Wave Dynamics on the Connectome: From the Graph Laplacian to its Square Root

The Adaptive Holographic Theory (AHT) models neural activity as a wave field Ψ(t) ∈ ℂⁿ evolving under Ψ̇ = −iLΨ − γΨ + S(t), where L is the graph Laplacian. We show that this starting point is physically unmotivated. The neurobiologically correct equation is the real, second-order wave equation ψ̈ = −Lψ, which via the complex mode-amplitude substitution aₖ = Φₖ + iΦ̇ₖ/ωₖ is exactly equivalent to Ψ̇ = −i√L Ψ − γΨ + S(t), with √L = U diag(√λₖ) Uᵀ. The Hebbian learning rule is unchanged.

We verify this numerically and examine its consequences for the four principal claims:

(1) Dispersion. Confirmed: ωₖ = √λₖ with √L to numerical precision (<10⁻⁴% error); ωₖ = λₖ with L — inconsistent with cortical EEG.

(2) Attractor formation (P1). Confirmed at scale: with L, Δ = +0.010 (n = 100,000, clustered connectome) vs. Δ = +0.925 with √L — a 92-fold improvement.

(3) Habituation. Critical finding: with L, the experience observable ℰ(t) = |Ψ̇| grows monotonically — habituation is impossible. With √L, ℰ(t) settles stably — habituation is recovered. This makes L → √L necessary, not merely preferable.

(4) Miller bound. The formula is structurally unchanged. At n = 100,000: ‖Ψ‖_eq = 9.30 with near-linear capacity scaling. The original estimate N_max ≈ 6.6 is confirmed within range.

(5) Semantic association (P1d). Under √L, bistable threshold-crossing recall fails because the wave operator is oscillatory. An amplitude-selectivity metric resolves this: mean B-amplitude under A-cue grows 2.47× after fiber-only Hebbian binding (before: 0.303; after: 0.749); B/C selectivity doubles (43.5 → 107.5), while unrelated cluster C is unchanged (Δ < 0.1%).