The finale keeps the exact run and boundary visible while stating the boundary of the claim. The mechanism is exact here; broader behavior is outside this rendered surface.
What is exact
The activations, margins, updates, and boundary states are exact integers for the displayed order. The final state is w=(2,-1), b=0.
exact finite update run \text{exact finite update run} exact finite update run Exact perceptron mechanics The table and boundary plot share the same fixed-order data. Exact perceptron run rule: y*a <= 0 updates t i x y old w,b a y*a upd new w,b 0 0 (1,0) +1 (0,0),0 0 0 yes (1,0),1 1 1 (1,1) +1 (1,0),1 2 2 no (1,0),1 2 2 (0,1) -1 (1,0),1 1 -1 yes (1,-1),0 3 3 (-1,0) -1 (1,-1),0 -1 1 no (1,-1),0 4 0 (1,0) +1 (1,-1),0 1 1 no (1,-1),0 5 1 (1,1) +1 (1,-1),0 0 0 yes (2,0),1 6 2 (0,1) -1 (2,0),1 1 -1 yes (2,-1),0 7 3 (-1,0) -1 (2,-1),0 -2 2 no (2,-1),0 8 0 (1,0) +1 (2,-1),0 2 2 no (2,-1),0 9 1 (1,1) +1 (2,-1),0 1 1 no (2,-1),0 10 2 (0,1) -1 (2,-1),0 -1 1 no (2,-1),0 11 3 (-1,0) -1 (2,-1),0 -2 2 no (2,-1),0 final w=(2,-1), b=0; separates shown data Boundary after each update Perceptron boundary evolution Training points with exact boundary lines after each update. 0:+1 1:+1 2:-1 3:-1 updates u1: w=(1,0), b=1 u2: w=(1,-1), b=0 u3: w=(2,0), b=1 u4: w=(2,-1), b=0 final boundary: w=(2,-1), b=0
What this surface stops at
The surface shows a finite run on a small signed dataset. A different order or dataset would need its own displayed recomputation.
shown run only \text{shown run only} shown run only Exact perceptron mechanics The table and boundary plot share the same fixed-order data. Exact perceptron run rule: y*a <= 0 updates t i x y old w,b a y*a upd new w,b 0 0 (1,0) +1 (0,0),0 0 0 yes (1,0),1 1 1 (1,1) +1 (1,0),1 2 2 no (1,0),1 2 2 (0,1) -1 (1,0),1 1 -1 yes (1,-1),0 3 3 (-1,0) -1 (1,-1),0 -1 1 no (1,-1),0 4 0 (1,0) +1 (1,-1),0 1 1 no (1,-1),0 5 1 (1,1) +1 (1,-1),0 0 0 yes (2,0),1 6 2 (0,1) -1 (2,0),1 1 -1 yes (2,-1),0 7 3 (-1,0) -1 (2,-1),0 -2 2 no (2,-1),0 8 0 (1,0) +1 (2,-1),0 2 2 no (2,-1),0 9 1 (1,1) +1 (2,-1),0 1 1 no (2,-1),0 10 2 (0,1) -1 (2,-1),0 -1 1 no (2,-1),0 11 3 (-1,0) -1 (2,-1),0 -2 2 no (2,-1),0 final w=(2,-1), b=0; separates shown data Boundary after each update Perceptron boundary evolution Training points with exact boundary lines after each update. 0:+1 1:+1 2:-1 3:-1 updates u1: w=(1,0), b=1 u2: w=(1,-1), b=0 u3: w=(2,0), b=1 u4: w=(2,-1), b=0 final boundary: w=(2,-1), b=0
Summary
This is not a convergence guarantee, not a claim of generalization to unseen points, and not a statement about any other order; it pins exact update mechanics on this toy run.
toy-data perceptron mechanics only \text{toy-data perceptron mechanics only} toy-data perceptron mechanics only Exact perceptron mechanics The table and boundary plot share the same fixed-order data. Exact perceptron run rule: y*a <= 0 updates t i x y old w,b a y*a upd new w,b 0 0 (1,0) +1 (0,0),0 0 0 yes (1,0),1 1 1 (1,1) +1 (1,0),1 2 2 no (1,0),1 2 2 (0,1) -1 (1,0),1 1 -1 yes (1,-1),0 3 3 (-1,0) -1 (1,-1),0 -1 1 no (1,-1),0 4 0 (1,0) +1 (1,-1),0 1 1 no (1,-1),0 5 1 (1,1) +1 (1,-1),0 0 0 yes (2,0),1 6 2 (0,1) -1 (2,0),1 1 -1 yes (2,-1),0 7 3 (-1,0) -1 (2,-1),0 -2 2 no (2,-1),0 8 0 (1,0) +1 (2,-1),0 2 2 no (2,-1),0 9 1 (1,1) +1 (2,-1),0 1 1 no (2,-1),0 10 2 (0,1) -1 (2,-1),0 -1 1 no (2,-1),0 11 3 (-1,0) -1 (2,-1),0 -2 2 no (2,-1),0 final w=(2,-1), b=0; separates shown data Boundary after each update Perceptron boundary evolution Training points with exact boundary lines after each update. 0:+1 1:+1 2:-1 3:-1 updates u1: w=(1,0), b=1 u2: w=(1,-1), b=0 u3: w=(2,0), b=1 u4: w=(2,-1), b=0 final boundary: w=(2,-1), b=0