arXiv subjects

Darren J. Edwards

Publications and source records attributed to Darren J. Edwards.

4 recordsLinked to original sources

arXiv2026

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

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.

cs.CC↗

arXiv2025

Shifted Partial Derivative Polynomial Rank and Codimension

Shifted partial derivative (SPD) methods are a central algebraic tool for circuit lower bounds, measuring the dimension of spaces of shifted derivatives of a polynomial. We develop the Shifted Partial Derivative Polynomial (SPDP) framework, packaging SPD into an explicit coefficient-matrix formalism. This turns shifted-derivative spans into concrete linear-algebraic objects and yields two dual measures: SPDP rank and SPDP codimension. We define the SPDP generating family, its span, and the SPDP matrix M_{k,l}(p) inside a fixed ambient coefficient space determined by the (k,l) regime, so rank is canonical and codimension is a deficit from ambient fullness. We prove structural properties needed for reuse: monotonicity in the shift/derivative parameters (with careful scoping for |S|=k versus |S|<=k conventions), invariance under admissible variable symmetries and basis changes, and robustness across standard Boolean/multilinear embeddings. We then give generic width-to-rank upper-bound templates for local circuit models via profile counting, separating the model-agnostic SPDP toolkit from additional compiled refinements used elsewhere. We illustrate the codimension viewpoint on representative examples.

cs.CC↗

arXiv2020

Developing Effective Community Network Analysis Tools According to Visualization Psychology

Visualization is a useful technology in health science, and especially for community network analysis. Because visualization applications in healthcare are typically risk-averse, health psychologists can play a significant role in ensuring appropriate and effective uses of visualization techniques in healthcare. In this paper, we examine the role of health psychologists in the triangle of "health science", "visualization technology", and "visualization psychology". We conclude that health psychologists can use visualization to aid data intelligence workflows in healthcare and health psychology, while researching into visualization psychology to aid the improvement and optimization of data visualization processes.

cs.HC↗

arXiv2019

Juxtaposing Controlled Empirical Studies in Visualization with Topic Developments in Psychology

Empirical studies form an integral part of visualization research. Not only can they facilitate the evaluation of various designs, techniques, systems, and practices in visualization, but they can also enable the discovery of the causalities explaining why and how visualization works. This state-of-the-art report focuses on controlled and semi-controlled empirical studies conducted in laboratories and crowd-sourcing environments. In particular, the survey provides a taxonomic analysis of over 129 empirical studies in the visualization literature. It juxtaposes these studies with topic developments between 1978 and 2017 in psychology, where controlled empirical studies have played a predominant role in research. To help appreciate this broad context, the paper provides two case studies in detail, where specific visualization-related topics were examined in the discipline of psychology as well as the field of visualization. Following a brief discussion on some latest developments in psychology, it outlines challenges and opportunities in making new discoveries about visualization through empirical studies.

cs.HC↗