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A total binary-valued function is diagonally noncomputable if it differs from each defined diagonal value in an effective enumeration of partial computable functions. Nonbinary diagonal outputs are automatically different. No such function is computable: its own program index would contradict the defining condition. Choosing the opposite bit at every defined binary diagonal value proves existence without supplying a computable procedure.

For an effective enumeration $({\phi }_e)$ of unary partial computable functions, admit a length-$n$ string $\sigma$ if it disagrees with every binary value of ${\phi }_e(e)$ observed within $n$ steps for $e<n$. The bounded halting predicate makes membership decidable, and the tests are prefix compatible. Its paths are exactly the binary diagonally noncomputable functions. The path space is nonempty and perfect because infinitely many indices of divergent programs leave arbitrarily late bits free; no path is computable. Dead ends in this decidable presentation are essential.

For a nonempty set $A$ of words, the following are equivalent: $A$ is computably enumerable, $A$ is the domain of a partial computable function, $A$ is the range of a partial computable function, and $A$ is the range of a total computable function. The nonempty hypothesis is needed only for the last formulation, since the range of a total function cannot be empty.

For every partial computable function $f(\mathbf{x})$, there are total primitive recursive functions $C,U$ such thatHere $C$ is zero exactly on codes of valid halting computation histories for the chosen program and input, and $U$ extracts the final output. Coding finite configurations and finite histories makes checking every local step a bounded primitive recursive operation. If a computation halts, some history code passes and every passing code has the same output; if it does not halt, none passes. This is the usual normal-form version with the program index fixed; a universal predicate can also retain that index as an input.

Every partial computable function is represented on Church numerals by a closed untyped lambda calculus term. First represent all primitive recursive functions using zero, successor, projections, composition and lambda definition of primitive recursion by pair iteration. Apply the Kleene normal form theorem and search successive history codes with a fixed-point combinator. Test the represented total predicate using a Church numeral zero test; return the decoded output only on the true branch. If no history passes, normal-order head reduction continues the search indefinitely and the term has no head normal form, so cannot be in beta equivalence with a Church numeral. This avoids the incorrect assumption that a lazy outer function must evaluate a divergent argument in an unguarded composition.