diff --git a/changes/fixed/55-sparse-integer-admission.md b/changes/fixed/55-sparse-integer-admission.md
new file mode 100644
index 0000000..f818157
--- /dev/null
+++ b/changes/fixed/55-sparse-integer-admission.md
@@ -0,0 +1 @@
+- Compact numeric table storage now admits half-dense integer keys regardless of their offset or insertion direction (#55).
diff --git a/corpus/D.manifest b/corpus/D.manifest
index 57ab1e7..bf117d5 100644
--- a/corpus/D.manifest
+++ b/corpus/D.manifest
@@ -155,4 +155,5 @@ operators: not u- # u~ + - * / // % ^ .. == ~= < <= > >= and or & | ~ << >>
 co_wrap_close_yieldable.lua unspecified-audit: ok
 tbl_table_value_type_matrix.lua unspecified-audit: ok
 tbl_sparse_geometric_keys.lua unspecified-audit: ok
+tbl_sparse_offset_descending.lua unspecified-audit: ok
 tbl_set_churn_bounded.lua unspecified-audit: ok
diff --git a/corpus/D/tbl_sparse_offset_descending.lua b/corpus/D/tbl_sparse_offset_descending.lua
new file mode 100644
index 0000000..9e950db
--- /dev/null
+++ b/corpus/D/tbl_sparse_offset_descending.lua
@@ -0,0 +1,38 @@
+-- surface: half-dense integer keys with an offset and descending insertion
+local function exercise(t, first, last)
+  assert(#t == 0)
+  local count, sum = 0, 0
+  for k, v in pairs(t) do
+    assert(k >= first and k <= last and k % 2 == 1)
+    count, sum = count + 1, sum + v
+  end
+  assert(count == 8 and sum == 36)
+
+  local k = next(t)
+  assert(k ~= nil)
+  local old = t[k]
+  t[k] = nil
+  assert(t[k] == nil)
+  t[k] = old
+  assert(t[k] == old)
+  assert(next(t) ~= nil)
+end
+
+local offset = {}
+for i = 1, 8 do offset[1001 + 2 * i] = i end
+exercise(offset, 1003, 1017)
+
+local descending = {}
+for i = 8, 1, -1 do descending[2000001 + 2 * i] = i end
+exercise(descending, 2000003, 2000017)
+
+-- Exercise both sides of the compact/hash admission boundary: the isolated
+-- key stays hashed, while the subsequent half-dense run is compact.
+local boundary = {}
+for i = 1, 8 do boundary[3001 + 2 * i] = i end
+boundary[9000001] = 9
+assert(boundary[9000001] == 9 and boundary[3003] == 1 and boundary[3017] == 8)
+boundary[9000001] = nil
+boundary[9000001] = 10
+assert(boundary[9000001] == 10)
+print("sparse-shapes-ok")
diff --git a/src/runtime.eigs b/src/runtime.eigs
index a56a318..34d28fc 100644
--- a/src/runtime.eigs
+++ b/src/runtime.eigs
@@ -8,7 +8,8 @@
 #   a: the array part, keys 1..len(a); holes are null, the last element is
 #      never null, and key len(a)+1 is never in o, so len(a) is a border;
 #   h: lazy string dictionary; o: lazy [encoded-key map, keys, values,
-#      numeric slots, free identity slots, live positive-integer hash keys].
+#      numeric slots, free identity slots, live positive-integer hash keys,
+#      first positive-integer hash key].
 # A Lua function takes one list of arguments and returns a list of results.
 # Errors are thrown as {"LUAERR": 1, "v": value}. The call-site stack lr_stk holds
 # the site id of every active call (0 = called from the runtime, "C"); the
@@ -852,13 +853,19 @@ define lr_rawset(t, k, v, site) as:
         # very large isolated integers still use the identity map below.
         local nek is lr_okey of (k)
         # Existing numeric slots remain addressable after density changes.
-        # For a new slot, k <= 2*(live integer hash keys + 1), so after the
-        # insertion padding is at most twice the live integer-key count.
-        # Grown identity capacity and non-integral numeric keys never count.
+        # Admit around the first integer rather than around zero.  Thus the
+        # same density rule applies to offset and descending insertion, while
+        # isolated integers still stay in the identity map.  Numeric slots
+        # remain absolute indices, so their padding is bounded by the fixed
+        # offset plus twice the live integer-key count.
         local numeric_old is null
         if t[4] != null and k == floor of k and k >= 1 and k <= len of t[4][3]:
             numeric_old is t[4][3][k - 1]
-        if numeric_old != null or (v != null and t[5] == null and k == floor of k and k > 1000 and t[4] != null and not (has_key of [t[4][0], nek]) and k <= 2 * (t[4][5] + 1)):
+        local numeric_admit is false
+        if v != null and t[5] == null and k == floor of k and k > 1000 and t[4] != null and t[4][6] != null and not (has_key of [t[4][0], nek]):
+            local radius is 2 * (t[4][5] + 1)
+            numeric_admit is k >= t[4][6] - radius and k <= t[4][6] + radius
+        if numeric_old != null or numeric_admit:
             local old is null
             if k <= len of t[4][3]:
                 old is t[4][3][k - 1]
@@ -890,7 +897,7 @@ define lr_rawset(t, k, v, site) as:
                 t[5][3] is t[5][3] - 1
     else:
         if t[4] == null:
-            t[4] is [{}, [], [], [], [], 0]
+            t[4] is [{}, [], [], [], [], 0, null]
         local slot is t[4][0][ek]
         if slot == null:
             lr_iter_insert of [t, k]
@@ -906,6 +913,8 @@ define lr_rawset(t, k, v, site) as:
             t[4][2][slot] is v
             if type of k == "num" and k == floor of k and k >= 1:
                 t[4][5] is t[4][5] + 1
+                if t[4][6] == null:
+                    t[4][6] is k
         else:
             t[4][2][slot] is v
     return null
diff --git a/tools/measure/run-table-memory.sh b/tools/measure/run-table-memory.sh
index 610e1fa..5074a9c 100755
--- a/tools/measure/run-table-memory.sh
+++ b/tools/measure/run-table-memory.sh
@@ -26,7 +26,7 @@ for n in 0 10000 100000 1000000; do
   measure lua empty "$n" "$REF" "$ROOT/tools/measure/table-memory.lua"
   measure eigenlua empty "$n" env EIGENLUA_ARGV0=eigenlua MEASURE_MODE=empty MEASURE_N="$n" "$EIGS" "$OUT/table-memory.eigs"
 done
-for mode in array strings sparse; do
+for mode in array strings sparse sparse_offset sparse_descending; do
   measure lua "$mode" 1000000 "$REF" "$ROOT/tools/measure/table-memory.lua"
   measure eigenlua "$mode" 1000000 env EIGENLUA_ARGV0=eigenlua MEASURE_MODE="$mode" MEASURE_N=1000000 "$EIGS" "$OUT/table-memory.eigs"
 done
diff --git a/tools/measure/table-memory.lua b/tools/measure/table-memory.lua
index cc9fec9..1404773 100644
--- a/tools/measure/table-memory.lua
+++ b/tools/measure/table-memory.lua
@@ -1,6 +1,7 @@
 -- EigenLua #31 table-memory workload.  Keep every allocation live until exit.
 -- Usage: lua table-memory.lua MODE N
--- MODE: empty, array, strings, sparse, small4, or baseline.
+-- MODE: empty, array, strings, sparse, sparse_offset, sparse_descending,
+-- small4, or baseline.
 -- EigenLua currently does not forward host argv to a compiled chunk, so the
 -- environment fallback makes the identical source runnable in both arms.
 local mode = assert(arg[1] or os.getenv("MEASURE_MODE"))
@@ -16,6 +17,10 @@ elseif mode == "strings" then
 elseif mode == "sparse" then
   -- Stay outside the array part in both implementations.
   for i = 1, n do root[2 * i] = i end
+elseif mode == "sparse_offset" then
+  for i = 1, n do root[1001 + 2 * i] = i end
+elseif mode == "sparse_descending" then
+  for i = n, 1, -1 do root[2 * i] = i end
 elseif mode == "small4" then
   for i = 1, n do
     root[i] = { tag = i, first = i + 1, second = i + 2, value = i + 3 }