All finite systems are to some degree controlled, engineered or governed and there is no way around this.
If we think about a bit (binary digit) as representing a truly random source, it is irrelevant if there should ever be a state of 10^10^130 ‘1’ bits (lots and lots) in a row before a ‘0’ bit. Adding an engineering requirement to any truly randomn source means that it is in fact not truly random.
True chaos is 100% in sync with pure randomness. Control of randomness is based on deliberately engineered decisions (even if decided by default). Theories of uncertainty, statistics, filters, etc. are all necessarily forms of that control (in fact the control can be considered as borrowed).
Stochastic Computing approaches computation using probabilities and controlled randomness to encode numbers as measures of time. SC multiplication is done with a single AND gate (for unipolar SC). Cost and power savings can be achieved by reducing the amount of parallel electronics and multiplexing a single gate (or groups of gates) via the time domain.
The number is encoded as the probability or proportion of ‘1’s in the unordered bitstream. The data is still strictly binary digital but as an encoded serial bitstream. This serial nature means accuracy is achieved progressively over time, defined by the Law of Large Numbers (the Latency-Accuracy Trade-off). The bottleneck in any bitstream is the top speed of the electronics, which can be mitigated via parallelism, with the right networking technology.
The AND gate, being a non-statistical physical device, is incapable of distinguishing between ‘two independent streams’ and ‘two copies of the same stream.’ When inputs are identical, the gate simply gives you the smaller of the two input numbers (or the common number, if they are the same) instead of their product. This instantly results in the wrong computational answer.
This independence requirement is an engineered one (governed, controlled). In this sense, the “independence” required by the gate is a highly dependent and deliberate architectural choice made by the engineer. It is a strict and expensive dependence on the randomness source and decorrelation hardware to fulfill the mathematical requirement of the gate.