Chapter 41
Massive Populations and Communication Degradation
41.1 The Initial Condition
We consider an initial condition consisting of a population \(\pop_0\) of agents in an interactive GSLT \(\GSLT\), with \(|\pop_0| \approx 10^{80}\) — a number comparable to the number of quarks in the observable universe.
The choice of scale is not arbitrary. We want a population large enough that every cluster of agents is itself a large enough system to implement a complex agent. The quark-scale population ensures this at every level of the hierarchy we are about to construct.
41.2 Communication Degradation
Let \(A, B \in \pop\) be two agents and let \(\gamma\) be a rewrite path representing the transmission of a message from \(A\) to \(B\). The fidelity of the transmission is \(|W(\gamma)|^2\), the squared modulus of the path amplitude defined in Part I. A transmission is coherent if its fidelity exceeds a threshold \(\epsilon > 0\).
The communication radius of an agent \(A \in \pop\) is the set \[B_\epsilon(A) = \{ B \in \pop \mid \exists \text{ coherent path } \gamma : A \to B \}\] of agents reachable from \(A\) with fidelity above \(\epsilon\).
For any interactive GSLT \(\GSLT\) with bounded step amplitudes, and for any \(\epsilon > 0\), there exists a scale \(N_\epsilon\) such that for any population \(\pop\) with \(|\pop| > N_\epsilon\): \[\exists A \in \pop \text{ such that } B_\epsilon(A) \subsetneq \pop.\] That is, no single agent can communicate coherently with the entire population.
The intuition is that message fidelity degrades along each transmission hop by a factor bounded away from \(1\). Over a path of length \(n\) the fidelity is at most \((1-\delta)^n\) for some \(\delta > 0\). For a population of diameter \(d\) (minimum path length between the most distant agents), fidelity falls below \(\epsilon\) when \(d > \log(1/\epsilon) / \log(1/(1-\delta))\). In a population of size \(10^{80}\) the diameter is enormous, and no \(\epsilon\)-coherent path can span it. Making this precise requires specifying the communication geometry of the GSLT — a gap noted in Section 45.
One might hope that error-correction protocols could restore global coherence. We argue this hope fails at sufficient scale. Any error-correction protocol is itself a computation in \(\GSLT\), consuming account resources and communicating over the same degraded channels. At population scales beyond \(N_\epsilon\), the overhead of error-correction exceeds the bandwidth of the channels it is trying to protect. The argument is self-referential: error-correction cannot bootstrap coherence it does not already have. This argument is currently informal; making it precise is a priority for future work.