd/unbottled.rb
+++ b/Library/Homebrew/dev-cmd/unbottled.rb
@@ -7,8 +7,6 @@ require "api"
 require "os/mac/xcode"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/unpack.rb b/Library/Homebrew/dev-cmd/unpack.rb
index b08e7a2aba..86a35544a0 100644
--- a/Library/Homebrew/dev-cmd/unpack.rb
+++ b/Library/Homebrew/dev-cmd/unpack.rb
@@ -6,8 +6,6 @@ require "formula"
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.unpack_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/dev-cmd/update-license-data.rb b/Library/Homebrew/dev-cmd/update-license-data.rb
index 05326a9ac1..1eb9344d00 100644
--- a/Library/Homebrew/dev-cmd/update-license-data.rb
+++ b/Library/Homebrew/dev-cmd/update-license-data.rb
@@ -6,7 +6,6 @@ require "utils/spdx"
 require "system_command"
 
 module Homebrew
-  extend T::Sig
   include SystemCommand::Mixin
 
   module_function
diff --git a/Library/Homebrew/dev-cmd/update-maintainers.rb b/Library/Homebrew/dev-cmd/update-maintainers.rb
index 98923595e3..dfac1f11ee 100644
--- a/Library/Homebrew/dev-cmd/update-maintainers.rb
+++ b/Library/Homebrew/dev-cmd/update-maintainers.rb
@@ -8,8 +8,6 @@ require "manpages"
 require "active_support/core_ext/hash/slice"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/update-python-resources.rb b/Library/Homebrew/dev-cmd/update-python-resources.rb
index 1e6e7bf96f..8d73ff7e8c 100644
--- a/Library/Homebrew/dev-cmd/update-python-resources.rb
+++ b/Library/Homebrew/dev-cmd/update-python-resources.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "utils/pypi"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/dev-cmd/update-sponsors.rb b/Library/Homebrew/dev-cmd/update-sponsors.rb
index 0ea4d480c6..52d491764d 100644
--- a/Library/Homebrew/dev-cmd/update-sponsors.rb
+++ b/Library/Homebrew/dev-cmd/update-sponsors.rb
@@ -5,8 +5,6 @@ require "cli/parser"
 require "utils/github"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   NAMED_MONTHLY_AMOUNT = 100
diff --git a/Library/Homebrew/dev-cmd/update-test.rb b/Library/Homebrew/dev-cmd/update-test.rb
index 4c92356af5..c9e66b55d3 100644
--- a/Library/Homebrew/dev-cmd/update-test.rb
+++ b/Library/Homebrew/dev-cmd/update-test.rb
@@ -3,8 +3,6 @@
 
 require "cli/parser"
 module Homebrew
-  extend T::Sig
-
   sig { returns(CLI::Parser) }
   def self.update_test_args
     Homebrew::CLI::Parser.new do
diff --git a/Library/Homebrew/dev-cmd/vendor-gems.rb b/Library/Homebrew/dev-cmd/vendor-gems.rb
index 5d3a49b9b8..da03d58f11 100644
--- a/Library/Homebrew/dev-cmd/vendor-gems.rb
+++ b/Library/Homebrew/dev-cmd/vendor-gems.rb
@@ -4,8 +4,6 @@
 require "cli/parser"
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   sig { returns(CLI::Parser) }
diff --git a/Library/Homebrew/development_tools.rb b/Library/Homebrew/development_tools.rb
index 6a7014f902..3ad70e3e93 100644
--- a/Library/Homebrew/development_tools.rb
+++ b/Library/Homebrew/development_tools.rb
@@ -6,8 +6,6 @@ require "version"
 # @private
 class DevelopmentTools
   class << self
-    extend T::Sig
-
     sig { params(tool: String).returns(T.nilable(Pathname)) }
     def locate(tool)
       # Don't call tools (cc, make, strip, etc.) directly!
diff --git a/Library/Homebrew/diagnostic.rb b/Library/Homebrew/diagnostic.rb
index 405b3b2aee..043c060305 100644
--- a/Library/Homebrew/diagnostic.rb
+++ b/Library/Homebrew/diagnostic.rb
@@ -48,8 +48,6 @@ module Homebrew
 
     # Diagnostic checks.
     class Checks
-      extend T::Sig
-
       def initialize(verbose: true)
         @verbose = verbose
       end
diff --git a/Library/Homebrew/download_strategy.rb b/Library/Homebrew/download_strategy.rb
index ba59dd32b4..e63bb23df5 100644
--- a/Library/Homebrew/download_strategy.rb
+++ b/Library/Homebrew/download_strategy.rb
@@ -27,8 +27,6 @@ using TimeRemaining
 #
 # @api private
 class AbstractDownloadStrategy
-  extend T::Sig
-
   extend Forwardable
   include FileUtils
   include Context
@@ -709,8 +707,6 @@ end
 #
 # @api public
 class SubversionDownloadStrategy < VCSDownloadStrategy
-  extend T::Sig
-
   def initialize(url, name, version, **meta)
     super
     @url = @url.sub("svn+http://", "")
@@ -1135,8 +1131,6 @@ end
 #
 # @api public
 class CVSDownloadStrategy < VCSDownloadStrategy
-  extend T::Sig
-
   def initialize(url, name, version, **meta)
     super
     @url = @url.sub(%r{^cvs://}, "")
@@ -1217,8 +1211,6 @@ end
 #
 # @api public
 class MercurialDownloadStrategy < VCSDownloadStrategy
-  extend T::Sig
-
   def initialize(url, name, version, **meta)
     super
     @url = @url.sub(%r{^hg://}, "")
@@ -1281,8 +1273,6 @@ end
 #
 # @api public
 class BazaarDownloadStrategy < VCSDownloadStrategy
-  extend T::Sig
-
   def initialize(url, name, version, **meta)
     super
     @url.sub!(%r{^bzr://}, "")
@@ -1346,8 +1336,6 @@ end
 #
 # @api public
 class FossilDownloadStrategy < VCSDownloadStrategy
-  extend T::Sig
-
   def initialize(url, name, version, **meta)
     super
     @url = @url.sub(%r{^fossil://}, "")
diff --git a/Library/Homebrew/env_config.rb b/Library/Homebrew/env_config.rb
index 94a3ca7f96..5884ea4b05 100644
--- a/Library/Homebrew/env_config.rb
+++ b/Library/Homebrew/env_config.rb
@@ -8,8 +8,6 @@ module Homebrew
   #
   # @api private
   module EnvConfig
-    extend T::Sig
-
     module_function
 
     ENVS = {
diff --git a/Library/Homebrew/exceptions.rb b/Library/Homebrew/exceptions.rb
index 7f8b4f9d6b..ff627360c9 100644
--- a/Library/Homebrew/exceptions.rb
+++ b/Library/Homebrew/exceptions.rb
@@ -6,8 +6,6 @@ require "utils"
 
 # Raised when a command is used wrong.
 class UsageError < RuntimeError
-  extend T::Sig
-
   attr_reader :reason
 
   def initialize(reason = nil)
@@ -83,8 +81,6 @@ end
 
 # Raised when neither a formula nor a cask with the given name is available.
 class FormulaOrCaskUnavailableError < RuntimeError
-  extend T::Sig
-
   attr_reader :name
 
   def initialize(name)
@@ -111,8 +107,6 @@ end
 
 # Raised when a formula or cask in a specific tap is not available.
 class TapFormulaOrCaskUnavailableError < FormulaOrCaskUnavailableError
-  extend T::Sig
-
   attr_reader :tap
 
   def initialize(tap, name)
@@ -130,8 +124,6 @@ end
 
 # Raised when a formula is not available.
 class FormulaUnavailableError < FormulaOrCaskUnavailableError
-  extend T::Sig
-
   attr_accessor :dependent
 
   sig { returns(T.nilable(String)) }
@@ -149,8 +141,6 @@ end
 #
 # @api private
 module FormulaClassUnavailableErrorModule
-  extend T::Sig
-
   attr_reader :path, :class_name, :class_list
 
   def to_s
@@ -195,8 +185,6 @@ end
 #
 # @api private
 module FormulaUnreadableErrorModule
-  extend T::Sig
-
   attr_reader :formula_error
 
   sig { returns(String) }
@@ -404,8 +392,6 @@ end
 
 # Raised when a formula conflicts with another one.
 class FormulaConflictError < RuntimeError
-  extend T::Sig
-
   attr_reader :formula, :conflicts
 
   def initialize(formula, conflicts)
@@ -466,8 +452,6 @@ end
 
 # Raised when an error occurs during a formula build.
 class BuildError < RuntimeError
-  extend T::Sig
-
   attr_reader :cmd, :args, :env
   attr_accessor :formula, :options
 
@@ -651,8 +635,6 @@ end
 
 # Raised by {Kernel#safe_system} in `utils.rb`.
 class ErrorDuringExecution < RuntimeError
-  extend T::Sig
-
   attr_reader :cmd, :status, :output
 
   def initialize(cmd, status:, output: nil, secrets: [])
diff --git a/Library/Homebrew/extend/ENV.rb b/Library/Homebrew/extend/ENV.rb
index b4c149a577..0a95b5ebb1 100644
--- a/Library/Homebrew/extend/ENV.rb
+++ b/Library/Homebrew/extend/ENV.rb
@@ -8,8 +8,6 @@ require "extend/ENV/std"
 require "extend/ENV/super"
 
 module Kernel
-  extend T::Sig
-
   sig { params(env: T.nilable(String)).returns(T::Boolean) }
   def superenv?(env)
     return false if env == "std"
@@ -24,8 +22,6 @@ end
 #  # @see https://rubydoc.info/stdlib/core/ENV ENV core documentation
 #  class ENV; end
 module EnvActivation
-  extend T::Sig
-
   sig { params(env: T.nilable(String)).void }
   def activate_extensions!(env: nil)
     if superenv?(env)
diff --git a/Library/Homebrew/extend/ENV/shared.rb b/Library/Homebrew/extend/ENV/shared.rb
index 6f337e543a..bef56363bc 100644
--- a/Library/Homebrew/extend/ENV/shared.rb
+++ b/Library/Homebrew/extend/ENV/shared.rb
@@ -11,8 +11,6 @@ require "development_tools"
 # @see Stdenv
 # @see https://www.rubydoc.info/stdlib/Env Ruby's ENV API
 module SharedEnvExtension
-  extend T::Sig
-
   include CompilerConstants
 
   CC_FLAG_VARS = %w[CFLAGS CXXFLAGS OBJCFLAGS OBJCXXFLAGS].freeze
diff --git a/Library/Homebrew/extend/ENV/std.rb b/Library/Homebrew/extend/ENV/std.rb
index 62b2692c43..8929f5f6c9 100644
--- a/Library/Homebrew/extend/ENV/std.rb
+++ b/Library/Homebrew/extend/ENV/std.rb
@@ -6,8 +6,6 @@ require "extend/ENV/shared"
 
 # @api private
 module Stdenv
-  extend T::Sig
-
   include SharedEnvExtension
 
   # @private
diff --git a/Library/Homebrew/extend/ENV/super.rb b/Library/Homebrew/extend/ENV/super.rb
index 977c30fd45..734eccb95d 100644
--- a/Library/Homebrew/extend/ENV/super.rb
+++ b/Library/Homebrew/extend/ENV/super.rb
@@ -15,8 +15,6 @@ require "development_tools"
 # 7. Simpler formulae that *just work*
 # 8. Build-system agnostic configuration of the toolchain
 module Superenv
-  extend T::Sig
-
   include SharedEnvExtension
 
   # @private
diff --git a/Library/Homebrew/extend/kernel.rb b/Library/Homebrew/extend/kernel.rb
index 8c9672ec4a..d575d3f888 100644
--- a/Library/Homebrew/extend/kernel.rb
+++ b/Library/Homebrew/extend/kernel.rb
@@ -4,8 +4,6 @@
 # Contains shorthand Homebrew utility methods like `ohai`, `opoo`, `odisabled`.
 # TODO: move these out of `Kernel`.
 module Kernel
-  extend T::Sig
-
   def require?(path)
     return false if path.nil?
 
diff --git a/Library/Homebrew/extend/module.rb b/Library/Homebrew/extend/module.rb
index b09433b483..62b97c2b01 100644
--- a/Library/Homebrew/extend/module.rb
+++ b/Library/Homebrew/extend/module.rb
@@ -2,6 +2,8 @@
 # frozen_string_literal: true
 
 class Module
+  include T::Sig
+
   def attr_rw(*attrs)
     attrs.each do |attr|
       module_eval <<-EOS, __FILE__, __LINE__+1
diff --git a/Library/Homebrew/extend/on_system.rb b/Library/Homebrew/extend/on_system.rb
index 2fc253904c..7d8670a65f 100644
--- a/Library/Homebrew/extend/on_system.rb
+++ b/Library/Homebrew/extend/on_system.rb
@@ -4,8 +4,6 @@
 require "simulate_system"
 
 module OnSystem
-  extend T::Sig
-
   ARCH_OPTIONS = [:intel, :arm].freeze
   BASE_OS_OPTIONS = [:macos, :linux].freeze
 
@@ -139,8 +137,6 @@ module OnSystem
   end
 
   module MacOSAndLinux
-    extend T::Sig
-
     sig { params(base: Class).void }
     def self.included(base)
       OnSystem.setup_arch_methods(base)
@@ -150,8 +146,6 @@ module OnSystem
   end
 
   module MacOSOnly
-    extend T::Sig
-
     sig { params(base: Class).void }
     def self.included(base)
       OnSystem.setup_arch_methods(base)
diff --git a/Library/Homebrew/extend/os/linux/cleaner.rb b/Library/Homebrew/extend/os/linux/cleaner.rb
index 8ac06950da..8a21e6e91d 100644
--- a/Library/Homebrew/extend/os/linux/cleaner.rb
+++ b/Library/Homebrew/extend/os/linux/cleaner.rb
@@ -4,7 +4,6 @@
 class Cleaner
   private
 
-  extend T::Sig
   sig { params(path: Pathname).returns(T.nilable(T::Boolean)) }
   def executable_path?(path)
     path.elf? || path.text_executable?
diff --git a/Library/Homebrew/extend/os/linux/cmd/update-report.rb b/Library/Homebrew/extend/os/linux/cmd/update-report.rb
index 5e9091a3e7..fd0ad1aeeb 100644
--- a/Library/Homebrew/extend/os/linux/cmd/update-report.rb
+++ b/Library/Homebrew/extend/os/linux/cmd/update-report.rb
@@ -2,8 +2,6 @@
 # frozen_string_literal: true
 
 module Homebrew
-  extend T::Sig
-
   module_function
 
   def no_changes_message
diff --git a/Library/Homebrew/extend/os/linux/dependency_collector.rb b/Library/Homebrew/extend/os/linux/dependency_collector.rb
index 785ca93326..e049afcaf6 100644
--- a/Library/Homebrew/extend/os/linux/dependency_collector.rb
+++ b/Library/Homebrew/extend/os/linux/dependency_collector.rb
@@ -4,8 +4,6 @@
 require "os/linux/glibc"
 
 class DependencyCollector
-  extend T::Sig
-
   undef gcc_dep_if_needed
   undef glibc_dep_if_needed
   undef init_global_dep_tree_if_needed!
diff --git a/Library/Homebrew/extend/os/linux/dev-cmd/update-test.rb b/Library/Homebrew/extend/os/linux/dev-cmd/update-test.rb
index 0f711eeafd..4072a84b44 100644
--- a/Library/Homebrew/extend/os/linux/dev-cmd/update-test.rb
+++ b/Library/Homebrew/extend/os/linux/dev-cmd/update-test.rb
@@ -2,8 +2,6 @@
 # frozen_string_literal: true
 
 module Homebrew
-  extend T::Sig
-
   class << self
     alias generic_git_tags git_tags
 
diff --git a/Library/Homebrew/extend/os/linux/development_tools.rb b/Library/Homebrew/extend/os/linux/development_tools.rb
index f5928260f4..d26b73f312 100644
--- a/Library/Homebrew/extend/os/linux/development_tools.rb
+++ b/Library/Homebrew/extend/os/linux/development_tools.rb
@@ -3,8 +3,6 @@
 
 class DevelopmentTools
   class << self
-    extend T::Sig
-
     sig { params(tool: String).returns(T.nilable(Pathname)) }
     def locate(tool)
       (@locate ||= {}).fetch(tool) do |key|
diff --git a/Library/Homebrew/extend/os/linux/extend/ENV/super.rb b/Library/Homebrew/extend/os/linux/extend/ENV/super.rb
index c0691c0001..2f151597d5 100644
--- a/Library/Homebrew/extend/os/linux/extend/ENV/super.rb
+++ b/Library/Homebrew/extend/os/linux/extend/ENV/super.rb
@@ -2,8 +2,6 @@
 # frozen_string_literal: true
 
 module Superenv
-  extend T::Sig
-
   # The location of Homebrew's shims on Linux.
   def self.shims_path
     HOMEBREW_SHIMS_PATH/"linux/super"
diff --git a/Library/Homebrew/extend/os/linux/formula_cellar_checks.rb b/Library/Homebrew/extend/os/linux/formula_cellar_checks.rb
index 11eb2a0db0..b08411fb1b 100644
--- a/Library/Homebrew/extend/os/linux/formula_cellar_checks.rb
+++ b/Library/Homebrew/extend/os/linux/formula_cellar_checks.rb
@@ -2,7 +2,6 @@
 # frozen_string_literal: true
 
 module FormulaCellarChecks
-  extend T::Sig
   sig { params(filename: Pathname).returns(T::Boolean) }
   def valid_library_extension?(filename)
     generic_valid_library_extension?(filename) || filename.basename.to_s.include?(".so.")
diff --git a/Library/Homebrew/extend/os/linux/requirements/xcode_requirement.rb b/Library/Homebrew/extend/os/linux/requirements/xcode_requirement.rb
index 8ebc4ecd72..296663fbca 100644
--- a/Library/Homebrew/extend/os/linux/requirements/xcode_requirement.rb
+++ b/Library/Homebrew/extend/os/linux/requirements/xcode_requirement.rb
@@ -4,8 +4,6 @@
 require "requirement"
 
 class XcodeRequirement < Requirement
-  extend T::Sig
-
   sig { returns(T::Boolean) }
   def xcode_installed_version
     true
diff --git a/Library/Homebrew/extend/os/linux/software_spec.rb b/Library/Homebrew/extend/os/linux/software_spec.rb
index 7f5b4d2890..fca4a667af 100644
--- a/Library/Homebrew/extend/os/linux/software_spec.rb
+++ b/Library/Homebrew/extend/os/linux/software_spec.rb
@@ -2,8 +2,6 @@
 # frozen_string_literal: true
 
 class BottleSpecification
-  extend T::Sig
-
   sig { params(tag: Utils::Bottles::Tag).returns(T::Boolean) }
   def skip_relocation?(tag: Utils::Bottles.tag)
     false
diff --git a/Library/Homebrew/extend/os/mac/development_tools.rb b/Library/Homebrew/extend/os/mac/development_tools.rb
index 353ae30428..0df45d805d 100644
--- a/Library/Homebrew/extend/os/mac/development_tools.rb
+++ b/Library/Homebrew/extend/os/mac/development_tools.rb
@@ -6,8 +6,6 @@ require "os/mac/xcode"
 # @private
 class DevelopmentTools
   class << self
-    extend T::Sig
-
     alias generic_locate locate
     undef installed?, default_compiler, curl_handles_most_https_certificates?,
           subversion_handles_most_https_certificates?
diff --git a/Library/Homebrew/extend/os/mac/extend/ENV/shared.rb b/Library/Homebrew/extend/os/mac/extend/ENV/shared.rb
index a13761a8b3..c696261f26 100644
--- a/Library/Homebrew/extend/os/mac/extend/ENV/shared.rb
+++ b/Library/Homebrew/extend/os/mac/extend/ENV/shared.rb
@@ -2,8 +2,6 @@
 # frozen_string_literal: true
 
 module SharedEnvExtension
-  extend T::Sig
-
   def setup_build_environment(formula: nil, cc: nil, build_bottle: false, bottle_arch: nil, testing_formula: false,
                               debug_symbols: false)
     generic_shared_setup_build_environment(formula: formula, cc: cc, build_bottle: build_bottle,
diff --git a/Library/Homebrew/extend/os/mac/extend/ENV/super.rb b/Library/Homebrew/extend/os/mac/extend/ENV/super.rb
index 36aac56489..3d0459a5a1 100644
--- a/Library/Homebrew/extend/os/mac/extend/ENV/super.rb
+++ b/Library/Homebrew/extend/os/mac/extend/ENV/super.rb
@@ -2,8 +2,6 @@
 # frozen_string_literal: true
 
 module Superenv
-  extend T::Sig
-
   class << self
     # The location of Homebrew's shims on macOS.
     def shims_path
diff --git a/Library/Homebrew/extend/os/mac/formula_support.rb b/Library/Homebrew/extend/os/mac/formula_support.rb
index 64e8d14726..563f652d74 100644
--- a/Library/Homebrew/extend/os/mac/formula_support.rb
+++ b/Library/Homebrew/extend/os/mac/formula_support.rb
@@ -2,7 +2,6 @@
 # frozen_string_literal: true
 
 class KegOnlyReason
-  extend T::Sig
   sig { returns(T::Boolean) }
   def applicable?
     true
diff --git a/Library/Homebrew/extend/os/mac/hardware.rb b/Library/Homebrew/extend/os/mac/hardware.rb
index 43adfa1dea..fb3273b7df 100644
--- a/Library/Homebrew/extend/os/mac/hardware.rb
+++ b/Library/Homebrew/extend/os/mac/hardware.rb
@@ -2,7 +2,6 @@
 # frozen_string_literal: true
 
 module Hardware
-  extend T::Sig
   sig { params(version: T.nilable(Version)).returns(Symbol) }
   def self.oldest_cpu(version = nil)
     version = if version
diff --git a/Library/Homebrew/extend/os/mac/missing_formula.rb b/Library/Homebrew/extend/os/mac/missing_formula.rb
index c7f7e701ed..8721980e76 100644
--- a/Library/Homebrew/extend/os/mac/missing_formula.rb
+++ b/Library/Homebrew/extend/os/mac/missing_formula.rb
@@ -7,9 +7,7 @@ require "cask/caskroom"
 
 module Homebrew
   module MissingFormula
-    extend T::Sig
     class << self
-      extend T::Sig
       sig { params(name: String).returns(T.nilable(String)) }
       def disallowed_reason(name)
         case name.downcase
diff --git a/Library/Homebrew/extend/os/mac/unpack_strategy/zip.rb b/Library/Homebrew/extend/os/mac/unpack_strategy/zip.rb
index cff64f5802..9bbdf4a33a 100644
--- a/Library/Homebrew/extend/os/mac/unpack_strategy/zip.rb
+++ b/Library/Homebrew/extend/os/mac/unpack_strategy/zip.rb
@@ -6,8 +6,6 @@ require "system_command"
 module UnpackStrategy
   class Zip
     module MacOSZipExtension
-      extend T::Sig
-
       include UnpackStrategy
       include SystemCommand::Mixin
 
diff --git a/Library/Homebrew/extend/os/mac/utils/analytics.rb b/Library/Homebrew/extend/os/mac/utils/analytics.rb
index 4d104e6dce..63f45f9f3b 100644
--- a/Library/Homebrew/extend/os/mac/utils/analytics.rb
+++ b/Library/Homebrew/extend/os/mac/utils/analytics.rb
@@ -4,8 +4,6 @@
 module Utils
   module Analytics
     class << self
-      extend T::Sig
-
       sig { returns(String) }
       def custom_prefix_label_google
         return generic_custom_prefix_label_google if Hardware::CPU.arm?
diff --git a/Library/Homebrew/extend/pathname.rb b/Library/Homebrew/extend/pathname.rb
index ca9144b27a..ee2d91f0b3 100644
--- a/Library/Homebrew/extend/pathname.rb
+++ b/Library/Homebrew/extend/pathname.rb
@@ -6,8 +6,6 @@ require "resource"
 require "metafiles"
 
 module DiskUsageExtension
-  extend T::Sig
-
   sig { returns(Integer) }
   def disk_usage
     return @disk_usage if defined?(@disk_usage)
@@ -78,8 +76,6 @@ end
 # Homebrew extends Ruby's `Pathname` to make our code more readable.
 # @see https://ruby-doc.org/stdlib-2.6.3/libdoc/pathname/rdoc/Pathname.html Ruby's Pathname API
 class Pathname
-  extend T::Sig
-
   include DiskUsageExtension
 
   # Moves a file from the original location to the {Pathname}'s.
@@ -498,11 +494,7 @@ require "extend/os/pathname"
 
 # @private
 module ObserverPathnameExtension
-  extend T::Sig
-
   class << self
-    extend T::Sig
-
     include Context
 
     sig { returns(Integer) }
diff --git a/Library/Homebrew/fetch.rb b/Library/Homebrew/fetch.rb
index 09b5614fd2..c71ebc8efe 100644
--- a/Library/Homebrew/fetch.rb
+++ b/Library/Homebrew/fetch.rb
@@ -4,8 +4,6 @@
 module Homebrew
   # @api private
   module Fetch
-    extend T::Sig
-
     sig { params(formula: Formula, args: CLI::Args).returns(T::Boolean) }
     def fetch_bottle?(formula, args:)
       bottle = formula.bottle
diff --git a/Library/Homebrew/formula.rb b/Library/Homebrew/formula.rb
index 10517613aa..d528ab402f 100644
--- a/Library/Homebrew/formula.rb
+++ b/Library/Homebrew/formula.rb
@@ -58,8 +58,6 @@ require "extend/api_hashable"
 #   end
 # end</pre>
 class Formula
-  extend T::Sig
-
   include FileUtils
   include Utils::Inreplace
   include Utils::Shebang
@@ -2726,7 +2724,7 @@ class Formula
   # The methods below define the formula DSL.
   class << self
     extend Predicable
-    extend T::Sig
+
     include BuildEnvironment::DSL
     include OnSystem::MacOSAndLinux
 
diff --git a/Library/Homebrew/formula_cellar_checks.rb b/Library/Homebrew/formula_cellar_checks.rb
index ed988b1d7c..285bd5e126 100644
--- a/Library/Homebrew/formula_cellar_checks.rb
+++ b/Library/Homebrew/formula_cellar_checks.rb
@@ -7,7 +7,6 @@ require "utils/shell"
 #
 # @api private
 module FormulaCellarChecks
-  extend T::Sig
   extend T::Helpers
 
   abstract!
diff --git a/Library/Homebrew/formula_creator.rb b/Library/Homebrew/formula_creator.rb
index 2e0ab2f35f..800f4a548f 100644
--- a/Library/Homebrew/formula_creator.rb
+++ b/Library/Homebrew/formula_creator.rb
@@ -9,8 +9,6 @@ module Homebrew
   #
   # @api private
   class FormulaCreator
-    extend T::Sig
-
     attr_reader :args, :url, :sha256, :desc, :homepage
     attr_accessor :name, :version, :tap, :path, :mode, :license
 
diff --git a/Library/Homebrew/formula_installer.rb b/Library/Homebrew/formula_installer.rb
index d69d38b367..187e85d3c5 100644
--- a/Library/Homebrew/formula_installer.rb
+++ b/Library/Homebrew/formula_installer.rb
@@ -27,8 +27,6 @@ require "service"
 #
 # @api private
 class FormulaInstaller
-  extend T::Sig
-
   include FormulaCellarChecks
   extend Predicable
 
diff --git a/Library/Homebrew/formula_support.rb b/Library/Homebrew/formula_support.rb
index 4859606c9b..fa570fd296 100644
--- a/Library/Homebrew/formula_support.rb
+++ b/Library/Homebrew/formula_support.rb
@@ -7,8 +7,6 @@ FormulaConflict = Struct.new(:name, :reason)
 # Used to annotate formulae that duplicate macOS-provided software
 # or cause conflicts when linked in.
 class KegOnlyReason
-  extend T::Sig
-
   attr_reader :reason
 
   def initialize(reason, explanation)
diff --git a/Library/Homebrew/formulary.rb b/Library/Homebrew/formulary.rb
index f16246e9d5..eba35694cc 100644
--- a/Library/Homebrew/formulary.rb
+++ b/Library/Homebrew/formulary.rb
@@ -14,8 +14,6 @@ require "active_support/core_ext/hash/deep_transform_values"
 #
 # @api private
 module Formulary
-  extend T::Sig
-
   extend Cachable
 
   URL_START_REGEX = %r{(https?|ftp|file)://}.freeze
@@ -486,8 +484,6 @@ module Formulary
 
   # Loads formulae from URLs.
   class FromUrlLoader < FormulaLoader
-    extend T::Sig
-
     attr_reader :url
 
     sig { params(url: T.any(URI::Generic, String), from: T.nilable(Symbol)).void }
diff --git a/Library/Homebrew/git_repository.rb b/Library/Homebrew/git_repository.rb
index 83a312c425..c50626f5fd 100644
--- a/Library/Homebrew/git_repository.rb
+++ b/Library/Homebrew/git_repository.rb
@@ -8,8 +8,6 @@ require "utils/popen"
 # @see Utils::Git
 # @api private
 class GitRepository
-  extend T::Sig
-
   sig { returns(Pathname) }
   attr_reader :pathname
 
diff --git a/Library/Homebrew/github_packages.rb b/Library/Homebrew/github_packages.rb
index 0cfec06f97..636eb0823c 100644
--- a/Library/Homebrew/github_packages.rb
+++ b/Library/Homebrew/github_packages.rb
@@ -9,8 +9,6 @@ require "zlib"
 #
 # @api private
 class GitHubPackages
-  extend T::Sig
-
   include Context
 
   URL_DOMAIN = "ghcr.io"
diff --git a/Library/Homebrew/github_releases.rb b/Library/Homebrew/github_releases.rb
index e59d268991..53dccb0dc8 100644
--- a/Library/Homebrew/github_releases.rb
+++ b/Library/Homebrew/github_releases.rb
@@ -8,8 +8,6 @@ require "json"
 #
 # @api private
 class GitHubReleases
-  extend T::Sig
-
   include Context
   include Utils::Curl
 
diff --git a/Library/Homebrew/github_runner.rb b/Library/Homebrew/github_runner.rb
index 0dbf845cdc..91da9f7454 100644
--- a/Library/Homebrew/github_runner.rb
+++ b/Library/Homebrew/github_runner.rb
@@ -5,8 +5,6 @@ require "linux_runner_spec"
 require "macos_runner_spec"
 
 class GitHubRunner < T::Struct
-  extend T::Sig
-
   const :platform, Symbol
   const :arch, Symbol
   const :spec, T.any(LinuxRunnerSpec, MacOSRunnerSpec)
diff --git a/Library/Homebrew/github_runner_matrix.rb b/Library/Homebrew/github_runner_matrix.rb
index 632c3b3c64..8f2e0c50dd 100644
--- a/Library/Homebrew/github_runner_matrix.rb
+++ b/Library/Homebrew/github_runner_matrix.rb
@@ -5,8 +5,6 @@ require "test_runner_formula"
 require "github_runner"
 
 class GitHubRunnerMatrix
-  extend T::Sig
-
   # FIXME: Enable cop again when https://github.com/sorbet/sorbet/issues/3532 is fixed.
   # rubocop:disable Style/MutableConstant
   MaybeStringArray = T.type_alias { T.nilable(T::Array[String]) }
diff --git a/Library/Homebrew/global.rb b/Library/Homebrew/global.rb
index 071eba618a..27f2015451 100644
--- a/Library/Homebrew/global.rb
+++ b/Library/Homebrew/global.rb
@@ -72,6 +72,7 @@ HOMEBREW_PULL_OR_COMMIT_URL_REGEX =
   %r[https://github\.com/([\w-]+)/([\w-]+)?/(?:pull/(\d+)|commit/[0-9a-fA-F]{4,40})].freeze
 HOMEBREW_BOTTLES_EXTNAME_REGEX = /\.([a-z0-9_]+)\.bottle\.(?:(\d+)\.)?tar\.gz$/.freeze
 
+require "extend/module"
 require "env_config"
 require "macos_versions"
 require "os"
@@ -134,7 +135,6 @@ require "extend/array"
 require "git_repository"
 require "extend/pathname"
 require "extend/predicable"
-require "extend/module"
 require "cli/args"
 
 require "PATH"
diff --git a/Library/Homebrew/hardware.rb b/Library/Homebrew/hardware.rb
index 02c6bec9df..22bebf626b 100644
--- a/Library/Homebrew/hardware.rb
+++ b/Library/Homebrew/hardware.rb
@@ -24,8 +24,6 @@ module Hardware
     INTEL_64BIT_OLDEST_CPU = :core2
 
     class << self
-      extend T::Sig
-
       def optimization_flags
         @optimization_flags ||= {
           native:             arch_flag("native"),
diff --git a/Library/Homebrew/installed_dependents.rb b/Library/Homebrew/installed_dependents.rb
index 7977dea71d..1b0a188ce5 100644
--- a/Library/Homebrew/installed_dependents.rb
+++ b/Library/Homebrew/installed_dependents.rb
@@ -7,8 +7,6 @@ require "cask_dependent"
 #
 # @api private
 module InstalledDependents
-  extend T::Sig
-
   module_function
 
   # Given an array of kegs, this method will try to find some other kegs
diff --git a/Library/Homebrew/keg.rb b/Library/Homebrew/keg.rb
index 8ea2c98f85..2d6f21cb91 100644
--- a/Library/Homebrew/keg.rb
+++ b/Library/Homebrew/keg.rb
@@ -10,8 +10,6 @@ require "extend/cachable"
 #
 # @api private
 class Keg
-  extend T::Sig
-
   extend Cachable
 
   # Error for when a keg is already linked.
@@ -40,8 +38,6 @@ class Keg
 
   # Error for when a file already exists or belongs to another keg.
   class ConflictError < LinkError
-    extend T::Sig
-
     sig { returns(String) }
     def suggestion
       conflict = Keg.for(dst)
@@ -72,8 +68,6 @@ class Keg
 
   # Error for when a directory is not writable.
   class DirectoryNotWritableError < LinkError
-    extend T::Sig
-
     sig { returns(String) }
     def to_s
       <<~EOS
diff --git a/Library/Homebrew/keg_relocate.rb b/Library/Homebrew/keg_relocate.rb
index 083529719b..59ffeab115 100644
--- a/Library/Homebrew/keg_relocate.rb
+++ b/Library/Homebrew/keg_relocate.rb
@@ -12,8 +12,6 @@ class Keg
   NULL_BYTE_STRING = "\\x00"
 
   class Relocation
-    extend T::Sig
-
     RELOCATABLE_PATH_REGEX_PREFIX = /(?:(?<=-F|-I|-L|-isystem)|(?<![a-zA-Z0-9]))/.freeze
 
     def initialize
diff --git a/Library/Homebrew/language/java.rb b/Library/Homebrew/language/java.rb
index c7ffdffd7c..64d42f975e 100644
--- a/Library/Homebrew/language/java.rb
+++ b/Library/Homebrew/language/java.rb
@@ -6,8 +6,6 @@ module Language
   #
   # @api public
   module Java
-    extend T::Sig
-
     sig { params(version: T.nilable(String)).returns(T.nilable(Formula)) }
     def self.find_openjdk_formula(version = nil)
       can_be_newer = version&.end_with?("+")
diff --git a/Library/Homebrew/language/node.rb b/Library/Homebrew/language/node.rb
index 4421b0f9e8..b5bb25389b 100644
--- a/Library/Homebrew/language/node.rb
+++ b/Library/Homebrew/language/node.rb
@@ -6,8 +6,6 @@ module Language
   #
   # @api public
   module Node
-    extend T::Sig
-
     sig { returns(String) }
     def self.npm_cache_config
       "cache=#{HOMEBREW_CACHE}/npm_cache"
diff --git a/Library/Homebrew/language/python.rb b/Library/Homebrew/language/python.rb
index b8d890e840..aac47e201c 100644
--- a/Library/Homebrew/language/python.rb
+++ b/Library/Homebrew/language/python.rb
@@ -124,8 +124,6 @@ module Language
 
     # Mixin module for {Formula} adding virtualenv support features.
     module Virtualenv
-      extend T::Sig
-
       # Instantiates, creates, and yields a {Virtualenv} object for use from
       # {Formula#install}, which provides helper methods for instantiating and
       # installing packages into a Python virtualenv.
diff --git a/Library/Homebrew/linkage_checker.rb b/Library/Homebrew/linkage_checker.rb
index ad3544ee50..17c4542b3c 100644
--- a/Library/Homebrew/linkage_checker.rb
+++ b/Library/Homebrew/linkage_checker.rb
@@ -10,8 +10,6 @@ require "fiddle"
 #
 # @api private
 class LinkageChecker
-  extend T::Sig
-
   attr_reader :undeclared_deps, :keg, :formula, :store
 
   def initialize(keg, formula = nil, cache_db:, rebuild_cache: false)
diff --git a/Library/Homebrew/linux_runner_spec.rb b/Library/Homebrew/linux_runner_spec.rb
index 533eb3034b..816f82d4e9 100644
--- a/Library/Homebrew/linux_runner_spec.rb
+++ b/Library/Homebrew/linux_runner_spec.rb
@@ -2,8 +2,6 @@
 # frozen_string_literal: true
 
 class LinuxRunnerSpec < T::Struct
-  extend T::Sig
-
   const :name, String
   const :runner, String
   const :container, T::Hash[Symbol, String]
diff --git a/Library/Homebrew/livecheck/livecheck.rb b/Library/Homebrew/livecheck/livecheck.rb
index 166beee14f..535d946c83 100644
--- a/Library/Homebrew/livecheck/livecheck.rb
+++ b/Library/Homebrew/livecheck/livecheck.rb
@@ -17,8 +17,6 @@ module Homebrew
   #
   # @api private
   module Livecheck
-    extend T::Sig
-
     module_function
 
     GITEA_INSTANCES = %w[
diff --git a/Library/Homebrew/livecheck/livecheck_version.rb b/Library/Homebrew/livecheck/livecheck_version.rb
index 9ae3d8d79f..bee235413d 100644
--- a/Library/Homebrew/livecheck/livecheck_version.rb
+++ b/Library/Homebrew/livecheck/livecheck_version.rb
@@ -7,8 +7,6 @@ module Homebrew
     #
     # @api private
     class LivecheckVersion
-      extend T::Sig
-
       include Comparable
 
       sig {
diff --git a/Library/Homebrew/livecheck/skip_conditions.rb b/Library/Homebrew/livecheck/skip_conditions.rb
index be5159cb97..fe8c52668b 100644
--- a/Library/Homebrew/livecheck/skip_conditions.rb
+++ b/Library/Homebrew/livecheck/skip_conditions.rb
@@ -10,8 +10,6 @@ module Homebrew
     #
     # @api private
     module SkipConditions
-      extend T::Sig
-
       module_function
 
       sig {
diff --git a/Library/Homebrew/livecheck/strategy.rb b/Library/Homebrew/livecheck/strategy.rb
index dc5446d160..65431ccf40 100644
--- a/Library/Homebrew/livecheck/strategy.rb
+++ b/Library/Homebrew/livecheck/strategy.rb
@@ -10,8 +10,6 @@ module Homebrew
     #
     # @api private
     module Strategy
-      extend T::Sig
-
       module_function
 
       # {Strategy} priorities informally range from 1 to 10, where 10 is the
diff --git a/Library/Homebrew/livecheck/strategy/apache.rb b/Library/Homebrew/livecheck/strategy/apache.rb
index 8cf811bc56..342eda6728 100644
--- a/Library/Homebrew/livecheck/strategy/apache.rb
+++ b/Library/Homebrew/livecheck/strategy/apache.rb
@@ -26,8 +26,6 @@ module Homebrew
       #
       # @api public
       class Apache
-        extend T::Sig
-
         # The `Regexp` used to determine if the strategy applies to the URL.
         URL_MATCH_REGEX = %r{
           ^https?://
diff --git a/Library/Homebrew/livecheck/strategy/bitbucket.rb b/Library/Homebrew/livecheck/strategy/bitbucket.rb
index 97e515352c..dd319ca6e2 100644
--- a/Library/Homebrew/livecheck/strategy/bitbucket.rb
+++ b/Library/Homebrew/livecheck/strategy/bitbucket.rb
@@ -28,8 +28,6 @@ module Homebrew
       #
       # @api public
       class Bitbucket
-        extend T::Sig
-
         # The `Regexp` used to determine if the strategy applies to the URL.
         URL_MATCH_REGEX = %r{
           ^https?://bitbucket\.org
diff --git a/Library/Homebrew/livecheck/strategy/cpan.rb b/Library/Homebrew/livecheck/strategy/cpan.rb
index 17ed6c98f4..751e5189b3 100644
--- a/Library/Homebrew/livecheck/strategy/cpan.rb
+++ b/Library/Homebrew/livecheck/strategy/cpan.rb
@@ -18,8 +18,6 @@ module Homebrew
       #
       # @api public
       class Cpan
-        extend T::Sig
-
         NICE_NAME = "CPAN"
 
         # The `Regexp` used to determine if the strategy applies to the URL.
diff --git a/Library/Homebrew/livecheck/strategy/electron_builder.rb b/Library/Homebrew/livecheck/strategy/electron_builder.rb
index 942ef21270..7f9d435262 100644
--- a/Library/Homebrew/livecheck/strategy/electron_builder.rb
+++ b/Library/Homebrew/livecheck/strategy/electron_builder.rb
@@ -12,8 +12,6 @@ module Homebrew
       #
       # @api private
       class ElectronBuilder
-        extend T::Sig
-
         NICE_NAME = "electron-builder"
 
         # A priority of zero causes livecheck to skip the strategy. We do this
diff --git a/Library/Homebrew/livecheck/strategy/extract_plist.rb b/Library/Homebrew/livecheck/strategy/extract_plist.rb
index 6266511970..21dc2c34a8 100644
--- a/Library/Homebrew/livecheck/strategy/extract_plist.rb
+++ b/Library/Homebrew/livecheck/strategy/extract_plist.rb
@@ -19,8 +19,6 @@ module Homebrew
       #
       # @api private
       class ExtractPlist
-        extend T::Sig
-
         # A priority of zero causes livecheck to skip the strategy. We do this
         # for {ExtractPlist} so we can selectively apply it when appropriate.
         PRIORITY = 0
@@ -43,8 +41,6 @@ module Homebrew
           :bundle_version,
           keyword_init: true,
         ) do
-          extend T::Sig
-
           extend Forwardable
 
           # @api public
diff --git a/Library/Homebrew/livecheck/strategy/git.rb b/Library/Homebrew/livecheck/strategy/git.rb
index 817bac687d..4a72b042f7 100644
--- a/Library/Homebrew/livecheck/strategy/git.rb
+++ b/Library/Homebrew/livecheck/strategy/git.rb
@@ -24,8 +24,6 @@ module Homebrew
       #
       # @api public
       class Git
-        extend T::Sig
-
         # The priority of the strategy on an informal scale of 1 to 10 (from
         # lowest to highest).
         PRIORITY = 8
diff --git a/Library/Homebrew/livecheck/strategy/github_latest.rb b/Library/Homebrew/livecheck/strategy/github_latest.rb
index d91f5074b4..d8a3d97df8 100644
--- a/Library/Homebrew/livecheck/strategy/github_latest.rb
+++ b/Library/Homebrew/livecheck/strategy/github_latest.rb
@@ -32,8 +32,6 @@ module Homebrew
       #
       # @api public
       class GithubLatest
-        extend T::Sig
-
         NICE_NAME = "GitHub - Latest"
 
         # A priority of zero causes livecheck to skip the strategy. We do this
diff --git a/Library/Homebrew/livecheck/strategy/gnome.rb b/Library/Homebrew/livecheck/strategy/gnome.rb
index 47ef45306f..a6612e6208 100644
--- a/Library/Homebrew/livecheck/strategy/gnome.rb
+++ b/Library/Homebrew/livecheck/strategy/gnome.rb
@@ -25,8 +25,6 @@ module Homebrew
       #
       # @api public
       class Gnome
-        extend T::Sig
-
         NICE_NAME = "GNOME"
 
         # The `Regexp` used to determine if the strategy applies to the URL.
diff --git a/Library/Homebrew/livecheck/strategy/gnu.rb b/Library/Homebrew/livecheck/strategy/gnu.rb
index afbe7b7d54..59c29e22e2 100644
--- a/Library/Homebrew/livecheck/strategy/gnu.rb
+++ b/Library/Homebrew/livecheck/strategy/gnu.rb
@@ -29,8 +29,6 @@ module Homebrew
       #
       # @api public
       class Gnu
-        extend T::Sig
-
         NICE_NAME = "GNU"
 
         # The `Regexp` used to determine if the strategy applies to the URL.
diff --git a/Library/Homebrew/livecheck/strategy/hackage.rb b/Library/Homebrew/livecheck/strategy/hackage.rb
index f483e8ab09..26c5ab6594 100644
--- a/Library/Homebrew/livecheck/strategy/hackage.rb
+++ b/Library/Homebrew/livecheck/strategy/hackage.rb
@@ -17,8 +17,6 @@ module Homebrew
       #
       # @api public
       class Hackage
-        extend T::Sig
-
         # A `Regexp` used in determining if the strategy applies to the URL and
         # also as part of extracting the package name from the URL basename.
         PACKAGE_NAME_REGEX = /(?<package_name>.+?)-\d+/i.freeze
diff --git a/Library/Homebrew/livecheck/strategy/header_match.rb b/Library/Homebrew/livecheck/strategy/header_match.rb
index 5f3ceb6e4f..a0d2729bb7 100644
--- a/Library/Homebrew/livecheck/strategy/header_match.rb
+++ b/Library/Homebrew/livecheck/strategy/header_match.rb
@@ -12,8 +12,6 @@ module Homebrew
       #
       # @api private
       class HeaderMatch
-        extend T::Sig
-
         NICE_NAME = "Header match"
 
         # A priority of zero causes livecheck to skip the strategy. We do this
diff --git a/Library/Homebrew/livecheck/strategy/json.rb b/Library/Homebrew/livecheck/strategy/json.rb
index 042dbe1644..3cf1c475ab 100644
--- a/Library/Homebrew/livecheck/strategy/json.rb
+++ b/Library/Homebrew/livecheck/strategy/json.rb
@@ -23,8 +23,6 @@ module Homebrew
       #
       # @api public
       class Json
-        extend T::Sig
-
         NICE_NAME = "JSON"
 
         # A priority of zero causes livecheck to skip the strategy. We do this
diff --git a/Library/Homebrew/livecheck/strategy/launchpad.rb b/Library/Homebrew/livecheck/strategy/launchpad.rb
index 5593f04a01..773870f5e0 100644
--- a/Library/Homebrew/livecheck/strategy/launchpad.rb
+++ b/Library/Homebrew/livecheck/strategy/launchpad.rb
@@ -23,8 +23,6 @@ module Homebrew
       #
       # @api public
       class Launchpad
-        extend T::Sig
-
         # The `Regexp` used to determine if the strategy applies to the URL.
         URL_MATCH_REGEX = %r{
           ^https?://(?:[^/]+?\.)*launchpad\.net
diff --git a/Library/Homebrew/livecheck/strategy/npm.rb b/Library/Homebrew/livecheck/strategy/npm.rb
index b72051c472..b152397196 100644
--- a/Library/Homebrew/livecheck/strategy/npm.rb
+++ b/Library/Homebrew/livecheck/strategy/npm.rb
@@ -17,8 +17,6 @@ module Homebrew
       #
       # @api public
       class Npm
-        extend T::Sig
-
         NICE_NAME = "npm"
 
         # The `Regexp` used to determine if the strategy applies to the URL.
diff --git a/Library/Homebrew/livecheck/strategy/page_match.rb b/Library/Homebrew/livecheck/strategy/page_match.rb
index 37a69b0c49..753b57adef 100644
--- a/Library/Homebrew/livecheck/strategy/page_match.rb
+++ b/Library/Homebrew/livecheck/strategy/page_match.rb
@@ -16,8 +16,6 @@ module Homebrew
       #
       # @api public
       class PageMatch
-        extend T::Sig
-
         NICE_NAME = "Page match"
 
         # A priority of zero causes livecheck to skip the strategy. We do this
diff --git a/Library/Homebrew/livecheck/strategy/pypi.rb b/Library/Homebrew/livecheck/strategy/pypi.rb
index 206585474b..99d495a20f 100644
--- a/Library/Homebrew/livecheck/strategy/pypi.rb
+++ b/Library/Homebrew/livecheck/strategy/pypi.rb
@@ -17,8 +17,6 @@ module Homebrew
       #
       # @api public
       class Pypi
-        extend T::Sig
-
         NICE_NAME = "PyPI"
 
         # The `Regexp` used to extract the package name and suffix (e.g., file
diff --git a/Library/Homebrew/livecheck/strategy/sourceforge.rb b/Library/Homebrew/livecheck/strategy/sourceforge.rb
index 8a08235fce..aa8d8d9d5c 100644
--- a/Library/Homebrew/livecheck/strategy/sourceforge.rb
+++ b/Library/Homebrew/livecheck/strategy/sourceforge.rb
@@ -31,8 +31,6 @@ module Homebrew
       #
       # @api public
       class Sourceforge
-        extend T::Sig
-
         NICE_NAME = "SourceForge"
 
         # The `Regexp` used to determine if the strategy applies to the URL.
diff --git a/Library/Homebrew/livecheck/strategy/sparkle.rb b/Library/Homebrew/livecheck/strategy/sparkle.rb
index 3a34de97f5..03a34676e8 100644
--- a/Library/Homebrew/livecheck/strategy/sparkle.rb
+++ b/Library/Homebrew/livecheck/strategy/sparkle.rb
@@ -14,8 +14,6 @@ module Homebrew
       #
       # @api private
       class Sparkle
-        extend T::Sig
-
         # A priority of zero causes livecheck to skip the strategy. We do this
         # for {Sparkle} so we can selectively apply it when appropriate.
         PRIORITY = 0
@@ -46,8 +44,6 @@ module Homebrew
           :bundle_version,
           keyword_init: true,
         ) do
-          extend T::Sig
-
           extend Forwardable
 
           # @api public
diff --git a/Library/Homebrew/livecheck/strategy/xml.rb b/Library/Homebrew/livecheck/strategy/xml.rb
index b1a59d40ab..06c9b09457 100644
--- a/Library/Homebrew/livecheck/strategy/xml.rb
+++ b/Library/Homebrew/livecheck/strategy/xml.rb
@@ -27,8 +27,6 @@ module Homebrew
       #
       # @api public
       class Xml
-        extend T::Sig
-
         NICE_NAME = "XML"
 
         # A priority of zero causes livecheck to skip the strategy. We do this
diff --git a/Library/Homebrew/livecheck/strategy/xorg.rb b/Library/Homebrew/livecheck/strategy/xorg.rb
index 513352d4a5..ed73d42bc6 100644
--- a/Library/Homebrew/livecheck/strategy/xorg.rb
+++ b/Library/Homebrew/livecheck/strategy/xorg.rb
@@ -38,8 +38,6 @@ module Homebrew
       #
       # @api public
       class Xorg
-        extend T::Sig
-
         NICE_NAME = "X.Org"
 
         # A `Regexp` used in determining if the strategy applies to the URL and
diff --git a/Library/Homebrew/livecheck/strategy/yaml.rb b/Library/Homebrew/livecheck/strategy/yaml.rb
index e96ca16608..2f75f57378 100644
--- a/Library/Homebrew/livecheck/strategy/yaml.rb
+++ b/Library/Homebrew/livecheck/strategy/yaml.rb
@@ -23,8 +23,6 @@ module Homebrew
       #
       # @api public
       class Yaml
-        extend T::Sig
-
         NICE_NAME = "YAML"
 
         # A priority of zero causes livecheck to skip the strategy. We do this
diff --git a/Library/Homebrew/locale.rb b/Library/Homebrew/locale.rb
index ac6b24c6a6..6c0fe89bf5 100644
--- a/Library/Homebrew/locale.rb
+++ b/Library/Homebrew/locale.rb
@@ -7,8 +7,6 @@
 #
 # @api private
 class Locale
-  extend T::Sig
-
   # Error when a string cannot be parsed to a `Locale`.
   class ParserError < StandardError
   end
diff --git a/Library/Homebrew/macos_runner_spec.rb b/Library/Homebrew/macos_runner_spec.rb
index eb3b5ff6c4..880e07e382 100644
--- a/Library/Homebrew/macos_runner_spec.rb
+++ b/Library/Homebrew/macos_runner_spec.rb
@@ -2,8 +2,6 @@
 # frozen_string_literal: true
 
 class MacOSRunnerSpec < T::Struct
-  extend T::Sig
-
   const :name, String
   const :runner, String
   const :cleanup, T::Boolean
diff --git a/Library/Homebrew/manpages.rb b/Library/Homebrew/manpages.rb
index 3787ef826e..8b53c571b4 100644
--- a/Library/Homebrew/manpages.rb
+++ b/Library/Homebrew/manpages.rb
@@ -12,8 +12,6 @@ module Homebrew
   #
   # @api private
   module Manpages
-    extend T::Sig
-
     Variables = Struct.new(
       :alumni,
       :commands,
diff --git a/Library/Homebrew/messages.rb b/Library/Homebrew/messages.rb
index 89e03b551a..b7f9a47a2f 100644
--- a/Library/Homebrew/messages.rb
+++ b/Library/Homebrew/messages.rb
@@ -4,8 +4,6 @@
 # A {Messages} object collects messages that may need to be displayed together
 # at the end of a multi-step `brew` command run.
 class Messages
-  extend T::Sig
-
   attr_reader :caveats, :package_count, :install_times
 
   sig { void }
diff --git a/Library/Homebrew/migrator.rb b/Library/Homebrew/migrator.rb
index eff91d77fc..89dd54c2af 100644
--- a/Library/Homebrew/migrator.rb
+++ b/Library/Homebrew/migrator.rb
@@ -9,8 +9,6 @@ require "tab"
 #
 # @api private
 class Migrator
-  extend T::Sig
-
   include Context
 
   # Error for when a migration is necessary.
diff --git a/Library/Homebrew/mktemp.rb b/Library/Homebrew/mktemp.rb
index 159c0b88df..4bbf14b5dd 100644
--- a/Library/Homebrew/mktemp.rb
+++ b/Library/Homebrew/mktemp.rb
@@ -4,8 +4,6 @@
 # Performs {Formula#mktemp}'s functionality, and tracks the results.
 # Each instance is only intended to be used once.
 class Mktemp
-  extend T::Sig
-
   include FileUtils
 
   # Path to the tmpdir used in this run, as a {Pathname}.
diff --git a/Library/Homebrew/options.rb b/Library/Homebrew/options.rb
index e1877729a0..17ff40168d 100644
--- a/Library/Homebrew/options.rb
+++ b/Library/Homebrew/options.rb
@@ -5,8 +5,6 @@
 #
 # @api private
 class Option
-  extend T::Sig
-
   attr_reader :name, :description, :flag
 
   def initialize(name, description = "")
@@ -44,8 +42,6 @@ end
 #
 # @api private
 class DeprecatedOption
-  extend T::Sig
-
   attr_reader :old, :current
 
   def initialize(old, current)
@@ -73,8 +69,6 @@ end
 #
 # @api private
 class Options
-  extend T::Sig
-
   include Enumerable
 
   def self.create(array)
diff --git a/Library/Homebrew/os.rb b/Library/Homebrew/os.rb
index da59954324..4743f6b727 100644
--- a/Library/Homebrew/os.rb
+++ b/Library/Homebrew/os.rb
@@ -7,8 +7,6 @@ require "version"
 #
 # @api private
 module OS
-  extend T::Sig
-
   # Check if the operating system is macOS.
   #
   # @api public
diff --git a/Library/Homebrew/os/linux.rb b/Library/Homebrew/os/linux.rb
index 0d7a7ca916..15d096bd97 100644
--- a/Library/Homebrew/os/linux.rb
+++ b/Library/Homebrew/os/linux.rb
@@ -6,8 +6,6 @@ require "utils"
 module OS
   # Helper module for querying system information on Linux.
   module Linux
-    extend T::Sig
-
     sig { returns(String) }
     def self.os_version
       if which("lsb_release")
diff --git a/Library/Homebrew/os/linux/glibc.rb b/Library/Homebrew/os/linux/glibc.rb
index 476723e741..440af7521d 100644
--- a/Library/Homebrew/os/linux/glibc.rb
+++ b/Library/Homebrew/os/linux/glibc.rb
@@ -7,8 +7,6 @@ module OS
     #
     # @api private
     module Glibc
-      extend T::Sig
-
       module_function
 
       sig { returns(Version) }
diff --git a/Library/Homebrew/os/linux/kernel.rb b/Library/Homebrew/os/linux/kernel.rb
index 026e01e41a..0fe09b3952 100644
--- a/Library/Homebrew/os/linux/kernel.rb
+++ b/Library/Homebrew/os/linux/kernel.rb
@@ -7,8 +7,6 @@ module OS
     #
     # @api private
     module Kernel
-      extend T::Sig
-
       module_function
 
       sig { returns(Version) }
diff --git a/Library/Homebrew/os/mac.rb b/Library/Homebrew/os/mac.rb
index 4b6eeffe23..3936b46d5f 100644
--- a/Library/Homebrew/os/mac.rb
+++ b/Library/Homebrew/os/mac.rb
@@ -9,8 +9,6 @@ require "os/mac/keg"
 module OS
   # Helper module for querying system information on macOS.
   module Mac
-    extend T::Sig
-
     ::MacOS = OS::Mac
 
     raise "Loaded OS::Mac on generic OS!" if ENV["HOMEBREW_TEST_GENERIC_OS"]
diff --git a/Library/Homebrew/os/mac/sdk.rb b/Library/Homebrew/os/mac/sdk.rb
index 3fadf31762..09cc68e61c 100644
--- a/Library/Homebrew/os/mac/sdk.rb
+++ b/Library/Homebrew/os/mac/sdk.rb
@@ -9,8 +9,6 @@ module OS
     #
     # @api private
     class SDK
-      extend T::Sig
-
       # 11.x SDKs are explicitly excluded - we want the MacOSX11.sdk symlink instead.
       VERSIONED_SDK_REGEX = /MacOSX(10\.\d+|\d+)\.sdk$/.freeze
 
@@ -35,7 +33,6 @@ module OS
     #
     # @api private
     class BaseSDKLocator
-      extend T::Sig
       extend T::Helpers
 
       abstract!
@@ -146,8 +143,6 @@ module OS
     #
     # @api private
     class XcodeSDKLocator < BaseSDKLocator
-      extend T::Sig
-
       sig { override.returns(Symbol) }
       def source
         :xcode
@@ -173,8 +168,6 @@ module OS
     #
     # @api private
     class CLTSDKLocator < BaseSDKLocator
-      extend T::Sig
-
       sig { override.returns(Symbol) }
       def source
         :clt
diff --git a/Library/Homebrew/os/mac/version.rb b/Library/Homebrew/os/mac/version.rb
index ce8ebb6880..6dcda2f571 100644
--- a/Library/Homebrew/os/mac/version.rb
+++ b/Library/Homebrew/os/mac/version.rb
@@ -11,8 +11,6 @@ module OS
     #
     # @api private
     class Version < ::Version
-      extend T::Sig
-
       sig { params(version: Symbol).returns(T.attached_class) }
       def self.from_symbol(version)
         str = MacOSVersions::SYMBOLS.fetch(version) { raise MacOSVersionError, version }
diff --git a/Library/Homebrew/os/mac/xcode.rb b/Library/Homebrew/os/mac/xcode.rb
index 97088b1f91..4cc61a5762 100755
--- a/Library/Homebrew/os/mac/xcode.rb
+++ b/Library/Homebrew/os/mac/xcode.rb
@@ -7,8 +7,6 @@ module OS
     #
     # @api private
     module Xcode
-      extend T::Sig
-
       DEFAULT_BUNDLE_PATH = Pathname("/Applications/Xcode.app").freeze
       BUNDLE_ID = "com.apple.dt.Xcode"
       OLD_BUNDLE_ID = "com.apple.Xcode"
@@ -259,8 +257,6 @@ module OS
     #
     # @api private
     module CLT
-      extend T::Sig
-
       # The original Mavericks CLT package ID
       EXECUTABLE_PKG_ID = "com.apple.pkg.CLTools_Executables"
       MAVERICKS_NEW_PKG_ID = "com.apple.pkg.CLTools_Base" # obsolete
diff --git a/Library/Homebrew/patch.rb b/Library/Homebrew/patch.rb
index aac65d171f..5d470f89c8 100644
--- a/Library/Homebrew/patch.rb
+++ b/Library/Homebrew/patch.rb
@@ -35,8 +35,6 @@ end
 #
 # @api private
 class EmbeddedPatch
-  extend T::Sig
-
   attr_writer :owner
   attr_reader :strip
 
@@ -67,8 +65,6 @@ end
 #
 # @api private
 class DATAPatch < EmbeddedPatch
-  extend T::Sig
-
   attr_accessor :path
 
   def initialize(strip)
@@ -110,8 +106,6 @@ end
 #
 # @api private
 class ExternalPatch
-  extend T::Sig
-
   extend Forwardable
 
   attr_reader :resource, :strip
diff --git a/Library/Homebrew/requirement.rb b/Library/Homebrew/requirement.rb
index 39bd77469e..525f6522ca 100644
--- a/Library/Homebrew/requirement.rb
+++ b/Library/Homebrew/requirement.rb
@@ -12,8 +12,6 @@ require "build_environment"
 #
 # @api private
 class Requirement
-  extend T::Sig
-
   include Dependable
   extend Cachable
 
@@ -164,8 +162,6 @@ class Requirement
   end
 
   class << self
-    extend T::Sig
-
     include BuildEnvironment::DSL
 
     attr_reader :env_proc, :build
diff --git a/Library/Homebrew/requirements/arch_requirement.rb b/Library/Homebrew/requirements/arch_requirement.rb
index 8c65b44fa6..17ce1c2701 100644
--- a/Library/Homebrew/requirements/arch_requirement.rb
+++ b/Library/Homebrew/requirements/arch_requirement.rb
@@ -7,8 +7,6 @@ require "requirement"
 #
 # @api private
 class ArchRequirement < Requirement
-  extend T::Sig
-
   fatal true
 
   attr_reader :arch
diff --git a/Library/Homebrew/requirements/codesign_requirement.rb b/Library/Homebrew/requirements/codesign_requirement.rb
index c1c18d0d2f..7a006472c7 100644
--- a/Library/Homebrew/requirements/codesign_requirement.rb
+++ b/Library/Homebrew/requirements/codesign_requirement.rb
@@ -5,8 +5,6 @@
 #
 # @api private
 class CodesignRequirement < Requirement
-  extend T::Sig
-
   fatal true
 
   def initialize(tags)
diff --git a/Library/Homebrew/requirements/linux_requirement.rb b/Library/Homebrew/requirements/linux_requirement.rb
index 36b70585ac..058f5a1166 100644
--- a/Library/Homebrew/requirements/linux_requirement.rb
+++ b/Library/Homebrew/requirements/linux_requirement.rb
@@ -5,8 +5,6 @@
 #
 # @api private
 class LinuxRequirement < Requirement
-  extend T::Sig
-
   fatal true
 
   satisfy(build_env: false) { OS.linux? }
diff --git a/Library/Homebrew/requirements/macos_requirement.rb b/Library/Homebrew/requirements/macos_requirement.rb
index b6047e1ea1..ddcba52f97 100644
--- a/Library/Homebrew/requirements/macos_requirement.rb
+++ b/Library/Homebrew/requirements/macos_requirement.rb
@@ -7,8 +7,6 @@ require "requirement"
 #
 # @api private
 class MacOSRequirement < Requirement
-  extend T::Sig
-
   fatal true
 
   attr_reader :comparator, :version
diff --git a/Library/Homebrew/requirements/xcode_requirement.rb b/Library/Homebrew/requirements/xcode_requirement.rb
index c75bea041e..097d0d3e07 100644
--- a/Library/Homebrew/requirements/xcode_requirement.rb
+++ b/Library/Homebrew/requirements/xcode_requirement.rb
@@ -7,8 +7,6 @@ require "requirement"
 #
 # @api private
 class XcodeRequirement < Requirement
-  extend T::Sig
-
   fatal true
 
   attr_reader :version
diff --git a/Library/Homebrew/resource.rb b/Library/Homebrew/resource.rb
index 475b7f320f..16579c8002 100644
--- a/Library/Homebrew/resource.rb
+++ b/Library/Homebrew/resource.rb
@@ -14,8 +14,6 @@ require "extend/on_system"
 #
 # @api private
 class Resource
-  extend T::Sig
-
   include Context
   include FileUtils
   include OnSystem::MacOSAndLinux
@@ -343,8 +341,6 @@ end
 #
 # @api private
 class ResourceStageContext
-  extend T::Sig
-
   extend Forwardable
 
   # The {Resource} that is being staged.
diff --git a/Library/Homebrew/rubocops.rb b/Library/Homebrew/rubocops.rb
index 3e5daa64c3..187b692368 100644
--- a/Library/Homebrew/rubocops.rb
+++ b/Library/Homebrew/rubocops.rb
@@ -2,5 +2,6 @@
 # frozen_string_literal: true
 
 require_relative "standalone"
+require_relative "extend/module"
 
 require "rubocops/all"
diff --git a/Library/Homebrew/rubocops/cask/ast/cask_header.rb b/Library/Homebrew/rubocops/cask/ast/cask_header.rb
index 01386a4667..ae2231c22a 100644
--- a/Library/Homebrew/rubocops/cask/ast/cask_header.rb
+++ b/Library/Homebrew/rubocops/cask/ast/cask_header.rb
@@ -7,8 +7,6 @@ module RuboCop
       # This class wraps the AST method node that represents the cask header. It
       # includes various helper methods to aid cops in their analysis.
       class CaskHeader
-        extend T::Sig
-
         def initialize(method_node)
           @method_node = method_node
         end
diff --git a/Library/Homebrew/rubocops/cask/mixin/cask_help.rb b/Library/Homebrew/rubocops/cask/mixin/cask_help.rb
index 66b8f04471..c4e76607cb 100644
--- a/Library/Homebrew/rubocops/cask/mixin/cask_help.rb
+++ b/Library/Homebrew/rubocops/cask/mixin/cask_help.rb
@@ -7,7 +7,7 @@ module RuboCop
       # Common functionality for cops checking casks.
       module CaskHelp
         extend T::Helpers
-        extend T::Sig
+
         abstract!
 
         sig { abstract.params(cask_block: RuboCop::Cask::AST::CaskBlock).void }
diff --git a/Library/Homebrew/rubocops/cask/no_dsl_version.rb b/Library/Homebrew/rubocops/cask/no_dsl_version.rb
index 63dbda85e6..4e8e7410fb 100644
--- a/Library/Homebrew/rubocops/cask/no_dsl_version.rb
+++ b/Library/Homebrew/rubocops/cask/no_dsl_version.rb
@@ -19,8 +19,6 @@ module RuboCop
       #     ...
       #   end
       class NoDslVersion < Base
-        extend T::Sig
-
         extend Forwardable
         extend AutoCorrector
         include CaskHelp
diff --git a/Library/Homebrew/rubocops/cask/no_overrides.rb b/Library/Homebrew/rubocops/cask/no_overrides.rb
index 8a72402cf1..0abc90794b 100644
--- a/Library/Homebrew/rubocops/cask/no_overrides.rb
+++ b/Library/Homebrew/rubocops/cask/no_overrides.rb
@@ -5,7 +5,6 @@ module RuboCop
   module Cop
     module Cask
       class NoOverrides < Base
-        extend T::Sig
         include CaskHelp
 
         ON_SYSTEM_METHODS = RuboCop::Cask::Constants::ON_SYSTEM_METHODS
diff --git a/Library/Homebrew/rubocops/dependency_order.rb b/Library/Homebrew/rubocops/dependency_order.rb
index 061b2a0633..d3964237fb 100644
--- a/Library/Homebrew/rubocops/dependency_order.rb
+++ b/Library/Homebrew/rubocops/dependency_order.rb
@@ -11,7 +11,6 @@ module RuboCop
       # precedence order:
       # build-time > test > normal > recommended > optional
       class DependencyOrder < FormulaCop
-        extend T::Sig
         extend AutoCorrector
 
         def audit_formula(_node, _class_node, _parent_class_node, body_node)
diff --git a/Library/Homebrew/rubocops/extend/formula_cop.rb b/Library/Homebrew/rubocops/extend/formula_cop.rb
index 38b0b0725d..86801438ab 100644
--- a/Library/Homebrew/rubocops/extend/formula_cop.rb
+++ b/Library/Homebrew/rubocops/extend/formula_cop.rb
@@ -9,7 +9,6 @@ module RuboCop
     #
     # @api private
     class FormulaCop < Base
-      extend T::Sig
       extend T::Helpers
       include RangeHelp
       include HelperFunctions
diff --git a/Library/Homebrew/rubocops/patches.rb b/Library/Homebrew/rubocops/patches.rb
index e28b42761d..10ef997e25 100644
--- a/Library/Homebrew/rubocops/patches.rb
+++ b/Library/Homebrew/rubocops/patches.rb
@@ -9,7 +9,6 @@ module RuboCop
       # This cop audits `patch`es in formulae.
       # TODO: Many of these could be auto-corrected.
       class Patches < FormulaCop
-        extend T::Sig
         extend AutoCorrector
 
         def audit_formula(node, _class_node, _parent_class_node, body)
diff --git a/Library/Homebrew/rubocops/shared/helper_functions.rb b/Library/Homebrew/rubocops/shared/helper_functions.rb
index 67229a2623..6f24b6667b 100644
--- a/Library/Homebrew/rubocops/shared/helper_functions.rb
+++ b/Library/Homebrew/rubocops/shared/helper_functions.rb
@@ -14,7 +14,6 @@ module RuboCop
     #
     # @api private
     module HelperFunctions
-      extend T::Sig
       include RangeHelp
 
       # Checks for regex match of pattern in the node and
diff --git a/Library/Homebrew/rubocops/urls.rb b/Library/Homebrew/rubocops/urls.rb
index 34d4a65e2b..e2048aaade 100644
--- a/Library/Homebrew/rubocops/urls.rb
+++ b/Library/Homebrew/rubocops/urls.rb
@@ -261,8 +261,6 @@ module RuboCop
       #
       # @api private
       class PyPiUrls < FormulaCop
-        extend T::Sig
-
         def audit_formula(_node, _class_node, _parent_class_node, body_node)
           return if body_node.nil?
 
diff --git a/Library/Homebrew/sandbox.rb b/Library/Homebrew/sandbox.rb
index 289ae6234c..6eda0073c5 100644
--- a/Library/Homebrew/sandbox.rb
+++ b/Library/Homebrew/sandbox.rb
@@ -10,8 +10,6 @@ require "tempfile"
 #
 # @api private
 class Sandbox
-  extend T::Sig
-
   SANDBOX_EXEC = "/usr/bin/sandbox-exec"
   private_constant :SANDBOX_EXEC
 
@@ -210,8 +208,6 @@ class Sandbox
 
   # Configuration profile for a sandbox.
   class SandboxProfile
-    extend T::Sig
-
     SEATBELT_ERB = <<~ERB
       (version 1)
       (debug deny) ; log all denied operations to /var/log/system.log
diff --git a/Library/Homebrew/service.rb b/Library/Homebrew/service.rb
index 588642960e..4fafa2c80e 100644
--- a/Library/Homebrew/service.rb
+++ b/Library/Homebrew/service.rb
@@ -8,7 +8,6 @@ module Homebrew
   # `service` block and stores related instance variables. Most of these methods
   # also return the related instance variable when no argument is provided.
   class Service
-    extend T::Sig
     extend Forwardable
     include OnSystem::MacOSAndLinux
 
diff --git a/Library/Homebrew/simulate_system.rb b/Library/Homebrew/simulate_system.rb
index 51be4690a6..741aa41b48 100644
--- a/Library/Homebrew/simulate_system.rb
+++ b/Library/Homebrew/simulate_system.rb
@@ -7,8 +7,6 @@ module Homebrew
   # @api private
   class SimulateSystem
     class << self
-      extend T::Sig
-
       attr_reader :arch, :os
 
       sig { params(new_os: Symbol).void }
diff --git a/Library/Homebrew/software_spec.rb b/Library/Homebrew/software_spec.rb
index d1c13f049c..800394cc6d 100644
--- a/Library/Homebrew/software_spec.rb
+++ b/Library/Homebrew/software_spec.rb
@@ -15,8 +15,6 @@ require "os/mac/version"
 require "extend/on_system"
 
 class SoftwareSpec
-  extend T::Sig
-
   extend Forwardable
   include OnSystem::MacOSAndLinux
 
@@ -293,8 +291,6 @@ end
 
 class Bottle
   class Filename
-    extend T::Sig
-
     attr_reader :name, :version, :tag, :rebuild
 
     def self.create(formula, tag, rebuild)
@@ -510,8 +506,6 @@ end
 class BottleSpecification
   RELOCATABLE_CELLARS = [:any, :any_skip_relocation].freeze
 
-  extend T::Sig
-
   attr_rw :rebuild
   attr_accessor :tap
   attr_reader :collector, :root_url_specs, :repository
diff --git a/Library/Homebrew/sorbet/parlour.rb b/Library/Homebrew/sorbet/parlour.rb
index 102df89819..6c64b82b53 100644
--- a/Library/Homebrew/sorbet/parlour.rb
+++ b/Library/Homebrew/sorbet/parlour.rb
@@ -1,6 +1,7 @@
 # typed: true
 # frozen_string_literal: true
 
+require_relative "../extend/module"
 require_relative "../warnings"
 Warnings.ignore :parser_syntax do
   require "parser/current"
@@ -9,8 +10,6 @@ end
 module Homebrew
   # Parlour type signature generator helper class for Homebrew.
   module Parlour
-    extend T::Sig
-
     ROOT_DIR = T.let(Pathname(__dir__).parent.realpath.freeze, Pathname).freeze
 
     sig { returns(T::Array[Parser::AST::Node]) }
diff --git a/Library/Homebrew/style.rb b/Library/Homebrew/style.rb
index c0de31b8c8..79d1a56ed2 100644
--- a/Library/Homebrew/style.rb
+++ b/Library/Homebrew/style.rb
@@ -325,8 +325,6 @@ module Homebrew
 
     # Source location of a style offense.
     class LineLocation
-      extend T::Sig
-
       attr_reader :line, :column
 
       def initialize(json)
diff --git a/Library/Homebrew/system_command.rb b/Library/Homebrew/system_command.rb
index 71a1badf22..9e7e98fabe 100644
--- a/Library/Homebrew/system_command.rb
+++ b/Library/Homebrew/system_command.rb
@@ -14,14 +14,10 @@ require "extend/time"
 #
 # @api private
 class SystemCommand
-  extend T::Sig
-
   using TimeRemaining
 
   # Helper functions for calling {SystemCommand.run}.
   module Mixin
-    extend T::Sig
-
     def system_command(executable, **options)
       SystemCommand.run(executable, **options)
     end
@@ -282,8 +278,6 @@ class SystemCommand
 
   # Result containing the output and exit status of a finished sub-process.
   class Result
-    extend T::Sig
-
     include Context
 
     attr_accessor :command, :status, :exit_status
diff --git a/Library/Homebrew/system_config.rb b/Library/Homebrew/system_config.rb
index a5a1e48198..36ce8c2ee5 100644
--- a/Library/Homebrew/system_config.rb
+++ b/Library/Homebrew/system_config.rb
@@ -12,8 +12,6 @@ require "system_command"
 # @api private
 module SystemConfig
   class << self
-    extend T::Sig
-
     include SystemCommand::Mixin
 
     def clang
diff --git a/Library/Homebrew/tab.rb b/Library/Homebrew/tab.rb
index b0cec2ec8c..544db29247 100644
--- a/Library/Homebrew/tab.rb
+++ b/Library/Homebrew/tab.rb
@@ -9,8 +9,6 @@ require "extend/cachable"
 
 # Rather than calling `new` directly, use one of the class methods like {Tab.create}.
 class Tab
-  extend T::Sig
-
   extend Cachable
 
   FILENAME = "INSTALL_RECEIPT.json"
diff --git a/Library/Homebrew/tap.rb b/Library/Homebrew/tap.rb
index cb4f3b080a..f72624c07e 100644
--- a/Library/Homebrew/tap.rb
+++ b/Library/Homebrew/tap.rb
@@ -14,8 +14,6 @@ require "settings"
 # {#user} represents the GitHub username and {#repo} represents the repository
 # name without the leading `homebrew-`.
 class Tap
-  extend T::Sig
-
   extend Cachable
 
   TAP_DIRECTORY = (HOMEBREW_LIBRARY/"Taps").freeze
@@ -849,8 +847,6 @@ end
 
 # A specialized {Tap} class for the core formulae.
 class CoreTap < Tap
-  extend T::Sig
-
   # @private
   sig { void }
   def initialize
@@ -1010,8 +1006,6 @@ end
 
 # Permanent configuration per {Tap} using `git-config(1)`.
 class TapConfig
-  extend T::Sig
-
   attr_reader :tap
 
   sig { params(tap: Tap).void }
diff --git a/Library/Homebrew/tap_auditor.rb b/Library/Homebrew/tap_auditor.rb
index 16bd6e1444..64d0f66a14 100644
--- a/Library/Homebrew/tap_auditor.rb
+++ b/Library/Homebrew/tap_auditor.rb
@@ -6,8 +6,6 @@ module Homebrew
   #
   # @api private
   class TapAuditor
-    extend T::Sig
-
     attr_reader :name, :path, :formula_names, :formula_aliases, :cask_tokens,
                 :tap_audit_exceptions, :tap_style_exceptions, :tap_pypi_formula_mappings, :problems
 
diff --git a/Library/Homebrew/test_runner_formula.rb b/Library/Homebrew/test_runner_formula.rb
index b3e2dded65..ce4ea2469a 100644
--- a/Library/Homebrew/test_runner_formula.rb
+++ b/Library/Homebrew/test_runner_formula.rb
@@ -4,8 +4,6 @@
 require "formula"
 
 class TestRunnerFormula
-  extend T::Sig
-
   sig { returns(String) }
   attr_reader :name
 
diff --git a/Library/Homebrew/unpack_strategy.rb b/Library/Homebrew/unpack_strategy.rb
index 8a29604b93..3f6f93d61f 100644
--- a/Library/Homebrew/unpack_strategy.rb
+++ b/Library/Homebrew/unpack_strategy.rb
@@ -7,7 +7,6 @@ require "system_command"
 #
 # @api private
 module UnpackStrategy
-  extend T::Sig
   extend T::Helpers
 
   include SystemCommand::Mixin
diff --git a/Library/Homebrew/unpack_strategy/air.rb b/Library/Homebrew/unpack_strategy/air.rb
index aac0881fa8..ca0137e872 100644
--- a/Library/Homebrew/unpack_strategy/air.rb
+++ b/Library/Homebrew/unpack_strategy/air.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking Adobe Air archives.
   class Air
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/bazaar.rb b/Library/Homebrew/unpack_strategy/bazaar.rb
index 62eb0c4946..f73902b3aa 100644
--- a/Library/Homebrew/unpack_strategy/bazaar.rb
+++ b/Library/Homebrew/unpack_strategy/bazaar.rb
@@ -6,8 +6,6 @@ require_relative "directory"
 module UnpackStrategy
   # Strategy for unpacking Bazaar archives.
   class Bazaar < Directory
-    extend T::Sig
-
     def self.can_extract?(path)
       super && (path/".bzr").directory?
     end
diff --git a/Library/Homebrew/unpack_strategy/bzip2.rb b/Library/Homebrew/unpack_strategy/bzip2.rb
index 82e9501b6a..26eeaa784e 100644
--- a/Library/Homebrew/unpack_strategy/bzip2.rb
+++ b/Library/Homebrew/unpack_strategy/bzip2.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking bzip2 archives.
   class Bzip2
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/cab.rb b/Library/Homebrew/unpack_strategy/cab.rb
index de2258af88..a7038df4d0 100644
--- a/Library/Homebrew/unpack_strategy/cab.rb
+++ b/Library/Homebrew/unpack_strategy/cab.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking Cabinet archives.
   class Cab
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/compress.rb b/Library/Homebrew/unpack_strategy/compress.rb
index 0cc11abbd7..0082312214 100644
--- a/Library/Homebrew/unpack_strategy/compress.rb
+++ b/Library/Homebrew/unpack_strategy/compress.rb
@@ -6,8 +6,6 @@ require_relative "tar"
 module UnpackStrategy
   # Strategy for unpacking compress archives.
   class Compress < Tar
-    extend T::Sig
-
     sig { returns(T::Array[String]) }
     def self.extensions
       [".Z"]
diff --git a/Library/Homebrew/unpack_strategy/directory.rb b/Library/Homebrew/unpack_strategy/directory.rb
index f20b9a566e..4a36198426 100644
--- a/Library/Homebrew/unpack_strategy/directory.rb
+++ b/Library/Homebrew/unpack_strategy/directory.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking directories.
   class Directory
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/dmg.rb b/Library/Homebrew/unpack_strategy/dmg.rb
index cb3ee47d12..21cc3326f7 100644
--- a/Library/Homebrew/unpack_strategy/dmg.rb
+++ b/Library/Homebrew/unpack_strategy/dmg.rb
@@ -6,14 +6,10 @@ require "tempfile"
 module UnpackStrategy
   # Strategy for unpacking disk images.
   class Dmg
-    extend T::Sig
-
     include UnpackStrategy
 
     # Helper module for listing the contents of a volume mounted from a disk image.
     module Bom
-      extend T::Sig
-
       DMG_METADATA = Set.new(%w[
         .background
         .com.apple.timemachine.donotpresent
@@ -79,8 +75,6 @@ module UnpackStrategy
 
     # Strategy for unpacking a volume mounted from a disk image.
     class Mount
-      extend T::Sig
-
       include UnpackStrategy
 
       def eject(verbose: false)
diff --git a/Library/Homebrew/unpack_strategy/executable.rb b/Library/Homebrew/unpack_strategy/executable.rb
index 6bf45599ed..4c5e712da8 100644
--- a/Library/Homebrew/unpack_strategy/executable.rb
+++ b/Library/Homebrew/unpack_strategy/executable.rb
@@ -6,8 +6,6 @@ require_relative "uncompressed"
 module UnpackStrategy
   # Strategy for unpacking executables.
   class Executable < Uncompressed
-    extend T::Sig
-
     sig { returns(T::Array[String]) }
     def self.extensions
       [".sh", ".bash"]
diff --git a/Library/Homebrew/unpack_strategy/fossil.rb b/Library/Homebrew/unpack_strategy/fossil.rb
index a09218cf77..beeeae8f9b 100644
--- a/Library/Homebrew/unpack_strategy/fossil.rb
+++ b/Library/Homebrew/unpack_strategy/fossil.rb
@@ -6,8 +6,6 @@ require "system_command"
 module UnpackStrategy
   # Strategy for unpacking Fossil repositories.
   class Fossil
-    extend T::Sig
-
     include UnpackStrategy
     extend SystemCommand::Mixin
 
diff --git a/Library/Homebrew/unpack_strategy/generic_unar.rb b/Library/Homebrew/unpack_strategy/generic_unar.rb
index a860f8ec6b..c537f5efca 100644
--- a/Library/Homebrew/unpack_strategy/generic_unar.rb
+++ b/Library/Homebrew/unpack_strategy/generic_unar.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking archives with `unar`.
   class GenericUnar
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/gzip.rb b/Library/Homebrew/unpack_strategy/gzip.rb
index dbc70d2766..22bdf33881 100644
--- a/Library/Homebrew/unpack_strategy/gzip.rb
+++ b/Library/Homebrew/unpack_strategy/gzip.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking gzip archives.
   class Gzip
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/jar.rb b/Library/Homebrew/unpack_strategy/jar.rb
index 21153d1c2b..892357b48b 100644
--- a/Library/Homebrew/unpack_strategy/jar.rb
+++ b/Library/Homebrew/unpack_strategy/jar.rb
@@ -6,8 +6,6 @@ require_relative "uncompressed"
 module UnpackStrategy
   # Strategy for unpacking Java archives.
   class Jar < Uncompressed
-    extend T::Sig
-
     sig { returns(T::Array[String]) }
     def self.extensions
       [".apk", ".jar"]
diff --git a/Library/Homebrew/unpack_strategy/lha.rb b/Library/Homebrew/unpack_strategy/lha.rb
index a8cdfdb4cf..ee26437eaa 100644
--- a/Library/Homebrew/unpack_strategy/lha.rb
+++ b/Library/Homebrew/unpack_strategy/lha.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking LHa archives.
   class Lha
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/lua_rock.rb b/Library/Homebrew/unpack_strategy/lua_rock.rb
index d5ce7de33e..a01fcd2f88 100644
--- a/Library/Homebrew/unpack_strategy/lua_rock.rb
+++ b/Library/Homebrew/unpack_strategy/lua_rock.rb
@@ -6,8 +6,6 @@ require_relative "uncompressed"
 module UnpackStrategy
   # Strategy for unpacking LuaRock archives.
   class LuaRock < Uncompressed
-    extend T::Sig
-
     sig { returns(T::Array[String]) }
     def self.extensions
       [".rock"]
diff --git a/Library/Homebrew/unpack_strategy/lzip.rb b/Library/Homebrew/unpack_strategy/lzip.rb
index d1f2c432af..8925de24e3 100644
--- a/Library/Homebrew/unpack_strategy/lzip.rb
+++ b/Library/Homebrew/unpack_strategy/lzip.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking lzip archives.
   class Lzip
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/lzma.rb b/Library/Homebrew/unpack_strategy/lzma.rb
index bfe04be2c9..11e7bdc664 100644
--- a/Library/Homebrew/unpack_strategy/lzma.rb
+++ b/Library/Homebrew/unpack_strategy/lzma.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking LZMA archives.
   class Lzma
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/microsoft_office_xml.rb b/Library/Homebrew/unpack_strategy/microsoft_office_xml.rb
index ad3940d86e..2ad2a40644 100644
--- a/Library/Homebrew/unpack_strategy/microsoft_office_xml.rb
+++ b/Library/Homebrew/unpack_strategy/microsoft_office_xml.rb
@@ -6,8 +6,6 @@ require_relative "uncompressed"
 module UnpackStrategy
   # Strategy for unpacking Microsoft Office documents.
   class MicrosoftOfficeXml < Uncompressed
-    extend T::Sig
-
     sig { returns(T::Array[String]) }
     def self.extensions
       [
diff --git a/Library/Homebrew/unpack_strategy/otf.rb b/Library/Homebrew/unpack_strategy/otf.rb
index 61ac40a249..7a75f78c4c 100644
--- a/Library/Homebrew/unpack_strategy/otf.rb
+++ b/Library/Homebrew/unpack_strategy/otf.rb
@@ -6,8 +6,6 @@ require_relative "uncompressed"
 module UnpackStrategy
   # Strategy for unpacking OpenType fonts.
   class Otf < Uncompressed
-    extend T::Sig
-
     sig { returns(T::Array[String]) }
     def self.extensions
       [".otf"]
diff --git a/Library/Homebrew/unpack_strategy/p7zip.rb b/Library/Homebrew/unpack_strategy/p7zip.rb
index 715c816076..fb32908f2d 100644
--- a/Library/Homebrew/unpack_strategy/p7zip.rb
+++ b/Library/Homebrew/unpack_strategy/p7zip.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking P7ZIP archives.
   class P7Zip
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/pax.rb b/Library/Homebrew/unpack_strategy/pax.rb
index c454b57ab4..cd16295c5b 100644
--- a/Library/Homebrew/unpack_strategy/pax.rb
+++ b/Library/Homebrew/unpack_strategy/pax.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking pax archives.
   class Pax
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/pkg.rb b/Library/Homebrew/unpack_strategy/pkg.rb
index 7151ba6078..b48a55244b 100644
--- a/Library/Homebrew/unpack_strategy/pkg.rb
+++ b/Library/Homebrew/unpack_strategy/pkg.rb
@@ -6,8 +6,6 @@ require_relative "uncompressed"
 module UnpackStrategy
   # Strategy for unpacking macOS package installers.
   class Pkg < Uncompressed
-    extend T::Sig
-
     sig { returns(T::Array[String]) }
     def self.extensions
       [".pkg", ".mkpg"]
diff --git a/Library/Homebrew/unpack_strategy/rar.rb b/Library/Homebrew/unpack_strategy/rar.rb
index 9a2be11326..c9e7d3679b 100644
--- a/Library/Homebrew/unpack_strategy/rar.rb
+++ b/Library/Homebrew/unpack_strategy/rar.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking RAR archives.
   class Rar
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/self_extracting_executable.rb b/Library/Homebrew/unpack_strategy/self_extracting_executable.rb
index a3ea1e8de1..b32cfa825c 100644
--- a/Library/Homebrew/unpack_strategy/self_extracting_executable.rb
+++ b/Library/Homebrew/unpack_strategy/self_extracting_executable.rb
@@ -6,8 +6,6 @@ require_relative "generic_unar"
 module UnpackStrategy
   # Strategy for unpacking self-extracting executables.
   class SelfExtractingExecutable < GenericUnar
-    extend T::Sig
-
     sig { returns(T::Array[String]) }
     def self.extensions
       []
diff --git a/Library/Homebrew/unpack_strategy/sit.rb b/Library/Homebrew/unpack_strategy/sit.rb
index d747cb3178..7d447eaf69 100644
--- a/Library/Homebrew/unpack_strategy/sit.rb
+++ b/Library/Homebrew/unpack_strategy/sit.rb
@@ -6,8 +6,6 @@ require_relative "generic_unar"
 module UnpackStrategy
   # Strategy for unpacking Stuffit archives.
   class Sit < GenericUnar
-    extend T::Sig
-
     sig { returns(T::Array[String]) }
     def self.extensions
       [".sit"]
diff --git a/Library/Homebrew/unpack_strategy/tar.rb b/Library/Homebrew/unpack_strategy/tar.rb
index 42ca323d33..0290a02dd8 100644
--- a/Library/Homebrew/unpack_strategy/tar.rb
+++ b/Library/Homebrew/unpack_strategy/tar.rb
@@ -6,8 +6,6 @@ require "system_command"
 module UnpackStrategy
   # Strategy for unpacking tar archives.
   class Tar
-    extend T::Sig
-
     include UnpackStrategy
     extend SystemCommand::Mixin
 
diff --git a/Library/Homebrew/unpack_strategy/ttf.rb b/Library/Homebrew/unpack_strategy/ttf.rb
index bd0da5ea76..d741378d29 100644
--- a/Library/Homebrew/unpack_strategy/ttf.rb
+++ b/Library/Homebrew/unpack_strategy/ttf.rb
@@ -6,8 +6,6 @@ require_relative "uncompressed"
 module UnpackStrategy
   # Strategy for unpacking TrueType fonts.
   class Ttf < Uncompressed
-    extend T::Sig
-
     sig { returns(T::Array[String]) }
     def self.extensions
       [".ttc", ".ttf"]
diff --git a/Library/Homebrew/unpack_strategy/uncompressed.rb b/Library/Homebrew/unpack_strategy/uncompressed.rb
index 08b72eeb83..6cb24becaa 100644
--- a/Library/Homebrew/unpack_strategy/uncompressed.rb
+++ b/Library/Homebrew/unpack_strategy/uncompressed.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking uncompressed files.
   class Uncompressed
-    extend T::Sig
-
     include UnpackStrategy
 
     sig {
diff --git a/Library/Homebrew/unpack_strategy/xar.rb b/Library/Homebrew/unpack_strategy/xar.rb
index 4707a602d5..e1f0649ccb 100644
--- a/Library/Homebrew/unpack_strategy/xar.rb
+++ b/Library/Homebrew/unpack_strategy/xar.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking xar archives.
   class Xar
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/xz.rb b/Library/Homebrew/unpack_strategy/xz.rb
index cc450d51d6..be6b7ca55f 100644
--- a/Library/Homebrew/unpack_strategy/xz.rb
+++ b/Library/Homebrew/unpack_strategy/xz.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking xz archives.
   class Xz
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/zip.rb b/Library/Homebrew/unpack_strategy/zip.rb
index 85ef33cbbb..de7c8b532e 100644
--- a/Library/Homebrew/unpack_strategy/zip.rb
+++ b/Library/Homebrew/unpack_strategy/zip.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking ZIP archives.
   class Zip
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unpack_strategy/zstd.rb b/Library/Homebrew/unpack_strategy/zstd.rb
index c5f0b2a6b9..0e51960472 100644
--- a/Library/Homebrew/unpack_strategy/zstd.rb
+++ b/Library/Homebrew/unpack_strategy/zstd.rb
@@ -4,8 +4,6 @@
 module UnpackStrategy
   # Strategy for unpacking zstd archives.
   class Zstd
-    extend T::Sig
-
     include UnpackStrategy
 
     sig { returns(T::Array[String]) }
diff --git a/Library/Homebrew/unversioned_cask_checker.rb b/Library/Homebrew/unversioned_cask_checker.rb
index 9da27738c1..b66c1f6139 100644
--- a/Library/Homebrew/unversioned_cask_checker.rb
+++ b/Library/Homebrew/unversioned_cask_checker.rb
@@ -11,9 +11,7 @@ module Homebrew
   #
   # @api private
   class UnversionedCaskChecker
-    extend T::Sig
-
-    sig {  returns(Cask::Cask) }
+    sig { returns(Cask::Cask) }
     attr_reader :cask
 
     sig { params(cask: Cask::Cask).void }
diff --git a/Library/Homebrew/utils.rb b/Library/Homebrew/utils.rb
index a620e38c69..e685e1e20b 100644
--- a/Library/Homebrew/utils.rb
+++ b/Library/Homebrew/utils.rb
@@ -24,7 +24,6 @@ require "extend/kernel"
 
 module Homebrew
   extend Context
-  extend T::Sig
 
   def self._system(cmd, *args, **options)
     pid = fork do
@@ -87,8 +86,6 @@ module Homebrew
 end
 
 module Utils
-  extend T::Sig
-
   # Removes the rightmost segment from the constant expression in the string.
   #
   #   deconstantize('Net::HTTP')   # => "Net"
diff --git a/Library/Homebrew/utils/analytics.rb b/Library/Homebrew/utils/analytics.rb
index 4ce8599335..f3ed90ede7 100644
--- a/Library/Homebrew/utils/analytics.rb
+++ b/Library/Homebrew/utils/analytics.rb
@@ -17,8 +17,6 @@ module Utils
     INFLUX_ORG = "9a707721bb47fc02"
 
     class << self
-      extend T::Sig
-
       include Context
 
       sig { params(type: Symbol, metadata: T::Hash[Symbol, T.untyped]).void }
diff --git a/Library/Homebrew/utils/ast.rb b/Library/Homebrew/utils/ast.rb
index 5531fa376e..116853d0f7 100644
--- a/Library/Homebrew/utils/ast.rb
+++ b/Library/Homebrew/utils/ast.rb
@@ -9,8 +9,6 @@ module Utils
   #
   # @api private
   module AST
-    extend T::Sig
-
     Node = RuboCop::AST::Node
     SendNode = RuboCop::AST::SendNode
     BlockNode = RuboCop::AST::BlockNode
@@ -79,7 +77,6 @@ module Utils
     #
     # @api private
     class FormulaAST
-      extend T::Sig
       extend Forwardable
       include AST
 
diff --git a/Library/Homebrew/utils/bottles.rb b/Library/Homebrew/utils/bottles.rb
index 77e28b2fbe..1ef0334cea 100644
--- a/Library/Homebrew/utils/bottles.rb
+++ b/Library/Homebrew/utils/bottles.rb
@@ -9,8 +9,6 @@ module Utils
   # @api private
   module Bottles
     class << self
-      extend T::Sig
-
       # Gets the tag for the running OS.
       def tag(symbol = nil)
         return Tag.from_symbol(symbol) if symbol.present?
@@ -124,8 +122,6 @@ module Utils
 
     # Denotes the arch and OS of a bottle.
     class Tag
-      extend T::Sig
-
       attr_reader :system, :arch
 
       sig { params(value: Symbol).returns(T.attached_class) }
@@ -245,8 +241,6 @@ module Utils
 
     # The specification for a specific tag
     class TagSpecification
-      extend T::Sig
-
       sig { returns(Utils::Bottles::Tag) }
       attr_reader :tag
 
@@ -265,8 +259,6 @@ module Utils
 
     # Collector for bottle specifications.
     class Collector
-      extend T::Sig
-
       sig { void }
       def initialize
         @tag_specs = T.let({}, T::Hash[Utils::Bottles::Tag, Utils::Bottles::TagSpecification])
diff --git a/Library/Homebrew/utils/curl.rb b/Library/Homebrew/utils/curl.rb
index 2687ad268d..faf8a423a5 100644
--- a/Library/Homebrew/utils/curl.rb
+++ b/Library/Homebrew/utils/curl.rb
@@ -10,8 +10,6 @@ module Utils
   #
   # @api private
   module Curl
-    extend T::Sig
-
     using TimeRemaining
 
     # This regex is used to extract the part of an ETag within quotation marks,
diff --git a/Library/Homebrew/utils/git.rb b/Library/Homebrew/utils/git.rb
index 10c36f2896..16044b6d24 100644
--- a/Library/Homebrew/utils/git.rb
+++ b/Library/Homebrew/utils/git.rb
@@ -7,8 +7,6 @@ module Utils
   # @see GitRepository
   # @api private
   module Git
-    extend T::Sig
-
     def self.available?
       version.present?
     end
diff --git a/Library/Homebrew/utils/git_repository.rb b/Library/Homebrew/utils/git_repository.rb
index 2af21e2989..8ad326348c 100644
--- a/Library/Homebrew/utils/git_repository.rb
+++ b/Library/Homebrew/utils/git_repository.rb
@@ -2,8 +2,6 @@
 # frozen_string_literal: true
 
 module Utils
-  extend T::Sig
-
   # Gets the full commit hash of the HEAD commit.
   sig {
     params(
diff --git a/Library/Homebrew/utils/github.rb b/Library/Homebrew/utils/github.rb
index e6d3525e5d..830c6c5a81 100644
--- a/Library/Homebrew/utils/github.rb
+++ b/Library/Homebrew/utils/github.rb
@@ -11,8 +11,6 @@ require "system_command"
 #
 # @api private
 module GitHub
-  extend T::Sig
-
   include SystemCommand::Mixin
 
   def self.check_runs(repo: nil, commit: nil, pull_request: nil)
diff --git a/Library/Homebrew/utils/github/actions.rb b/Library/Homebrew/utils/github/actions.rb
index ed27283bb3..b6e3d6b768 100644
--- a/Library/Homebrew/utils/github/actions.rb
+++ b/Library/Homebrew/utils/github/actions.rb
@@ -9,8 +9,6 @@ module GitHub
   #
   # @api private
   module Actions
-    extend T::Sig
-
     sig { params(string: String).returns(String) }
     def self.escape(string)
       # See https://github.community/t/set-output-truncates-multiline-strings/16852/3.
@@ -39,8 +37,6 @@ module GitHub
 
     # Helper class for formatting annotations on GitHub Actions.
     class Annotation
-      extend T::Sig
-
       ANNOTATION_TYPES = [:notice, :warning, :error].freeze
 
       sig { params(path: T.any(String, Pathname)).returns(T.nilable(Pathname)) }
diff --git a/Library/Homebrew/utils/github/api.rb b/Library/Homebrew/utils/github/api.rb
index 2e3e21d6e4..120d9a2507 100644
--- a/Library/Homebrew/utils/github/api.rb
+++ b/Library/Homebrew/utils/github/api.rb
@@ -31,8 +31,6 @@ module GitHub
   #
   # @api private
   module API
-    extend T::Sig
-
     # Generic API error.
     class Error < RuntimeError
       attr_reader :github_message
diff --git a/Library/Homebrew/utils/gzip.rb b/Library/Homebrew/utils/gzip.rb
index f31951fbf1..a7df3e6db4 100644
--- a/Library/Homebrew/utils/gzip.rb
+++ b/Library/Homebrew/utils/gzip.rb
@@ -8,8 +8,6 @@ GZIP_BUFFER_SIZE = 64 * 1024
 module Utils
   # Helper functions for creating gzip files.
   module Gzip
-    extend T::Sig
-
     sig {
       params(
         path:      T.any(String, Pathname),
diff --git a/Library/Homebrew/utils/inreplace.rb b/Library/Homebrew/utils/inreplace.rb
index 50ff78f80b..9e3118a430 100644
--- a/Library/Homebrew/utils/inreplace.rb
+++ b/Library/Homebrew/utils/inreplace.rb
@@ -8,8 +8,6 @@ module Utils
   #
   # @api private
   module Inreplace
-    extend T::Sig
-
     # Error during text replacement.
     class Error < RuntimeError
       def initialize(errors)
diff --git a/Library/Homebrew/utils/pypi.rb b/Library/Homebrew/utils/pypi.rb
index c65576a186..c3be9afbeb 100644
--- a/Library/Homebrew/utils/pypi.rb
+++ b/Library/Homebrew/utils/pypi.rb
@@ -5,8 +5,6 @@
 #
 # @api private
 module PyPI
-  extend T::Sig
-
   PYTHONHOSTED_URL_PREFIX = "https://files.pythonhosted.org/packages/"
   private_constant :PYTHONHOSTED_URL_PREFIX
 
@@ -14,8 +12,6 @@ module PyPI
   #
   # @api private
   class Package
-    extend T::Sig
-
     attr_accessor :name, :extras, :version
 
     sig { params(package_string: String, is_url: T::Boolean).void }
diff --git a/Library/Homebrew/utils/shebang.rb b/Library/Homebrew/utils/shebang.rb
index 7ce5a56284..95daadcd6d 100644
--- a/Library/Homebrew/utils/shebang.rb
+++ b/Library/Homebrew/utils/shebang.rb
@@ -6,16 +6,12 @@ module Utils
   #
   # @api private
   module Shebang
-    extend T::Sig
-
     module_function
 
     # Specification on how to rewrite a given shebang.
     #
     # @api private
     class RewriteInfo
-      extend T::Sig
-
       attr_reader :regex, :max_length, :replacement
 
       sig { params(regex: Regexp, max_length: Integer, replacement: T.any(String, Pathname)).void }
diff --git a/Library/Homebrew/utils/shell.rb b/Library/Homebrew/utils/shell.rb
index 4fe444be02..6a903d4c88 100644
--- a/Library/Homebrew/utils/shell.rb
+++ b/Library/Homebrew/utils/shell.rb
@@ -3,8 +3,6 @@
 
 module Utils
   module Shell
-    extend T::Sig
-
     module_function
 
     # Take a path and heuristically convert it to a shell name,
diff --git a/Library/Homebrew/utils/string_inreplace_extension.rb b/Library/Homebrew/utils/string_inreplace_extension.rb
index ab49f00a92..d04ed40dfd 100644
--- a/Library/Homebrew/utils/string_inreplace_extension.rb
+++ b/Library/Homebrew/utils/string_inreplace_extension.rb
@@ -5,8 +5,6 @@
 #
 # @api private
 class StringInreplaceExtension
-  extend T::Sig
-
   attr_accessor :errors, :inreplace_string
 
   sig { params(string: String).void }
diff --git a/Library/Homebrew/utils/svn.rb b/Library/Homebrew/utils/svn.rb
index 5c4269ddd6..71a1d9ddf2 100644
--- a/Library/Homebrew/utils/svn.rb
+++ b/Library/Homebrew/utils/svn.rb
@@ -9,8 +9,6 @@ module Utils
   # @api private
   module Svn
     class << self
-      extend T::Sig
-
       include SystemCommand::Mixin
 
       sig { returns(T::Boolean) }
diff --git a/Library/Homebrew/utils/topological_hash.rb b/Library/Homebrew/utils/topological_hash.rb
index cac774c857..726eab599e 100644
--- a/Library/Homebrew/utils/topological_hash.rb
+++ b/Library/Homebrew/utils/topological_hash.rb
@@ -6,8 +6,6 @@ require "tsort"
 module Utils
   # Topologically sortable hash map.
   class TopologicalHash < Hash
-    extend T::Sig
-
     include TSort
 
     sig {
diff --git a/Library/Homebrew/utils/tty.rb b/Library/Homebrew/utils/tty.rb
index 97ee940256..4897c92fb3 100644
--- a/Library/Homebrew/utils/tty.rb
+++ b/Library/Homebrew/utils/tty.rb
@@ -40,8 +40,6 @@ module Tty
   CODES = COLOR_CODES.merge(STYLE_CODES).freeze
 
   class << self
-    extend T::Sig
-
     sig { params(stream: T.any(IO, StringIO), _block: T.proc.params(arg0: T.any(IO, StringIO)).void).void }
     def with(stream, &_block)
       previous_stream = @stream
diff --git a/Library/Homebrew/utils/user.rb b/Library/Homebrew/utils/user.rb
index c2db57e637..cb758ef47e 100644
--- a/Library/Homebrew/utils/user.rb
+++ b/Library/Homebrew/utils/user.rb
@@ -10,8 +10,6 @@ require "system_command"
 #
 # @api private
 class User < SimpleDelegator
-  extend T::Sig
-
   include SystemCommand::Mixin
 
   # Return whether the user has an active GUI session.
diff --git a/Library/Homebrew/version.rb b/Library/Homebrew/version.rb
index 7b76ed791f..af12755d83 100644
--- a/Library/Homebrew/version.rb
+++ b/Library/Homebrew/version.rb
@@ -10,8 +10,6 @@ require "version/parser"
 #
 # @api private
 class Version
-  extend T::Sig
-
   include Comparable
 
   sig { params(name: T.any(String, Symbol), full: T::Boolean).returns(Regexp) }
@@ -21,7 +19,6 @@ class Version
 
   # A part of a {Version}.
   class Token
-    extend T::Sig
     extend T::Helpers
     abstract!
 
@@ -105,8 +102,6 @@ class Version
 
   # A pseudo-token representing the absence of a token.
   class NullToken < Token
-    extend T::Sig
-
     sig { override.returns(NilClass) }
     attr_reader :value
 
@@ -174,7 +169,6 @@ class Version
   # A token consisting of only numbers.
   class NumericToken < Token
     PATTERN = /[0-9]+/i.freeze
-    extend T::Sig
 
     sig { override.returns(Integer) }
     attr_reader :value
diff --git a/Library/Homebrew/version/head.rb b/Library/Homebrew/version/head.rb
index 2607c91517..0edeb47470 100644
--- a/Library/Homebrew/version/head.rb
+++ b/Library/Homebrew/version/head.rb
@@ -7,8 +7,6 @@ class Version
   #
   # @api private
   class HeadVersion < Version
-    extend T::Sig
-
     sig { returns(T.nilable(String)) }
     attr_reader :commit
 
diff --git a/Library/Homebrew/version/null.rb b/Library/Homebrew/version/null.rb
index b27e0074df..80fa6f686c 100644
--- a/Library/Homebrew/version/null.rb
+++ b/Library/Homebrew/version/null.rb
@@ -8,8 +8,6 @@ class Version
   #
   # @api private
   class NullVersion < Version
-    extend T::Sig
-
     include Comparable
     include Singleton
 
diff --git a/Library/Homebrew/version/parser.rb b/Library/Homebrew/version/parser.rb
index c5127dd99e..d8cf50d9b8 100644
--- a/Library/Homebrew/version/parser.rb
+++ b/Library/Homebrew/version/parser.rb
@@ -4,7 +4,6 @@
 class Version
   # @api private
   class Parser
-    extend T::Sig
     extend T::Helpers
     abstract!
 
@@ -14,7 +13,6 @@ class Version
 
   # @api private
   class RegexParser < Parser
-    extend T::Sig
     extend T::Helpers
     abstract!
 
@@ -43,8 +41,6 @@ class Version
 
   # @api private
   class UrlParser < RegexParser
-    extend T::Sig
-
     sig { override.params(spec: Pathname).returns(String) }
     def self.process_spec(spec)
       spec.to_s
@@ -53,8 +49,6 @@ class Version
 
   # @api private
   class StemParser < RegexParser
-    extend T::Sig
-
     SOURCEFORGE_DOWNLOAD_REGEX = %r{(?:sourceforge\.net|sf\.net)/.*/download$}.freeze
     NO_FILE_EXTENSION_REGEX = /\.[^a-zA-Z]+$/.freeze
 
diff --git a/docs/Typechecking.md b/docs/Typechecking.md
index e2f0a408a8..f11dce7ef5 100644
--- a/docs/Typechecking.md
+++ b/docs/Typechecking.md
@@ -8,7 +8,7 @@ The [Sorbet Documentation] is a good place to get started if you want to dive de
 
 ### Inline Type Annotations
 
-To add type annotations to a class or module, we extend it with the `T::Sig` module (read this as `Type::Signature`). This adds the `sig` method which is used to annotate method signatures. Here's a simple example:
+The `sig` method is used to annotate method signatures. Here's a simple example:
 
 ```ruby
 class MyClass