Directional Fractional Dimension: Angular Structure, Occupation, and Scaling—Analysis, Direction Bundles, and Canonical Radial Reduction

This paper studies measured direction states in real projective spaces and Grassmannians. A geometric extractor assigns chord, tangent, segment, or higher-order direction states. For a measurable state \(Dⁿ⁻¹\) and an observing probability \(μ\), the occupation and scaling coordinates are\[OccDimμ(D)=1+(n-1)μ(D),(D)=1+HD.\]They record respectively the protocol mass and projective scaling of \(D\). In the plane, a connected projective arc occupying one quarter of \(P^1\) has occupation dimension \(5/4\); a disconnected state of the same mass has the same scalar value and a different direction topology. For a geometric set \(E\), the central invariant is the occupation triad\[OccDirμ,(E)=(OccDimμ( Dch(E)),OccDimμ( Dₜₐₙ(E)),OccDimμ( Dseg^(E))).\]The three states are generated respectively by point pairs, local limiting secants, and contained affine intervals. The segment coordinate is bounded above by both the chord and tangent coordinates. Under normalized planar angular measure, a nondegenerate unit segment, the unit circle, and the filled unit square have respective triads, for \(0<≤1\),\[(1,1,1),(2,2,1),(2,2,2).\]For nonempty sets, the full triad state is recovered exactly from its family of Dirac-protocol occupation triads. On the class of all sets, the nonemptiness indicator is adjoined. An angular protocol \(ρ\) refines occupation by carrying direction weights and the principal-angle kernel. Its moment matrix gives\[AngDir(ρ)=1+r_ρ(1-tr(M_ρ^2)),EffDir(ρ)=tr(M_ρ^2)⁻¹.\]These quantities are invariant under rotations and independent translations of direction-bearing components; every parallel-line configuration has angular direction dimension one. The full pair-angle law and weighted angle matrix retain distinctions compressed by the scalar readings. The theory establishes direction-first factorization and exact union defects. It also develops the direction measure algebra, graded states, protocol duality, direction channels, finite-resolution recovery, and higher-order Grassmannian extensions. The second moment\[A_μ=∫ vvT\,dμ([v])\]generates anisotropic elliptic, spectral, heat, wave, Fourier, and variational theories. For an occupied event \(D\), the unnormalized moment \(B_μ(D)\) satisfies\[trB_μ(D)=μ(D).\]Direction bundles retain base–direction incidence and separate carrier topology from protocol mass. For compact \(D\), canonical radial reduction satisfies\[H R(D)=1+HD.\]The full segment-direction class reduces to the unit ball and has directional value \(n\). On the bi-Lipschitz radial realization class \(E=f( R(D))\), one has\[HE=1+HD.\]For segment-full sets, ambient equality is known in dimensions two and three and remains open in dimensions at least four. Direction, scale, and multiplicity are complete in the declared positional quotient, and exact global chord-direction preservation forces a similarity. Keywords **direction states; projective space; occupation dimension; angular protocols; direction bundles; radial reduction.**