PAC–μ⁴ Materials: A Spectral-Compression Paradigm for Designing Function, Stability, and Information Efficiency

We develop a materials-design paradigm grounded in the PAC–μ⁴ framework (Perception–Action/Agency–Compression), which models any material as a spectral object generated by a positive self-adjoint operator K≥0K\ge 0K≥0. Dynamics is bridged by Ω:=K1/2\Omega:=K^{1/2}Ω:=K1/2 with H=ℏΩH=\hbar\OmegaH=ℏΩ. Two descriptors—spectral compression rate μ\muμ and scale-dependent spectral dimension ds(ℓ)d_s(\ell)ds(ℓ)—jointly quantify (i) how compactly a material’s relevant modes encode its functionality and (ii) how microstructure and topology distribute those modes across scales. We formalize CSS (Causal–Safe Shaping) constraints as a feasibility cone unifying stability and controllability, and introduce performance metrics—net compression gain Δμ\Delta\muΔμ, μ\muμ-curvature κμ\kappa_\muκμ (robustness), control efficiency εμctrl\varepsilon_\mu^{\rm ctrl}εμctrl, and spectral mismatch δspec\delta_{\rm spec}δspec. Building on these, we propose three disruptive material programs: μ-optimal metamaterials, CSS-stable quantum materials, and spectrally programmable matter. We give a unified high-throughput spectral-screening and AI inverse-design workflow; derive compact, task-agnostic figures of merit linking {μX,ds,Πedge}\{\mu_X,d_s,\Pi_{\rm edge}\}{μX,ds,Πedge} to superconductivity, thermoelectrics, ion conduction, and fracture toughness; and specify falsification criteria. Five work packages (W1–W5) provide immediate experimental roadmaps and KPIs. This establishes a reproducible, system-level methodology for designing materials that simultaneously optimize function, stability, and information efficiency.