diff --git a/.trinity/seals/ternary_GftSgdStep.json b/.trinity/seals/ternary_GftSgdStep.json
new file mode 100644
index 000000000..a1e68cd0f
--- /dev/null
+++ b/.trinity/seals/ternary_GftSgdStep.json
@@ -0,0 +1,11 @@
+{
+  "gen_hash_c": "sha256:5515d0ea89d404f1f3b4cf2a3cddff4696c15054a9805cdfd22714c42884e36b",
+  "gen_hash_rust": "sha256:5385f7872c428856cf7dd8404a3624ddbbb556fcea1829963ec4f6ea31e70b2a",
+  "gen_hash_verilog": "sha256:688a6551ea442bf503fcc1d7baa3a7532f2474d774b0b59142cc0d5cf9dd20fc",
+  "gen_hash_zig": "sha256:395ba1306710fa36de41e2e57910c0c0dd262040d49f69c5bb9719b99e85d576",
+  "module": "GftSgdStep",
+  "ring": 12,
+  "sealed_at": "2026-08-06T17:05:04Z",
+  "spec_hash": "sha256:21db83ed35b6b0627dc77997430eaf7739126aeff07ea0120b917421eda29108",
+  "spec_path": "specs/ternary/gft_sgd_step.t27"
+}
\ No newline at end of file
diff --git a/bootstrap/tests/gft_sgd_step.rs b/bootstrap/tests/gft_sgd_step.rs
new file mode 100644
index 000000000..17df580f7
--- /dev/null
+++ b/bootstrap/tests/gft_sgd_step.rs
@@ -0,0 +1,73 @@
+// ============================================================================
+// Check for the spec-first GF-T SGD weight update (specs/ternary/gft_sgd_step.t27):
+// w' = w - eta*g, the final brick of an on-device training step. Bit-exact to the
+// integer oracle over 500 vectors (tests/gft_sgd_step_vectors.txt).
+// ============================================================================
+
+use std::env;
+use std::fs;
+use std::path::PathBuf;
+use std::process::Command;
+
+fn t27c() -> &'static str { env!("CARGO_BIN_EXE_t27c") }
+fn spec_path() -> PathBuf {
+    PathBuf::from(env!("CARGO_MANIFEST_DIR")).join("..").join("specs").join("ternary").join("gft_sgd_step.t27")
+}
+fn vectors_path() -> PathBuf {
+    PathBuf::from(env!("CARGO_MANIFEST_DIR")).join("tests").join("gft_sgd_step_vectors.txt")
+}
+fn tool_available(t: &str) -> bool {
+    Command::new(t).arg("-V").output().map(|o| o.status.success()).unwrap_or(false)
+}
+
+#[test]
+fn spec_first_gft_sgd_step_matches_oracle() {
+    let gen = Command::new(t27c()).arg("gen-verilog").arg(spec_path()).output().expect("gen-verilog");
+    assert!(gen.status.success(), "gen-verilog failed:\n{}", String::from_utf8_lossy(&gen.stderr));
+    let v = String::from_utf8_lossy(&gen.stdout).into_owned();
+    assert!(
+        v.contains("input  wire [31:0] w") && v.contains("input  wire [31:0] eta")
+            && v.contains("output wire [31:0] result"),
+        "sgd_step missing w/g/eta -> result interface:\n{}", v
+    );
+    if !tool_available("iverilog") || !tool_available("vvp") {
+        eprintln!("SKIP: iverilog/vvp not on PATH; skipping sgd_step check");
+        return;
+    }
+    let vectors = fs::read_to_string(vectors_path()).expect("read vectors");
+    let n = vectors.lines().filter(|l| !l.trim().is_empty()).count();
+    let dir = env::temp_dir().join(format!("t27_sgd_{}", std::process::id()));
+    let _ = fs::remove_dir_all(&dir);
+    fs::create_dir_all(&dir).unwrap();
+    fs::write(dir.join("spec.v"), &gen.stdout).unwrap();
+    fs::write(dir.join("vec.txt"), &vectors).unwrap();
+    let tb = format!(
+        r#"`timescale 1ns/1ps
+module tb;
+  reg [31:0] w,g,eta; wire [31:0] y; integer fails,nn,fd,code; reg [31:0] exp;
+  GftSgdStep dut(.clk(1'b0),.rst_n(1'b1),.en(1'b1),.w(w),.g(g),.eta(eta),.ready(),.result(y));
+  initial begin
+    fails=0; nn=0; fd=$fopen("{}","r");
+    while(!$feof(fd)) begin code=$fscanf(fd,"%h %h %h %h\n",w,g,eta,exp);
+      if(code==4) begin #1; nn=nn+1;
+        if(y!==exp) begin fails=fails+1; if(fails<=6)$display("FAIL w=%h g=%h eta=%h y=%h exp=%h",w,g,eta,y,exp); end
+      end end
+    $fclose(fd);
+    if(fails==0)$display("ALL_PASS %0d",nn); else $display("FAILED %0d/%0d",fails,nn);
+    $finish;
+  end
+endmodule
+"#,
+        dir.join("vec.txt").to_str().unwrap()
+    );
+    fs::write(dir.join("tb.v"), tb).unwrap();
+    let vvp = dir.join("sim.vvp");
+    let compile = Command::new("iverilog").args(["-g2012", "-o", vvp.to_str().unwrap()])
+        .arg(dir.join("spec.v")).arg(dir.join("tb.v")).output().unwrap();
+    assert!(compile.status.success(), "iverilog compile failed:\n{}", String::from_utf8_lossy(&compile.stderr));
+    let run = Command::new("vvp").arg(&vvp).output().unwrap();
+    let stdout = String::from_utf8_lossy(&run.stdout).into_owned();
+    let _ = fs::remove_dir_all(&dir);
+    assert!(stdout.contains(&format!("ALL_PASS {}", n)), "sgd_step differs from the oracle:\n{}", stdout);
+    assert!(!stdout.contains("FAIL"), "sgd_step mismatch:\n{}", stdout);
+}
diff --git a/bootstrap/tests/gft_sgd_step_vectors.txt b/bootstrap/tests/gft_sgd_step_vectors.txt
new file mode 100644
index 000000000..f95bc417d
--- /dev/null
+++ b/bootstrap/tests/gft_sgd_step_vectors.txt
@@ -0,0 +1,500 @@
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diff --git a/docs/NOW.md b/docs/NOW.md
index af6997694..4e459abdf 100644
--- a/docs/NOW.md
+++ b/docs/NOW.md
@@ -1,3 +1,20 @@
+# NOW — feat(spec): GF-T SGD weight update — training loop closed (2026-08-07)
+
+Last updated: 2026-08-07
+
+## feat(spec): GF-T SGD step w' = w − η·g — the on-device training loop is complete (Refs #1764)
+
+- Branch: `feat/gft-sgd-step` (stacked on `feat/gft-softmax-grad`)
+
+### Что легло
+- `specs/ternary/gft_sgd_step.t27` (`GftSgdStep`): a GF-T **SGD weight update** `w' = w − η·g` — the final brick of an on-device training step. `η` positive learning rate, `g` the (signed) gradient, `w` the (signed) weight. Composes a **signed** multiply `smul` (sign = XOR of signs, magnitude = the verified RNE magnitude mul) with subtract (`sadd`+`neg`). Bit-exact to the integer oracle **500/500** (iverilog); accuracy to GF-T16 precision (≤1 ULP, ~0.03 abs at the largest magnitudes). Spot: `w=1,g=0.5,η=0.5 → 0.75` exact; `g=0 → w` unchanged; `w=1,g=−1,η=1 → 2.0` (ascent) exact.
+- Fresh seal for `GftSgdStep` (`seal --verify` MATCH). No compiler change.
+
+**The full on-device training loop is now expressible spec-first on GF-T:**
+`logits → softmax → prob → NLL loss` (forward) → `gradient p−y` (backward) → `w' = w − η·g` (**update**). Every stage iverilog-verified bit-exact. Combined with the BitNet×GF-T layers/MLP/classifier, GF-T now spans a complete train+infer stack in synthesizable spec-first hardware.
+
+---
+
 # NOW — feat(spec): GF-T softmax+cross-entropy gradient (backward pass) (2026-08-07)
 
 Last updated: 2026-08-07
diff --git a/specs/ternary/gft_sgd_step.t27 b/specs/ternary/gft_sgd_step.t27
new file mode 100644
index 000000000..d851f3632
--- /dev/null
+++ b/specs/ternary/gft_sgd_step.t27
@@ -0,0 +1,121 @@
+module GftSgdStep;
+// #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;
+}
+
+// w' = w - eta*g.
+fn on_comb(w: u32, g: u32, eta: u32) -> u32 {
+    var delta : u32 = smul(eta, g);
+    return sadd(w, neg(delta));
+}
+
+// w=+1.0, g=+0.5, eta=0.5 -> 1 - 0.25 = 0.75 (offset39,mant256 = 0x4f00 = 20224).
+test step { assert_eq(on_comb(20480, 19968, 19968), 20224); }
+// g=0 -> weight unchanged.
+test nograd { assert_eq(on_comb(20480, 0, 19968), 20480); }
+// w=+1.0, g=-1.0, eta=1.0 -> 1 - (1*-1) = 2.0 (offset41 = 0x5200 = 20992).
+test ascend { assert_eq(on_comb(20480, 86016, 20480), 20992); }
+endmodule