This repository contains the manuscript and foundational framework for a self-contained, constructive solution to the classical Graph Reconstruction Conjecture, originally proposed by Kelly (1942) and Ulam (1960). While traditional approaches to the reconstruction problem rely on extensive combinatorial case analysis and graph-isomorphism invariants, this work introduces an algebraic paradigm shift by encoding graph topologies as binary edge-labeled relational systems. By assigning binary values ₔₕ 0, 1 to all vertex pairs, the global structure of a simple connected graph G on n 3 vertices is mapped to a weighted adjacency system. The core breakthrough of this paper lies in bridging the gap between discrete graph decks and continuous linear algebraic systems without introducing loss of information. The reconstruction algorithm proceeds via two main phases: 1. Local and Global Feature Recovery: From the multiset of vertex-deleted subgraphs (the deck), we uniquely recover the local flux (vertex degrees) and global cycle information (wave curvatures along discrete loops). 2. Gauge-Fixed Inversion: To resolve the non-flat correction terms arising from cycle inconsistencies, we introduce a reference spanning tree to systematically fix the discrete gauge degrees of freedom, uniquely isolating local edge corrections. By applying these parameters to a pinned discrete Laplacian system, we establish a Pinned Laplacian Equation (Lₚ = b). Crucially, we prove the Discrete Consistency and Integrability Theorem, demonstrating that the continuous potential solution automatically and exactly integerizes into the binary domain. The inversion process guarantees a 100% accurate, exact reconstruction of the original graph's adjacency matrix from its deck, utilizing only elementary linear algebra and discrete potential theory. This framework serves as a foundational showcase for the broader principles of Wave-Point Geometry, wherein spatial and structural properties emerge from phase consistency over relational networks.
Ren Matsuoka (Sat,) studied this question.