The email arrived. The documentation page loaded. The Slack message appeared. The lecture's audio was clear. The bits traversed the channel and arrived intact. Shannon's theory guarantees this — given sufficient encoding, any message can be transmitted reliably through a noisy channel. Modern channels approach this guarantee routinely. Messages arrive. And yet communication fails. A junior developer reads the senior architect's design document three times and cannot connect the caching strategy to anything they know. A patient hears the physician's explanation of their diagnosis and walks away confused. A student reads the textbook paragraph about eigenvalues and finds it impenetrable despite every word being spelled correctly. A new hire reads the onboarding wiki and finds it simultaneously too detailed in some areas and too sparse in others. The channel worked. The bits arrived. The failure lives at the endpoints — in the cost of transforming thoughts into symbols at the sender, and especially in the cost of transforming symbols into understanding at the receiver. These costs exist outside Shannon's framework. His 1948 formalization drew a clean boundary: source produces symbols, encoder transforms them for the channel, channel transmits with noise, decoder recovers symbols, destination receives. He formalized the middle three stages with full mathematical rigor. What the source does before producing symbols and what the destination does after receiving them, he explicitly excluded. This paper formalizes the excluded territory — the endpoint costs — and shows that for most real-world communication, they dominate the channel cost Shannon formalized. The channel is not where communication fails. The endpoints are. Optimizing the channel is necessary but insufficient. Optimizing the endpoints requires reasoning about what the sender and receiver have dissolved and what they haven't, which requires a theory of processing that Shannon declared out of scope. The vocabulary builds in order. Processing is what any system does when it must act on information — a CPU executing instructions, a physician diagnosing, a developer debugging, a student learning. The unit of processing is the **op**: one irreducible transformation by one processor. Processing entropy is the op count a specific processor requires for a specific task — it is receiver-dependent, unlike Shannon's entropy which is a property of the source regardless of who receives the message. When processing entropy reaches zero through repeated engagement under consistent conditions, the task is **dissolved**: handled structurally without consuming the processor's scarce sequential pipeline. The processor's capacity is bounded by one inequality: total ops multiplied by average op duration must not exceed the available time budget.
Geoffrey Howland (Mon,) studied this question.