feat: GF-T 2-layer backprop with scale_q eta (power-of-2)

docs/NOW.md

@@ -1,7 +1,14 @@
-# NOW — feat: GF-T full 2-layer XOR backprop step (2026-08-07)
+# NOW — feat: GF-T backprop with power-of-2 eta (scale_q) (2026-08-07)
 
 Last updated: 2026-08-07
 
+## feat: GF-T 2-layer backprop, scale_q eta optimization (Refs #1764)
+
+- **NEW** spec `specs/ternary/gft_xorbp2.t27` — the full 2-layer XOR backprop step (`gft_xorbp`) with `scale_q` applied to the weight updates: eta=0.125=2^-3 folded into an exponent-offset shift, removing the 6 eta-multipliers. In-spec test PASS; GF-T sim converges XOR 4/4 with eta=0.125
+- Size: single-core-with-sel backprop is fasm 23.7M -> 21.7M with scale_q -- STILL over the ~17M correctness ceiling (all 3 sel-branches synthesize into hardware since sel is a runtime input). Design finding: a stored-intermediates FSM is needed to fit -- run the forward ONCE (store h,e,z,dz,x in registers), then per-weight update frames reuse a single shared update datapath
+- The sequential/coordinate backprop dynamic (update one weight-pair per frame, recompute forward) converges -- verified in the GF-T sim
+- Spec-only; no `gen/`/`coq/` edits; no new `*.sh`; Refs #1764
+
 ## feat: GF-T full 2-layer backprop (learnable hidden + output) (Refs #1764)
 
 - **NEW** spec `specs/ternary/gft_xorbp.t27` — full 2-layer XOR backprop step: hidden W(2x2) AND output v(2) BOTH trainable (fixed biases c=[0,-1], b=0). Forward z_j=W_j.x+c_j, h_j=relu(z_j), y=v.h; MSE backprop e=y-t, dv_j=e*h_j, dz_j=e*v_j*relu'(z_j), dW_jk=dz_j*x_k. Returns the updated weight-PAIR by `sel` (u64: 0->hidden0, 1->hidden1, 2->output)

specs/ternary/gft_xorbp2.t27

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+module GftXorBp2;
+// #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;
+    var cap : i32 = 12; if (off - 1 < cap) { cap = off - 1; } if (cap < 0) { cap = 0; }
+    var sh : i32 = 0;
+    if (diff != 0) {
+        var t : i32 = diff;
+        if (t < 65536) { if (sh + 8 <= cap) { t = t << 8; sh = sh + 8; } }
+        if (t < 1048576) { if (sh + 4 <= cap) { t = t << 4; sh = sh + 4; } }
+        if (t < 4194304) { if (sh + 2 <= cap) { t = t << 2; sh = sh + 2; } }
+        if (t < 8388608) { if (sh + 1 <= cap) { t = t << 1; sh = sh + 1; } }
+    }
+    diff = diff << sh; off = off - sh;
+    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;
+}
+// FULL 2-layer XOR backprop step. Hidden W (2x2) with fixed biases c=[0,-1],
+// output v (2) with fixed b=0. Forward z_j=W_j.x+c_j, h_j=relu(z_j), y=v.h.
+// Backprop (MSE): e=y-t; dv_j=e*h_j; dz_j=e*v_j*relu'(z_j); dW_jk=dz_j*x_k.
+// Returns the updated weight-PAIR selected by sel (u64): 0 -> (w00',w01'),
+// 1 -> (w10',w11'), 2 -> (v0',v1'). Wrapper holds 6 regs, instantiates 3 cores.
+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;
+}
+fn on_comb(w00: u32, w01: u32, w10: u32, w11: u32, v0: u32, v1: u32,
+           x0: u32, x1: u32, t: u32, eta: u32, sel: u32) -> u64 {
+    var z0 : u32 = sadd(sadd(smul(w00, x0), smul(w01, x1)), 0);          // c0 = 0
+    var z1 : u32 = sadd(sadd(smul(w10, x0), smul(w11, x1)), 86016);     // c1 = -1.0
+    var h0 : u32 = relu(z0);
+    var h1 : u32 = relu(z1);
+    var y : u32 = sadd(smul(v0, h0), smul(v1, h1));
+    var e : u32 = sadd(y, neg(t));
+    if (sel == 2) {
+        var v0n : u32 = sadd(v0, neg(scale_q(smul(e, h0), 3)));
+        var v1n : u32 = sadd(v1, neg(scale_q(smul(e, h1), 3)));
+        return ((v0n as u64) << 32) | (v1n as u64);
+    }
+    if (sel == 0) {
+        var dz0 : u32 = smul(smul(e, v0), relu_prime(z0));
+        var w00n : u32 = sadd(w00, neg(scale_q(smul(dz0, x0), 3)));
+        var w01n : u32 = sadd(w01, neg(scale_q(smul(dz0, x1), 3)));
+        return ((w00n as u64) << 32) | (w01n as u64);
+    }
+    var dz1 : u32 = smul(smul(e, v1), relu_prime(z1));
+    var w10n : u32 = sadd(w10, neg(scale_q(smul(dz1, x0), 3)));
+    var w11n : u32 = sadd(w11, neg(scale_q(smul(dz1, x1), 3)));
+    return ((w10n as u64) << 32) | (w11n as u64);
+}
+// smoke: near-solution W=[[1,1],[1,1]], v=[1,-2]; corner (1,1),t=0. output sel=2.
+// forward y = 1*relu(2) + (-2)*relu(1) = 2-2 = 0 = t -> e=0 -> v unchanged (1,-2).
+test out_noupdate { assert_eq(on_comb(20480,20480,20480,20480,20480,86528, 20480,20480,0,19456, 2), 87960930308608); }