The finale states the boundary: exact discrete spine, named continuous magnitudes, and no claim beyond mechanics.
What is exact
The discrete spine is exact: token IDs, wiring, residual adds, argmax, and the output token. The output token is a.
exact spine → a \text{exact spine}\rightarrow a exact spine → a Transformer honesty boundary Exact discrete spine with named continuous magnitudes. Transformer honesty boundary Exact discrete spine with named continuous magnitudes. 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 layernorm sqrt boundary this transformer's 2-D layernorms are exact here; in general layernorm's square root is named named 3-vector v=(1,2,3); mean=2 centered=(-1,0,1) var=2/3; std=√(2/3); register=named normalized=(-1,0,1)/√(2/3) degenerate exact 3-vector v=(2,2,2); mean=2 centered=(0,0,0) var=0; std=0; register=exact normalized=zero-variance exact std; normalization not divided
What is named
Softmax and layernorm's square root are the named boundary. The rendered three-vector row shows the square-root case explicitly.
softmax , var = named boundary \operatorname{softmax},\sqrt{\operatorname{var}}=\text{named boundary} softmax , var = named boundary Transformer honesty boundary Exact discrete spine with named continuous magnitudes. Transformer honesty boundary Exact discrete spine with named continuous magnitudes. 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 layernorm sqrt boundary this transformer's 2-D layernorms are exact here; in general layernorm's square root is named named 3-vector v=(1,2,3); mean=2 centered=(-1,0,1) var=2/3; std=√(2/3); register=named normalized=(-1,0,1)/√(2/3) degenerate exact 3-vector v=(2,2,2); mean=2 centered=(0,0,0) var=0; std=0; register=exact normalized=zero-variance exact std; normalization not divided
Summary
The discrete spine is bit-exact: tokens, wiring, residual adds, argmax, and the output token. Softmax and layernorm's square root are the named boundary. This pins the full transformer mechanics by hand on a tiny dim two model; it is not learning, not meaning, and not a real-scale model. Every real transformer runs these same steps, just bigger and in floating point.
full transformer mechanics by hand \text{full transformer mechanics by hand} full transformer mechanics by hand Transformer honesty boundary Exact discrete spine with named continuous magnitudes. Transformer honesty boundary Exact discrete spine with named continuous magnitudes. 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 layernorm sqrt boundary this transformer's 2-D layernorms are exact here; in general layernorm's square root is named named 3-vector v=(1,2,3); mean=2 centered=(-1,0,1) var=2/3; std=√(2/3); register=named normalized=(-1,0,1)/√(2/3) degenerate exact 3-vector v=(2,2,2); mean=2 centered=(0,0,0) var=0; std=0; register=exact normalized=zero-variance exact std; normalization not divided