Consider computation as a thin, linear pipeline: symbol in, symbol out. But in a setting with feedback, branching recursion, and layered state spaces, computation becomes expressive in exactly the way biological systems are: through repeated structure, constraint, and variation, or with one word: algorithmic entropy.

TLDR; In a proof theoretic intuitionist^[1] setting, we hint at the possibility that there are problems that are neither solvable, nor the opposite, more precisely:

Once we make the isomorphism between provability and solvability explicit, certain decision problem-forms become unstable when they quantify over the full class of problems they can themselves represent. Under such conditions, self-reference (diagonalization) inevitably produces an instance whose status no uniform method can determine. The resulting problem is neither solvable nor unsolvable in the ordinary sense; rather, impredicative^[2].

The TSP and the optimal Rubik’s Cube solution set permit injective representations in which each *colored sticker* behaves as a constrained traveler; the cube moves enact globally consistent permutations analogous to TSP transitions under structural constraints: The two problems are predicative.

Consider how a sentence would look that would follow a Gödel-Rosser^[3] type diagonal construction that attacks the totality of a solvability predicate insofar it classifies in polynomial and at least super-polynomial.

What we present is neither an answer, nor a meta-theoretical dissolution, but a limitative fixed point theorem of problem adequacy in the sense of Poincaré, Russel^[4] and others. Our first argument demonstrates plausibility by reformulating a theory of provability to one of solvability. A stronger formal argument through Gödel Numbering over problem solvability can be attained at: https://arxiv.org/abs/2511.14665.

We push Brouwer-Heyting-Kolmogorov (BHK) semantics^[1],

$\begin{matrix} conj & A \land B : a \in A, b \in B disj & A \lor B : 0 + a \in A or 1 + b \in B impl & A \to B : (A \to B) map absurdity & \neg A : A \to ⊥ exist & \exists x A (x) : (x, a) forall & \forall x A (x) : x \mapsto a \end{matrix}$

into a domain where it is seldom invoked, and then notice that the failure of admissibility of some class separation in Heyting Arithmetic corresponds exactly to the same reflective trap^[5] identified by Russel, Gödel, and others.

Consider how in complexity theory, uniform assertions like

$\begin{matrix} \forall L \subseteq Σ^{*}, [L \in N P ⟹ L \in P] . \end{matrix}$

are treated as ordinary set-theore...

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