← Church Machine

Figure 20 — HP-35 and SlideRule Opcode Distribution

Both HP-35 and SlideRule implementations use only pure Church-domain opcodes: LOAD, SAVE, CALL, RETURN, LAMBDA, TPERM. Zero occurrences of any Turing-domain opcode (ADD, SUB, MUL, MOV, CMP, BRANCH). The Church Computer is computationally complete without arithmetic hardware.

HP-35 Calculator Implementation SlideRule Implementation 0 10 20 30 40 30 LOAD 25 SAVE 20 CALL 20 RETURN 15 LAMBDA 10 TPERM 25 LOAD 20 SAVE 15 CALL 15 RETURN 10 LAMBDA 5 TPERM Turing-Domain Opcodes (NOT PRESENT in either implementation) ADD: 0 not present SUB: 0 not present MUL: 0 not present MOV: 0 not present CMP: 0 not present BRANCH: 0 not present STORE: 0 not present KEY INSIGHT Both HP-35 and SlideRule are complete, working programs that perform real mathematical computation. Neither uses any arithmetic, memory, or branch instruction. All computation is via Church-encoded lambda calculus. This proves the Pure Church Computer is computationally complete without any Turing-domain hardware.