feat: GF-T 2-neuron hidden layer on-chip trainer (proven on AX7203)

docs/NOW.md

@@ -1,7 +1,14 @@
-# NOW — feat: GF-T nonlinear (ReLU) on-chip trainer (2026-08-07)
+# NOW — feat: GF-T 2-neuron hidden layer trainer (2026-08-07)
 
 Last updated: 2026-08-07
 
+## feat: GF-T 2-neuron hidden layer on-chip trainer — proven on AX7203 (Refs #1764)
+
+- **NEW** spec `specs/ternary/gft_hidden2.t27` — on-chip SGD step of a 2-neuron hidden layer `y=relu(w0*x0)+relu(w1*x1)` (fixed unit output weights): two INDEPENDENT nonlinear units, each with its own relu gate; per-neuron gated grad `dw_j=e*relu'(z_j)*x_j`; updates `w_j'=w_j-eta*dw_j`; returns `(w0'<<32)|w1'`. Reuses `smul/sadd/neg/relu/relu_prime/mag*`
+- `test` blocks (both active, one-dead) PASS via `icarus-simulate` per L4
+- Proven on a live AX7203 (`uart_hidden2.v`): the two units gate INDEPENDENTLY — killing one neuron's activation (its input <0) freezes only that neuron's weight while the other keeps learning (MOVED vs FROZEN across 3 phases)
+- Spec-only; no `gen/`/`coq/` edits; no new `*.sh`; Refs #1764
+
 ## feat: GF-T nonlinear (ReLU) on-chip trainer — proven on AX7203 (Refs #1764)
 
 - **NEW** spec `specs/ternary/gft_train2relu.t27` — on-chip SGD step of a NONLINEAR 2-input neuron `y=relu(w0*x0+w1*x1)`: forward with `relu(z)`, error, gradient GATED by `relu'(z)` (`d=e*relu_prime(z)`, `g_i=d*x_i`), updates `w_i'=w_i-eta*g_i`, returns `(w0'<<32)|w1'`. Reuses `smul/sadd/neg/mag*` from `gft_sgd_step`

specs/ternary/gft_hidden2.t27

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+module GftHidden2;
+// #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;
+}
+
+// GF-T ReLU: max(0,z) — zero if z is zero or has the sign bit set, else z.
+fn relu(z: u32) -> u32 {
+    if (z == 0) { return 0; }
+    if (((z >> 16) & 1) == 1) { return 0; }
+    return z;
+}
+// ReLU derivative as a GF-T gate: 1.0 (=20480) if z>0, else 0.
+fn relu_prime(z: u32) -> u32 {
+    if (z == 0) { return 0; }
+    if (((z >> 16) & 1) == 1) { return 0; }
+    return 20480;
+}
+// One on-chip SGD step of a 2-NEURON HIDDEN LAYER with fixed unit output weights:
+//   h0=relu(w0*x0), h1=relu(w1*x1), y=h0+h1 ; error e=y-t ;
+//   per-neuron gated grad dw_j = e*relu'(z_j)*x_j ; update w_j'=w_j-eta*dw_j.
+// Each hidden unit has its OWN relu gate -> one can learn while the other is dead.
+// Returns (w0'<<32)|w1'.
+fn on_comb(w0: u32, w1: u32, x0: u32, x1: u32, t: u32, eta: u32) -> u64 {
+    var z0 : u32 = smul(w0, x0);
+    var z1 : u32 = smul(w1, x1);
+    var y : u32 = sadd(relu(z0), relu(z1));
+    var e : u32 = sadd(y, neg(t));
+    var d0 : u32 = smul(e, relu_prime(z0));
+    var d1 : u32 = smul(e, relu_prime(z1));
+    var g0 : u32 = smul(d0, x0);
+    var g1 : u32 = smul(d1, 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);
+}
+// both active: w=[.5,.5],x=[1,1],t=1.5 -> z0=z1=.5>0, h0=h1=.5, y=1.0, e=-.5,
+//   d_j=-.5, g_j=-.5, w'=[.625,.625]. (placeholder 0 -> PROBE prints truth)
+test both { assert_eq(on_comb(19968,19968,20480,20480,20736,19456), 86311662800512); }
+// neuron1 dead: x1 negative so z1<0 -> h1=0, only w0 learns, w1 frozen.
+test onedead { assert_eq(on_comb(19968,19968,20480,86016,20736,19456), 86861418614272); }