From Cognitive Discovery to an Executable World: Mathematical Construction—State-Machine Intermediary Layer

Fills in the intermediary layer missing from the original “P/NP Dynamic Rate” series: once an agent has “understood” something, how does that understanding actually become a thing that can run repeatedly? This paper splits that path into six stages — cognitive discovery, formal specification, mathematical construction, substrate encoding and implementation, state transition and execution, and verification and stabilization — and expands the original three-stage time model (search/execution/verification) into six time terms. Core reminder: “cognitive collapse” does not automatically equal algorithm completion; any argument claiming that “the search has been eliminated” must account for whether the eliminated cost has actually vanished, been amortized, or simply been shifted into representation, construction, precision, memory, or substrate. This paper's role is to complete a bridge, not to cross the river ahead of schedule — it does not claim to have solved P/NP, but rather supplies a shared resource-accounting language for the 24 rounds of dual-hypothesis rehearsal that follow.