This capstone traces a complete tiny transformer forward pass. It shows the entire exact-or-named pipeline before focusing on each stage and the boundary where named magnitudes appear.
The whole transformer, by hand
This book traces a complete tiny transformer forward pass and the next token by exact arithmetic where possible. The configuration is vocab a,b,c, dim 2, one head, one block, and tied unembed. The point is to thread the pieces together and mark exactly where the named boundaries appear.
tiny transformer: dim 2 \text{tiny transformer: dim }2 tiny transformer: dim 2 Tiny transformer forward Exact spine with named boundaries. Tiny transformer forward Exact spine with named boundaries. tiny transformer exact-or-named forward discrete spine exact; softmax/layernorm sqrt become named only at the boundary input: a b; E[a]=(1,0), E[b]=(0,1), E[c]=(1,1); P0=(0,0), P1=(1,0) weights: Wq=Wk=Wv=I; MLP=I+ReLU+I; gamma=(1,1), beta=(0,0); unembed tied to E position 0: fully exact path x0=(1,0); Q0=K0=V0=(1,0) score S00=1; softmax=[1] exact attn0=(1,0); residual1=(2,0) ln1 mean=1; centered=(1,-1); var=1; std=1 ln1 output=(1,-1) MLP ReLU=(1,0); mlp=(1,0) residual2=(2,-1) ln2 mean=1/2; centered=(3/2,-3/2); var=9/4; std=3/2 ln2 output=(1,-1) logits: a=1, b=-1, c=0 argmax=a; output token=a position 1: named softmax boundary x1=(1,1); Q1=(1,1) K0=(1,0); K1=(1,1) scores=[1, 2] softmax=[e^1/(e^1+e^2), e^2/(e^1+e^2)] named after multi-entry softmax ordered pipeline tokens -> embed -> +pos -> attention -> +residual -> layernorm -> MLP -> +residual -> layernorm -> unembed -> logits -> argmax -> output token pos0 remains exact; pos1 stops at named softmax
Ordered pipeline
The ordered flow is tokens, embeddings, position table, attention, residual add, layernorm, MLP, residual add, layernorm, unembed, logits, argmax. That order is the discrete spine of the pass: each stage consumes the previous stage's displayed output.
tokens → logits → argmax \text{tokens}\rightarrow\text{logits}\rightarrow\operatorname{argmax} tokens → logits → argmax Tiny transformer forward Exact spine with named boundaries. Tiny transformer forward Exact spine with named boundaries. tiny transformer exact-or-named forward discrete spine exact; softmax/layernorm sqrt become named only at the boundary input: a b; E[a]=(1,0), E[b]=(0,1), E[c]=(1,1); P0=(0,0), P1=(1,0) weights: Wq=Wk=Wv=I; MLP=I+ReLU+I; gamma=(1,1), beta=(0,0); unembed tied to E position 0: fully exact path x0=(1,0); Q0=K0=V0=(1,0) score S00=1; softmax=[1] exact attn0=(1,0); residual1=(2,0) ln1 mean=1; centered=(1,-1); var=1; std=1 ln1 output=(1,-1) MLP ReLU=(1,0); mlp=(1,0) residual2=(2,-1) ln2 mean=1/2; centered=(3/2,-3/2); var=9/4; std=3/2 ln2 output=(1,-1) logits: a=1, b=-1, c=0 argmax=a; output token=a position 1: named softmax boundary x1=(1,1); Q1=(1,1) K0=(1,0); K1=(1,1) scores=[1, 2] softmax=[e^1/(e^1+e^2), e^2/(e^1+e^2)] named after multi-entry softmax ordered pipeline tokens -> embed -> +pos -> attention -> +residual -> layernorm -> MLP -> +residual -> layernorm -> unembed -> logits -> argmax -> output token pos0 remains exact; pos1 stops at named softmax
Summary
The capstone combines the exact forward pass, exact ReLU block, named softmax boundary, named layernorm square-root boundary, and greedy output token. It is the whole pipeline in miniature, not a real-scale model.
exact spine plus named boundaries \text{exact spine plus named boundaries} exact spine plus named boundaries Tiny transformer forward Exact spine with named boundaries. Tiny transformer forward Exact spine with named boundaries. tiny transformer exact-or-named forward discrete spine exact; softmax/layernorm sqrt become named only at the boundary input: a b; E[a]=(1,0), E[b]=(0,1), E[c]=(1,1); P0=(0,0), P1=(1,0) weights: Wq=Wk=Wv=I; MLP=I+ReLU+I; gamma=(1,1), beta=(0,0); unembed tied to E position 0: fully exact path x0=(1,0); Q0=K0=V0=(1,0) score S00=1; softmax=[1] exact attn0=(1,0); residual1=(2,0) ln1 mean=1; centered=(1,-1); var=1; std=1 ln1 output=(1,-1) MLP ReLU=(1,0); mlp=(1,0) residual2=(2,-1) ln2 mean=1/2; centered=(3/2,-3/2); var=9/4; std=3/2 ln2 output=(1,-1) logits: a=1, b=-1, c=0 argmax=a; output token=a position 1: named softmax boundary x1=(1,1); Q1=(1,1) K0=(1,0); K1=(1,1) scores=[1, 2] softmax=[e^1/(e^1+e^2), e^2/(e^1+e^2)] named after multi-entry softmax ordered pipeline tokens -> embed -> +pos -> attention -> +residual -> layernorm -> MLP -> +residual -> layernorm -> unembed -> logits -> argmax -> output token pos0 remains exact; pos1 stops at named softmax