feat: GF-T 2-input on-chip trainer primitive (proven on AX7203)

docs/NOW.md

@@ -1,7 +1,15 @@
-# NOW — feat: GF-T on-chip trainer primitive (2026-08-07)
+# NOW — feat: GF-T 2-input on-chip trainer (2026-08-07)
 
 Last updated: 2026-08-07
 
+## feat: GF-T 2-input on-chip trainer — proven on AX7203 (Refs #1764)
+
+- **NEW** spec `specs/ternary/gft_train2.t27` — `on_comb(w0,w1,x0,x1,t,eta) -> u64` runs one full on-chip SGD step of a 2-input linear neuron `y=w0*x0+w1*x1`: forward → error `e=y-t` → grads `g_i=e*x_i` → updates `w_i'=w_i-eta*g_i`, returning both updated weights packed `(w0'<<32)|w1'` (reuses `smul/sadd/neg/mag*` from `gft_sgd_step`)
+- `test` block PASS via `icarus-simulate` per L4
+- On-device training: with both weights in registers the whole forward+backward+update runs on-chip; host streams only `(x0,x1,t)`
+- Proven on a live AX7203 (`uart_train2.v`): streaming a hidden 2-weight function `w*=(1.0,0.5)`, the board discovers BOTH — w0 0.25→1.03, w1 0.25→0.48
+- Spec-only; no `gen/`/`coq/` edits; no new `*.sh`; Refs #1764
+
 ## feat: GF-T on-chip trainer primitive — proven on AX7203 (Refs #1764)
 
 - **NEW** spec `specs/ternary/gft_train1.t27` — `on_comb(w,x,t,eta)` runs one full SGD step of a 1-weight linear neuron combinationally: forward `y=w*x` → error `e=y-t` → gradient `g=e*x` → update `w'=w-eta*g` (reuses the verified `smul/sadd/neg/mag*` helpers from `gft_sgd_step`)

specs/ternary/gft_train2.t27

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+module GftTrain2;
+// #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;
+}
+
+// On-chip training step of a 2-input linear neuron y = w0*x0 + w1*x1.
+//   forward y ; error e=y-t ; grads g0=e*x0,g1=e*x1 ; updates w_i'=w_i-eta*g_i.
+// Returns both updated weights packed: (w0' << 32) | w1'. The whole
+// forward+backward+update runs on the FPGA; the host streams only (x0,x1,t).
+fn on_comb(w0: u32, w1: u32, x0: u32, x1: u32, t: u32, eta: u32) -> u64 {
+    var y : u32 = sadd(smul(w0, x0), smul(w1, x1));
+    var e : u32 = sadd(y, neg(t));
+    var g0 : u32 = smul(e, x0);
+    var g1 : u32 = smul(e, x1);
+    var w0n : u32 = sadd(w0, neg(smul(eta, g0)));
+    var w1n : u32 = sadd(w1, neg(smul(eta, g1)));
+    return ((w0n as u64) << 32) | (w1n as u64);
+}
+// w=[.5,.5], x=[1,1], t=1.5, eta=.25 -> y=1.0,e=-.5,g=-.5, w'=[.625,.625]=20096.
+// packed = (20096<<32)|20096 = 86311662800512.
+test learn { assert_eq(on_comb(19968,19968,20480,20480,20736,19456), 86311662800512); }