The Whole Pipeline

Tokens to Vectors

The input tokens become exact vectors through token and position tables. No sinusoidal position encoding appears in this tiny model.

highlighted = computed this step

Tokens to vectors

The input is a then b, with IDs 0 and 1. Token vectors and position vectors are displayed as exact integer tables.

a↦0,b↦1a\mapsto 0,\quad b\mapsto 1
Tokens to vectorsExact token and position tables feed the pass.Tokens to vectorsExact token and position tables feed the pass.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

Add the position table

For position zero, E[a]+P zero gives x zero = (1,0). For position one, E[b]+P one gives x one = (1,1).

xzero=(1,0),xone=(1,1)x_{\text{zero}}=(1,0),\quad x_{\text{one}}=(1,1)
Tokens to vectorsExact token and position tables feed the pass.Tokens to vectorsExact token and position tables feed the pass.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

Summary

Tokenization and vector lookup are discrete and exact here. The position table is exact, not sinusoidal.

lookup tables are exact\text{lookup tables are exact}
Tokens to vectorsExact token and position tables feed the pass.Tokens to vectorsExact token and position tables feed the pass.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