Directional Probe Energy in Banded Fractional Hessians Fractional stiffness, angular completeness, spectral mixing, and the geometric-coercivity problem

Abstract

We introduce a banded fractional-Hessian framework for measuring the scale-dependent energy of tube-localized directional probes. For an -normalized probe concentrated in a unit-length tube of transverse radius , a band with fractional stiffness index contributes energy of order An -separated direction net on contains directions, so the sum of the separately normalized probe energies grows as The exponent is therefore a directional energy-growth exponent. It is not, by itself, a Minkowski or Hausdorff dimension. Converting the energy law into a geometric support theorem requires an additional bridge: a lower frame estimate for the collective probe family together with an upper bound relating collective Hessian energy to covering number, local capacity, or incidence multiplicity. We formulate that bridge explicitly, show how a correction exponent enters the conditional dimension bound, and separate it from the standard fractional-capacity implication . We then extend the framework to anisotropic and interacting-band operators. In the interacting case, the optimized probe cost is controlled by the smallest eigenvalue of a matrix-valued principal symbol. Its logarithmic slope defines a running effective stiffness exponent. Scale-dependent coupling and eigenmode competition can generate intermediate finite-window exponents, exact or avoided crossings, and plateau structures. These are spectral energy phenomena unless a separate geometric-coercivity theorem is proved. Introduction and Claim Boundary Directional abundance and fine-scale localization create two distinct mathematical costs. Directional abundance contributes the cardinality of an angular net, while fine-scale localization contributes the high-frequency cost of the operator used to measure the probe. The present paper isolates these two effects in a constructive model. Let be a finite-dimensional internal Hilbert space decomposed into orthogonal bands. Each band carries a positive fractional elliptic operator. We place normalized wave packets in unit-length tubes of radius , calculate their individual Hessian energies, and sum those energies over an -net of directions. The second implication is not automatic because many directionally distinct tubes may overlap in the same region. Their diagonal energies do not determine the energy of a collective superposition, and neither quantity is automatically bounded by the covering number of the support. The missing information is encoded in frame coercivity, cross terms, local energy capacity, and tube-incidence structure. The paper therefore has two aims. First, it establishes the operator-theoretic energy laws cleanly. Second, it states the exact additional hypotheses required before a dimension theorem can be claimed.