Figure 6 — Lambda Calculus to CM Hardware Mapping

How Church's lambda calculus concepts map directly to CM hardware elements via the LAMBDA instruction

Lambda Calculus Notation CM Hardware Permission/Domain Cycle Cost Abstraction (λx.body) λx.M "bind x in body M" Golden Token in CRn (X perm) X (Execute) Turing domain 0 cycles GT already loaded Variable (x) x "bound parameter" Data Register DRn (argument) R/W (Read/Write) Turing domain 0 cycles already in register Application (f x) (λx.M) N "apply f to arg" LAMBDA CRn, x the instruction X checked on CRn bridges Church → Turing ~1 cycle machine status only Substitution (M[x:=N]) M[x:=N] "replace x with N" Code Body executes at CR14+x X (in scope) DRn supplies the value N cycles body instruction count Result (value) β-reduction "computed value" Data Register DRn (result) R/W (Read/Write) Turing domain 0 cycles result in register CONCRETE EXAMPLE: (λx. x+1) 5 → 6 Lambda Calculus (λx. x+1) 5 λx = bind parameter x x+1 = body (add 1 to x) 5 = argument (substituted for x) result = 6 (β-reduced value) maps to CM Hardware MOV DR1, #5 ; x = 5 LAMBDA CR2, addOne ; apply λ ; -- body at offset addOne -- ADD DR1, DR1, #1 ; x+1 RETURN ; DR1 = 6 DOMAIN BRIDGE LAMBDA bridges Church (capability) and Turing (computation) domains Church Domain Golden Tokens (GTs) Capabilities & permissions L (Load), S (Save), E (Enter) CR0-CR7 hold GTs LAMBDA CRn, x X permission = bridge ~1 cycle, no stack GT exec Turing Domain Data values & computation Arithmetic, logic, flow R (Read), W (Write), X (Execute) DR0-DR15 hold values