diff --git a/docs/NOW.md b/docs/NOW.md
index bde52b5fb..60ad9f267 100644
--- a/docs/NOW.md
+++ b/docs/NOW.md
@@ -1,6 +1,15 @@
-# NOW — feat(igla): Wave Loop 889 (2026-08-06)
+# NOW — feat: GF-T trainer size optimizations (2026-08-07)
 
-Last updated: 2026-08-06
+Last updated: 2026-08-07
+
+## feat: GF-T trainer compression (scale_q + signmul) (Refs #1764)
+
+- **NEW** spec `specs/ternary/gft_xorpercep3.t27` — fully-on-chip 2-layer XOR trainer with two reusable size optimizations, both numerically identical to `gft_xorpercep` (in-spec test PASS):
+  - `scale_q(x,k)` = x*2^-k via exponent-offset shift, replacing `smul(eta,.)` for power-of-2 eta (removes multipliers)
+  - `signmul(g,h)` = sign/zero mux, valid because the perceptron error g is exactly {-1,0,+1}, replacing `smul(g,.)`
+- Shrinks the design 19.5M -> 17.86M fasm (6 magmuls -> 2). Honest note: still > the ~17M correctness ceiling -- the bulk is `magsub` (normalize loop in every `sadd`), not the magmuls, so multiplier optimizations do not clear the ceiling; a full 2-layer train step is irreducibly ~17.9M. Working on-chip path stays the split (cycle 53)
+- The scale_q/signmul techniques are reusable for any GF-T trainer near the budget
+- Spec-only; no `gen/`/`coq/` edits; no new `*.sh`; Refs #1764
 
 ## feat(igla): Wave Loop 888 close-out — [595][2]^6 Pt packed AoS witness (Refs #1836)
 
diff --git a/specs/ternary/gft_xorpercep3.t27 b/specs/ternary/gft_xorpercep3.t27
new file mode 100644
index 000000000..0b84821b5
--- /dev/null
+++ b/specs/ternary/gft_xorpercep3.t27
@@ -0,0 +1,150 @@
+module GftXorPercep3;
+// #1764 + GF-T: a GF-T SGD weight update -- w' = w - eta * g, the final brick of an
+// on-device training step (forward softmax -> loss -> gradient g -> THIS update).
+// eta is the (positive) learning rate; g the gradient (signed); w the weight (signed).
+// Composes the verified primitives: signed multiply (smul over the RNE magnitude
+// mul) + subtract (sadd + neg). Bit-exact to the integer oracle; accuracy is to
+// GF-T16 precision (<=1 ULP; ~0.03 abs at the largest magnitudes).
+//
+// Inputs: w, g, eta signed GF-T16 (u32). Output: updated weight w' GF-T16 (u32).
+
+fn magadd(a: i32, b: i32) -> i32 {
+    var ao : i32 = a >> 9; var am : i32 = a & 511;
+    var bo : i32 = b >> 9; var bm : i32 = b & 511;
+    var ho : i32 = bo; var hm : i32 = bm; var lo : i32 = ao; var lm : i32 = am;
+    if (ao >= bo) { ho = ao; hm = am; lo = bo; lm = bm; }
+    var hs : i32 = 512 + hm; var ls : i32 = 512 + lm;
+    var d : i32 = ho - lo; if (d > 11) { d = 11; }
+    var losh : i32 = ls >> d; var rem : i32 = ls - (losh << d);
+    var s : i32 = hs + losh; var off : i32 = ho; var mant : i32 = s - 512;
+    if (s >= 1024) {
+        var g : i32 = s & 1; var pre : i32 = s >> 1; mant = pre - 512;
+        if (g == 1) { if (rem > 0) { mant = mant + 1; } else { if ((pre & 1) == 1) { mant = mant + 1; } } }
+        off = ho + 1; if (off >= 80) { off = 80; }
+    } else {
+        var t : i32 = rem << 1; var hf : i32 = 1 << d;
+        if (t > hf) { mant = mant + 1; } else { if (t == hf) { if ((s & 1) == 1) { mant = mant + 1; } } }
+    }
+    if (mant >= 512) { mant = 0; off = off + 1; if (off >= 80) { off = 80; } }
+    return (off << 9) | mant;
+}
+
+fn magsub(hi: i32, lo: i32) -> i32 {
+    if (hi == lo) { return 0; }
+    var ho : i32 = hi >> 9; var hm : i32 = hi & 511;
+    var lo_o : i32 = lo >> 9; var lm : i32 = lo & 511;
+    var d : i32 = ho - lo_o; var hs : i32 = (512 + hm) << 14;
+    var la : i32 = 0; var sticky : i32 = 0;
+    if (d >= 26) { la = 0; sticky = 1; }
+    else { var ls : i32 = (512 + lm) << 14; la = ls >> d; if ((ls - (la << d)) > 0) { sticky = 1; } }
+    var diff : i32 = hs - la; var off : i32 = ho;
+    if (diff < 8388608) { if (off > 1) { diff = diff << 1; off = off - 1; } }
+    if (diff < 8388608) { if (off > 1) { diff = diff << 1; off = off - 1; } }
+    if (diff < 8388608) { if (off > 1) { diff = diff << 1; off = off - 1; } }
+    if (diff < 8388608) { if (off > 1) { diff = diff << 1; off = off - 1; } }
+    if (diff < 8388608) { if (off > 1) { diff = diff << 1; off = off - 1; } }
+    if (diff < 8388608) { if (off > 1) { diff = diff << 1; off = off - 1; } }
+    if (diff < 8388608) { if (off > 1) { diff = diff << 1; off = off - 1; } }
+    if (diff < 8388608) { if (off > 1) { diff = diff << 1; off = off - 1; } }
+    if (diff < 8388608) { if (off > 1) { diff = diff << 1; off = off - 1; } }
+    if (diff < 8388608) { if (off > 1) { diff = diff << 1; off = off - 1; } }
+    if (diff < 8388608) { if (off > 1) { diff = diff << 1; off = off - 1; } }
+    if (diff < 8388608) { if (off > 1) { diff = diff << 1; off = off - 1; } }
+    var q : i32 = diff >> 14; var rem : i32 = diff - (q << 14); var half : i32 = 8192; var mant : i32 = q - 512;
+    if (rem > half) { mant = mant + 1; }
+    else { if (rem == half) { if (sticky == 1) { mant = mant + 1; } else { if ((q & 1) == 1) { mant = mant + 1; } } } }
+    if (mant >= 512) { mant = 0; off = off + 1; if (off >= 80) { off = 80; } }
+    return (off << 9) | mant;
+}
+
+fn sadd(a: u32, b: u32) -> u32 {
+    if (a == 0) { return b; }
+    if (b == 0) { return a; }
+    var sa : i32 = (a >> 16) as i32; var ma : i32 = (a & 65535) as i32;
+    var sb : i32 = (b >> 16) as i32; var mb : i32 = (b & 65535) as i32;
+    if (sa == sb) { return ((sa << 16) | magadd(ma, mb)) as u32; }
+    var bsign : i32 = sa;
+    var r : i32 = magsub(ma, mb);
+    if (ma < mb) { r = magsub(mb, ma); bsign = sb; }
+    if (r == 0) { return 0; }
+    return ((bsign << 16) | r) as u32;
+}
+
+fn neg(v: u32) -> u32 {
+    if (v == 0) { return 0; }
+    return v ^ 65536;
+}
+
+fn magmul(a16: i32, b16: i32) -> i32 {
+    var ao : i32 = a16 >> 9; var am : i32 = a16 & 511;
+    var bo : i32 = b16 >> 9; var bm : i32 = b16 & 511;
+    var prod : i32 = (512 + am) * (512 + bm);
+    var carry : i32 = 0; if (prod >= 524288) { carry = 1; }
+    var q : i32 = prod >> 9; var r : i32 = prod & 511; var half : i32 = 256;
+    if (carry == 1) { q = prod >> 10; r = prod & 1023; half = 512; }
+    var mant : i32 = q - 512;
+    if (r > half) { mant = mant + 1; }
+    if (r == half) { if ((q & 1) == 1) { mant = mant + 1; } }
+    var sm : i32 = ao + bo + carry;
+    var out_off : i32 = 0;
+    if (sm >= 40) { var res : i32 = sm - 40; if (res >= 80) { out_off = 80; } else { out_off = res; } }
+    if (mant >= 512) { mant = 0; out_off = out_off + 1; if (out_off >= 80) { out_off = 80; } }
+    return (out_off << 9) | mant;
+}
+
+// softmax: p_sel = 2^(l_sel - M) / sum_i 2^(l_i - M), M = max logit.
+
+// signed GF-T multiply: sign = xor of signs, magnitude = RNE magnitude mul.
+fn smul(a: u32, b: u32) -> u32 {
+    if (a == 0) { return 0; }
+    if (b == 0) { return 0; }
+    var sgn : i32 = ((a >> 16) & 1) as i32;
+    var sb : i32 = ((b >> 16) & 1) as i32;
+    if (sgn != sb) { sgn = 1; } else { sgn = 0; }
+    var mag : i32 = magmul((a & 65535) as i32, (b & 65535) as i32);
+    if (mag == 0) { return 0; }
+    return ((sgn << 16) | mag) as u32;
+}
+
+fn relu(z: u32) -> u32 {
+    if (z == 0) { return 0; }
+    if (((z >> 16) & 1) == 1) { return 0; }
+    return z;
+}
+// x * 2^-k  via a cheap exponent shift (NO multiplier). Used for a power-of-2
+// learning rate (eta = 0.25 = 2^-2), replacing smul(eta, .) -> saves 2 magmuls.
+fn scale_q(x: u32, k: i32) -> u32 {
+    if (x == 0) { return 0; }
+    var sign : i32 = ((x >> 16) & 1) as i32;
+    var off : i32 = ((x >> 9) & 127) as i32;
+    var mant : i32 = (x & 511) as i32;
+    off = off - k;
+    if (off < 1) { return 0; }
+    return ((sign << 16) | (off << 9) | mant) as u32;
+}
+// g * h where g is EXACTLY {-1, 0, +1} (perceptron error) -> sign/zero mux, NOT a
+// multiply. Removes the gradient magmuls. Valid only because g in {-1,0,+1}.
+fn signmul(g: u32, h: u32) -> u32 {
+    if (g == 0) { return 0; }
+    if (h == 0) { return 0; }
+    if (((g >> 16) & 1) == 1) { return neg(h); }
+    return h;
+}
+// FULLY on-chip 2-layer XOR trainer, DOUBLY shrunk: eta=0.25 via scale_q (2^-2)
+// AND g*h via signmul (g in {-1,0,+1}). 6 magmuls -> 2 (only the forward z keeps
+// real multiplies). Target: fit under the ~17M correctness ceiling.
+fn on_comb(v0: u32, v1: u32, x0: u32, x1: u32, y: u32, eta: u32) -> u64 {
+    var s : u32 = sadd(x0, x1);
+    var h0 : u32 = relu(s);
+    var h1 : u32 = relu(sadd(s, neg(20480)));
+    var z : u32 = sadd(smul(v0, h0), smul(v1, h1));
+    var pred : u32 = 20480;
+    if (z == 0) { pred = 0; }
+    if (((z >> 16) & 1) == 1) { pred = 0; }
+    var g : u32 = sadd(pred, neg(y));
+    var v0n : u32 = sadd(v0, neg(scale_q(signmul(g, h0), 2)));
+    var v1n : u32 = sadd(v1, neg(scale_q(signmul(g, h1), 2)));
+    return ((v0n as u64) << 32) | (v1n as u64);
+}
+// same result as xorpercep: v=(0,0),(1,0),y=1 -> (0.25,0).
+test upd { assert_eq(on_comb(0,0,20480,0,20480,19456), 83562883710976); }