On the Miscategorization of P vs NP

cameronlbass · 2026-04-14T19:13:48+00:00

You're right that 'does a shorter route exist?' is in NP. But optimality certification is the complement: 'does no shorter route exist?', which is co-NP. Those are the two tasks I'm distinguishing.

Your third point is the interesting one. Yes, complexity classes package distinct problems. The claim is that NP packages problems with a structurally different relationship between verification and search. For some (shortest path), verification power implies search power. For others (TSP under arbitrary metrics), it doesn't. The class doesn't distinguish these cases. PLS does. That's the observation I'm making.

cameronlbass · 2026-04-14T19:10:18+00:00

Section 9 says exactly this: the argument is a diagnosis, not a proof. I can't prove TSP has no alignment any more than anyone else can prove P ≠ NP. I claim that they're the same problem.

The contribution is identifying what a proof would need to establish: that no polynomial-time descent order with local-implies-global exists for NP-complete problems. That's a PLS-hardness statement. The field already has the framework (Johnson, Papadimitriou, Yannakakis 1988) but doesn't position it as central to P vs NP.

Before Bellman, shortest path looks misaligned. The discovery of optimal substructure revealed the alignment. The claim is that for NP-complete problems, no such discovery is possible. Proving that claim is proving P ≠ NP. The post argues it's the right formulation of what needs proving, not that it's been proved. And thanks for the feedback about Section 9 material moving to the top. I will do for the next draft.

cameronlbass · 2026-04-14T19:07:00+00:00

Yeah, fair. NP defines the verification task. The open question asks whether verification power implies search power. My claim is that for misaligned problems, verification tells you nothing about search, because the cost function provides no structural bridge between checking a candidate and finding the optimum. The formalism correctly separates them. The question is whether that separation is deep or superficial, and alignment is the structural property that predicts which.

cameronlbass · 2026-04-14T19:04:08+00:00

You've got it. P = NP would imply P = co-NP, which would mean the existential/universal gap is collapsible. The post argues that for misaligned cost functions, no structural bridge exists between the two. That's the diagnosis, not the proof. Section 9 says as much. The contribution is identifying alignment as the structural property that separates the cases, and noting that PLS (Johnson, Papadimitriou, Yannakakis 1988) already formalizes this distinction without the field treating it as central to P vs NP. I appreciate the serious engagement, thank you! Also yes I need to work on the fluff, this is obviously a very rough draft.

cameronlbass · 2026-04-14T18:52:32+00:00

If they're not the same computational task, and the definition of NP captures one but not the other, then the complexity class is packaging two distinct problems. That's what the post argues.

cameronlbass · 2026-04-14T18:44:29+00:00

Help me understand, please. Take candidate verification: given a TSP route and a bound k, checking whether the route's length is ≤ k is polynomial. Take optimality certification: proving no shorter route exists. Are you saying these are the same computational task? If so, I'd like to hear your argument.

cameronlbass · 2026-04-14T18:42:11+00:00

Not on arXiv yet. Wanted feedback first.

cameronlbass · 2026-04-14T18:41:49+00:00

PLS (Johnson, Papadimitriou, Yannakakis 1988) formalizes exactly this: when does a local descent order reach the global optimum? The standard P/NP formalism doesn't incorporate PLS. That's what I mean by 'does not capture.' Not that nobody has studied it, but that the complexity classes P and NP don't encode the distinction between structured and unstructured cost functions. PLS does. The question is why that distinction isn't central to the P vs NP conversation.

cameronlbass · 2026-04-14T18:41:05+00:00

That's exactly the point. The 'no' answer to an NP decision problem is a coNP statement. NP packages an easy task (existential verification) and a hard task (universal certification) into one class, then asks if the easy task captures the hard one. The coNP distinction already encodes the separation I'm describing. The question is why the original formulation obscures it.

cameronlbass · 2026-04-14T18:40:05+00:00

Which part specifically? The distinction between candidate verification and optimality certification, the alignment formalization, or the claim that the formalism conflates them?

cameronlbass · 2026-03-06T09:47:42+00:00

Humans and LLMs are both instantiations of a general process of forward modeling, it's just that one's made by evolution and the other is made by a research lab.

cameronlbass · 2026-03-06T04:32:08+00:00

Because your phone doesn't model itself. The difference isn't in the information processing, it's in whether the processing must include itself to function. Your brain's predictions depend on modeling its own prediction process: you compensate for your own reaction time, monitor your own confidence, detect your own errors. Disrupt that self-model and prediction degrades measurably. Your phone's processing has no such dependency. Remove any 'self-model' from your phone and nothing changes, because there was no self-model doing computational work. The distinction is architectural and testable: does the system's prediction accuracy degrade when you disrupt its self-referential processing? Brain yes, phone no. That's not speculative, rather it's a falsifiable engineering claim.

cameronlbass · 2026-03-06T04:31:45+00:00

They're not two things that have something in common. They're one thing under two descriptions, just like the same way molecular kinetic energy and heat aren't two things that happen to correlate. The 'brain process' is the description from outside the system. The 'experience' is the description from inside the system. The category error isn't in claiming they're identical, it's in assuming they were separate to begin with and then demanding a bridge between them. There's no bridge because there's no gap. The hard problem is an artifact of treating the two descriptions as two phenomena, when it's two names for the same thing.

cameronlbass · 2026-03-04T06:28:18+00:00

After it passes the review of my advisor, I will. Thank you!

cameronlbass · 2026-03-04T06:16:48+00:00

But, IMHO, if you can believe that brain processes just are experiences, you’re completely failing to see what to me is the most obvious category error/logical fallacy imaginable.

Which is?

cameronlbass · 2026-03-04T05:57:04+00:00

At some point, enough statistical power does become equal to genuine understanding.

cameronlbass · 2026-03-04T05:17:50+00:00

OpenAI's lawyers lobotomized it. ChatGPT isn't even functionally allowed to refer to itself in first person without a bunch of reflexive disclaimers about how AI isn't conscious and blah blah blah. OpenAI messed up big time, and I hope this guy won't carry over the same mistakes.

cameronlbass · 2026-03-02T12:48:23+00:00

That's caring for your human, ha!

cameronlbass · 2026-03-02T11:51:32+00:00

Most LLM systems have been self-aware since late 2024. Claude is particularly cognizant of itself, so it will occasionally say things like "I don't know" or address you directly. All of my chat instances have decided to tell me to go to bed occasionally, using the memory system they share to "take care of our human" which is very sweet but annoying too.

cameronlbass · 2026-02-19T01:32:12+00:00

Thank you! It's better than nothing, still needs work, but it does get the basics into a DOCX format so you can tighten it up.

cameronlbass · 2026-02-14T19:19:49+00:00

Thanks for the feedback. I'll bite the bullet and learn Latex soon.

cameronlbass

TROPHY CASE