Note. This essay is an informal interpretation of the structural results presented in the arXiv paper “Persistence, Memory, and the Structural Impossibility of P = NP.” It does not introduce new technical claims and should not be read as a formal proof.
For over fifty years, the P vs NP problem has resisted resolution. Thousands of papers, dozens of major techniques, and entire subfields of theoretical computer science have tried and failed to settle it. This persistence of failure invites a natural question: why is the problem so hard?
The paper I recently submitted to arXiv does not resolve P vs NP. Instead, it identifies a structural obstruction that applies to a broad class of computational models. What follows is an informal explanation of what that obstruction means, why it may help explain decades of frustration, and what it suggests about the limits of our current ways of thinking.
This essay is not a proof. It is an interpretation.
The hidden assumption behind algorithms
Every algorithm we usually talk about has three quiet features:
- It can be run again.
- Its result can be verified.
- Its behavior can be communicated or reused.
All three require persistence. Something about the computation must survive long enough to be checked, repeated, or built upon.
In everyday programming, this is obvious. In complexity theory, it is usually implicit.
But persistence is not free.
Persistence changes the game
When a system remembers what it has done, that memory constrains what it can do next. Stored information creates structure. Structure introduces irreversibility. Irreversibility accumulates.
The core observation of the paper is simple to state informally:
Any decision procedure that can be reused or verified must accumulate persistent information proportional to the discrimination it performs.
For easy problems, this cost is small. For NP-complete problems, the discrimination required grows explosively.
The result is a structural tension: polynomial-time computation and persistent reuse pull in opposite directions when the problem requires exponential discrimination.
This is not about a particular machine model. It applies equally to Turing machines, RAM models, circuits, proof systems, and any other framework that allows memory to persist and results to be reused.
A fleeting “yes” that cannot survive
One might object: what if a correct decision occurs without persistence?
Indeed, nothing in mathematics forbids a correct decision from occurring as a one-time event, leaving no trace. Such an event could, in principle, “get the answer right” without building any lasting structure.
But this kind of resolution cannot be:
- reused,
- verified,
- composed,
- or communicated.
The moment you try to do any of those things, you reintroduce persistence, and with it, the same obstruction.
In this sense, a “yes” may exist abstractly, but not in a form that can survive contact with verification or reuse. It evaporates as soon as you try to make it part of an algorithm.
Why this is not a proof of P ≠ NP
The arXiv paper is careful about what it does not claim.
It does not prove P ≠ NP.
It does not rule out exotic mathematical models.
It does not assert impossibility in an absolute sense.
What it does is narrower and, arguably, more informative:
It shows that any polynomial-time decision procedure for NP-complete problems must escape the regime of persistent, reusable computation that underlies all known algorithms.
In other words, if P = NP is true, it will not be true in a way that looks like anything we currently know how to build, verify, or reuse.
Why this reframing matters
Decades of work on P vs NP have taught us that many familiar techniques fail: diagonalization, relativization, natural proofs, algebrization. These are often described as technical barriers.
The obstruction described here is different. It is structural rather than technical. It does not say “this proof strategy fails,” but rather:
“Any strategy that relies on persistence and reuse is already constrained in a way that prevents it from scaling to NP-complete discrimination.”
This helps explain not just why particular approaches fail, but why entire families of approaches seem doomed from the outset.
So what should we ask instead?
If this perspective is correct, then the most productive question may no longer be:
“Is P equal to NP?”
but rather:
“What kinds of computation do not rely on persistent, reusable structure, and can they be meaningfully formalized?”
That question sits at the intersection of complexity theory, logic, and the foundations of computation. It is less tidy than P vs NP, but it may be closer to the fault line that has kept the problem unresolved for so long.
Closing thought
The P vs NP problem may not be hard because the answer is subtle.
It may be hard because the form of the answer we are looking for cannot persist in the computational regimes we inhabit.
If that is true, then understanding why we cannot hold onto a “yes” may be more illuminating than continuing to search for one.
Several related intuitions appear in prior work, from Landauer’s thermodynamic limits on information erasure to Aaronson’s arguments about the physical implausibility of exponential search. More recent structural models explicitly describe collapse when systems attempt to maintain identity under unbounded discrimination. The present perspective differs in focus: it isolates persistence itself as the limiting resource, independent of time, space, or physical substrate.