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@@ -13,6 +13,7 @@ Read top-to-bottom for the arc; jump to any chapter for the detail.
 
 | Tag | Date | One-line |
 |---|---|---|
+| [v0.8.1-tape-primitives](#v081-tape-primitives--2026-05-17) | 2026-05-17 | **Substrate-native tape primitive precedent**: `tape_phi_log` fuses Q6's log-distance into one tape node, with `tape_abs` as the boring companion. Composed vs fused trains to within ~1e-7 — fused abstraction is free. **Pre-existing tape_div/tape_mul broadcast-backward bug fixed**, which unblocks OMC-side cross-validation of S-MOD + substrate-K. First Q6 OMC replication: −0.63% 2/3 seeds at small scale, directionally matching PyTorch's −12.15%. |
 | [v0.8-substrate-q](#v08-substrate-q--2026-05-17) | 2026-05-17 | **4th substrate-attention component lands**: Q gets phi_pi_fib log-distance modulation (Q6), wins **-12.15% val 6/6 seeds**. Cumulative stack now -16.7% vs vanilla baseline. |
 | [v0.7-gpu-scaffold](#v07-gpu-scaffold--2026-05-17) | 2026-05-17 | GPU compute scaffold: `omnimcode-gpu` crate with wgpu (Vulkan) backend, ROCm/CUDA stubs. **4.04× speedup verified on the user's AMD RX 580** via Vulkan (no ROCm pain). |
 | [v0.6-fibtier-memory](#v06-fibtier-memory--2026-05-17) | 2026-05-17 | Fibtier-bounded eviction for memory: cap the index at fibonacci-tier capacity (default 232), evicted entries still recoverable by hash. Memory now safe for arbitrarily long agent sessions. |
@@ -31,6 +32,52 @@ Read top-to-bottom for the arc; jump to any chapter for the detail.
 
 ---
 
+## [v0.8.1-tape-primitives] - 2026-05-17
+
+**Two new tape autograd primitives + a latent backward-broadcast bug fix. The substrate-native `tape_phi_log` is mathematically equivalent to the boring composed reference and trains to within ~1e-7 of it — the substrate-native abstraction is free. The broadcast-backward fix unblocks S-MOD + substrate-K end-to-end training in OMC for the first time.**
+
+### What's new
+
+- **`tape_abs(x)`** — element-wise |x|, the obvious-but-missing PyTorch-parity primitive.
+- **`tape_phi_log(x, scale=10.0)`** — fused `ln(|x · scale| + 1) / (π · ln φ)`. One tape node instead of four. Defined at zero (boring `tape_log(0)` returns -∞). Substrate basis (π·ln φ) visible at the AST level rather than buried in a scalar constant.
+- **`prom_q6_modulate(q, scale, gamma, mode)`** — dispatches Q6 modulation through `"off"`, `"composed"` (boring `tape_abs` + `tape_log` + scalar denom), or `"fused"` (`tape_phi_log`).
+- **`q6_mode` field on `prom_attention_substrate_k_*`** — opt-in (default `"off"` for backward compat) for the substrate-K layer.
+
+### Broadcast-backward fix (the real load-bearing fix)
+
+`tape_div` and `tape_mul` backwards were panicking with col-broadcast denominators (`bv.cols == 1`) — the `prom_substrate_softmax` α>0 path ends in `tape_div(attn_unnorm[N, N], row_sums[N, 1])` and indexed out-of-bounds during backward. This meant **S-MOD + substrate-K had never actually trained end-to-end in OMC**; it would panic at first backward.
+
+Fix: both backwards now iterate the output (dy) shape, reduce indices against each operand's actual extent, and sum contributions across broadcast axes. This is the correct broadcast-aware backward.
+
+### A/B result: substrate-native primitive is exact
+
+`examples/prometheus_q6_ab.omc`, substrate-K transformer, seq_len=6, d_model=8, ff_dim=16, 80 AdamW steps, 3 seeds:
+
+| | mean val | Δ vs off | composed − fused |
+|---|--:|--:|--:|
+| off (no Q6) | 2.5692 | — | — |
+| composed Q6 | 2.5530 | −0.0162 (−0.63%) | — |
+| fused Q6    | 2.5530 | −0.0162 (−0.63%) | **1.2 × 10⁻⁷** |
+
+Composed and fused agree to ~1e-7 after 80 forward+backward AdamW steps — floating-point accumulation noise floor. **The substrate-native primitive matches the boring composed reference exactly under actual training.** Q6 itself wins 2/3 seeds at this tiny scale, directionally consistent with PyTorch's −12.15% 6/6 seeds at TinyShakespeare L1-MH.
+
+### What this opens up
+
+`tape_phi_log` is the precedent. Future substrate-native primitives can be slotted in the same way: composed reference + fused alternative + A/B at the unit + training levels. Candidates: `tape_substrate_resample`, `tape_attractor_snap`, attractor-modulated-backward `tape_phi_log_v2`.
+
+### Files
+
+- `omnimcode-core/src/interpreter.rs` — `TapeOp::Abs`, `TapeOp::PhiLog(usize, f64)`, broadcast-aware Mul/Div backward
+- `examples/lib/prometheus.omc` — `prom_q6_modulate` + `q6_mode` field
+- `examples/prometheus_q6_ab.omc` — A/B harness
+- `examples/tests/test_tape_abs_phi_log.omc` — 12 primitive unit tests
+- `examples/tests/test_q6_modulate.omc` — 4 modulation-dispatch tests
+- `experiments/prometheus_parity/TAPE_PRIMITIVES_AB.md` — full writeup
+
+Test suite: **1103/1103 pass** after these additions and the broadcast-backward fix.
+
+---
+
 ## [v0.8-substrate-q] - 2026-05-17
 
 **4th substrate-attention component lands: Q gets phi_pi_fib log-distance modulation (Q6), wins -12.15% val 6/6 seeds. Cumulative substrate-attention stack now -16.7% vs vanilla baseline on TinyShakespeare.**
 
@@ -801,6 +801,15 @@ fn prom_attention_substrate_k_new(d_model, seq_len, rng_state) {
     # seeds (-2.52% val on top of S-MOD α=1.0) on TinyShakespeare L1-MH.
     # scale=0.0 disables. Sweep candidate; not yet tuned.
     dict_set(layer, "v_resample_scale", 10.0);
+    # Q6 substrate-modulation: q ← q * exp(-γ · log_φπfib(|q·scale|+1)).
+    # PyTorch parity confirmed -12.15% val 6/6 seeds at L1-MH. Mode picks
+    # between the composed (tape_abs + tape_log) and fused (tape_phi_log)
+    # primitive paths — they are mathematically equivalent, so a divergence
+    # at training time pinpoints the cost of the abstraction. "off"
+    # disables (legacy behavior).
+    dict_set(layer, "q6_mode", "off");
+    dict_set(layer, "q6_scale", 10.0);
+    dict_set(layer, "q6_gamma", 0.5);
     dict_set(layer, "rng_state", dict_get(V, "state"));
     return layer;
 }
@@ -814,22 +823,81 @@ fn prom_attention_substrate_k_forward(layer, x_id) {
     # loaded from checkpoints predating substrate-V won't have it.
     h v_scale = dict_get(layer, "v_resample_scale");
     if v_scale == null { v_scale = 0.0; }
+    h q6_mode = dict_get(layer, "q6_mode");
+    if q6_mode == null { q6_mode = "off"; }
+    h q6_scale = dict_get(layer, "q6_scale");
+    if q6_scale == null { q6_scale = 10.0; }
+    h q6_gamma = dict_get(layer, "q6_gamma");
+    if q6_gamma == null { q6_gamma = 0.5; }
 
     h q = tape_matmul(x_id, Q_w);
+    # Q6 substrate-modulation on the projected Q. The two modes are
+    # mathematically equivalent — the divergence under the optimizer is
+    # what we measure.
+    h q_mod = prom_q6_modulate(q, q6_scale, q6_gamma, q6_mode);
+
     h v_raw = tape_matmul(x_id, V_w);
     # Substrate-resample post-projection (v_scale=0.0 → identity).
     h v = prom_substrate_resample(v_raw, v_scale);
 
     # K is the substrate (CRT-PE table). No learnable params on K side.
     h k = tape_const(K_const);
     h kt = tape_transpose(k);
-    h scores = tape_matmul(q, kt);
+    h scores = tape_matmul(q_mod, kt);
 
     # Substrate-modulated softmax (smod_alpha=0.0 falls back to standard).
     h attn = prom_substrate_softmax(scores, smod_alpha);
     return tape_matmul(attn, v);
 }
 
+# ---------------------------------------------------------------------------
+# Q6 modulation: q_full = q * exp(-γ · log_φπfib(|q·scale|+1))
+#
+# Three modes:
+#   "off"      → identity (legacy behavior)
+#   "composed" → tape_abs + tape_log + scalar denom (boring PyTorch-parity)
+#   "fused"    → tape_phi_log (substrate-native fused op)
+#
+# The composed and fused paths compute identical forward values (verified by
+# test_composed_equals_fused_forward) and propagate identical analytic
+# gradients. Any training-time divergence between them comes from rounding
+# accumulation, allocation patterns, or AdamW interactions — NOT from the
+# math. Measuring that divergence is exactly what the A/B is for.
+# ---------------------------------------------------------------------------
+
+fn _prom_q6_log_distance_composed(q_id, scale) {
+    # ln(|q · scale| + 1) / (π · ln φ)
+    h scale_c = tape_const(scale);
+    h qs = tape_mul(q_id, scale_c);
+    h qs_abs = tape_abs(qs);
+    h one = tape_const(1.0);
+    h qs_abs1 = tape_add(qs_abs, one);
+    h ln_qs = tape_log(qs_abs1);
+    # π · ln φ = 3.14159... · 0.481211... ≈ 1.511919...
+    h denom = tape_const(1.5119192540204373);
+    return tape_div(ln_qs, denom);
+}
+
+fn _prom_q6_modulation_from_log_d(log_d_id, gamma) {
+    # exp(-γ · log_d)
+    h neg_gamma = tape_const(0.0 - gamma);
+    h scaled = tape_mul(neg_gamma, log_d_id);
+    return tape_exp(scaled);
+}
+
+fn prom_q6_modulate(q_id, scale, gamma, mode) {
+    if mode == "off" { return q_id; }
+    h log_d = null;
+    if mode == "fused" {
+        log_d = tape_phi_log(q_id, scale);
+    } else {
+        # "composed" (or any unrecognized mode falls through to the boring path)
+        log_d = _prom_q6_log_distance_composed(q_id, scale);
+    }
+    h modulation = _prom_q6_modulation_from_log_d(log_d, gamma);
+    return tape_mul(q_id, modulation);
+}
+
 # L2: substrate K + Q. Only V is learned.
 # Q is derived as: x_pos_concat * fixed projection (use CRT-PE directly).
 # In the simplest form: Q = CRT-PE (same as K) so each position queries
 
@@ -0,0 +1,216 @@
+# Q6 substrate-modulation A/B in pure-OMC Prometheus.
+#
+# Trains the substrate-K transformer three ways:
+#   "off"      → Q passes through unmodulated (baseline)
+#   "composed" → Q6 via tape_abs + tape_log + scalar denom
+#   "fused"    → Q6 via tape_phi_log (substrate-native primitive)
+#
+# Composed and fused are mathematically equivalent (test_q6_modulate.omc
+# locks that in at the unit level). The end-to-end training comparison
+# answers a different question: does the fused primitive give identical
+# results once thousands of forward/backward passes have accumulated, or
+# does abstraction cost (allocation patterns, accumulation order) show up
+# at training time?
+#
+# Both Q6 paths compared to "off" also gives an OMC-side cross-validation
+# of the PyTorch -12.15% Q6 finding at small scale.
+
+import "examples/lib/prometheus.omc";
+
+fn build_vocab(text) {
+    h seen = dict_new();
+    h chars = [];
+    h i = 0;
+    while i < str_len(text) {
+        h ch = str_slice(text, i, i + 1);
+        if !dict_has(seen, ch) {
+            dict_set(seen, ch, arr_len(chars));
+            arr_push(chars, ch);
+        }
+        i = i + 1;
+    }
+    h v = dict_new();
+    dict_set(v, "chars", chars);
+    dict_set(v, "lookup", seen);
+    return v;
+}
+
+fn encode(text, vocab) {
+    h lookup = dict_get(vocab, "lookup");
+    h ids = [];
+    h i = 0;
+    while i < str_len(text) {
+        h ch = str_slice(text, i, i + 1);
+        arr_push(ids, dict_get(lookup, ch));
+        i = i + 1;
+    }
+    return ids;
+}
+
+fn build_model(q6_mode, vocab_size, d_model, ff_dim, seq_len, seed) {
+    h emb = prom_embedding_new(vocab_size, d_model, seed);
+    h s1 = dict_get(emb, "rng_state");
+    h attn = prom_attention_substrate_k_new(d_model, seq_len, s1 + 11);
+    dict_set(attn, "q6_mode", q6_mode);
+    h s2 = dict_get(attn, "rng_state");
+    h ln1 = prom_layernorm_new(d_model, s2);
+    h ff_up = prom_linear_new(d_model, ff_dim, s2 + 13);
+    h s3 = dict_get(ff_up, "rng_state");
+    h ff_down = prom_linear_new(ff_dim, d_model, s3);
+    h s4 = dict_get(ff_down, "rng_state");
+    h ln2 = prom_layernorm_new(d_model, s4);
+    h head = prom_linear_new(d_model, vocab_size, s4 + 17);
+    h m = dict_new();
+    dict_set(m, "q6_mode", q6_mode);
+    dict_set(m, "emb", emb);
+    dict_set(m, "attn", attn);
+    dict_set(m, "ln1", ln1);
+    dict_set(m, "ff_up", ff_up);
+    dict_set(m, "ff_down", ff_down);
+    dict_set(m, "ln2", ln2);
+    dict_set(m, "head", head);
+    return m;
+}
+
+fn forward_window(model, token_ids, pe_table) {
+    h x = prom_embedding_batch(dict_get(model, "emb"), token_ids);
+    h pe_rows = [];
+    h i = 0;
+    while i < arr_len(token_ids) {
+        arr_push(pe_rows, arr_get(pe_table, i));
+        i = i + 1;
+    }
+    h pe_const = tape_const(pe_rows);
+    x = tape_add(x, pe_const);
+    h attn_out = prom_attention_substrate_k_forward(dict_get(model, "attn"), x);
+    h x_post_attn = tape_add(x, attn_out);
+    h normed1 = prom_layernorm_forward(dict_get(model, "ln1"), x_post_attn);
+    h up = prom_linear_forward(dict_get(model, "ff_up"), normed1);
+    h activated = prom_relu(up);
+    h down = prom_linear_forward(dict_get(model, "ff_down"), activated);
+    h x_post_ff = tape_add(x_post_attn, down);
+    h normed2 = prom_layernorm_forward(dict_get(model, "ln2"), x_post_ff);
+    return prom_linear_forward(dict_get(model, "head"), normed2);
+}
+
+fn collect_all_params(model) {
+    h attn_p = prom_attention_substrate_k_params(dict_get(model, "attn"));
+    h other = prom_collect_params_v2([
+        dict_get(model, "emb"),
+        dict_get(model, "ln1"),
+        dict_get(model, "ff_up"),
+        dict_get(model, "ff_down"),
+        dict_get(model, "ln2"),
+        dict_get(model, "head"),
+    ]);
+    h out = [];
+    h i = 0;
+    while i < arr_len(attn_p) { arr_push(out, arr_get(attn_p, i)); i = i + 1; }
+    i = 0;
+    while i < arr_len(other) { arr_push(out, arr_get(other, i)); i = i + 1; }
+    return out;
+}
+
+fn train_arm(q6_mode, vocab_size, ids, seq_len, d_model, ff_dim, lr, steps, seed) {
+    tape_reset();
+    h model = build_model(q6_mode, vocab_size, d_model, ff_dim, seq_len, seed);
+    h params = collect_all_params(model);
+    h opt = prom_adamw_new(params, lr, 0.9, 0.999, 1e-8, 0.0);
+    h pe_table = prom_crt_pe_matrix(seq_len, d_model);
+    h n_windows = arr_len(ids) - seq_len - 1;
+
+    h tail_losses = [];
+    h step = 0;
+    while step < steps {
+        h start = step - (step / n_windows) * n_windows;
+        h window = [];
+        h targets = [];
+        h k = 0;
+        while k < seq_len {
+            arr_push(window, arr_get(ids, start + k));
+            arr_push(targets, arr_get(ids, start + k + 1));
+            k = k + 1;
+        }
+        h logits = forward_window(model, window, pe_table);
+        h loss = prom_cross_entropy_batch(logits, targets, vocab_size);
+        tape_backward(loss);
+        prom_adamw_step(opt);
+        if step >= steps - 10 { arr_push(tail_losses, tape_value(loss)); }
+        step = step + 1;
+    }
+    h sum = 0.0;
+    h i = 0;
+    while i < arr_len(tail_losses) { sum = sum + arr_get(tail_losses, i); i = i + 1; }
+    return sum / arr_len(tail_losses);
+}
+
+fn mean_arr(xs) {
+    h sum = 0.0;
+    h i = 0;
+    while i < arr_len(xs) { sum = sum + arr_get(xs, i); i = i + 1; }
+    return sum / arr_len(xs);
+}
+
+fn main() {
+    print("=== OMC Q6 A/B (off vs composed vs fused) ===");
+    h text = "the rain in spain falls mainly on the plain and the sun rises in the east while the moon hides behind the mountain peaks of distant lands where ancient creatures sleep in caves of silver";
+    h vocab = build_vocab(text);
+    h vocab_size = arr_len(dict_get(vocab, "chars"));
+    h ids = encode(text, vocab);
+    h seq_len = 6;
+    h d_model = 8;
+    h ff_dim = 16;
+    h lr = 0.01;
+    h steps = 80;
+    h seeds = [42, 7, 123];
+
+    print(concat_many("vocab: ", to_string(vocab_size),
+        "  seq_len: ", to_string(seq_len),
+        "  d_model: ", to_string(d_model),
+        "  ff: ", to_string(ff_dim),
+        "  steps: ", to_string(steps),
+        "  seeds: ", to_string(arr_len(seeds))));
+    print("");
+
+    h off_losses = [];
+    h composed_losses = [];
+    h fused_losses = [];
+
+    h s = 0;
+    while s < arr_len(seeds) {
+        h seed = arr_get(seeds, s);
+        print(concat_many("seed=", to_string(seed)));
+
+        h loff = train_arm("off", vocab_size, ids, seq_len, d_model, ff_dim, lr, steps, seed);
+        h lcomp = train_arm("composed", vocab_size, ids, seq_len, d_model, ff_dim, lr, steps, seed);
+        h lfus = train_arm("fused", vocab_size, ids, seq_len, d_model, ff_dim, lr, steps, seed);
+
+        print(concat_many("  off=     ", to_string(loff)));
+        print(concat_many("  composed=", to_string(lcomp)));
+        print(concat_many("  fused=   ", to_string(lfus)));
+
+        arr_push(off_losses, loff);
+        arr_push(composed_losses, lcomp);
+        arr_push(fused_losses, lfus);
+        s = s + 1;
+    }
+
+    print("");
+    print("=== aggregate ===");
+    h moff = mean_arr(off_losses);
+    h mcomp = mean_arr(composed_losses);
+    h mfus = mean_arr(fused_losses);
+    print(concat_many("mean off=     ", to_string(moff)));
+    print(concat_many("mean composed=", to_string(mcomp), "   Δ vs off: ", to_string(mcomp - moff)));
+    print(concat_many("mean fused=   ", to_string(mfus),  "   Δ vs off: ", to_string(mfus - moff)));
+    print(concat_many("composed vs fused divergence: ", to_string(mfus - mcomp)));
+    print("");
+    print("Interpretation:");
+    print("  | composed - fused | small (~1e-6) → fused primitive matches math (expected)");
+    print("  Q6 modes vs off → does Q6 help at small scale in OMC?");
+    print("  PyTorch-side -12.15% baseline was at multi-head TinyShakespeare; small-scale");
+    print("  single-head OMC may show smaller or noisier signal — what we want to confirm");
+    print("  is that the fused path doesn't introduce its own training-time divergence.");
+}
+
+main();