Pregeometric Relational Dynamics and Universal Causal Structure. Spectral Constraints and Conditions for a Common Low-Energy Cone

This work develops a concrete spectral framework for pregeometric relational dynamics, asking whether quasilocality, spatial geometry, and a universal causal structure can arise as effective properties of a closed system whose fundamental description contains relations rather than a pre-existing spacetime. The central problem is not the numerical value of the speed of light, but something deeper: why physically distinct low-energy excitations should share the same characteristic cone. Within finite weighted networks, the analysis turns this question into explicit spectral and matrix conditions and, at the same time, rules out several apparently natural mechanisms. In particular, rearranging a common network cannot equalize the propagation speeds of independent fields when their kinetic coefficients remain distinct: c₁²/c₂² = α₁/α₂ is unaffected by changes of the common graph Laplacian. Likewise, a homogeneous positively coupled Landau potential does not by itself generate stable relational sectors. A key result is a quantitative formulation of spectral quasilocality. Weakly coupled relational sectors remain distinguishable when a gap separates a low-frequency subspace from the rest of the spectrum, with its stability controlled by ‖P − P₀‖ ≤ ε/(γ − ε). The paper then studies interacting multicomponent systems and shows how coupling can suppress relative degrees of freedom, leaving a collective massless excitation. In the exactly solvable two-component model, the infrared branch takes the form ω₋²(ν) = v₀²ν − Kν² + O(ν³): a finite universal low-energy mode appears even when the microscopic component speeds differ. Crucially, however, this represents a reduction to one collective massless branch, not an equalization of several independent massless fields. The central universality theorem addresses the genuinely multibranch case. After kinetic normalization and elimination of gapped modes, the leading propagation of the r-dimensional massless sector is governed by the compressed stiffness operator D₀₀. All r massless modes possess the same leading characteristic cone if and only if D₀₀ = v² Iᵣ. Equivalently, for any basis Q of ker U, the condition is QᵀBQ = v² QᵀAQ. If a spatial continuum limit exists with ν(k) = βhⁱʲkᵢkⱼ + O(|k|⁴), the resulting common cone satisfies −ω² + c²hⁱʲkᵢkⱼ = 0, with c² = βv². This converts the emergence of a universal low-energy causal structure from a qualitative expectation into a precise and testable spectral criterion. The framework also closes the dynamics by allowing the network relations themselves to evolve: energy can be exchanged between field degrees of freedom and relational degrees of freedom while the total Hamiltonian remains conserved. At the same time, the analysis identifies an important boundary of the construction. On a finite network, analytic response functions generally develop nonzero propagation tails, so a common infrared cone is not equivalent to strict microscopic causality. Lorentzian geometry therefore remains an emergent, conditional interpretation requiring a controlled continuum limit, while the origins of time orientation, dimensionality, realistic field symmetries, and gravitational dynamics remain open. The significance of the result is thus both constructive and restrictive: it isolates the mathematical conditions under which universal causal kinematics can emerge and sharply separates them from the additional dynamical ingredients still required for spacetime and gravity.