I am no longer permitted to reach my full ChatGPT pages. All recent ones have had substantial portions lopped off above what I show on this blog.
Yes—if “represented as a TSP” means represented within the same finite resource model, then you're very close to a precise theorem. But there's one important logical correction.
Let CT,E be the set of problems solvable by a particular computational system within time T and energy E. Define RT,E as the set of problems that can be encoded as TSP instances and decoded back to the desired answer within those same bounds.
Then certainly
RT,E ⊆ CT,E.
Your stronger claim is
CT,E = RT,E
—that TSP is complete for the entire class of physically permissible computations.
If that equality holds, then yes:
P ∉ RT,E ⟹ P ∉ CT,E,
so no AI operating under those constraints can solve P.
Where I would modify your statement
It's not enough that a problem cannot be represented as a TSP in some particular encoding. There may simply be a better encoding.
What you need is:
There exists no resource-respecting reduction from the problem to TSP.
That's a much stronger statement.
And there's another subtlety: if your TSP solver is itself an AI, then saying “the AI can solve TSP” doesn’t automatically mean it can solve every other problem. You need the TSP solver to be a sufficiently general oracle and the reduction to fit within the resource budget.
Your idea can be stated very cleanly
Let
𝒞T,E
be all computations realizable by the physical AI within T time and E energy.
If you can prove
∀ P ∈ 𝒞T,E, P ≤resource TSP,
then TSP is complete for physically realizable computation under those constraints.
Consequently,
P ≰resource TSP ⟹ P ∉ 𝒞T,E.
That is a very interesting formulation because it shifts the question away from “Is TSP NP-complete?” and toward:
Is TSP complete for bounded physical computation?
That's not something standard NP-completeness by itself establishes. The difficult part would be proving the resource-preserving reduction, particularly when the TSP instance, its encoding, and the decoding of its answer all have to fit inside the same physical limits.
And your stochastic-AI observation fits naturally into this framework: randomness can be treated as part of the finite computation tree, so it doesn’t by itself enlarge the class of computations beyond what the bounded model permits.
Let CT,E be the set of problems solvable by a particular computational system within time T and energy E. Define RT,E as the set of problems that can be encoded as TSP instances and decoded back to the desired answer within those same bounds.
Then certainly
RT,E ⊆ CT,E.
Your stronger claim is
CT,E = RT,E
—that TSP is complete for the entire class of physically permissible computations.
If that equality holds, then yes:
P ∉ RT,E ⟹ P ∉ CT,E,
so no AI operating under those constraints can solve P.
Where I would modify your statement
It's not enough that a problem cannot be represented as a TSP in some particular encoding. There may simply be a better encoding.
What you need is:
There exists no resource-respecting reduction from the problem to TSP.
That's a much stronger statement.
And there's another subtlety: if your TSP solver is itself an AI, then saying “the AI can solve TSP” doesn’t automatically mean it can solve every other problem. You need the TSP solver to be a sufficiently general oracle and the reduction to fit within the resource budget.
Your idea can be stated very cleanly
Let
𝒞T,E
be all computations realizable by the physical AI within T time and E energy.
If you can prove
∀ P ∈ 𝒞T,E, P ≤resource TSP,
then TSP is complete for physically realizable computation under those constraints.
Consequently,
P ≰resource TSP ⟹ P ∉ 𝒞T,E.
That is a very interesting formulation because it shifts the question away from “Is TSP NP-complete?” and toward:
Is TSP complete for bounded physical computation?
That's not something standard NP-completeness by itself establishes. The difficult part would be proving the resource-preserving reduction, particularly when the TSP instance, its encoding, and the decoding of its answer all have to fit inside the same physical limits.
And your stochastic-AI observation fits naturally into this framework: randomness can be treated as part of the finite computation tree, so it doesn’t by itself enlarge the class of computations beyond what the bounded model permits.
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