diff --git a/Library/Homebrew/livecheck/strategy.rb b/Library/Homebrew/livecheck/strategy.rb
index f00a455a64e6fd..b952a44f0ce31a 100644
--- a/Library/Homebrew/livecheck/strategy.rb
+++ b/Library/Homebrew/livecheck/strategy.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 module Homebrew
@@ -97,9 +97,9 @@ module Strategy
       # @return [Hash]
       sig { returns(T::Hash[Symbol, T.untyped]) }
       def strategies
-        return @strategies if defined? @strategies
+        @strategies ||= T.let({ uninitialized: true }, T.nilable(T::Hash[Symbol, T.untyped]))
+        return @strategies unless @strategies[:uninitialized]
 
-        @strategies = {}
         Strategy.constants.sort.each do |const_symbol|
           constant = Strategy.const_get(const_symbol)
           next unless constant.is_a?(Class)
@@ -107,6 +107,7 @@ def strategies
           key = Utils.underscore(const_symbol).to_sym
           @strategies[key] = constant
         end
+        @strategies.delete(:uninitialized)
         @strategies
       end
       private_class_method :strategies
diff --git a/Library/Homebrew/rubocops/livecheck.rb b/Library/Homebrew/rubocops/livecheck.rb
index 7a974fdcf8fa7d..69c5b083d31624 100644
--- a/Library/Homebrew/rubocops/livecheck.rb
+++ b/Library/Homebrew/rubocops/livecheck.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "rubocops/extend/formula_cop"
@@ -11,6 +11,14 @@ module FormulaAudit
       class LivecheckSkip < FormulaCop
         extend AutoCorrector
 
+        sig {
+          override.params(
+            _node:              RuboCop::AST::ClassNode,
+            _class_node:        RuboCop::AST::ConstNode,
+            _parent_class_node: RuboCop::AST::ConstNode,
+            body_node:          RuboCop::AST::Node,
+          ).void
+        }
         def audit_formula(_node, _class_node, _parent_class_node, body_node)
           livecheck_node = find_block(body_node, :livecheck)
           return if livecheck_node.blank?
@@ -39,6 +47,14 @@ def audit_formula(_node, _class_node, _parent_class_node, body_node)
 
       # This cop ensures that a `url` is specified in the `livecheck` block.
       class LivecheckUrlProvided < FormulaCop
+        sig {
+          override.params(
+            _node:              RuboCop::AST::ClassNode,
+            _class_node:        RuboCop::AST::ConstNode,
+            _parent_class_node: RuboCop::AST::ConstNode,
+            body_node:          RuboCop::AST::Node,
+          ).void
+        }
         def audit_formula(_node, _class_node, _parent_class_node, body_node)
           livecheck_node = find_block(body_node, :livecheck)
           return unless livecheck_node
@@ -62,6 +78,14 @@ def audit_formula(_node, _class_node, _parent_class_node, body_node)
       class LivecheckUrlSymbol < FormulaCop
         extend AutoCorrector
 
+        sig {
+          override.params(
+            _node:              RuboCop::AST::ClassNode,
+            _class_node:        RuboCop::AST::ConstNode,
+            _parent_class_node: RuboCop::AST::ConstNode,
+            body_node:          RuboCop::AST::Node,
+          ).void
+        }
         def audit_formula(_node, _class_node, _parent_class_node, body_node)
           livecheck_node = find_block(body_node, :livecheck)
           return if livecheck_node.blank?
@@ -117,6 +141,14 @@ def audit_formula(_node, _class_node, _parent_class_node, body_node)
       class LivecheckRegexParentheses < FormulaCop
         extend AutoCorrector
 
+        sig {
+          override.params(
+            _node:              RuboCop::AST::ClassNode,
+            _class_node:        RuboCop::AST::ConstNode,
+            _parent_class_node: RuboCop::AST::ConstNode,
+            body_node:          RuboCop::AST::Node,
+          ).void
+        }
         def audit_formula(_node, _class_node, _parent_class_node, body_node)
           livecheck_node = find_block(body_node, :livecheck)
           return if livecheck_node.blank?
@@ -144,6 +176,14 @@ class LivecheckRegexExtension < FormulaCop
 
         TAR_PATTERN = /\\?\.t(ar|(g|l|x)z$|[bz2]{2,4}$)(\\?\.((g|l|x)z)|[bz2]{2,4}|Z)?$/i
 
+        sig {
+          override.params(
+            _node:              RuboCop::AST::ClassNode,
+            _class_node:        RuboCop::AST::ConstNode,
+            _parent_class_node: RuboCop::AST::ConstNode,
+            body_node:          RuboCop::AST::Node,
+          ).void
+        }
         def audit_formula(_node, _class_node, _parent_class_node, body_node)
           livecheck_node = find_block(body_node, :livecheck)
           return if livecheck_node.blank?
@@ -171,6 +211,14 @@ def audit_formula(_node, _class_node, _parent_class_node, body_node)
       # This cop ensures that a `regex` is provided when `strategy :page_match` is specified
       # in the `livecheck` block.
       class LivecheckRegexIfPageMatch < FormulaCop
+        sig {
+          override.params(
+            _node:              RuboCop::AST::ClassNode,
+            _class_node:        RuboCop::AST::ConstNode,
+            _parent_class_node: RuboCop::AST::ConstNode,
+            body_node:          RuboCop::AST::Node,
+          ).void
+        }
         def audit_formula(_node, _class_node, _parent_class_node, body_node)
           livecheck_node = find_block(body_node, :livecheck)
           return if livecheck_node.blank?
@@ -199,6 +247,14 @@ class LivecheckRegexCaseInsensitive < FormulaCop
 
         MSG = "Regexes should be case-insensitive unless sensitivity is explicitly required for proper matching."
 
+        sig {
+          override.params(
+            _node:              RuboCop::AST::ClassNode,
+            _class_node:        RuboCop::AST::ConstNode,
+            _parent_class_node: RuboCop::AST::ConstNode,
+            body_node:          RuboCop::AST::Node,
+          ).void
+        }
         def audit_formula(_node, _class_node, _parent_class_node, body_node)
           return if tap_style_exception? :regex_case_sensitive_allowlist