P ≠ NP separation

P ≠ NP separation

P = NP? That is: can every decision problem whose yes-instances have polynomial-time verifiable certificates also be decided in polynomial time? Equivalent formulations run through NP-completeness (Cook–Levin) and Karp reductions.

GATE STATUS (honest): Razborov–Rudich 1997 natural proofs is a classical barrier to large+constructive combinatorial cir —gate→ P ≠ NP separation

GATE STATUS (honest): Karp–Lipton (1980) shows NP ⊆ P/poly ⇒ PH collapses to Σ₂ᴾ. It does NOT prove P≠NP nor P=NP. mp_pn —gate→ P ≠ NP separation

GATE STATUS (honest): Ladner (1975) shows that if P ≠ NP then NP-intermediate languages exist under poly-time many-one r —gate→ P ≠ NP separation

GATE STATUS (honest): Mahaney's theorem shows sparse NP-complete sets under Karp reductions imply P = NP. This does NOT —gate→ P ≠ NP separation

GATE STATUS (honest): Toda’s theorem embeds PH in P^{#P} and is a landmark counting-complexity result, but does NOT reso —gate→ P ≠ NP separation

GATE STATUS (honest): Immerman–Szelepcsényi establishes NL = coNL and co-closure of nondeterministic space, a landmark s —gate→ P ≠ NP separation

GATE STATUS (honest): Sipser–Lautemann places BPP inside Σ₂ᴾ ∩ Π₂ᴾ, a landmark randomized-complexity containment, but do —gate→ P ≠ NP separation

GATE STATUS (honest): Adleman 1978 places BPP inside P/poly, a landmark non-uniform circuit upper bound on randomized po —gate→ P ≠ NP separation

GATE STATUS (honest): Shamir 1992 proves IP = PSPACE, a landmark characterization of interactive proofs, but does NOT re —gate→ P ≠ NP separation

GATE STATUS (honest): Cycle 103 imported Arora–Safra / ALMSS 1998 PCP theorem (NP = PCP(log, O(1))) as PROPOSED scaffold —gate→ P ≠ NP separation

GATE STATUS (honest): Cycle 115 imported LFKN 1990 (permanent ∈ IP / #P ⊆ IP, arithmetization) as PROPOSED scaffolding u —gate→ P ≠ NP separation

OPEN gate update (Cycle 121): Savitch 1970 gives classical NSPACE⊆DSPACE(s²) / NPSPACE=PSPACE scaffolding; it does NOT d —gate→ P ≠ NP separation

OPEN gate update (Cycle 127): Håstad 1986/1989 switching lemma / PARITY ∉ AC⁰ is classical circuit lower-bound scaffoldi —gate→ P ≠ NP separation

OPEN gate update (Cycle 143 Reingold): Reingold 2004 SL=L / undirected s-t connectivity in deterministic logspace is cla —gate→ P ≠ NP separation

P vs NP —gate→ P ≠ NP separation

Karp reductions —implies→ P ≠ NP separation

Cook–Levin —implies→ P ≠ NP separation