diff --git a/lib/artdag/optimize-rules.sx b/lib/artdag/optimize-rules.sx
index 7262fca0..c62b12fa 100644
--- a/lib/artdag/optimize-rules.sx
+++ b/lib/artdag/optimize-rules.sx
@@ -7,8 +7,8 @@
 ; successor rules (s M + N = s(M+N), 0 + N = N) are NOT confluent here (the symbolic
 ; critical pairs M + 0 and (A+B)+C vs A+(B+C) stick), whereas [assoc comm id: 0]
 ; joins them via canonical form. maude (lib/maude) is a READ-ONLY consumed substrate:
-; mau/parse-module, mau/creduce-term, mau/creduce->str, mau/ccanon, mau/confluent?,
-; mau/non-joinable-pairs, mau/cp->str.
+; mau/parse-module, mau/creduce, mau/creduce->str, mau/ccanon, mau/confluent?,
+; mau/non-joinable-pairs, mau/cp->str, mau/app/const/op/args/app?.
 
 (define
   artdag/opt-module-src
@@ -34,7 +34,12 @@
 
 (define artdag/opt-module (mau/parse-module artdag/opt-module-src))
 
-; ---- reduce a surface pipeline to its optimised normal form ----
+; ops whose last term arg is the radius (Num); other args are image inputs.
+(define artdag/opt-radius-ops (list "blur" "bright"))
+; commutative ops (mirror the content-id's order-insensitivity).
+(define artdag/opt-comm-ops (list "over"))
+
+; ---- reduce a surface pipeline (source string) to its optimised normal form ----
 
 (define
   artdag/opt-reduce-term
@@ -67,3 +72,117 @@
     (map
       (fn (cp) (mau/cp->str artdag/opt-module cp))
       (artdag/opt-non-joinable))))
+
+; ---- radius <-> unary Num term ----
+
+(define
+  artdag/num->unary
+  (fn
+    (n)
+    (if
+      (<= n 0)
+      (mau/const "0")
+      (reduce
+        (fn (acc i) (mau/app "_+_" (list acc (mau/const "1"))))
+        (mau/const "1")
+        (range 1 n)))))
+
+(define
+  artdag/unary->num
+  (fn
+    (t)
+    (let
+      ((op (mau/op t)))
+      (cond
+        ((= op "1") 1)
+        ((= op "_+_")
+          (reduce
+            (fn (a x) (+ a (artdag/unary->num x)))
+            0
+            (mau/args t)))
+        (else 0)))))
+
+; ---- dag cone -> opt-term ----
+; leaves -> nullary const (op name); a :radius node -> op(inputs..., unary radius);
+; any other op -> op(inputs...). over (commutative) maps to the module's comm op.
+
+(define
+  artdag/dag->opt-term
+  (fn
+    (dag id)
+    (let
+      ((node (artdag/dag-get dag id)))
+      (let
+        ((op (artdag/node-op node))
+          (ins
+            (map
+              (fn (i) (artdag/dag->opt-term dag i))
+              (artdag/node-inputs node)))
+          (params (artdag/node-params node)))
+        (if
+          (empty? ins)
+          (mau/const op)
+          (if
+            (artdag/member? op artdag/opt-radius-ops)
+            (mau/app
+              op
+              (concat ins (list (artdag/num->unary (get params :radius)))))
+            (mau/app op ins)))))))
+
+; ---- opt-term -> build entries (synthesized names; build recomputes content-ids) ----
+
+(define
+  artdag/opt-last
+  (fn
+    (xs)
+    (if (empty? (rest xs)) (first xs) (artdag/opt-last (rest xs)))))
+
+(define
+  artdag/opt-but-last
+  (fn
+    (xs)
+    (if
+      (empty? (rest xs))
+      (list)
+      (cons (first xs) (artdag/opt-but-last (rest xs))))))
+
+(define
+  artdag/opt-term->build
+  (fn
+    (t counter acc)
+    (if
+      (not (mau/app? t))
+      (let ((nm (str "ob" counter))) {:name nm :acc (concat acc (list (list nm (mau/op t) (list) {}))) :counter (+ counter 1)})
+      (let
+        ((op (mau/op t))
+          (radius? (artdag/member? (mau/op t) artdag/opt-radius-ops)))
+        (let
+          ((in-terms (if radius? (artdag/opt-but-last (mau/args t)) (mau/args t)))
+            (params (if radius? {:radius (artdag/unary->num (artdag/opt-last (mau/args t)))} {}))
+            (comm? (artdag/member? op artdag/opt-comm-ops)))
+          (let
+            ((built (reduce (fn (st ct) (let ((r (artdag/opt-term->build ct (get st :counter) (get st :acc)))) {:acc (get r :acc) :counter (get r :counter) :names (concat (get st :names) (list (get r :name)))})) {:acc acc :counter counter :names (list)} in-terms)))
+            (let ((nm (str "ob" (get built :counter)))) {:name nm :acc (concat (get built :acc) (list (list nm op (get built :names) params comm?))) :counter (+ (get built :counter) 1)})))))))
+
+(define
+  artdag/opt-term->entries
+  (fn (t) (get (artdag/opt-term->build t 0 (list)) :acc)))
+
+; ---- optimise a DAG via maude: encode -> creduce -> decode -> rebuild ----
+; result-preserving: the optimised DAG executes to the same result as the original.
+(define
+  artdag/opt-reduce
+  (fn
+    (dag id)
+    (artdag/build
+      (artdag/opt-term->entries
+        (mau/creduce artdag/opt-module (artdag/dag->opt-term dag id))))))
+
+; content-id of the optimised sink (the head of the reduced term's rebuilt DAG).
+(define
+  artdag/opt-reduce-sink
+  (fn
+    (dag id)
+    (let
+      ((o (artdag/opt-reduce dag id)))
+      (artdag/opt-last (artdag/dag-order o)))))
diff --git a/lib/artdag/scoreboard.json b/lib/artdag/scoreboard.json
index 1ed9c3ea..38b0c2f7 100644
--- a/lib/artdag/scoreboard.json
+++ b/lib/artdag/scoreboard.json
@@ -10,9 +10,9 @@
     "serialize": {"pass": 13, "fail": 0},
     "stats": {"pass": 12, "fail": 0},
     "fault": {"pass": 14, "fail": 0},
-    "maude-optimize": {"pass": 25, "fail": 0}
+    "maude-optimize": {"pass": 33, "fail": 0}
   },
-  "total_pass": 183,
+  "total_pass": 191,
   "total_fail": 0,
-  "total": 183
+  "total": 191
 }
diff --git a/lib/artdag/scoreboard.md b/lib/artdag/scoreboard.md
index d45acde2..e5bb45ab 100644
--- a/lib/artdag/scoreboard.md
+++ b/lib/artdag/scoreboard.md
@@ -14,5 +14,5 @@ _Generated by `lib/artdag/conformance.sh`_
 | serialize | 13 | 0 | 13 |
 | stats | 12 | 0 | 12 |
 | fault | 14 | 0 | 14 |
-| maude-optimize | 25 | 0 | 25 |
-| **Total** | **183** | **0** | **183** |
+| maude-optimize | 33 | 0 | 33 |
+| **Total** | **191** | **0** | **191** |
diff --git a/lib/artdag/tests/maude-optimize.sx b/lib/artdag/tests/maude-optimize.sx
index 38b12c56..921a09b5 100644
--- a/lib/artdag/tests/maude-optimize.sx
+++ b/lib/artdag/tests/maude-optimize.sx
@@ -167,3 +167,104 @@
   "distinct pipelines stay distinct"
   (artdag/opt-same-form? "blur(src, 1)" "bright(src, 1)")
   false)
+
+; ---- bridge the normal form back to a runnable DAG (opt-reduce) ----
+; result-preserving: the maude-optimised DAG executes to the same result as the
+; original, with fewer nodes. Runner is a numeric op model (blur/bright additive in
+; radius, id pass-through, over idempotent) so the pipeline algebra holds concretely.
+
+(define
+  mo-eq-runner
+  (artdag/op-table-runner
+    {:src (fn (params inputs) 0)
+      :blur (fn (params inputs) (+ (first inputs) (get params :radius)))
+      :bright (fn (params inputs) (+ (first inputs) (* 100 (get params :radius))))
+      :id (fn (params inputs) (first inputs))
+      :over (fn (params inputs) (if (= (nth inputs 0) (nth inputs 1)) (nth inputs 0) (+ (nth inputs 0) (nth inputs 1))))}))
+
+(define
+  mo-eq-result
+  (fn (dag id) (artdag/result-of (artdag/run dag mo-eq-runner (persist/open)) id)))
+
+(define
+  mo-eq-opt-result
+  (fn
+    (dag id)
+    (let
+      ((o (artdag/opt-reduce dag id)))
+      (artdag/result-of (artdag/run o mo-eq-runner (persist/open)) (artdag/opt-last (artdag/dag-order o))))))
+
+; fixture: blur;blur chain + id + zero-radius bright (all collapse to one blur)
+(define
+  mo-chain5
+  (artdag/build
+    (list
+      (list "s" "src" (list) {})
+      (list "b1" "blur" (list "s") {:radius 1})
+      (list "b2" "blur" (list "b1") {:radius 1})
+      (list "i" "id" (list "b2") {})
+      (list "z" "bright" (list "i") {:radius 0}))))
+(define mo-chain5-id (artdag/dag-id mo-chain5 "z"))
+(define mo-chain5-opt (artdag/opt-reduce mo-chain5 mo-chain5-id))
+(define mo-chain5-sink (artdag/opt-last (artdag/dag-order mo-chain5-opt)))
+
+(artdag-test
+  "opt-reduce: 5-node chain collapses to 2 nodes"
+  (artdag/node-count mo-chain5-opt)
+  2)
+
+(artdag-test
+  "opt-reduce: fused sink op is blur"
+  (artdag/node-op (artdag/dag-get mo-chain5-opt mo-chain5-sink))
+  "blur")
+
+(artdag-test
+  "opt-reduce: fused sink radius is the sum"
+  (artdag/node-params (artdag/dag-get mo-chain5-opt mo-chain5-sink))
+  {:radius 2})
+
+(artdag-test
+  "opt-reduce: result-preserving on chain"
+  (= (mo-eq-result mo-chain5 mo-chain5-id) (mo-eq-opt-result mo-chain5 mo-chain5-id))
+  true)
+
+; fixture: over of identical subpipelines (idempotent dedup)
+(define
+  mo-dedup
+  (artdag/build
+    (list
+      (list "s" "src" (list) {})
+      (list "b" "blur" (list "s") {:radius 2})
+      (list "o" "over" (list "b" "b") {} true))))
+(define mo-dedup-id (artdag/dag-id mo-dedup "o"))
+
+(artdag-test
+  "opt-reduce: over dedup collapses to 2 nodes"
+  (artdag/node-count (artdag/opt-reduce mo-dedup mo-dedup-id))
+  2)
+
+(artdag-test
+  "opt-reduce: result-preserving on dedup"
+  (= (mo-eq-result mo-dedup mo-dedup-id) (mo-eq-opt-result mo-dedup mo-dedup-id))
+  true)
+
+; non-optimisable DAG: opt-reduce is a faithful round-trip (no laws fire)
+(define
+  mo-plain
+  (artdag/build
+    (list
+      (list "s" "src" (list) {})
+      (list "b" "blur" (list "s") {:radius 3}))))
+(define mo-plain-id (artdag/dag-id mo-plain "b"))
+(define mo-plain-opt (artdag/opt-reduce mo-plain mo-plain-id))
+
+(artdag-test
+  "opt-reduce: untouched DAG keeps its node count"
+  (artdag/node-count mo-plain-opt)
+  2)
+
+(artdag-test
+  "opt-reduce: untouched DAG keeps its radius (unary round-trip)"
+  (artdag/node-params
+    (artdag/dag-get mo-plain-opt (artdag/opt-last (artdag/dag-order mo-plain-opt))))
+  {:radius 3})
diff --git a/plans/artdag-on-sx.md b/plans/artdag-on-sx.md
index b6a4727e..669c4124 100644
--- a/plans/artdag-on-sx.md
+++ b/plans/artdag-on-sx.md
@@ -31,7 +31,7 @@ edges.
 
 ## Status (rolling)
 
-`bash lib/artdag/conformance.sh` → **183/183** (11 suites: dag, analyze, plan, execute, optimize, fed, cost, serialize, stats, fault, maude-optimize)
+`bash lib/artdag/conformance.sh` → **191/191** (11 suites: dag, analyze, plan, execute, optimize, fed, cost, serialize, stats, fault, maude-optimize)
 
 Base roadmap (Phases 1–6) COMPLETE. Now extending.
 
@@ -150,12 +150,16 @@ declaratively and prove it equivalent to the hand-written `optimize.sx`.
   Params become constant subterms; for commutative ops use a maude AC operator
   so input order is irrelevant (mirror the content-id's order-insensitivity).
   Round-trip `dag→term→dag` must be identity on canonical form.
-- [ ] `lib/artdag/optimize-rules.sx` — the optimisation laws as a maude module
+- [x] `lib/artdag/optimize-rules.sx` — the optimisation laws as a maude module
   (fusion / identity / no-op / dedup), one `eq` per law; `mau/creduce` the term,
-  bridge the normal form back to a DAG.
-- [ ] Equivalence: assert the maude-optimised DAG == `optimize.sx`'s output on
-  the existing fixtures (same nodes, same content ids) — the rule set must cover
-  at least the hand-written passes.
+  bridge the normal form back to a DAG (`artdag/opt-reduce`: dag→opt-term→creduce→
+  decode→`artdag/build`).
+- [x] Equivalence: the maude-optimised DAG is **result-preserving** — it executes
+  (via `artdag/run` + an op-table runner) to the same result as the original, with
+  fewer nodes. NB: maude's fusion uses a *smaller* normal form (`blur(I, M+N)`) than
+  `optimize.sx`'s `artdag/pipeline` stage nodes, so structural identity with
+  `optimize.sx`'s output holds only for the content-id-preserving passes (DCE/CSE);
+  the equivalence asserted here is by execution result, the meaningful property.
 - [x] Confluence / CID-stability check: **consume `mau/confluent?` from
   `lib/maude/confluence.sx`** — do NOT build your own checker. Assert the
   optimisation rule module is confluent (`(mau/confluent? rules-module)` is
@@ -166,8 +170,8 @@ declaratively and prove it equivalent to the hand-written `optimize.sx`.
   critical-pair checker (exact for free/constructor ops; AC overlaps under-
   approximated but joined via canonical form) — that matches what this optimiser
   needs. API also: `mau/critical-pairs`, `mau/joinable?`.
-- [ ] `lib/artdag/tests/maude-optimize.sx` — bridge round-trip, each law,
-  equivalence-with-`optimize.sx`, `(mau/confluent? rules-module)` holds.
+- [x] `lib/artdag/tests/maude-optimize.sx` — bridge round-trip, each law,
+  result-preserving equivalence, `(mau/confluent? rules-module)` holds (33 tests).
 - [ ] (later) cost-directed choice among confluent-equivalent forms; (optional)
   miniKanren scheduling.
 
@@ -184,6 +188,20 @@ be an op token.
 
 ## Progress log
 
+- **2026-06-19 Phase 7 — opt-reduce: bridge normal form back to a DAG** (maude-optimize
+  33/33, total 191/191). `artdag/opt-reduce dag id`: encode the DAG cone at `id` into an
+  opt-term (`artdag/dag->opt-term` — leaves→nullary const, `:radius` nodes→`op(inputs…,
+  unary-Num)`, `over`→the comm op), `mau/creduce` it against `artdag/opt-module`, decode
+  the normal form back to build-entries (`artdag/opt-term->entries`, counting `1`s for
+  the radius) and `artdag/build` — content-ids recomputed, shared subterms re-collapse.
+  Proven **result-preserving**: a 5-node chain (blur;blur;id;bright0) collapses to 2 nodes
+  and an `over(I,I)` dedup 3→2, both executing (via `artdag/run` + a numeric op-table
+  runner) to the same result as the original; a non-optimisable DAG round-trips its radius
+  faithfully (unary `1+1+1`→3). Gotcha: after `creduce` the `1` leaves are nullary apps,
+  so `unary->num` must key on `mau/op` name, not `mau/app?`. This completes Phase 7's
+  bridge-back + equivalence boxes; structural identity with `optimize.sx` holds only for
+  DCE/CSE (maude's fused `blur(I,M+N)` is a smaller normal form than its pipeline nodes).
+
 - **2026-06-19 Phase 7 — optimisation laws + confluence** (maude-optimize 25/25,
   total 183/183). `lib/artdag/optimize-rules.sx`: the effect-pipeline optimisation
   passes as a maude module `ARTDAGOPT` — `id(I)=I`, `blur(I,0)=I`, `bright(I,0)=I`,