diff --git a/src/dispatch.jl b/src/dispatch.jl
index f8b70c4..d74aa55 100644
--- a/src/dispatch.jl
+++ b/src/dispatch.jl
@@ -13,6 +13,16 @@
 
 abstract type AbstractMutable end
 
+# Zero arithmetic methods for AbstractMutable types.
+# The main Zero arithmetic is defined in rewrite.jl with Number/AbstractArray;
+# these methods extend it to AbstractMutable.
+Base.:*(z::Zero, ::AbstractMutable) = z
+Base.:*(::AbstractMutable, z::Zero) = z
+Base.:+(::Zero, x::AbstractMutable) = copy_if_mutable(x)
+Base.:+(x::AbstractMutable, ::Zero) = copy_if_mutable(x)
+Base.:-(::Zero, x::AbstractMutable) = operate(-, x)
+Base.:-(x::AbstractMutable, ::Zero) = copy_if_mutable(x)
+
 function Base.sum(
     a::AbstractArray{T};
     dims = :,
diff --git a/src/rewrite.jl b/src/rewrite.jl
index 485cd89..f37ca58 100644
--- a/src/rewrite.jl
+++ b/src/rewrite.jl
@@ -58,20 +58,31 @@ broadcast!!(::Union{typeof(add_mul),typeof(+)}, ::Zero, x) = copy_if_mutable(x)
 broadcast!!(::typeof(add_mul), ::Zero, x, y) = x * y
 
 # Needed in `@rewrite(1 .+ sum(1 for i in 1:0) * 1^2)`
-Base.:*(z::Zero, ::Any) = z
-Base.:*(::Any, z::Zero) = z
+# These methods are narrowed to `Number` and `AbstractArray` to avoid invalidating
+# the very broad `+(x, y)`, `*(x, y)` fallbacks in Base, which causes thousands of
+# method invalidations across the ecosystem. Downstream packages that define custom
+# types participating in MutableArithmetics rewrites should define their own
+# `+(::MyType, ::Zero)` etc. methods.
+Base.:*(z::Zero, ::Number) = z
+Base.:*(::Number, z::Zero) = z
+Base.:*(z::Zero, ::AbstractArray) = z
+Base.:*(::AbstractArray, z::Zero) = z
 Base.:*(z::Zero, ::Zero) = z
-Base.:+(::Zero, x::Any) = x
-Base.:+(x::Any, ::Zero) = x
+Base.:+(::Zero, x::Number) = x
+Base.:+(x::Number, ::Zero) = x
+Base.:+(::Zero, x::AbstractArray) = x
+Base.:+(x::AbstractArray, ::Zero) = x
 Base.:+(z::Zero, ::Zero) = z
-Base.:-(::Zero, x::Any) = -x
-Base.:-(x::Any, ::Zero) = x
+Base.:-(::Zero, x::Number) = -x
+Base.:-(x::Number, ::Zero) = x
+Base.:-(::Zero, x::AbstractArray) = -x
+Base.:-(x::AbstractArray, ::Zero) = x
 Base.:-(z::Zero, ::Zero) = z
 Base.:-(z::Zero) = z
 Base.:+(z::Zero) = z
 Base.:*(z::Zero) = z
 
-function Base.:/(z::Zero, x::Any)
+function Base.:/(z::Zero, x::Number)
     if iszero(x)
         throw(DivideError())
     else