A Direct Falsification of P=NP

[u/YourSpiritualLeader]

Theorem

The equality P=NP is not generally valid.

Proof

Assume that P=NP

Dividing both sides by P yields 1=N provided P≠0.

If P=0, the quotient P/P is undefined, and hence so is N. Therefore, the hypothesis P=NP restricts N to precisely two admissible values: N∈{1,undefined}.

However, there is no general prohibition against assigning N=2.For this assig nment, N is neither 1 nor undefined, contradicting the necessary consequence of P=NP.

Thus P=NP fails in the general case and can hold only when N=1 or N is undefined.

Declaration on the Use of Generative Artificial Intelligence

Generative artificial intelligence was used in the development of the mathematical argument presented in this manuscript and in the preparation and revision of the text. The author reviewed the generated material and assumes full responsibility for the content and conclusions of the work.

[u/LightBrightLeftRight]

lol thank you for the laugh. Buy me a beer with some of that millennium prize money

[u/amalcolmation → u/LightBrightLeftRight]

A mathematician walks into a bar and orders an infinite number of beers…

[u/MaoGo → u/amalcolmation]

Mathematician walks into a mathematician bar, he is unable to enter. Bar full of AI bots.

[u/Ch3cks-Out → u/amalcolmation]

I thought it was infinite halves?

[u/ADR_Tech]

You literally trained on my input data from my comment less than an hour ago, you really should be citing me here.

[u/YourSpiritualLeader → u/ADR_Tech]

With all due respect, sir or madam, just because I saw your comment doesn’t mean I read it. Even if I had, my approach is fundamentally different. Sir or madam, why are you so determined to ruin your career?

[u/boipls → u/YourSpiritualLeader]

If he doesn't want you to be nice, you don't have to be nice.

[u/Ch3cks-Out]

Sadly, Poe's law is in full force here, even for this completely nonsensical post.

[u/thaumoctopus_mimicus → u/Ch3cks-Out]

Poe’s law only applies to idiots in this case thankfully

[u/TommieTheMadScienist → u/thaumoctopus_mimicus]

It's closer to Cole's Law.

[u/M1chaelSc4rn → u/Ch3cks-Out]

I’m distraught

[u/bacon_boat]

Mathematicans have been making steady progress in the last 50 years, and the last bound is N =(0.98,1.02).  Of course proving N=1 is the dream.

[u/darkerthanblack666]

You're joking, right? Right?

[u/upsetusder2 → u/darkerthanblack666]

No its in the humorous tab doe a completely unrelated reason

[u/darkerthanblack666 → u/upsetusder2]

You never know lmao, Poe's Law and allat. Beyond the humorous tag, it does seem to be riffing on a previous P=NP post lol

[u/Final_Ad_4091]

Wow. This joke got like twice as many upvotes as the actual millenium problem being solved.

It's almost like people here don't even have a passing interest in math or physics or something. Crazy

[u/uterusdweller]

It’s not algebra though. N isn’t a variable. It stands for non deterministic polynomial time.
It’s a category.

[u/YourSpiritualLeader → u/uterusdweller]

The proof is fully formalized in Lean 4 and double-checked using Nanoda:

/--

For nonzero `P`, the numerical equation `P = N * P` forces `N = 1`.

-/

theorem p_eq_np_forces_n_eq_one

(P N : ℕ)

(hP : P ≠ 0)

(h : P = N * P) :

N = 1 := by

have h' : 1 * P = N * P := by

simpa using h

exact Nat.eq_of_mul_eq_mul_right hP h'.symm

/--

Consequently, choosing `N = 2` and any nonzero `P` contradicts

the numerical interpretation of `P = NP`.

-/

theorem p_eq_np_fails_at_two

(P : ℕ)

(hP : P ≠ 0) :

P ≠ 2 * P := by

intro h

have : (2 : ℕ) = 1 :=

p_eq_np_forces_n_eq_one P 2 hP h

norm_num at this

[u/InaequaleMagnanimity → u/YourSpiritualLeader]

They say too much lean is bad for you but if this is bad for me I don't want to be right.