diff --git a/FIX.org b/FIX.org
index f3c64e6..c7303e3 100644
--- a/FIX.org
+++ b/FIX.org
@@ -5198,3 +5198,91 @@ Anything that pinned a hash re-pins it once.
   and even ~(u64 2935910691)~ are refused for not fitting in an i32, while the
   literal written straight into a u64 operand is fine. That is why the
   multiplier sits in the multiply rather than in a let.
+
+* Chained comparisons, 2026-09-21
+
+From the game, at sand.flan:45:7:
+
+: < takes 2 arguments, given 3
+
+and the ruling:
+
+#+begin_quote
+dispatch an agent that's going to do a pass through the stdlib and make
+functions variadic, lisp style. We should have +, *, -, and < > et all should
+be variadic
+#+end_quote
+
+Most of that was already true. [+ - * /], [bit-and], [bit-or], [bit-xor],
+[min] and [max] have taken two operands or more since [fold_left_prim] went
+in; the six comparisons had not, and they are the ones the game hit. So the
+change is one arm: [= != < <= > >=] take two operands or more, [= < <= > >=]
+chaining and [!=] asking about every pair — the ruling below.
+
+(< a b c) asks whether a is below b and b is below c. The left fold the
+folding operators use would compare a bool against a number, so there was
+never a second reading to choose between.
+
+The middle operand is the whole of the difficulty. Two links name it, and the
+spelling anyone would reach for by hand — (and (< a b) (< b c)) — evaluates it
+twice, which is wrong the moment it is a call. So the chain binds every
+operand to a slot first, in source order, and compares the slots. The links
+then stop early with nothing observable riding on it: by the time any link
+runs, every operand has already been evaluated.
+
+The refusals stayed refusals. (< x) would have to be true whatever it was
+handed, which is a typo that carries a value, so it is refused beside (+) and
+(- x) — see the note above [fold_arity] for why those two are refused, which
+this follows rather than reopens. Lisp answers true there; this language does
+not, and the message says what to write instead.
+
+[%] and the shifts are still two operands, for the reasons already written
+against them: a chain of remainders has no agreed reading, and (<< x 30 30)
+would be two legal shifts that between them shift the value away.
+
+There is no three-way join, and none was invented. The first pair joins the
+way any pair of operands joins — the widening rule of 2026-09-20 — and every
+operand after it is checked against the type that join produced, so a third
+operand that does not fit is refused the way a second one would be. That is
+[fold_left_prim]'s rule exactly, and the comparisons now answer the question
+the same way the arithmetic does rather than a second way of their own.
+
+!= was built as a chain first and the author ruled against it:
+
+#+begin_quote
+adjacent chaining makes little sense to me, (!= 1 2 1) should be false
+#+end_quote
+
+So != is all-distinct, Common Lisp's /=: (!= a b c) is true when every operand
+differs from every other. The other five keep adjacent chaining.
+
+The two differ because they are different questions. Asking whether a sequence
+is increasing is a question about neighbours — c has nothing to say about a,
+and (< a b c) is done when it has looked at two pairs. Asking whether a set of
+values are all different is a question about the set, and the pair chaining
+would never look at, first against last, is exactly the one (!= 1 2 1) turns
+on. Only the first question is a chain.
+
+The cost of that is pairwise: n operands means n(n-1)/2 comparisons rather
+than n-1. Fine at the sizes anyone writes — four operands is six compares of
+values already sitting in slots — and invisible to every program there is
+today, because at two operands the two readings are one pair and the
+two-operand form does not go through the n-ary lowering at all. It emits what
+it always did.
+
+Single evaluation is unchanged by the switch, and is the reason the pairwise
+reading costs nothing worse than compares: every operand is in its slot before
+any pair is looked at, so the extra pairs read slots. The conjunction still
+stops at the first pair that fails, having already run everything.
+
+The slots are slots of whatever type the operands have, which strings are what
+says: = and != admit them and the orderings do not, so (= "a" "a" "a") is the
+one row here that is not about machine words. A dyn ordering carries the trap
+site every pair of it, because the whole comparison is written at one place
+and a trap from any of its pairs happened there.
+
+test/programs/chain.flan, on both backends and at -O0 and -O2. What it asserts
+that a checker test cannot is the tag transcripts: a chain whose first link is
+already false still prints abc, a != whose *first* pair already says no still
+prints abcd, and the call counts (10, 12, 15) are what an operand evaluated
+once per pair that names it would break — the 12 would be 24.
diff --git a/lib/check.ml b/lib/check.ml
index 9df534b..d18e346 100644
--- a/lib/check.ml
+++ b/lib/check.ml
@@ -5695,11 +5695,13 @@ and arity _ctx loc name n args =
     fail loc "%s takes %d argument%s, given %d" name n
       (if n = 1 then "" else "s") (List.length args)
 
-(* The operators that fold: [+ - * /], [min]/[max] and the three bitwise
-   combining operators all take two operands or more, and mean the same thing
-   applied left to right. [%] and the shifts are not in that set — a chain of
-   remainders or of shifts has no reading a reader would agree on in advance,
-   so there the arity error is the useful answer.
+(* The operators that take two operands or more: [+ - * /], [min]/[max], the
+   three bitwise combining operators, and the six comparisons. The first ten
+   fold left; the ordered comparisons chain and [!=] asks about every pair,
+   which [cmp_over] and the two pair-pickers under it explain. [%] and the
+   shifts are not in that set — a chain of remainders or of shifts has no
+   reading a reader would agree on in advance, so there the arity error is the
+   useful answer.
 
    Two is the floor, and the two missing cases are refused rather than
    invented. Zero operands would have to mean an identity element, 0 for + and
@@ -5708,7 +5710,13 @@ and arity _ctx loc name n args =
    for [/], and this language has no unary minus anywhere: the prelude writes
    every negation as [(- 0 n)] or [(- 0.0 x)], and [(- x)] meaning something
    else than the [-] two lines above it is a rule a reader has to carry rather
-   than see. *)
+   than see. Integer division makes the reciprocal worse still: [(/ 3)] would
+   be 0.
+
+   A one-operand comparison would have to be [true] — there is no pair to
+   disagree, and nothing for a lone value to be distinct from — and a test
+   that is true whatever it is handed is a typo with a value, which is the
+   worst kind. So it is refused here with the rest. *)
 and fold_arity loc name args =
   match args with
   | _ :: _ :: _ -> ()
@@ -5720,9 +5728,21 @@ and fold_arity loc name args =
     fail loc
       "/ takes two arguments or more, given 1 — there is no reciprocal; \
        write (/ 1.0 x)"
+  | [ _ ] when String.equal name "!=" ->
+    fail loc
+      "!= takes two arguments or more, given 1 — a test for distinctness \
+       needs something to be distinct from, as in (!= x y)"
+  | [ _ ] when is_comparison name ->
+    fail loc
+      "%s takes two arguments or more, given 1 — a comparison needs a second \
+       value to compare against, as in (%s x y)" name name
   | _ ->
     fail loc "%s takes two arguments or more, given %d" name (List.length args)
 
+and is_comparison = function
+  | "=" | "!=" | "<" | "<=" | ">" | ">=" -> true
+  | _ -> false
+
 (* The first two operands decide the type — [binary] picks which of them is
    allowed to, and that decision is not re-made per pair — and every operand
    after them is checked against it. *)
@@ -5872,6 +5892,61 @@ and dyn_fold ctx ~want loc name first rest =
   in
   expect ctx loc ~want acc
 
+(* A comparison over three operands or more asks about more than one pair, and
+   every operand is bound to a slot before any pair is looked at. That is what
+   makes "left to right, exactly once" true of the lowering and not only of
+   the source: an operand two pairs name is written down once. The spelling a
+   reader would reach for, [(and (< a b) (< b c))], evaluates b twice, which
+   is wrong the moment b is a call — that is the whole reason this is a form
+   the compiler builds rather than a macro.
+
+   The pairs are then required to hold, and the conjunction stops at the first
+   one that does not. That costs nothing observable, because by the time any
+   pair is compared every operand has already been evaluated — stopping skips
+   a machine compare, never a call.
+
+   [pairs] says which pairs this operator asks about and [link] builds one
+   comparison, so the two readings below and the dyn lowering of each are the
+   same code with two arguments changed. *)
+and cmp_over ctx loc ty ~pairs ~link ops =
+  let binds = List.map (fun (e : Tast.expr) -> fresh_slot ctx ty, e) ops in
+  let locals = List.map (fun (s, _) -> mk loc ty (Tast.Local s)) binds in
+  let rec conj = function
+    | [ t ] -> t
+    | t :: rest ->
+      mk loc Types.Bool
+        (Tast.If (t, conj rest, mk loc Types.Bool (Tast.Bool false)))
+    | [] -> assert false
+  in
+  let tests = List.map (fun (a, b) -> link a b) (pairs locals) in
+  mk loc Types.Bool (Tast.Let (binds, [ conj tests ]))
+
+(* The ordered comparisons chain: [(< a b c)] asks whether a is below b and b
+   is below c. The other reading, the left fold [(< (< a b) c)], compares a
+   bool against a number, and there is no program that meant it. So the pairs
+   are the adjacent ones, n-1 of them, and the operand in the middle is the
+   one two of them share. *)
+and adjacent_pairs xs =
+  match xs with
+  | a :: (b :: _ as rest) -> (a, b) :: adjacent_pairs rest
+  | _ -> []
+
+(* [!=] is the one that does not chain. "Is this sequence increasing" and "are
+   these values all different" are different questions, and only the first is
+   about adjacent pairs: under chaining [(!= 1 2 1)] would be true, because
+   each neighbour differs from the next, while the thing anyone means by it is
+   false. So [!=] asks about *every* pair — Common Lisp's [/=] — which is
+   n(n-1)/2 comparisons rather than n-1.
+
+   That growth is fine at the sizes anyone writes: four operands is six
+   compares of values already in slots. It is also invisible to every program
+   there is today, because two operands is one pair either way and does not
+   come through here at all. *)
+and all_pairs xs =
+  match xs with
+  | [] -> []
+  | x :: rest -> List.map (fun y -> (x, y)) rest @ all_pairs rest
+
 (* ── Allocation failure, spec-memory.md ────────────────────────────────
    No allocating operation returns an error and none can fail silently. When
    the allocator cannot satisfy a request the operation signals
@@ -6284,8 +6359,21 @@ and named_call ?(qualified = false) ctx ~want loc name args =
       | "=" -> Tast.Eq | "!=" -> Tast.Ne | "<" -> Tast.Lt
       | "<=" -> Tast.Le | ">" -> Tast.Gt | _ -> Tast.Ge
     in
-    arity ctx loc name 2 args;
-    let a, b = binary ctx ~dyn_ok:true name loc ~want:None args in
+    fold_arity loc name args;
+    let x, y, rest =
+      match args with x :: y :: rest -> x, y, rest | _ -> assert false
+    in
+    (* The first pair decides the type, and whether this is a dyn comparison
+       at all, exactly as it does for the folding operators: [binary] joins
+       the two, and every operand after them is checked against the answer.
+       Past the first pair nothing widens, which is [fold_left_prim]'s rule
+       and not a second one. *)
+    let a, b = binary ctx ~dyn_ok:true name loc ~want:None [ x; y ] in
+    (* Which pairs this operator asks about. Every one but [!=] chains, and
+       [!=] asks about all of them — see [all_pairs]. At two operands the two
+       readings are one pair and the same answer, which is why the two-operand
+       path below is the same code it always was. *)
+    let pairs = if String.equal name "!=" then all_pairs else adjacent_pairs in
     (* A comparison with a dyn operand answers a *bool*, not a dyn, even though
        the runtime's own entry point answers a dyn holding one. The reason is
        where the result goes: a comparison is overwhelmingly the test of an
@@ -6307,18 +6395,30 @@ and named_call ?(qualified = false) ctx ~want loc name args =
         | ">" -> "flan_dyn_gt" | _ -> "flan_dyn_ge"
       in
       (* [eq] never traps and takes no site; the four orderings do, and get
-         one, for the reason [dyn_fold] gives. *)
+         one, for the reason [dyn_fold] gives. Every pair of a chain gets the
+         same site — the whole comparison is written at one place, and a trap
+         from any of its pairs happened there. *)
       let site = if String.equal sym "flan_dyn_eq" then [] else [ here loc ] in
-      let cmp =
-        unbox loc Types.Bool
-          (rt loc Types.Dyn sym ([ box loc a; box loc b ] @ site))
-      in
       (* [!=] has no entry point of its own: there is one structural equality
          and the negation is an [i1] flip the backend folds away. *)
-      let r =
-        if String.equal name "!=" then mk loc Types.Bool (Tast.Prim (Tast.Not, [ cmp ]))
+      let link u v =
+        let cmp =
+          unbox loc Types.Bool (rt loc Types.Dyn sym ([ u; v ] @ site))
+        in
+        if String.equal name "!=" then
+          mk loc Types.Bool (Tast.Prim (Tast.Not, [ cmp ]))
         else cmp
       in
+      let r =
+        match rest with
+        | [] -> link (box loc a) (box loc b)
+        | _ ->
+          let ops =
+            box loc a :: box loc b
+            :: map_lr (fun e -> box loc (check ctx ~want:Types.Dyn e)) rest
+          in
+          cmp_over ctx loc Types.Dyn ~pairs ~link ops
+      in
       expect ctx loc ~want r
     end else begin
     (* [=] and [!=] admit types [<] does not. A handle is one: a pair of
@@ -6349,7 +6449,13 @@ and named_call ?(qualified = false) ctx ~want loc name args =
          fail loc
            "%s orders machine numbers and enums, and %s is neither" name
              (Types.to_string a.Tast.ty));
-    prim p Types.Bool [ a; b ]
+    match rest with
+    | [] -> prim p Types.Bool [ a; b ]
+    | _ ->
+      let ty = a.Tast.ty in
+      let link u v = mk loc Types.Bool (Tast.Prim (p, [ u; v ])) in
+      let ops = a :: b :: map_lr (fun e -> check ctx ~want:ty e) rest in
+      expect ctx loc ~want (cmp_over ctx loc ty ~pairs ~link ops)
     end
   | "not" ->
     arity ctx loc name 1 args;
@@ -9114,18 +9220,23 @@ let builtins : (string * string * string) list =
     ("%", "% [numeric? numeric?] numeric?",
      "Remainder, and it stays at two operands: (% a b c) would mean \
       (% (% a b) c), which is a thing nobody writes on purpose.");
-    ("=", "= [equal? equal?] bool",
-     "Equality. It admits two types < does not: a handle, where \"the same \
-      entity\" is the question the type exists to answer, and a string, \
-      compared bytewise by content rather than ordered.");
-    ("!=", "!= [equal? equal?] bool",
-     "Inequality, over everything = accepts.");
-    ("<", "< [ordered? ordered?] bool",
-     "Less than. Machine numbers only: ordering a handle would order a \
-      free-list slot index, which means nothing.");
-    ("<=", "<= [ordered? ordered?] bool", "Less than or equal.");
-    (">", "> [ordered? ordered?] bool", "Greater than.");
-    (">=", ">= [ordered? ordered?] bool", "Greater than or equal.");
+    ("=", "= [equal? ...] bool",
+     "Equality, chained over two operands or more: (= a b c) is a = b and \
+      b = c, which is every operand alike. It admits two types < does not: a \
+      handle, where \"the same entity\" is the question the type exists to \
+      answer, and a string, compared bytewise by content rather than \
+      ordered.");
+    ("!=", "!= [equal? ...] bool",
+     "All different: (!= a b c) is true when every operand differs from every \
+      other, so (!= 1 2 1) is false. Over everything = accepts.");
+    ("<", "< [ordered? ...] bool",
+     "Less than, chained: (< a b c) is a < b and b < c, and every operand is \
+      evaluated once. Machine numbers and enums only — ordering a handle \
+      would order a free-list slot index, which means nothing.");
+    ("<=", "<= [ordered? ...] bool", "Less than or equal, chained like <.");
+    (">", "> [ordered? ...] bool", "Greater than, chained like <.");
+    (">=", ">= [ordered? ...] bool",
+     "Greater than or equal, chained like <.");
     ("not", "not [bool] bool",
      "Negates a bool. Nothing else in this language is a truth value.");
     ("bit-and", "bit-and [int ...] int",
diff --git a/test/programs/chain.flan b/test/programs/chain.flan
new file mode 100644
index 0000000..74fc61d
--- /dev/null
+++ b/test/programs/chain.flan
@@ -0,0 +1,173 @@
+;;;; Chained comparisons, and the property no checker test can assert: how
+;;;; many times each operand runs.
+;;;;
+;;;; (< a b c) means a < b and b < c. The reading a compiler could otherwise
+;;;; have picked, the left fold ((a < b) < c), compares a bool against a
+;;;; number and nothing means it. The interesting half is the middle operand:
+;;;; two links mention b, and the spelling a reader would write by hand,
+;;;; (and (< a b) (< b c)), evaluates it twice. So every operand here comes
+;;;; through [mark], which prints its tag, and the printed tags are what say
+;;;; each operand ran once and in source order.
+;;;;
+;;;; The folding operators are marked the same way, because "left to right,
+;;;; exactly once" is one claim and it is made about both families.
+
+(defonce calls i32)
+
+;; Prints its tag and answers its value. Every operand below is one of these,
+;; so the tag line is a transcript of the evaluation.
+(defn mark [tag string v i32] i32
+  (set calls (+ calls 1))
+  (print tag)
+  v)
+
+(defn markf [tag string v f64] f64
+  (print tag)
+  v)
+
+;; A generic chain, admitted by the predicate its signature carries. The same
+;; three-link body as everything above, over a type the function does not know.
+(defn between [a $t b $t c $t] bool {:where (ordered? $t)}
+  (< a b c))
+
+;; Three unannotated parameters are three dyn ones, so this is the chain the
+;; runtime compares rather than the machine.
+(defn dyn-rising [a b c] bool
+  (< a b c))
+
+;; And the all-pairs reading over dyn operands, which asks the runtime the
+;; same question three times rather than twice.
+(defn dyn-distinct [a b c] bool
+  (!= a b c))
+
+;; Prints y when its two arguments agree and N when they do not. Two bools
+;; cannot be compared with = — bool is not an ordered or an equatable type —
+;; so the agreement between a chain and the spelling it stands for is said
+;; with the operators bool does have.
+(defn agree [a bool b bool] ()
+  (if (and (or a b) (not (and a b))) (print "N") (print "y")))
+
+(defn line [b bool] ()
+  (print " -> ")
+  (print b)
+  (println ""))
+
+(defn main [] i32
+  ;; ── the chain itself ──────────────────────────────────────────────
+  (print (< 1 2 3)) (print " ")            ; true
+  (print (< 1 5 3)) (print " ")            ; false: the second link
+  (print (< 5 1 3)) (print " ")            ; false: the first link
+  (print (< 1 2 3 4)) (print " ")          ; true, four operands
+  (print (< 1 2 4 3)) (println "")         ; false, four operands
+
+  ;; Every member of the family, chained. The middle link is the false one in
+  ;; each of the second column.
+  (print (<= 1 1 2)) (print " ") (print (<= 1 2 1)) (print " ")
+  (print (> 3 2 1))  (print " ") (print (> 3 1 2))  (print " ")
+  (print (>= 3 3 1)) (print " ") (print (>= 3 1 2)) (print " ")
+  (print (= 2 2 2))  (print " ") (print (= 2 3 2))
+  (println "")
+
+  ;; != is the one that does not chain: it asks whether the operands are all
+  ;; different, so the pair it looks at is every pair and not only the
+  ;; neighbouring ones. (!= 1 2 1) has no two neighbours alike and is still
+  ;; false, which is the whole of the difference.
+  (print (!= 1 2 3)) (print " ")           ; true
+  (print (!= 1 1 2)) (print " ")           ; false, the adjacent pair
+  (print (!= 1 2 1)) (print " ")           ; false, the pair chaining misses
+  (print (!= 1 2 3 4)) (print " ")         ; true, six pairs
+  (print (!= 1 2 3 1)) (print " ")         ; false, first against last
+  (print (!= 1 2 3 2)) (println "")        ; false, a pair in the middle
+
+  ;; The chain and the spelling it stands for agree, everywhere both are
+  ;; legal. Written out rather than trusted, because it is the whole claim.
+  (agree (< 1 2 3) (and (< 1 2) (< 2 3)))
+  (agree (< 1 5 3) (and (< 1 5) (< 5 3)))
+  (agree (< 5 1 3) (and (< 5 1) (< 1 3)))
+  (agree (<= 1 1 2) (and (<= 1 1) (<= 1 2)))
+  (agree (> 3 1 2) (and (> 3 1) (> 1 2)))
+  (agree (= 2 3 2) (and (= 2 3) (= 3 2)))
+  (agree (< 1 2 3 4) (and (< 1 2) (and (< 2 3) (< 3 4))))
+  ;; != stands for the all-pairs spelling, not the adjacent one.
+  (agree (!= 1 2 1) (and (!= 1 2) (and (!= 2 1) (!= 1 1))))
+  (agree (!= 1 2 3) (and (!= 1 2) (and (!= 1 3) (!= 2 3))))
+  (println "")
+
+  ;; ── one evaluation of each operand, in order ──────────────────────
+  ;; The first link is true here, so the chain goes on. Three tags.
+  (set calls 0)
+  (line (< (mark "a" 1) (mark "b" 2) (mark "c" 3)))
+  ;; abc -> true
+
+  ;; The first link is *false* here, and the tags still say abc: stopping
+  ;; early stops comparing, not running. b appears once although two links
+  ;; name it, which is the bug the slots exist to prevent.
+  (line (< (mark "a" 9) (mark "b" 1) (mark "c" 5)))
+  ;; abc -> false
+
+  ;; Four operands, false in the middle, and every one of them runs.
+  (line (< (mark "a" 1) (mark "b" 2) (mark "c" 0) (mark "d" 9)))
+  ;; abcd -> false
+
+  ;; Ten marks across three chains is the count, and a middle operand
+  ;; evaluated twice would make it eleven.
+  (print calls) (println "")               ; 10
+
+  ;; != makes the same promise over its larger set of pairs. All different,
+  ;; six comparisons over four operands, four tags.
+  (set calls 0)
+  (line (!= (mark "a" 1) (mark "b" 2) (mark "c" 3) (mark "d" 4)))
+  ;; abcd -> true
+
+  ;; The *first* pair it looks at already says no, and the other three
+  ;; operands run anyway: every operand is in its slot before any pair is
+  ;; compared, so stopping early stops comparing and nothing else.
+  (line (!= (mark "a" 1) (mark "b" 1) (mark "c" 3) (mark "d" 4)))
+  ;; abcd -> false
+
+  ;; The pair chaining would never have looked at — first against last.
+  (line (!= (mark "a" 1) (mark "b" 2) (mark "c" 3) (mark "d" 1)))
+  ;; abcd -> false
+
+  ;; Twelve, and each of the three lines above contributed four. An operand
+  ;; named by three pairs evaluated once per pair would say twenty-four.
+  (print calls) (println "")               ; 12
+
+  ;; The folding operators make the same promise.
+  (set calls 0)
+  (print (+ (mark "p" 1) (mark "q" 2) (mark "r" 3))) (println "")   ; pqr6
+  (print (- (mark "p" 10) (mark "q" 3) (mark "r" 2))) (println "")  ; pqr5
+  (print (* (mark "p" 2) (mark "q" 3) (mark "r" 4))) (println "")   ; pqr24
+  (print (min (mark "p" 5) (mark "q" 2) (mark "r" 8))) (println "") ; pqr2
+  (print (max (mark "p" 5) (mark "q" 2) (mark "r" 8))) (println "") ; pqr8
+  (print calls) (println "")               ; 15
+
+  ;; ── the types a chain admits ──────────────────────────────────────
+  ;; One width throughout: the first pair decides, the rest are checked
+  ;; against it, and nothing widens implicitly at three operands any more
+  ;; than it does at two.
+  (let [x (u8 1) y (u8 2) z (u8 3)]
+    (print (< x y z)) (print " "))         ; true
+  (let [x (i64 -5) y (i64 0) z (i64 5)]
+    (print (< x y z)) (print " "))         ; true
+  (print (< (markf "f" 1.0) (markf "g" 2.0) (markf "h" 3.0))) (println "")
+  ;; true, and fgh printed first
+
+  ;; Strings, which = and != admit and the orderings do not. They are the one
+  ;; operand type here that is not a machine word, so they are what says the
+  ;; slots a chain binds are slots of whatever type the operands have.
+  (print (= "a" "a" "a")) (print " ")      ; true
+  (print (= "a" "b" "a")) (print " ")      ; false
+  (print (!= "a" "b" "c")) (print " ")     ; true
+  (print (!= "a" "b" "a")) (println "")    ; false, the non-adjacent pair
+
+  ;; The generic chain, at two types.
+  (print (between (i32 1) (i32 2) (i32 3))) (print " ")   ; true
+  (print (between (f64 3.0) (f64 2.0) (f64 1.0))) (println "") ; false
+
+  ;; And the dyn chain, where the runtime does the comparing.
+  (print (dyn-rising 1 2 3)) (print " ")   ; true
+  (print (dyn-rising 1 5 3)) (print " ")   ; false
+  (print (dyn-distinct 1 2 3)) (print " ") ; true
+  (print (dyn-distinct 1 2 1)) (println "") ; false, the non-adjacent pair
+  0)
diff --git a/test/test_acceptance.ml b/test/test_acceptance.ml
index 8af1369..7364e66 100644
--- a/test/test_acceptance.ml
+++ b/test/test_acceptance.ml
@@ -380,6 +380,37 @@ let () =
     outputs "value semantics" "programs/values.flan" values_out;
     outputs "machine surface" "programs/machine.flan" machine_out;
     outputs "unit main exits 0" "programs/unit-main.flan" "ok\n";
+    (* Comparisons over three operands and more. The lines that carry the
+       whole claim are the tag transcripts: [abc -> false] is a chain whose
+       *first* link already decided the answer and whose middle operand —
+       named by two links — still ran exactly once, and the counts (10, 12
+       and 15) are what a second evaluation of a shared operand would break.
+       The third line is the one [!=] is here for: it is all-pairs, so
+       (!= 1 2 1) is false although no two neighbours are alike. Both
+       backends and both optimisation levels, because the lowering is a Let
+       of slots and a conjunction of Ifs, which is the one shape a backend
+       could get right at -O2 by accident.
+
+       The string line is here because a string is the one operand type in
+       this file that is not a machine word: the slots a comparison binds are
+       slots of whatever type the operands have, and nothing else would say
+       so. *)
+    let chain_out =
+      "true false false true false\n\
+       true false true false true false true false\n\
+       true false false true false false\n\
+       yyyyyyyyy\n\
+       abc -> true\nabc -> false\nabcd -> false\n10\n\
+       abcd -> true\nabcd -> false\nabcd -> false\n12\n\
+       pqr6\npqr5\npqr24\npqr2\npqr8\n15\n\
+       true true fghtrue\ntrue false true false\n\
+       true false\ntrue false true false\n"
+    in
+    outputs "chained comparisons" "programs/chain.flan" chain_out;
+    outputs ~opt:"-O0" "chained comparisons, -O0" "programs/chain.flan"
+      chain_out;
+    outputs ~x86:true "chained comparisons, --x86" "programs/chain.flan"
+      chain_out;
     (* A global of move-only type, which this compiler used to refuse outright.
        What the numbers assert is the half of the rule no checker test can: the
        global is loaded once and *stays* loaded across a second entry, which is
diff --git a/test/test_flan.ml b/test/test_flan.ml
index 3660dc1..b2a9f4f 100644
--- a/test/test_flan.ml
+++ b/test/test_flan.ml
@@ -1271,6 +1271,102 @@ let () =
   accepts "a typed if with a bare keyword condition now compiles"
     "(defn f [] i32 (if :kw 1 2))";
 
+  (* ── Two operands or more, and the two counts below it ─────────── *)
+  (* The operators that take a run of operands, each at three and at four, so
+     the floor and the arms above it are both pinned. The comparisons are the
+     block after this one. *)
+  infers "+ at three"        "(+ 1 2 3)" "i32";
+  infers "+ at four"         "(+ 1 2 3 4)" "i32";
+  infers "- at four"         "(- 10 1 2 3)" "i32";
+  infers "* at four"         "(* 1 2 3 4)" "i32";
+  infers "/ at three"        "(/ 100 5 2)" "i32";
+  infers "/ at four"         "(/ 1000 5 2 2)" "i32";
+  infers "bit-and at three"  "(bit-and 7 6 4)" "i32";
+  infers "bit-or at four"    "(bit-or 1 2 4 8)" "i32";
+  infers "bit-xor at three"  "(bit-xor 1 2 4)" "i32";
+  infers "min at four"       "(min 4 1 3 2)" "i32";
+  infers "max at four"       "(max 4 1 3 2)" "i32";
+  (* And the two counts below the floor. Zero would have to mean an identity
+     element and one a unary operator this language does not have; both are a
+     typo far more often than an intent, so both are refused by name. *)
+  rejects_check "a sum with no terms"
+    "(defn f [] i32 (+))" ~needle:"+ takes two arguments or more, given 0";
+  rejects_check "a product with no factors"
+    "(defn f [] i32 (*))" ~needle:"* takes two arguments or more, given 0";
+  rejects_check "there is no unary minus"
+    "(defn f [] i32 (- 1))" ~needle:"there is no unary minus";
+  rejects_check "there is no reciprocal"
+    "(defn f [] f64 (/ 2.0))" ~needle:"there is no reciprocal";
+  rejects_check "one operand is not a bitwise and"
+    "(defn f [] i32 (bit-and 7))"
+    ~needle:"bit-and takes two arguments or more, given 1";
+  rejects_check "no operands are not a bitwise or"
+    "(defn f [] i32 (bit-or))"
+    ~needle:"bit-or takes two arguments or more, given 0";
+  rejects_check "one operand is not a min"
+    "(defn f [] i32 (min 7))" ~needle:"min takes two arguments or more, given 1";
+
+  (* ── Chained comparisons ───────────────────────────────────────── *)
+  (* (< a b c) is a < b and b < c. The left fold — ((a < b) < c) — would be
+     comparing a bool against a number, so there is one reading and this is
+     it. What the run-time half means is asserted in programs/chain.flan. *)
+  infers "a three-way comparison is still bool" "(< 1 2 3)" "bool";
+  infers "a four-way comparison is still bool" "(< 1 2 3 4)" "bool";
+  accepts "every comparison takes three"
+    "(defn f [] bool (and (= 1 1 1) (!= 1 2 3) (<= 1 2 2) (> 3 2 1) \
+     (>= 3 3 2)))";
+  (* [!=] is the one that does not chain. "Is this sequence increasing" and
+     "are these all different" are different questions, and only the first is
+     about neighbours: under chaining (!= 1 2 1) would be true. It asks about
+     every pair instead. The answers themselves are asserted where they can
+     be run, in programs/chain.flan. *)
+  infers "all-distinct is still bool" "(!= 1 2 1)" "bool";
+  accepts "!= at four operands" "(defn f [] bool (!= 1 2 3 4))";
+  (* Strings, which = and != admit and the orderings do not. The slots a
+     comparison binds are slots of the operands' own type, so this is the one
+     row that is not about machine words. *)
+  infers "= over strings at three" "(= \"a\" \"a\" \"a\")" "bool";
+  infers "!= over strings at three" "(!= \"a\" \"b\" \"c\")" "bool";
+  rejects_check "the orderings still refuse strings at three"
+    "(defn f [] bool (< \"a\" \"b\" \"c\"))" ~needle:"orders machine numbers";
+  (* The operands after the first pair are checked against the type that pair
+     decided, and nothing widens on the way — the same rule two operands have
+     had all along, applied one more time. *)
+  accepts "a chain over one width" "(defn f [x u8 y u8 z u8] bool (< x y z))";
+  rejects_check "a chain does not widen its third operand"
+    "(defn f [x u8 y u8 z i64] bool (< x y z))" ~needle:"expected u8";
+  rejects_check "a chain does not widen its fourth operand"
+    "(defn f [x i32 y i32 z i32 w f64] bool (< x y z w))"
+    ~needle:"expected i32";
+  (* A generic operand is admitted by the predicate its signature carries, at
+     three operands as at two. *)
+  rejects_check "all-distinct does not widen its third operand either"
+    "(defn f [x u8 y u8 z i64] bool (!= x y z))" ~needle:"expected u8";
+  accepts "a chain over a type variable"
+    "(defn between [a $t b $t c $t] bool {:where (ordered? $t)} (< a b c))";
+  accepts "all-distinct over a type variable"
+    "(defn three [a $t b $t c $t] bool {:where (equal? $t)} (!= a b c))";
+  rejects_check "a chain still wants the right predicate"
+    ~needle:"nothing here says t is ordered?"
+    "(defn between [a $t b $t c $t] bool {:where (equal? $t)} (< a b c))";
+  (* One operand and none. Both would have to be [true] whatever they were
+     handed, which is a typo carrying a value. *)
+  rejects_check "one operand is not a comparison"
+    "(defn f [] bool (< 1))" ~needle:"needs a second value to compare against";
+  rejects_check "no operands are not a comparison"
+    "(defn f [] bool (<))" ~needle:"< takes two arguments or more, given 0";
+  rejects_check "the same for equality"
+    "(defn f [] bool (= 1))" ~needle:"needs a second value to compare against";
+  rejects_check "one value is distinct from nothing"
+    "(defn f [] bool (!= 1))" ~needle:"needs something to be distinct from";
+  rejects_check "no values are not a distinctness test"
+    "(defn f [] bool (!=))" ~needle:"!= takes two arguments or more, given 0";
+  (* Chaining does not reach the operators that are not comparisons. *)
+  rejects_check "a chain of remainders is still refused"
+    "(defn f [] i32 (% 10 3 2))" ~needle:"% takes 2 arguments";
+  rejects_check "a chain of shifts is still refused"
+    "(defn f [] i32 (<< 1 2 3))" ~needle:"<< takes 2 arguments";
+
   (* ── Unknown types ─────────────────────────────────────────────── *)
   (* A lowercase name is a type variable (plan.org, Types), so a mistyped
      primitive would otherwise be reported as unimplemented generics and send