Chapter 58: The τ

Chapter 54 defined the E₂ computational agent abstractly: a code C paired with a decoder D such that D(C) produces another code C’, and the cycle stays within E₂ . This chapter builds the concrete machine model that instantiates the abstract specification. The τ-Tower Machine (TTM) is a register machine whose instruction set is derived from the five generators {α, π, γ, η, ω} of Category τ. Its registers hold τ-addresses—elements of the NF tower—not binary strings. Three architectural features distinguish it from classical models: (i) the number of registers is fixed (O(1)), but each register carries an unbounded τ-address; (ii) a TTM program is itself a τ-address, so programs, data, and decoders are objects of the same type; (iii) the transition function reads a bounded observation window of constant width K_M, yielding a Cook–Levin tableau with constant row width. These three features—multiplicity bounded, self-referential, constant-width observable—are the structural prerequisites for the collapse established in Chapter 56.