Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2512.11820

Toward P vs NP: An Observer-Theoretic Separation via SPDP Rank and a ZFC-Equivalent Foundation within the N-Frame Model

Darren J. Edwards
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a self-contained separation framework for P vs NP developed entirely within ZFC. The approach consists of: (i) a deterministic, radius-1 compilation from uniform polynomial-time Turing computation to local sum-of-squares (SoS) polynomials with polylogarithmic contextual entanglement width (CEW); (ii) a formal Width-to-Rank upper bound for the resulting SPDP matrices at matching parameters; (iii) an NP-side identity-minor lower bound in the same encoding; and (iv) a rank-monotone, instance-uniform extraction map from the compiled P-side polynomials to the NP family. Together these yield a contradiction under the assumption P = NP, establishing a separation. We develop a correspondence between CEW, viewed as a quantitative measure of computational contextuality, and SPDP rank, yielding a unified criterion for complexity separation. We prove that bounded-CEW observers correspond to polynomial-rank computations (the class P), while unbounded CEW characterizes the class NP. This implies that exponential SPDP rank for #3SAT and related hard families forces P != NP within the standard framework of complexity theory. Key technical components include: (1) constructive lower bounds on SPDP rank via Ramanujan-Tseitin expander families; (2) a non-circular reduction from Turing-machine computation to low-rank polynomial evaluation; (3) a codimension-collapse lemma ensuring that rank amplification cannot occur within polynomial resources; and (4) proofs of barrier immunity against relativization, natural proofs, and algebrization. The result is a complete ZFC proof architecture whose primitives and compositions are fully derived, with community verification and machine-checked formalization left as future work.

Explore related subjects

cs.CCcs.DM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Darren J. Edwards. 2026-01-08. Toward P vs NP: An Observer-Theoretic Separation via SPDP Rank and a ZFC-Equivalent Foundation within the N-Frame Model. https://arxiv.org/abs/2512.11820

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Counting Small Induced Subgraphs: Hardness of Symmetry-Based Properties

Jerrum and Meeks (TOCT, JCSS 2015) introduced the counting problems $\text{IndSub}(Φ)$ for fixed graph properties $Φ$: Given an input graph $G$ and $k\in\mathbb N$, count the $k$-vertex subsets $S \subseteq V(G)$ such that the induced subgraph $G[S]$ satisfies $Φ$. For recursively enumerable $Φ$, it is known that $\text{IndSub}(Φ)$ is either #W[1]-hard or fixed-parameter tractable. A direct classification depending on $Φ$ however still remains open. In particular, the status was open for the property of graphs without nontrivial automorphisms, also mentioned in a very recent survey on parameterized counting by Roth (Comput.~Sci.~Rev.~2026). This is a natural property that evades all currently known techniques for proving #W[1]-hardness, including a general toolkit based on Fourier analysis that was very recently introduced by Curticapean and Neuen (SODA~2025). In this paper, we show that counting induced $k$-vertex graphs without nontrivial automorphisms is #W[1]-hard by constructing ``clique scaffolds'', i.e., problem-specific restrictions of the property that enable a reduction from the $k$-clique problem. More generally, we show that for every finite group $Q$, counting $k$-vertex induced subgraphs with automorphism group $Q$ is #W[1]-hard.

cs.CC↗
arXiv2026

Improved Algorithms for the Remote Point Problem

The Remote Point Problem (RPP) is an algorithmic problem that asks, given a linear subspace $L \subseteq \mathbb{F}^n$ of dimension $k$, to deterministically find a vector $v \in \mathbb{F}^n$ far in Hamming distance from $L$. This problem was introduced by Alon, Panigrahy and Yekhanin [APY09], motivated in part by the matrix rigidity approach for proving circuit lower bounds. An algorithm is said to achieve remoteness $d$ if it finds a vector $v$ whose Hamming distance from $L$ is at least $d$. We observe that over the rational numbers, the problem admits a deterministic polynomial-time algorithm that achieves optimal remoteness $n-k$. Over finite fields, we obtain a (modest) improvement of a result of Alon, Panigrahy and Yekhanin [APY09], and give an algorithm that achieves remoteness $Ω\left(\frac{n}{\max\{k, \log n\}} \log n\right)$.

cs.CC↗
arXiv2026

Strong Selective and List-Decoding Direct Product Theorems for Quantum Query Complexity

Quantum strong direct-product theorems for specific functions have been known for nearly two decades. These have been extended to general results for function computation and state generation. The proofs of these results use a version of the multiplicative adversary method that does not naturally extend to relations. Standard strong direct-product theorems apply when algorithms must correctly answer every given question. Prior work extended them to equivalent threshold direct-product theorems, which require answers to all questions but only require that most answers are correct. We focus on two further generalizations. Strong selective direct-products apply to algorithms that adaptively choose, based on what they learn from queries, which questions from a large list to answer. This generalization is relational and useful for proving time-space tradeoffs. We prove a quantum strong selective direct-product theorem for all functions using a new multiplicative adversary formulation for relations that satisfies a strong selective direct product property while being strong enough to capture any query lower bound for functions proven by negative-weights adversaries. This was not previously known even without selectivity. The second generalization is list-decoding direct product problems introduced by Ben-David and Blais for classical randomized query complexity. These allow an algorithm to produce a large list of possible output vectors such that one of them is fully correct. They proved that such theorems hold for classical randomized complexity of all Boolean functions. We prove a quantum analogue of this theorem for all partial Boolean functions. We show that strong list-decoding direct-product theorems are implied by a special case of multiplicative adversaries which we show, via a new reduction, can be obtained from negative-weights adversaries for any Boolean-valued function.

cs.CC↗