Is This a Proof? Yes — but Conditional
At its core, this work does present a proof—but not in the narrow, traditional sense people often expect from the P vs NP problem. It is a conditional proof. That means the conclusion follows rigorously if a clearly stated condition holds. Importantly, that condition is not an arbitrary mathematical assumption—it is a principle grounded in how computation works in the physical world.
The condition is simple to state:
Information (or entropy) cannot be eliminated without being dispersed through local, irreversible interactions.
If that principle holds—and it already underlies thermodynamics, information theory, and real computation—then the paper shows that polynomial-time discovery of NP solutions is impossible for a large and natural class of hard problems. Verification remains easy. Discovery does not.
What the Paper Proves Unconditionally
Most of the heavy lifting in the paper is unconditional. It rigorously establishes that:
- Any successful NP solver must eliminate a large amount of uncertainty (search entropy).
- Producing a reliable, classical “fact” (a recorded solution) necessarily involves irreversibility.
- Irreversibility implies entropy must be discarded somewhere (Landauer’s principle).
- Under emergent time, entropy reduction must happen incrementally, step by step.
- All known mechanisms that could reduce entropy incrementally—logical deduction (resolution) or statistical correlation (low-degree/SQ methods)—are provably blocked on hard NP instances.
These pieces do not rely on conjectures about P vs NP. They are established results from proof complexity, learning theory, information theory, and physics.
The Single Conditional Step
What remains is one sharply isolated condition:
There is no way to reduce search entropy “all at once,” without local dispersion.
If such a shortcut existed, it would allow uncertainty to collapse globally—without passing through identifiable, local interactions with the problem instance. But no such mechanism exists in physics. It would be the informational equivalent of extracting work from a system in perfect thermal equilibrium.
So the proof is conditional in the following sense:
- If entropy reduction must obey the same locality and irreversibility constraints that govern physical processes,
- then polynomial-time NP discovery is impossible on hard instances.
This is not a hidden assumption equivalent to “P ≠ NP.” It is a statement about how entropy behaves, not about complexity classes.
Why This Kind of Conditional Proof Matters
Many foundational results in computer science are conditional. Cryptography relies on the assumption that one-way functions exist. Quantum advantage relies on assumptions about circuit complexity and randomness. What makes this work different is where the condition comes from.
The remaining condition is not mathematical convenience—it is physical admissibility. To violate it would require a new law of nature: a way for information to be destroyed without heat, time, or local interaction.
So while the paper does not claim to have “solved P vs NP” in the formal Clay sense, it does something arguably more explanatory. It shows that if P = NP were true, it would not just overturn complexity theory—it would overturn our understanding of entropy and computation itself.
That is the sense in which this is a conditional proof—and why the condition is so powerful.
Why having a simple answer doesn’t make a problem easy
It’s natural to think:
“If a problem has a clear, simple answer, surely there must be a fast way to find it.”
Surprisingly, that’s often not true.
What matters is not how simple the answer is, but whether the problem gives you clues as you search.
A locked box with no hints
Imagine a locked box with a number code.
- There is exactly one correct code
- The code itself is short and simple
- Every wrong attempt gives the same response: “No”
You try one code after another, but you never learn whether you’re getting closer.
Even though:
- the answer exists
- the answer is simple
finding it still takes an enormous number of guesses.
The problem isn’t the answer — it’s the lack of feedback.
Why feedback matters
Now imagine a different version of the same game.
Instead of silence, you’re told:
- “You’re getting warmer”
- or “You’re getting colder”
Suddenly, you have direction.
You can rule things out quickly and converge on the answer.
The difference is information flow.
Some problems give clues, others don’t
Many everyday puzzles work because they leak information:
- Each step tells you something
- Wrong choices narrow the field
- Uncertainty shrinks steadily
But some problems are designed so that:
- Every wrong move looks identical
- No partial attempt tells you anything useful
- The uncertainty never decreases
These are the problems your paper is about.
They don’t hide the answer — they hide the path.
Why this matters
Your paper argues that for certain hard problems:
- There is a correct answer
- The answer may be short and simple
- But no method gets directional information while searching
Without direction, the only option is repeated guessing — and that grows explosively as the problem size increases.
No clever shortcut helps, because there’s nothing to follow.
The core idea in one sentence
A problem is hard not because the answer is complex, but because the problem gives you no way to tell whether you’re getting closer.
That’s the intuition behind the paper’s conclusion — and why the result is conditional on deep physical and informational limits, not on the ingenuity of algorithms.