Sequential composition collapses to the constant one

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Composition acts from right to left. The increment and decrement cancel, the identity operations preserve the current value, and f₅ then resets it to one; the following identity and reciprocal leave one unchanged. The final output is one for every ordinary finite real initial input, so the chain does not preserve a seven-stage diversity of values.

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Original context

6.1 Proposition 1: the sequential loop is a constant map

For every finite real initial value in the ordinary interpretation,

f7∘f6∘f5∘f4∘f3∘f2∘f1z=1.

Proof. Addition and subtraction cancel. The next two operators preserve the value. The fifth maps every finite real input to one. Division by one and the reciprocal of one both return one. Therefore the complete composition is constant. No asymptotic argument, critical circulation speed, or activation threshold is needed.

Starting from two, the exact path is

2→3→2→2→2→1→1→1.

\left({f}_{7}∘{f}_{6}∘{f}_{5}∘{f}_{4}∘{f}_{3}∘{f}_{2}∘{f}_{1}\right)\left(z\right)=1.