Accumulator approaches one from below

Read the equation

With zero initial state, retention 0.9, and input 0.1, the exact value is 1−0.9ⁿ. It remains below one at every finite nonnegative n and converges to one. A finite threshold crossing depends on the chosen threshold, even though exact equality is never reached in ideal arithmetic.

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Original context

solving the geometric series gives

An=q1−ρn1−ρ.

By contrast, for an imposed nonleaky rule An+1=An+0.1, threshold two is reached at step 20. That result follows from the nonleakage assumption. It does not establish that all physical or computational loops implement a nonleaky accumulator.

Even positive increments need not produce unbounded accumulation: summable increments such as 2−n have a finite total. A valid crossing theorem needs a condition on the cumulative sum, not just an assertion that data keeps moving.

7.3 Speed can alter timing without creating a new capability

{A}_{n}=1-{0.9}^{n}<1