The -1-eigenspace of the coprime graph of integers: prime-cube modes and an exact multiplicity

Let ΓN be the coprime graph on 1, …, N (vertices i, j adjacent iff gcd (i, j) = 1). Sander and Sander (2009) determined its kernel, the 0-eigenspace: its dimension equals the number of non-squarefree integers up to N, which is (1 - 6/π²) N + O (√N), with a simple twin-difference basis. Banerjee (2025) began the study of the -1-eigenspace, proving mult (-1, ΓN) ≥ π (N) - π (N/2) from a dominating true-twin clique, and posed the exact multiplicity as an open problem. This paper determines the -1-eigenspace. By complementation it is, away from the all-ones vector, the nullspace K of the conspiracy graph GN (vertices i, j adjacent iff gcd (i, j) > 1). We prove that K is spanned, in every odd order ω, by alternating prime-cube modes. Hence mult (-1, ΓN) = (π (N) - π (N/2) ) + dim K, making Banerjee's bound an equality. The proof is a single induction over a closed family of thin multiplicative slabs; its engine is a one-cell eigenvector identity by which the kernel annihilates exactly its window-edge cells, so the elimination requires no input on the distribution of primes in short intervals, in any dimension. The structural result is unconditional; only the asymptotic size of dim K is governed by the Hardy–Littlewood k-tuple conjecture. The box-generation machinery is related to switching components in discrete tomography and to lattice bases for the no-three-way-interaction model of algebraic statistics; the new contributions are the reduction of the coprime graph's -1-eigenspace to this windowed, order-constrained problem, and the keystone identity that confines the switching to the prime window. Prepared with assistance from a large language model (Claude, Anthropic) for organisation and extensive symbolic and numerical verification; all statements, proofs, and their interpretation are the author's and were checked independently.