diff --git a/docs/specs/2026-05-21-fieldtest-operator-routing-eq-ord.md b/docs/specs/2026-05-21-fieldtest-operator-routing-eq-ord.md
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@@ -0,0 +1,241 @@
+# Fieldtest — operator-routing-eq-ord — 2026-05-21
+
+**Status:** Draft — awaiting orchestrator triage
+**Author:** ailang-fieldtester (dispatched by skills/fieldtest)
+
+## Scope
+
+The `operator-routing-eq-ord` milestone removed the surface comparator
+names `==` / `!=` / `<` / `<=` / `>` / `>=` from the language. Equality
+and ordering are now class methods that resolve via `prelude.Eq` /
+`prelude.Ord` class-dispatch — `eq` / `ne` / `lt` / `le` / `gt` / `ge` /
+`compare` over the primitive instances (Int, Bool, Str, Unit) plus any
+user ADT with a hand-written instance. Float has no Eq/Ord instance by
+design; the six monomorphic prelude fns `float_eq` / `float_ne` /
+`float_lt` / `float_le` / `float_gt` / `float_ge` are the explicit
+named-fn surface for IEEE-aware Float comparison, each lowered to a
+single `fcmp` via the codegen intercept with `alwaysinline`.
+
+## Examples
+
+### `examples/fieldtest/eqord_1_fizzbuzz.ail` — FizzBuzz over [1..15]
+
+- **What it does:** classic FizzBuzz, dispatching equality on
+  `(app eq (app % n K) 0)` for K in {3, 5, 15}, plus the empty-string
+  guard `(app eq label "")`, plus a tail-recursive loop driver bounded
+  by `(app gt i stop)`.
+- **Why it fits:** exercises the primitive Eq path at Int and Str
+  simultaneously, plus the Ord-class free helper `gt` at Int — the
+  three most-reached-for shapes in everyday integer code.
+- **Outcome:** compiles, runs, matches expected stdout
+  (`1\n2\nFizz\n4\nBuzz\nFizz\n7\n8\nFizz\nBuzz\n11\nFizz\n13\n14\nFizzBuzz\n`).
+
+### `examples/fieldtest/eqord_2_rational_eq.ail` — Rational with hand-written Eq
+
+- **What it does:** defines `data Rational (ctor MkRational Int Int)`,
+  writes `instance prelude.Eq Rational` using cross-multiplication
+  (`a/b == c/d` iff `a*d == b*c`), drives four comparisons in `main`.
+- **Why it fits:** Axis 2 north-star. The user-instance body's inner
+  `(app eq (app * a d) (app * b c))` dispatches to `prelude.eq__Int`,
+  while the outer `(app eq r1 r2)` from main dispatches to this
+  module's `eq__Rational`. End-to-end: user-instance → prelude class
+  method → primitive intercept.
+- **Outcome:** compiles, runs, matches expected stdout
+  (`true\nfalse\ntrue\nfalse\n`).
+
+### `examples/fieldtest/eqord_3_newton_sqrt.ail` — Newton's method via float_lt / float_eq
+
+- **What it does:** Newton-Raphson iteration `x_{k+1} = (x_k + n/x_k) / 2`
+  to compute sqrt. Convergence loop guards on
+  `(app float_lt (app fabs (app - xnew x)) tol)`; pathological-zero
+  guard on `(app float_eq n 0.0)`. `fabs` implemented inline as
+  `if (app float_lt x 0.0) then -x else x`.
+- **Why it fits:** Axis 3 north-star. Demonstrates the LLM-natural
+  shape of the Float-comparison named-fn family in numerical code.
+  Mirrors how Rust/Python/Swift guide users toward bespoke Float-aware
+  methods (`f64::abs`, `math.isclose`).
+- **Outcome:** compiles, runs, matches expected stdout
+  (`2.0\n3.16228\n0.0\n1.0\n`). The Show Float rendering of
+  `3.16228` is libc-`%g`-shaped per `float-semantics.md` (the exact
+  IEEE-correct value 3.1622776601683795 truncates in the default
+  Show width). Acceptable.
+
+### `examples/fieldtest/eqord_4_float_ord_must_fail.ail` — Ord at Float rejection
+
+- **What it does:** single-statement program
+  `(app print (if (app lt 1.5 2.5) 1 0))`. Must fail at `ail check`.
+- **Why it fits:** Axis 4 north-star. Verifies the diagnostic that
+  steers an LLM-author from the polymorphic `lt` (an Ord-class free
+  helper) toward the named-fn `float_lt` when the inferred operand
+  type is Float.
+- **Outcome:** `ail check` exits 1 with stderr:
+  ```
+  error: [no-instance] main: fn `main` calls `compare` requiring
+  `instance prelude.Ord Float`, but no such instance exists in the
+  workspace — Float has no Eq/Ord instance by design (partial
+  orderability per IEEE-754); use float_eq / float_lt (and siblings
+  float_ne / float_le / float_gt / float_ge) for explicit IEEE-aware
+  comparison. See design/contracts/float-semantics.md.
+  ```
+
+### `examples/fieldtest/eqord_5_float_eq_must_fail.ail` — Eq at Float rejection (companion)
+
+- **What it does:** single-statement program
+  `(app print (if (app eq 1.5 1.5) 1 0))`. Must fail at `ail check`.
+- **Why it fits:** Axis 4 companion. Verifies the Eq-class half of the
+  Float diagnostic.
+- **Outcome:** `ail check` exits 1 with stderr substituting `Eq` for
+  `Ord` and `eq` for `compare`; rest of the addendum identical.
+
+## Findings
+
+### [working] Primitive Eq/Ord at Int/Bool/Str/Unit is friction-free
+
+- **Example(s):** eqord_1_fizzbuzz (Int eq, Str eq, Int gt),
+  eqord_2_rational_eq (Int eq inside ADT-instance body).
+- **What happened:** every `(app eq …)` and `(app gt …)` call I
+  wrote compiled on first try, lowered correctly, and produced the
+  expected output. No friction in the surface shape, no diagnostic
+  needed.
+- **Why it's `working`:** the milestone's central thesis (class-dispatch
+  is the surface for all comparison) holds — there is no second
+  thought during authoring about which spelling to use, and no
+  ambiguity about which primitive instance dispatches. The FizzBuzz
+  fixture is the natural form an LLM would emit on the prompt
+  "write FizzBuzz in AILang".
+- **Recommended downstream action:** carry-on. Worth keeping the
+  fixtures around as regression protection against any future
+  re-introduction of operator-name surface.
+
+### [working] User-ADT Eq with nested primitive eq calls Just Works
+
+- **Example(s):** eqord_2_rational_eq.
+- **What happened:** the `instance prelude.Eq Rational` body wrote
+  itself naturally — match both operands, cross-multiply, compare
+  with inner `(app eq …)` on Int. No coercion ceremony, no manual
+  qualifier on the inner call (the inner `eq` resolves through the
+  same class-dispatch as the outer, just at a different inferred
+  type). The end-to-end run produced the correct cross-multiplication
+  results.
+- **Why it's `working`:** this is the milestone's clause-1 evidence —
+  the form was structurally impossible before the milestone (the old
+  built-in `==` rejected user ADTs at codegen with a generic error;
+  even after writing `instance Eq Point`, the inner `(app eq a1 a2)`
+  on Int would route via the dead polymorphic-`==` path with no
+  visible gain). Post-milestone, the LLM-author's mental model
+  ("classes do equality") matches the implementation surface 1:1.
+- **Recommended downstream action:** carry-on.
+
+### [working] Float-comparison named-fn surface is LLM-natural
+
+- **Example(s):** eqord_3_newton_sqrt.
+- **What happened:** writing `(app float_lt x 0.0)`, `(app float_eq n 0.0)`,
+  and `(app float_lt (app fabs …) tol)` felt entirely natural while
+  composing the Newton iteration. The named-fn surface is no more
+  verbose than the deleted operator surface (`(app < x 0.0)` was three
+  tokens, `(app float_lt x 0.0)` is three tokens), and the explicit
+  `float_` prefix is an active reminder during authoring that this is
+  an IEEE-partial operation rather than a structural-equality one.
+- **Why it's `working`:** the spec's clause-3 framing — "the
+  NaN-comparison anti-pattern stays visible in code as `float_eq`
+  rather than hiding behind `eq`" — held up in practice. Newton's
+  method is exactly the kind of numerical code where mistaking
+  Float-eq for structural-eq would cause real bugs; the explicit
+  spelling made me pause and think about whether `float_eq n 0.0` was
+  the right zero-test (it is, because `-0.0 == 0.0` per IEEE, which
+  is the correct guard).
+- **Recommended downstream action:** carry-on.
+
+### [working] NoInstance Eq/Ord Float diagnostic is informative and actionable
+
+- **Example(s):** eqord_4_float_ord_must_fail, eqord_5_float_eq_must_fail.
+- **What happened:** both `ail check` invocations exit 1; both
+  diagnostics name the right class (`Eq` vs `Ord`), the right method
+  call site (`fn main calls eq` / `calls compare` — note the Ord
+  helper `lt` is reported via its desugar to `compare`), the named-fn
+  family `float_eq / float_lt (and siblings …)`, the IEEE-754 design
+  rationale, and the contract path
+  `design/contracts/float-semantics.md`. An LLM-author who hits this
+  diagnostic learns the entire replacement surface in one read.
+- **Why it's `working`:** this is the milestone's clause-2 evidence —
+  the diagnostic carries enough information for an LLM that has only
+  the stderr to converge on the right call site without re-reading
+  the spec. Acceptance-criterion #3 ("Float-aware hint pointing at
+  `float_eq`") is comfortably met.
+- **Recommended downstream action:** carry-on. The diagnostic is
+  the load-bearing surface for clean-break removal of the operator
+  names; protect it with the existing `eq_float_noinstance.rs` test.
+
+### [spec_gap] Spec's own example program uses bare `(class Eq)` but the language requires `(class prelude.Eq)`
+
+- **Example(s):** confirmed by a /tmp probe; my own
+  `eqord_2_rational_eq.ail` writes `(class prelude.Eq)` because I
+  copied the pattern from the existing `eq_ord_user_adt.ail`, not
+  from the spec.
+- **What happened:** the design spec at
+  `docs/specs/2026-05-20-operator-routing-eq-ord.md:113` writes
+  `(class Eq)` (bare) in the §"Concrete code shapes" north-star
+  example program. An LLM-author who copies that pattern verbatim
+  hits the diagnostic:
+  ```
+  error: [bare-cross-module-class-ref] module `…` contains bare class
+  name `Eq` that does not resolve to a local class; candidates from
+  imports: ["prelude.Eq"]
+  ```
+  The diagnostic is excellent (lists the qualified candidate), but
+  the spec primed the wrong pattern.
+- **Why it's `spec_gap`:** the milestone spec is the primary
+  reference for the milestone-defining surface; if the spec's own
+  example doesn't typecheck verbatim, then either the spec is wrong
+  (most likely) or bare class names are a feature the language should
+  in fact support (unlikely — qualified is the established style per
+  every existing fixture). The honest action is to update the spec's
+  example to use `(class prelude.Eq)`. This is one-line of doc
+  drift, but it is the highest-information reference an LLM-author
+  consults when picking up the milestone surface.
+- **Recommended downstream action:** tighten the design ledger — fix
+  the example in `docs/specs/2026-05-20-operator-routing-eq-ord.md`
+  (lines 109-133) to use `(class prelude.Eq)`, and consider whether
+  the same drift exists in `design/contracts/prelude-classes.md` or
+  `design/models/typeclasses.md`. A single forward-fix doc commit.
+
+### [friction] `(app compare …)` at Float diagnostic does not acknowledge there is no `float_compare`
+
+- **Example(s):** /tmp probe writing
+  `(app compare 1.5 2.5)` inside a match-on-Ordering.
+- **What happened:** the diagnostic is identical to the `(app lt …)`
+  diagnostic — names `float_eq / float_lt (and siblings …)`. None of
+  the suggested replacements give the three-way LT/EQ/GT result that
+  `compare` returns. An LLM-author who specifically reached for
+  `compare` because they want to drive an `Ordering`-match on Float
+  reads this diagnostic and is left guessing whether they should
+  pattern on `float_lt` + `float_eq` themselves to reconstruct the
+  three-way decision, or whether some other surface exists.
+- **Why it's `friction`:** the spec at lines 433-436 acknowledged
+  this case explicitly ("`compare`/`Ord` use case names `float_lt` /
+  `float_compare` (the latter doesn't ship as a fn …) — so the hint
+  says 'use float_lt / float_eq for explicit IEEE-aware comparison'").
+  So the absence of `float_compare` is deliberate. But the
+  diagnostic doesn't tell the author *that* — it just lists the
+  replacements without explaining the gap. A one-line addendum
+  ("note: there is no `float_compare` because IEEE-754 ordering is
+  partial; build the three-way decision from `float_lt` + `float_eq`
+  if you need it") would close the gap without changing any code
+  path.
+- **Recommended downstream action:** plan a tidy iteration that
+  extends the float-aware diagnostic to differentiate on the called
+  method — when the call site is `compare`, append the
+  "no float_compare exists, build it from float_lt + float_eq"
+  sentence. Scoped to one branch in the diagnostic addendum, no
+  contract change.
+
+## Recommendation summary
+
+| Finding | Class | Action |
+|---|---|---|
+| Primitive Eq/Ord at Int/Bool/Str/Unit friction-free | working | carry-on |
+| User-ADT Eq with nested primitive eq Just Works | working | carry-on |
+| Float-comparison named-fn surface is LLM-natural | working | carry-on |
+| NoInstance Eq/Ord Float diagnostic is informative | working | carry-on |
+| Spec example uses bare `(class Eq)` but language needs `(class prelude.Eq)` | spec_gap | tighten the design ledger (doc fix) |
+| `(app compare …)` at Float diagnostic doesn't name the missing `float_compare` | friction | plan a tidy iteration (diagnostic addendum) |
diff --git a/examples/fieldtest/eqord_1_fizzbuzz.ail b/examples/fieldtest/eqord_1_fizzbuzz.ail
new file mode 100644
index 0000000..73e1e33
--- /dev/null
+++ b/examples/fieldtest/eqord_1_fizzbuzz.ail
@@ -0,0 +1,73 @@
+; Fieldtest fixture — Axis 1: equality on primitive Int via `(app eq …)`.
+;
+; FizzBuzz over [1..15]:
+;   if n % 15 == 0 print "FizzBuzz"
+;   else if n % 3  == 0 print "Fizz"
+;   else if n % 5  == 0 print "Buzz"
+;   else print (show n)
+;
+; Equality dispatches through prelude.Eq.eq at Int (intercept arm
+; eq__Int → `icmp eq i64`). Demonstrates the LLM-natural shape of
+; `(app eq …)` on the primitive type that dominates most numerical
+; code.
+;
+; Expected stdout (one per line):
+;   1
+;   2
+;   Fizz
+;   4
+;   Buzz
+;   Fizz
+;   7
+;   8
+;   Fizz
+;   Buzz
+;   11
+;   Fizz
+;   13
+;   14
+;   FizzBuzz
+
+(module eqord_1_fizzbuzz
+
+  (fn classify
+    (doc "Return the FizzBuzz label for n, or the empty string if n is a plain number.")
+    (type
+      (fn-type
+        (params (con Int))
+        (ret (con Str))))
+    (params n)
+    (body
+      (if (app eq (app % n 15) 0)
+        "FizzBuzz"
+        (if (app eq (app % n 3) 0)
+          "Fizz"
+          (if (app eq (app % n 5) 0)
+            "Buzz"
+            "")))))
+
+  (fn emit_one
+    (doc "Print the FizzBuzz label or the number itself.")
+    (type (fn-type (params (con Int)) (ret (con Unit)) (effects IO)))
+    (params n)
+    (body
+      (let label (app classify n)
+        (if (app eq label "")
+          (app print n)
+          (app print label)))))
+
+  (fn loop
+    (doc "Tail-recursive driver over [i..stop].")
+    (type (fn-type (params (con Int) (con Int)) (ret (con Unit)) (effects IO)))
+    (params i stop)
+    (body
+      (if (app gt i stop)
+        (lit-unit)
+        (seq
+          (app emit_one i)
+          (tail-app loop (app + i 1) stop)))))
+
+  (fn main
+    (type (fn-type (params) (ret (con Unit)) (effects IO)))
+    (params)
+    (body (app loop 1 15))))
diff --git a/examples/fieldtest/eqord_2_rational_eq.ail b/examples/fieldtest/eqord_2_rational_eq.ail
new file mode 100644
index 0000000..94254a1
--- /dev/null
+++ b/examples/fieldtest/eqord_2_rational_eq.ail
@@ -0,0 +1,56 @@
+; Fieldtest fixture — Axis 2: user-ADT Eq instance, hand-written.
+;
+; Rational numbers compared by cross-multiplication:
+;   a/b == c/d  iff  a*d == b*c
+; (assumes positive denominators; the constructor enforces it.)
+;
+; The interesting bit for the milestone is the *nested* equality call
+; inside the user instance body: the inner `(app eq (app * a d) (app * b c))`
+; dispatches to prelude.eq__Int (the primitive instance, intercept-
+; lowered to `icmp eq i64`), while the outer `(app eq r1 r2)` from main
+; dispatches to this module's own `eq__Rational`. End-to-end the chain
+; is: user-instance method → prelude class method → primitive intercept.
+;
+; Expected stdout (one per line):
+;   true     ; 1/2 == 2/4
+;   false    ; 1/2 == 1/3
+;   true     ; 3/4 == 6/8
+;   false    ; 5/6 == 4/5
+
+(module eqord_2_rational_eq
+
+  (data Rational
+    (doc "Numerator + (positive) denominator. No normalisation.")
+    (ctor MkRational (con Int) (con Int)))
+
+  (instance
+    (class prelude.Eq)
+    (type (con Rational))
+    (doc "Cross-multiplication equality. Inner eq dispatches to prelude.eq__Int.")
+    (method eq
+      (body
+        (lam
+          (params (typed r1 (con Rational)) (typed r2 (con Rational)))
+          (ret (con Bool))
+          (body
+            (match r1
+              (case (pat-ctor MkRational a b)
+                (match r2
+                  (case (pat-ctor MkRational c d)
+                    (app eq (app * a d) (app * b c)))))))))))
+
+  (fn main
+    (type (fn-type (params) (ret (con Unit)) (effects IO)))
+    (params)
+    (body
+      (let r_1_2 (term-ctor Rational MkRational 1 2)
+        (let r_2_4 (term-ctor Rational MkRational 2 4)
+          (let r_1_3 (term-ctor Rational MkRational 1 3)
+            (let r_3_4 (term-ctor Rational MkRational 3 4)
+              (let r_6_8 (term-ctor Rational MkRational 6 8)
+                (let r_5_6 (term-ctor Rational MkRational 5 6)
+                  (let r_4_5 (term-ctor Rational MkRational 4 5)
+                    (seq (app print (app eq r_1_2 r_2_4))
+                    (seq (app print (app eq r_1_2 r_1_3))
+                    (seq (app print (app eq r_3_4 r_6_8))
+                         (app print (app eq r_5_6 r_4_5)))))))))))))))
diff --git a/examples/fieldtest/eqord_3_newton_sqrt.ail b/examples/fieldtest/eqord_3_newton_sqrt.ail
new file mode 100644
index 0000000..62bdb4a
--- /dev/null
+++ b/examples/fieldtest/eqord_3_newton_sqrt.ail
@@ -0,0 +1,61 @@
+; Fieldtest fixture — Axis 3: Float comparison via the named-fn surface.
+;
+; Newton's method for sqrt: iterate x_{k+1} = (x_k + n/x_k) / 2 until
+;   |x_{k+1} - x_k| < tol
+; Reaches for `float_lt` (loop convergence) and `float_eq` (zero-arg
+; guard against pathological 0.0 input). Also exercises Float
+; arithmetic + Show Float via `print`.
+;
+; The interesting bit for the milestone is that there is no
+; polymorphic `eq` / `lt` at Float — those would fail at typecheck
+; with `NoInstance Eq Float`. The LLM-author has to reach for the
+; named family `float_eq` / `float_lt` / `float_gt`. The shape mirrors
+; how Rust's `f64::abs` / partial_cmp guides users toward bespoke
+; Float-aware methods.
+;
+; abs implemented inline as `if x < 0 then -x else x` using float_lt.
+;
+; Expected stdout (one per line):
+;   2 (sqrt 4.0)             ; 2.0
+;   ~3.16 (sqrt 10.0)        ; 3.1622776601683795 — exact IEEE
+;   0 (sqrt 0.0)             ; 0.0, short-circuit
+;   1 (sqrt 1.0)             ; 1.0
+
+(module eqord_3_newton_sqrt
+
+  (fn fabs
+    (doc "Float absolute value via the named-fn comparison surface.")
+    (type (fn-type (params (con Float)) (ret (con Float))))
+    (params x)
+    (body
+      (if (app float_lt x 0.0)
+        (app - 0.0 x)
+        x)))
+
+  (fn iterate
+    (doc "Tail-recursive Newton iteration. Stops when |xnew - x| < tol.")
+    (type (fn-type (params (con Float) (con Float) (con Float)) (ret (con Float))))
+    (params n x tol)
+    (body
+      (let xnew (app / (app + x (app / n x)) 2.0)
+        (if (app float_lt (app fabs (app - xnew x)) tol)
+          xnew
+          (tail-app iterate n xnew tol)))))
+
+  (fn sqrt
+    (doc "sqrt n via Newton with initial guess 1.0 and tolerance 1e-10. Returns 0.0 for n == 0.0; assumes n >= 0.")
+    (type (fn-type (params (con Float)) (ret (con Float))))
+    (params n)
+    (body
+      (if (app float_eq n 0.0)
+        0.0
+        (app iterate n 1.0 0.0000000001))))
+
+  (fn main
+    (type (fn-type (params) (ret (con Unit)) (effects IO)))
+    (params)
+    (body
+      (seq (app print (app sqrt 4.0))
+      (seq (app print (app sqrt 10.0))
+      (seq (app print (app sqrt 0.0))
+           (app print (app sqrt 1.0))))))))
diff --git a/examples/fieldtest/eqord_4_float_ord_must_fail.ail b/examples/fieldtest/eqord_4_float_ord_must_fail.ail
new file mode 100644
index 0000000..0f7a634
--- /dev/null
+++ b/examples/fieldtest/eqord_4_float_ord_must_fail.ail
@@ -0,0 +1,23 @@
+; Fieldtest fixture — Axis 4: rejection diagnostic for Ord at Float.
+;
+; This fixture is the discriminator for whether the post-milestone
+; NoInstance diagnostic steers an LLM-author who reached for a
+; polymorphic Ord-helper at Float toward the `float_lt` named-fn
+; replacement.
+;
+; Today (per spec §Error handling) the diagnostic at Eq Float says
+;   "use float_eq for explicit IEEE-aware comparison"
+; The spec also promises the Ord variant says
+;   "use float_lt / float_eq for explicit IEEE-aware comparison"
+; so an LLM-author seeing this fail learns the named-fn surface
+; without needing to read the prelude.
+;
+; Expected behaviour: `ail check` exits non-zero. stderr contains:
+;   - NoInstance Ord Float (or similar Ord-at-Float wording)
+;   - the literal string `float_lt` somewhere in the addendum
+
+(module eqord_4_float_ord_must_fail
+  (fn main
+    (type (fn-type (params) (ret (con Unit)) (effects IO)))
+    (params)
+    (body (app print (if (app lt 1.5 2.5) 1 0)))))
diff --git a/examples/fieldtest/eqord_5_float_eq_must_fail.ail b/examples/fieldtest/eqord_5_float_eq_must_fail.ail
new file mode 100644
index 0000000..b00d2b3
--- /dev/null
+++ b/examples/fieldtest/eqord_5_float_eq_must_fail.ail
@@ -0,0 +1,16 @@
+; Fieldtest fixture — Axis 4 companion: rejection diagnostic for Eq at Float.
+;
+; Twin of eqord_4_float_ord_must_fail.ail for the Eq variant.
+; Verifies the diagnostic produced when an LLM-author tries the
+; polymorphic `eq` at Float — what the milestone spec calls the
+; Klausel-3 discriminator.
+;
+; Expected behaviour: `ail check` exits non-zero. stderr contains:
+;   - `NoInstance` mention of `Eq` and `Float`
+;   - the literal string `float_eq` somewhere in the addendum
+
+(module eqord_5_float_eq_must_fail
+  (fn main
+    (type (fn-type (params) (ret (con Unit)) (effects IO)))
+    (params)
+    (body (app print (if (app eq 1.5 1.5) 1 0)))))