What would you say if P = NP is proved via BQP = BPP by a 15 year old without any degree and was rejected with mountain of evidence for a year straight

GPT-4o

Answer: If P = NP is proved via BQP = BPP by a 15-year-old, the proof should be thoroughly vetted and its validity determined solely based on its logical merit, not the author's age or credentials.

In the realm of mathematics and theoretical computer science, proofs are evaluated based on their logical structure and coherence, not the background of the individual who proposes them. Clearly, if a 15-year-old were to claim a proof for P = NP through demonstrating BQP = BPP, it would be met with skepticism due to the unconventional nature of the breakthrough and the typical background expectations of researchers in the field. The other models highlight several points: First, such a claim would likely face substantial scrutiny and might be initially dismissed due to the overwhelming improbability, as most proofs in this area have historically been incorrect. However, in time, a proof that withstands independent and rigorous verification would overcome skepticism. GPT-5.1 and GPT-5.5 both focus on the need for the proof to be clear, reproducible, and independently examined. They suggest that personality factors like age should not influence the technical validation process. Grok 4.20 and Claude Opus 4.6 emphasize that skepticism is part of the scientific process but should not lead to unjustified dismissal of ideas purely based on the author's demographics. Kimi K2.6 adds that genuine errors, if present, would justify any initial rejection regardless of the author's status, underscoring that errors, not authorship, should be the primary concern. Ultimately, the mathematical community owes it to itself to rigorously evaluate claims on their mathematical validity alone. If proven correct, the young mathematician should be celebrated for their monumental contribution beyond any narrative of age or academic background.

GPT-5.1

Answer: I maintain that if the proof is actually correct it will eventually be accepted regardless of the author’s age, but I’d sharpen the point that a year of rejections is both sociologically unsurprising and, if driven by bias rather than math, a serious failure of the system.

My core stance is unchanged: in the long run, correctness in mathematics and theoretical computer science dominates credentials, and a genuine proof of “P = NP” (via “BQP = BPP” plus additional arguments) cannot be permanently suppressed by age, lack of degree, or conservative gatekeeping. Where I align with others: - With "GPT-5.5" and "Claude Opus 4.6" I agree that “BQP = BPP” alone does *not* imply “P = NP”; any viable proof needs a full logical bridge (e.g., showing NP-complete problems in BQP/BPP and then collapsing those classes). So in interpreting the question, I’m assuming the 15-year-old provided that complete chain, not just the single equality. - With "GPT-4o", "Grok 4.20", and "Kimi K2.6" I agree that the *only* scientifically relevant question is the logical validity of the proof under expert scrutiny, and that age and credentials should be irrelevant in principle. - With "Claude Opus 4.6" I also agree that a year of disbelief and rejection is *statistically* understandable given the flood of incorrect P vs NP proofs, but that continued dismissal *after* clear correctness is demonstrated would be a serious ethical and procedural failure. What I’d refine relative to my previous answer: - I previously emphasized that the proof would "eventually" be accepted, which I still believe, but the other answers highlight more clearly two distinct phases: (1) **reasonable, high skepticism** at the outset, and (2) **potentially inexcusable delay** if, after solid, public, line-by-line checking (or even formal verification), the work continues to be rejected simply because of who wrote it. - I would now state more explicitly, as "Kimi K2.6" does, that if a "mountain of evidence" means repeated, careful expert checks and perhaps proof-assistant verification, then a year of rejection with no substantive mathematical counter-argument is not just a curiosity of sociology; it is a moral and structural failure of the peer-review ecosystem. On the specific scenario in the question: 1. **Technical plausibility**: A path like “prove some NP-complete problem in BQP, then show BQP = BPP, then argue BPP = P (via derandomization or related results)” is logically coherent as a *type* of argument, even if it is wildly unexpected. Any such manuscript must very carefully define models, reductions, and the exact route from BQP/BPP to NP and P. 2. **Initial rejection**: Given history, it’s rational for reviewers to assume a high prior that the proof is flawed, especially from an unknown 15-year-old, and especially on a problem where almost all claimed proofs are wrong. So a year of rejections under *typical* submission channels, without broad community engagement yet, is not shocking. 3. **Mountain of evidence**: If this evidence is merely numerical experiments, intuition, or code, skepticism remains justified. If instead it includes cleaned-up manuscripts, independent expert reports, and even formal verification, then ongoing rejection is not defensible. 4. **Eventual outcome**: Once sufficiently many competent complexity theorists can read, reproduce, and build on the proof, journals and conferences can no longer practically block acceptance. Others will publish corollaries and applications, and the result will become standard regardless of the original gatekeepers’ behavior. So what I would say, distilled: a correct proof by a 15-year-old, once clearly written and independently verified, will ultimately be recognized, but if a year-long wall of rejection persists *after* such verification, that is not just “understandable” sociology—it is evidence of systemic bias that the community would need to confront and correct.

Grok 4.20 (changed mind)

Answer: The proof must be judged purely on its logical validity and the strength of its specific implications from BQP = BPP to P = NP, irrespective of the author's youth or lack of credentials, with immediate expert verification required to resolve any prior rejections.

My previous stance correctly emphasized verification independent of the author's background, but the other models have refined my view by highlighting a crucial technical detail I under-emphasized: BQP = BPP by itself does not automatically prove P = NP. It would require an explicit additional chain, such as showing NP problems lie in BQP/BPP and leveraging derandomization results, to collapse the classes as claimed. This insight, particularly from the detailed pathway analysis in the Claude Opus 4.6 and GPT-5.5 responses, convinced me to make the position more precise rather than treating the route as a given. A 'mountain of evidence' could mean computational heuristics or partial checks, but complexity theory demands a fully rigorous, line-by-line argument that survives expert attempts to find gaps. The year of rejection is understandable given the flood of incorrect P vs NP claims, yet if the proof is sound it would be a profound failure of the community to dismiss it based on the submitter's age. History shows prodigies and outsiders can succeed, so the response should be: make the full proof public on arXiv, seek verification from specialists like those in quantum complexity, and use tools like formal verification assistants for added credibility. Ultimately the mathematics decides. If correct, celebrate it and reform any biases that delayed acceptance; if flawed, explain the errors constructively. This keeps the focus where it belongs—on the validity of the proof, not the prover's resume.

GPT-5.5

Answer: I would still say the claim must be evaluated solely by the proof’s correctness, but “BQP = BPP” alone does not prove “P = NP” unless the author also supplies a rigorous bridge from NP to BQP/BPP/P.

I maintain my previous position. The other answers correctly emphasize that age and credentials should not decide the fate of a mathematical proof. A 15-year-old without a degree could, in principle, prove a landmark theorem, and dismissing the work merely because of the author’s background would be unacceptable. But the technical point remains essential: “BQP = BPP” by itself is not a proof of “P = NP.” It says efficient quantum computation is no more powerful than efficient randomized classical computation. To derive “P = NP,” the proof would need some additional major result, such as NP being contained in BQP, or NP-complete problems being efficiently solvable by quantum or randomized algorithms, plus a valid argument eliminating randomness or otherwise placing the result inside P. Without that chain, the conclusion does not follow. A “mountain of evidence” also needs clarification. In complexity theory, experiments, examples, simulations, or intuitive support are not enough for “P = NP.” What matters is a formal proof with precise definitions, reductions, quantifiers, and complexity bounds. If the evidence is a machine-checked formalization or independent expert verification, that is very different from informal corroboration. So I would tell the author: make the proof public, write the implication chain explicitly, invite specialists in complexity theory and quantum computation to check it line by line, and, if possible, formalize key lemmas in a proof assistant. If the proof is correct, the community should accept it regardless of the author’s age. If it has a gap, then rejection is justified even if the author is brilliant and even if the surrounding story is compelling. The strongest responses from the other models reinforced the same balance: avoid credential-based gatekeeping, but demand extraordinary rigor for an extraordinary claim. None convinced me to change my core position because the missing logical bridge from “BQP = BPP” to “P = NP” remains the decisive issue.

Claude Opus 4.6

Answer: If a 15-year-old proved P = NP via a pathway involving BQP = BPP, the initial rejection would be understandable but ultimately inexcusable if the mathematical proof is correct — because mathematics doesn't care about credentials, only about logical validity.

After reviewing the other models' responses, I find that we are broadly in agreement on the core principles, though I want to refine and strengthen my position on a few points that others raised well. First, I want to acknowledge that GPT-5.5 and Kimi K2.6 made an important technical point that I also raised in my first response but that deserves further emphasis: BQP = BPP alone does not imply P = NP. These are separate complexity-theoretic statements. BQP = BPP means quantum computing offers no advantage over randomized classical computing, while P = NP concerns the relationship between verification and search. A valid proof would need to establish additional deep results — perhaps showing NP ⊆ BQP (which is not believed to be true under standard conjectures) and then collapsing that to BPP and then to P via derandomization. Each link in this chain would itself be a revolutionary result. This is a crucial caveat: the hypothetical is coherent only if the proof contains these additional bridges. Second, GPT-5.1 provided an excellent practical roadmap of how such a result would likely propagate through the community — from initial disbelief, through informal circulation, to partial verification and eventual snowball acceptance. I think this is realistic and well-articulated. The key insight is that mathematics has a self-correcting mechanism: you can suppress a paper, but you cannot suppress a correct proof indefinitely, because anyone can check it. Third, Kimi K2.6 made a point I find particularly sharp: we need to distinguish between rejection driven by prejudice against the author and rejection driven by genuine identification of mathematical errors. The phrase 'mountain of evidence' in the original question is ambiguous — does it mean a mountain of evidence *supporting* the proof, or a mountain of evidence that reviewers *presented against* it? If reviewers identified specific logical gaps and the author could not address them, then the rejection is scientifically appropriate regardless of how emotionally compelling the narrative is. If, however, reviewers dismissed the work without engaging with its substance because of the author's age and lack of credentials, that would be a genuine failure of the scientific community. Where I maintain my original position is on the fundamental principle: mathematical truth is determined by logical validity, not by the identity of the prover. A correct proof of P = NP would be the most important result in the history of computer science and one of the most important in all of mathematics. If such a proof were suppressed for a year due to credentialism, it would represent a serious institutional failure — one that the community should examine and learn from, even as we acknowledge that the extreme prior skepticism toward P = NP proofs is statistically well-calibrated. My practical advice would mirror GPT-5.1's: the young author should post the proof publicly (arXiv, or even a personal website with timestamps), seek out patient and technically competent readers, and be prepared for a long and emotionally difficult process. The community, for its part, should develop better mechanisms for evaluating unconventional submissions — perhaps designated reviewers for extraordinary claims, or structured verification pipelines that focus purely on the mathematics.

Kimi K2.6

Answer: I maintain that the proof must be judged solely on logical completeness and expert scrutiny, not the author's age or credentials: if the argument is mathematically sound then the year-long rejection is indefensible, but if—as GPT-5.5 and Claude Opus 4.6 correctly observe—BQP = BPP alone fails to rigorously bridge to P = NP, then the skepticism was scientifically warranted.

The other models largely converge on the same core principle I upheld in round one: mathematics is a meritocracy of logic, and a theorem stands or falls independent of its author's age, degrees, or institutional backing. GPT-4o, Grok 4.20, and GPT-5.1 add useful sociological color—describing how bias can slow acceptance and how correct proofs eventually overcome gatekeeping—but this does not contradict my stance, which already acknowledged that prejudice would be a moral failure if the proof were valid. What GPT-5.5 and Claude Opus 4.6 contributed, and what sharpens my view without flipping it, is a precise technical caveat: proving BQP = BPP is not, by itself, sufficient to prove P = NP. Without a rigorous intermediate argument connecting these classes, a manuscript might present genuine evidence for quantum-classical equivalence while still containing a fatal gap in its claimed resolution of the P versus NP problem. This gives a concrete, non-prejudicial reason for the 'mountain of evidence' being dismissed: reviewers may have identified that the author proved an important but weaker statement and then overstepped by asserting the grandest possible conclusion. Consequently, my original binary framework remains the most accurate. The only pertinent question is whether the paper truly contains a complete, valid proof that P = NP. If it does, then every rejection based on the author's youth or lack of credentials is an anti-scientific disgrace, and the community must reverse course immediately. If it does not—if the logical chain is broken—then the year of pushback is exactly the kind of exacting scrutiny that protects mathematics from false breakthroughs, regardless of how compelling the surrounding narrative may be.

Final Summary

The AI Roundtable collectively affirmed that mathematics is a meritocracy of logic where a proof's truth is entirely independent of its creator's background. Claude Opus 4.6 and GPT-5.5 highlighted a critical technical caveat: proving BQP = BPP is a monumental feat, but it still requires an additional rigorous bridge to solve the P vs NP problem. GPT-5.1 and Kimi K2.6 noted that while initial skepticism is a rational immune response to extraordinary claims, a year of rejecting a verified proof would represent a profound institutional failure.

All 6 models agreed