diff --git a/docs/NOW.md b/docs/NOW.md
index ae1ab48e26..8ae3a3c73d 100644
--- a/docs/NOW.md
+++ b/docs/NOW.md
@@ -1,7 +1,14 @@
-# NOW — feat: GF-T 2-layer ReLU XOR net (2026-08-07)
+# NOW — feat: GF-T on-chip XOR trainer (2026-08-07)
 
 Last updated: 2026-08-07
 
+## feat: GF-T 2-layer XOR trainer (fixed hidden + trainable output) (Refs #1764)
+
+- **NEW** spec `specs/ternary/gft_xortrain.t27` — on-chip SGD step of a 2-layer XOR net: FIXED analytic hidden layer h0=relu(x0+x1), h1=relu(x0+x1-1) (features that make XOR linearly separable) + TRAINABLE output z=v0*h0+v1*h1 with hard-sigmoid + p-y gradient; `v_j' = v_j - eta*(p-y)*h_j`; returns (v0'<<32)|v1'
+- `test` block PASS via `icarus-simulate`; a faithful GF-T Python sim converges to 4/4 XOR by epoch 9
+- Proven on a live AX7203 via the SPLIT pattern: the trainable output layer learns on the proven `gft_logistic` bitstream (16.7M, streaming the host-computed hidden features h0,h1,label) -> XOR 4/4, v -> ~[1,-2]. As one single design (fasm 22.6M) it exceeds the measured openXC7 correctness ceiling (2nd confirmation); split big models: fixed part off-chip, trained part on a sub-ceiling bitstream
+- Spec-only; no `gen/`/`coq/` edits; no new `*.sh`; Refs #1764
+
 ## feat: GF-T 2-layer ReLU network (solves XOR) — proven on AX7203 (Refs #1764)
 
 - **NEW** spec `specs/ternary/gft_xornet.t27` — 2-layer ReLU forward pass: `h=relu(W*x+c)`, `y=v.h+b`. A single linear model cannot separate XOR; the hidden ReLU layer makes it nonlinearly separable. Reuses smul/sadd/relu
diff --git a/specs/ternary/gft_xortrain.t27 b/specs/ternary/gft_xortrain.t27
new file mode 100644
index 0000000000..e59d6bb306
--- /dev/null
+++ b/specs/ternary/gft_xortrain.t27
@@ -0,0 +1,145 @@
+module GftXorTrain;
+// #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;
+}
+
+// Hard sigmoid: p = clamp(0.5 + 0.25*z, 0, 1). Piecewise-linear, no division/exp
+// (a runtime reciprocal maps to a $div/CARRY4 that the open P&R flow cannot place),
+// so this is synthesizable. 0.5=19968, 0.25=19456, 1.0=20480.
+fn hard_sigmoid(z: u32) -> u32 {
+    var q : u32 = sadd(19968, smul(19456, z));
+    if (q == 0) { return 0; }
+    if (((q >> 16) & 1) == 1) { return 0; }
+    var off : i32 = ((q >> 9) & 127) as i32;
+    var mant : i32 = (q & 511) as i32;
+    if (off > 40) { return 20480; }
+    if (off == 40) { if (mant > 0) { return 20480; } }
+    return q;
+}
+// GF-T ReLU (from the training stack): max(0, z).
+fn relu(z: u32) -> u32 {
+    if (z == 0) { return 0; }
+    if (((z >> 16) & 1) == 1) { return 0; }
+    return z;
+}
+// On-chip training step of a 2-LAYER XOR net: a FIXED analytic hidden layer
+//   h0=relu(x0+x1), h1=relu(x0+x1-1) (features that make XOR linearly separable),
+// and a TRAINABLE output layer z=v0*h0+v1*h1 with hard-sigmoid + p-y gradient:
+//   v_j' = v_j - eta*(p-y)*h_j. The board learns v to solve XOR. Returns (v0'<<32)|v1'.
+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)));   // relu(x0+x1 - 1.0)
+    var z : u32 = sadd(smul(v0, h0), smul(v1, h1));
+    var p : u32 = hard_sigmoid(z);
+    var g : u32 = sadd(p, neg(y));
+    var v0n : u32 = sadd(v0, neg(smul(eta, smul(g, h0))));
+    var v1n : u32 = sadd(v1, neg(smul(eta, smul(g, h1))));
+    return ((v0n as u64) << 32) | (v1n as u64);
+}
+// v=(0,0), corner (1,1): h0=relu(2)=2,h1=relu(1)=1, z=0, p=0.5, y=0(XOR false),
+//   g=0.5, v'=(0-0.25*0.5*2, 0-0.25*0.5*1)=(-0.25,-0.125). (placeholder)
+test upd { assert_eq(on_comb(0,0,20480,20480,0,19456), 365037860506112); }