diff --git a/CMakeLists.txt b/CMakeLists.txt
index 0a88fd44fc013550a1b1b499c816c86bd3f1b6dd..1116228e4748dd5132a207fdc38c3724755f5916 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -23,11 +23,11 @@ OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
 SOFTWARE.
 ]]
 
-set(CMAKE_MAXIMUM_SUPPORTED_VERSION 3.31)
+set(CMAKE_MAXIMUM_SUPPORTED_VERSION 4.0)
 
 cmake_minimum_required(VERSION 3.12...${CMAKE_MAXIMUM_SUPPORTED_VERSION})
 
-project(AccelerateLAPACK VERSION 1.4.0 LANGUAGES C)
+project(AccelerateLAPACK VERSION 1.5.0 LANGUAGES C)
 
 string(COMPARE EQUAL "${CMAKE_PROJECT_NAME}" "${PROJECT_NAME}" IS_TOP_LEVEL)
 
@@ -41,16 +41,22 @@ if (CMAKE_HOST_SYSTEM_NAME MATCHES "Darwin")
     # The available binary Accelerate version is determined by the MacOS system
     message(STATUS "The Darwin kernel version is: ${CMAKE_HOST_SYSTEM_VERSION}")
 
-    if (CMAKE_HOST_SYSTEM_VERSION VERSION_GREATER_EQUAL 24.0)
+    if (CMAKE_HOST_SYSTEM_VERSION VERSION_GREATER_EQUAL 24.5)
+        # Mac OS X 15.5 Sequoia or later
+        message(STATUS "The MacOS version is: >=15.5")
+        set(MINIMUM_MACOS_SDK_VERSION 15.5)
+        set(ACCELERATE_NEW_LAPACK_VERSION 3.12.0)
+    elseif (CMAKE_HOST_SYSTEM_VERSION VERSION_GREATER_EQUAL 24.0)
         # Mac OS X 15.0 Sequoia or later
-        message(STATUS "The MacOS version is: >=15.0")
+        message(STATUS "The MacOS version is: >=15.0,<15.5")
         set(MINIMUM_MACOS_SDK_VERSION 15.0)
+        set(VALID_MACOS_SDK_VERSIONS 15.4 15.2 15.1 ${MINIMUM_MACOS_SDK_VERSION})
         set(ACCELERATE_NEW_LAPACK_VERSION 3.11.0)
     elseif (CMAKE_HOST_SYSTEM_VERSION VERSION_GREATER_EQUAL 22.4)
         # Mac OS X 13.3 Ventura or later
         message(STATUS "The MacOS version is: >=13.3,<15.0")
         set(MINIMUM_MACOS_SDK_VERSION 13.3)
-        set(VALID_MACOS_SDK_VERSIONS 14.5 14.4 14.2 14.0 13.3)
+        set(VALID_MACOS_SDK_VERSIONS 14.5 14.4 14.2 14.0 ${MINIMUM_MACOS_SDK_VERSION})
         set(ACCELERATE_NEW_LAPACK_VERSION 3.9.1)
     else ()
         # Before Mac OS X 13.3 Ventura
diff --git a/README.md b/README.md
index 76546ff8b61fa05fbf11bcbb68f7e60aac5e3ad7..0c3d0b8d16b09b37ca61d60321487a17f80da954 100644
--- a/README.md
+++ b/README.md
@@ -38,6 +38,13 @@ Unfortunately, there is no mention of it in the [MacOS 15.0 Sequoia Release
 Notes][macos15-release-notes], but the note in the [Accelerate BLAS
 docs][accelerate-docs] has been updated accordingly.
 
+**Update2**: With the release of MacOS 15.5 Sequoia, Apple updated the
+Accelerate framework to be compatible with [Reference LAPACK
+v3.12.0][lapack-v3.12.0]. Unfortunately, there is no mention of it in the [MacOS
+15.5 Sequoia Release Notes][macos15.5-release-notes], but the note in the
+[Accelerate BLAS docs][accelerate-docs] has been updated accordingly, even
+though it erroneously says, v3.12.0 will be supported with MacOS 16.
+
 These new interfaces are hidden behind the preprocessor defines
 `ACCELERATE_NEW_LAPACK` and `ACCELERATE_LAPACK_ILP64` and they only work, if you
 include the Accelerate C/C++ headers.
@@ -47,7 +54,9 @@ include the Accelerate C/C++ headers.
 [lapack-v3.2.1]: https://netlib.org/lapack/#_lapack_version_3_2_1
 [lapack-v3.9.1]: https://github.com/Reference-LAPACK/lapack/releases/tag/v3.9.1
 [lapack-v3.11.0]: https://github.com/Reference-LAPACK/lapack/releases/tag/v3.11.0
+[lapack-v3.12.0]: https://github.com/Reference-LAPACK/lapack/releases/tag/v3.12.0
 [macos15-release-notes]: https://developer.apple.com/documentation/macos-release-notes/macos-15-release-notes
+[macos15.5-release-notes]: https://developer.apple.com/documentation/macos-release-notes/macos-15_5-release-notes
 
 - [The Problem](#the-problem)
 - [The Solution](#the-solution)
@@ -85,7 +94,7 @@ The reason for this is, that the binary symbol provided by the Accelerate
 framework is called `_dgeqrt$NEWLAPACK`, literally. This is a symbol, that no
 Fortran compiler will probably ever emit voluntarily. So, what to do?
 
-[dgeqrt-ref]: https://github.com/Reference-LAPACK/lapack/blob/v3.11.0/CMakeLists.txt#L365-L366
+[dgeqrt-ref]: https://github.com/Reference-LAPACK/lapack/blob/v3.12.0/CMakeLists.txt#L295-L296
 
 ## The Solution ##
 
@@ -179,9 +188,9 @@ include(FetchContent)
 FetchContent_Declare(
     AccelerateLAPACK
     GIT_REPOSITORY "https://github.com/lepus2589/accelerate-lapack.git"
-    GIT_TAG v1.4.0
+    GIT_TAG v1.5.0
     SYSTEM
-    FIND_PACKAGE_ARGS 1.4.0 CONFIG NAMES AccelerateLAPACK
+    FIND_PACKAGE_ARGS 1.5.0 CONFIG NAMES AccelerateLAPACK
 )
 set(AccelerateLAPACK_INCLUDE_PACKAGING TRUE)
 FetchContent_MakeAvailable(AccelerateLAPACK)