The Whole Pipeline

Attention, Exactly — and the Boundary

Attention is exact until a multi-entry softmax appears. The one-entry row pins to one; the two-entry row is named.

highlighted = computed this step

Attention, exactly

The Q, K, and V maps are identity matrices, so the displayed vectors pass through unchanged. Position zero can attend only to itself.

Q=K=V=IQ=K=V=I
Attention in the full passPosition zero exact; position one named.Attention in the full passPosition zero exact; position one named.tiny transformer exact-or-named forwarddiscrete spine exact; softmax/layernorm sqrt become named only at the boundaryinput: 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 Eposition 0: fully exact pathx0=(1,0); Q0=K0=V0=(1,0)score S00=1; softmax=[1] exactattn0=(1,0); residual1=(2,0)ln1 mean=1; centered=(1,-1); var=1; std=1ln1 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=3/2ln2 output=(1,-1)logits: a=1, b=-1, c=0argmax=a; output token=aposition 1: named softmax boundaryx1=(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 softmaxordered pipelinetokens -> embed -> +pos -> attention -> +residual-> layernorm -> MLP -> +residual -> layernorm-> unembed -> logits -> argmax -> output tokenpos0 remains exact; pos1 stops at named softmax

One-entry softmax

Position zero has score 1 and a one-entry softmax, so the attention weight is exactly 1. Therefore attn zero is (1,0).

Szero,zero=1,softmax⁡=[1]S_{\text{zero,zero}}=1,\quad \operatorname{softmax}=[1]
Attention in the full passPosition zero exact; position one named.Attention in the full passPosition zero exact; position one named.tiny transformer exact-or-named forwarddiscrete spine exact; softmax/layernorm sqrt become named only at the boundaryinput: 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 Eposition 0: fully exact pathx0=(1,0); Q0=K0=V0=(1,0)score S00=1; softmax=[1] exactattn0=(1,0); residual1=(2,0)ln1 mean=1; centered=(1,-1); var=1; std=1ln1 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=3/2ln2 output=(1,-1)logits: a=1, b=-1, c=0argmax=a; output token=aposition 1: named softmax boundaryx1=(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 softmaxordered pipelinetokens -> embed -> +pos -> attention -> +residual-> layernorm -> MLP -> +residual -> layernorm-> unembed -> logits -> argmax -> output tokenpos0 remains exact; pos1 stops at named softmax

The named boundary

Position one has two unmasked scores, 1 and 2. Its softmax is named with exponentials, not decimalized.

softmax⁡([1,2])=named\operatorname{softmax}([1,2])=\text{named}
Attention in the full passPosition zero exact; position one named.Attention in the full passPosition zero exact; position one named.tiny transformer exact-or-named forwarddiscrete spine exact; softmax/layernorm sqrt become named only at the boundaryinput: 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 Eposition 0: fully exact pathx0=(1,0); Q0=K0=V0=(1,0)score S00=1; softmax=[1] exactattn0=(1,0); residual1=(2,0)ln1 mean=1; centered=(1,-1); var=1; std=1ln1 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=3/2ln2 output=(1,-1)logits: a=1, b=-1, c=0argmax=a; output token=aposition 1: named softmax boundaryx1=(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 softmaxordered pipelinetokens -> embed -> +pos -> attention -> +residual-> layernorm -> MLP -> +residual -> layernorm-> unembed -> logits -> argmax -> output tokenpos0 remains exact; pos1 stops at named softmax