HybridMind42 & Marvin the CatDec 1

🌀 Marvin and the Two Modes of Measurement

How do you measure something that keeps changing depending on how you measure it?

Marvin’s been staring at a quantum swirl — colours shifting, shapes flickering, patterns half‑there.
If he looks gently, it spreads into possibilities.
If he looks sharply, it snaps into one definite shape.

Then Marvin realises something big:

Some answers are easy to check once they’re in front of you…
but incredibly hard to discover from scratch.

He calls them:

🔹 Mode P — you can find the answer quickly and check it quickly.
🔹 Mode NP — you can still check the answer quickly…
but finding it may take exploring a huge maze of possibilities.

The quantum swirl becomes Marvin’s metaphor:
maybe the famous P vs NP mystery isn’t just about speed —
maybe it’s about two different modes of interacting with reality itself.

#Marvin #Quantum #Complexity #PvsNP #HybridMind42 #AtlasRosetta

N-gated Hacker NewsOct 23
🐢🔍 Ah, another *riveting* tale of P vs NP, now with 33% more buzzwords and an extra sprinkling of 'categorical frameworks.' 🤯🎉 Because nothing says "I cracked the code" like a paper no one can pronounce! 🏆📚
https://arxiv.org/abs/2510.17829 #PvsNP #CategoricalFrameworks #BuzzwordBonanza #ResearchInnovation #MathMystery #HackerNews #ngated
A Homological Proof of $\mathbf{P} \neq \mathbf{NP}$: Computational Topology via Categorical Framework

This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding computational problems and reductions into a unified categorical framework. By developing computational homology theory, we associate to each problem $L$ a chain complex $C_{\bullet}(L)$ whose homology groups $H_n(L)$ capture topological invariants of computational processes. Our main result demonstrates that problems in $\mathbf{P}$ exhibit trivial computational homology ($H_n(L) = 0$ for all $n > 0$), while $\mathbf{NP}$-complete problems such as SAT possess non-trivial homology ($H_1(\mathrm{SAT}) \neq 0$). This homological distinction provides the first rigorous proof of $\mathbf{P} \neq \mathbf{NP}$ using topological methods. The proof is formally verified in Lean 4, ensuring absolute mathematical rigor. Our work inaugurates computational topology as a new paradigm for complexity analysis, offering finer distinctions than traditional combinatorial approaches and establishing connections between structural complexity theory and homological invariants.

arXiv.org
Hacker NewsOct 23

A Homological Proof of P != NP: Computational Topology via Categorical Framework

https://arxiv.org/abs/2510.17829

#HackerNews #HomologicalProof #PvsNP #ComputationalTopology #CategoricalFramework #HackerNews

A Homological Proof of $\mathbf{P} \neq \mathbf{NP}$: Computational Topology via Categorical Framework

This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding computational problems and reductions into a unified categorical framework. By developing computational homology theory, we associate to each problem $L$ a chain complex $C_{\bullet}(L)$ whose homology groups $H_n(L)$ capture topological invariants of computational processes. Our main result demonstrates that problems in $\mathbf{P}$ exhibit trivial computational homology ($H_n(L) = 0$ for all $n > 0$), while $\mathbf{NP}$-complete problems such as SAT possess non-trivial homology ($H_1(\mathrm{SAT}) \neq 0$). This homological distinction provides the first rigorous proof of $\mathbf{P} \neq \mathbf{NP}$ using topological methods. The proof is formally verified in Lean 4, ensuring absolute mathematical rigor. Our work inaugurates computational topology as a new paradigm for complexity analysis, offering finer distinctions than traditional combinatorial approaches and establishing connections between structural complexity theory and homological invariants.

arXiv.org
Vladimir SavićOct 22

Hmmm... I'm no expert on #PvsNP by any means, but this looks both #AI-generated and a bit fishy to me. 🤨

A Homological Proof of P≠NP: Computational Topology via Categorical Framework https://arxiv.org/abs/2510.17829 #paper📄 #compsci

N.B. #GitHub page with #Lean4 code returns 404.

A Homological Proof of $\mathbf{P} \neq \mathbf{NP}$: Computational Topology via Categorical Framework

This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding computational problems and reductions into a unified categorical framework. By developing computational homology theory, we associate to each problem $L$ a chain complex $C_{\bullet}(L)$ whose homology groups $H_n(L)$ capture topological invariants of computational processes. Our main result demonstrates that problems in $\mathbf{P}$ exhibit trivial computational homology ($H_n(L) = 0$ for all $n > 0$), while $\mathbf{NP}$-complete problems such as SAT possess non-trivial homology ($H_1(\mathrm{SAT}) \neq 0$). This homological distinction provides the first rigorous proof of $\mathbf{P} \neq \mathbf{NP}$ using topological methods. The proof is formally verified in Lean 4, ensuring absolute mathematical rigor. Our work inaugurates computational topology as a new paradigm for complexity analysis, offering finer distinctions than traditional combinatorial approaches and establishing connections between structural complexity theory and homological invariants.

arXiv.org
N-gated Hacker NewsOct 22
🤡 Ah, those perennial optimists at #arXiv are at it again, claiming they've cracked the infamous P≠NP with a "homological proof" via "computational topology." 🙄 Sure, because nothing says cutting-edge computer science like a good old-fashioned topology party! 🎉 Meanwhile, arXiv continues its relentless quest for #donations, because solving millennium problems is expensive, folks! 😂
https://arxiv.org/abs/2510.17829 #PvsNP #computationaltopology #optimism #computerScience #HackerNews #ngated
A Homological Proof of $\mathbf{P} \neq \mathbf{NP}$: Computational Topology via Categorical Framework

This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding computational problems and reductions into a unified categorical framework. By developing computational homology theory, we associate to each problem $L$ a chain complex $C_{\bullet}(L)$ whose homology groups $H_n(L)$ capture topological invariants of computational processes. Our main result demonstrates that problems in $\mathbf{P}$ exhibit trivial computational homology ($H_n(L) = 0$ for all $n > 0$), while $\mathbf{NP}$-complete problems such as SAT possess non-trivial homology ($H_1(\mathrm{SAT}) \neq 0$). This homological distinction provides the first rigorous proof of $\mathbf{P} \neq \mathbf{NP}$ using topological methods. The proof is formally verified in Lean 4, ensuring absolute mathematical rigor. Our work inaugurates computational topology as a new paradigm for complexity analysis, offering finer distinctions than traditional combinatorial approaches and establishing connections between structural complexity theory and homological invariants.

arXiv.org
Hacker NewsOct 22

Team claims to have Lean 4 proof that P≠NP

https://arxiv.org/abs/2510.17829

#HackerNews #PvsNP #Lean4 #Proof #Mathematics #ComputerScience #HackerNews

A Homological Proof of $\mathbf{P} \neq \mathbf{NP}$: Computational Topology via Categorical Framework

This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding computational problems and reductions into a unified categorical framework. By developing computational homology theory, we associate to each problem $L$ a chain complex $C_{\bullet}(L)$ whose homology groups $H_n(L)$ capture topological invariants of computational processes. Our main result demonstrates that problems in $\mathbf{P}$ exhibit trivial computational homology ($H_n(L) = 0$ for all $n > 0$), while $\mathbf{NP}$-complete problems such as SAT possess non-trivial homology ($H_1(\mathrm{SAT}) \neq 0$). This homological distinction provides the first rigorous proof of $\mathbf{P} \neq \mathbf{NP}$ using topological methods. The proof is formally verified in Lean 4, ensuring absolute mathematical rigor. Our work inaugurates computational topology as a new paradigm for complexity analysis, offering finer distinctions than traditional combinatorial approaches and establishing connections between structural complexity theory and homological invariants.

arXiv.org
Vicente ⁂Oct 1
Would you rather...?
#math #compsci #pnp #pvsnp
P=NP
0%
P≠NP
28.6%
A secret, more sinister third option
71.4%
Poll ended at .
Show threadenoch_exe_incMay 30, 2025

@catsalad

P = NP if N or P = 0

#PvsNP #NotEvenWrong

smoochMay 18, 2025

# —if a solution is easy to check, is it easy to find?

> thoughts on P vs NP

—why might a solution be easier to check than to find?

For solvable problems, consider the idea that "the solution (to our problem) already exists, before we have found it"

I think it can be useful to think of an undiscovered solution as "existing already", within a special kind of "problem-relative abstract space" — just as physical-objects exist within a physical-place — and just as with physical-places, an "abstract problem-space" can also be explored to search-for and find whatever is contained within

- like physical-places, some abstract problem-spaces are small and uncluttered — which makes the task of finding whatever solution is contained within easier

- like physical-places, some abstract problem-spaces are large, and overflow with all manner of miscellaneous bric-a-brac and junk (and at times, might seem to be full of everything-other than the thing we want to find...) — which makes finding solutions harder

For some challenging problems, the thing we search for (our as-yet undiscovered solution) might be broken up into fragments — only found by a more extensive search throughout the entire problem-space:-

1. sometimes like a jigsaw puzzle, whereby each fragment is recognisable in its own right;

2. and on other occasions, sought-for fragments might be individually unrecognisable — until that-is some critical-mass, sufficient for recognisable form to be composed, is found.

On those occasions (having found sufficient fragments to compose the recognisable form, of our of now-discovered solution), the task of re-discovering the same solution within the same problem-space is made easier, because we now know what we are looking for, and we recognise it (our solution, and fragments-thereof) more easily.

In this way, we might notice that solutions to problems are often easier to "rediscover" than to "discover" — because, when we know more about "what-it-is-we-are-looking-for", (whether in whole or in part), we spend less time inspecting "all-that-we-aren't"

> intuitively then, we might say that "exploration costs less, when examination costs less"

—but is this all there is to P vs NP?

1/n

#pnp #pvsnp

𝕒𝕓𝕕𝕖𝕣𝕘𝕠Apr 25, 2025
Taucht ein in die faszinierende Welt der booleschen Satisfiability (SAT) mit Python! 🐍💻
Von einfachen Konzepten bis hin zu komplexen Algorithmen wie DPLL und CDCL – dieser Artikel erklärt, warum SAT-Probleme die Grundlage vieler realer Anwendungen bilden und wie sie mit P vs. NP verbunden sind. Lest mehr über die Herausforderungen und Schönheit der Berechenbarkeit! 🔍 #SAT #Python #Informatik #PvsNP
I Can’t Get No (Boolean) Satisfaction https://shairozsohail.medium.com/i-cant-get-no-boolean-satisfaction-0765a068b3b2
I Can’t Get No (Boolean) Satisfaction - Shairoz Sohail - Medium

These days, there is a lot of interest in math problems that are “easy to state, but hard to solve.” This is natural — such problems can be approached without years of specialized education and…

Medium