Fixed Points and Local Stability

Slope Less Than One Is a Local Contract

A local stability label must cite the fixed-point source and exact slope. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Stability cites the fixed-point source

The fixed point is reused, then the local slope one half is checked at that same map.

∣f′(x∗)∣=∣12∣=12|f'(x^*)|=\left|\frac{1}{2}\right|=\frac{1}{2}
Local slope contractThe slope must match the checked map and candidate.slope=1/2slopeMagnitude=1/2acceptedBit=1 bit

Less than one is the accepted local contract

Since one half is less than 1, the accepted bit is 1.

12<1⇒1\frac{1}{2}<1\Rightarrow 1
Accepted local stabilityThis is local finite classification, not a convergence proof.slope=1/2slopeMagnitude=1/2acceptedBit=1 bit