diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index 8426e69..6e41895 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -694,6 +694,10 @@ jobs:
           #
           #   openkal = "0.5.2"            a released version. The question is
           #                                whether it is this one.
+          #   openkal = "^0.24.0"          a compatible range (a caret: the
+          #                                same 0.y, or the same major above 0,
+          #                                and not older). The question is
+          #                                whether this version is in it.
           #   openkal = { git = …, branch = … }
           #                                a development line. The version is
           #                                not stated and cannot be; what makes
@@ -705,7 +709,21 @@ jobs:
           # empty string --- which named neither the cause nor the symptom.
           here="$(sed -n 's/^version[[:space:]]*=[[:space:]]*"\([^"]*\)".*/\1/p' mcpp.toml | head -1)"
           there="$(sed -n 's/^openkal[[:space:]]*=[[:space:]]*"\([^"]*\)".*/\1/p' .impl/mcpp.toml | head -1)"
-          if [ -n "$there" ]; then
+          in_range() {  # in_range VERSION ^FLOOR
+            local v="$1" floor="${2#^}"
+            IFS=. read -r vmaj vmin _ <<<"$v"
+            IFS=. read -r fmaj fmin _ <<<"$floor"
+            [ "$vmaj" = "$fmaj" ] || return 1
+            [ "$fmaj" != 0 ] || [ "$vmin" = "$fmin" ] || return 1
+            [ "$(printf '%s\n%s\n' "$floor" "$v" | sort -V | head -1)" = "$floor" ]
+          }
+          if [ -n "$there" ] && [ "${there#^}" != "$there" ]; then
+            if ! in_range "$here" "$there"; then
+              echo "::error::this is openkal $here and ${{ matrix.implementation }} is written against openkal $there, a range without it. They are not in step." >&2
+              exit 1
+            fi
+            echo "openkal $here is in ${{ matrix.implementation }}'s range $there"
+          elif [ -n "$there" ]; then
             if [ "$here" != "$there" ]; then
               echo "::error::this is openkal $here and ${{ matrix.implementation }} is written against openkal $there. Nothing is wrong with either; they are not in step. Re-run after the implementation's branch has the matching version." >&2
               exit 1
diff --git a/SPEC.md b/SPEC.md
index 4dfe153..548bcb9 100644
--- a/SPEC.md
+++ b/SPEC.md
@@ -2152,6 +2152,7 @@ the operations a loader needs and a program could not otherwise reach
 | imports | only names this specification declares, and names other images export; a name of a C library or of an environment is refused. A weak import is optional: it is bound when the implementation or an image the image needs provides it, and is zero otherwise |
 | entry | an executable image's ELF entry is a function of no parameters returning `int`, in the image's convention; its value is the image's status. An image that another loads has no entry |
 | calling convention | the architecture's, fixed for every environment: the System V convention on x86-64, the procedure call standard on AArch64; an implementation whose environment uses another presents entries in this one to the image (12.3) |
+| floating-point state | the convention's at the entry and in each context: on x86-64 the x87 control word `0x037F` (extended precision, every exception masked) and MXCSR `0x1F80`, on AArch64 FPCR zero; an image's C library computes in that state (a `long double` of x86-64 is the x87's extended format) |
 | platform register | an AArch64 image does not use `x18`, which two environments reserve |
 | stack | a frame larger than a page is touched page by page, in order: one environment commits a context's stack as its guard page is touched, and a frame that steps over the guard ends the program |
 | thread-local storage | through the emulated form, in which a variable is found by a function the image itself supplies; an image does not address the thread pointer, which one environment does not let a program set and another does not let it change |
@@ -2206,7 +2207,12 @@ Where an environment's own convention differs from the image's --- on x86-64,
 one environment passes arguments in other registers --- the loader presents each
 entry in the image's convention, and a function the image passes to the
 implementation (`kal_task_start`'s entry, `kal_space_start`'s) is entered in the
-image's convention. The presentation is mechanical and is generated from
+image's convention. The convention includes the floating-point state (12.2): the
+loader establishes it at the entry, for the initialisers and finalisers, and in
+each context the image starts, and a call that a program of the environment makes
+into an image's function is made in it, the caller's own restored after it; one
+environment begins a thread with the x87 at double precision. The presentation is
+mechanical and is generated from
 `SURFACE.txt` and the declarations, so that it cannot omit a name. It is the
 loader's and not the implementation's: an implementation is written in its
 environment's convention, as every program of that environment is, and a