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The manuscript constructs a continuum and lattice framework for chiral matter governed by exact or emergent multipole conservation. A central no-go theorem proves that exact scalar dipole conservation for a charged local fermion theory is incompatible with microscopic Lorentz invariance unless extra spatial structures, such as a foliation normal or crystalline tensor, are explicitly introduced into the background. To analyze the transport consequences, the text contrasts standard chiral pumping with fractonic kinematics. In a standard relativistic plasma, the Adler-Bell-Jackiw anomaly governs the chiral magnetic effect and is expressed as

\partial_{\mu}J_{5}^{\mu}=\frac{q^{2}}{16\pi^{2}}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}

. However, in a fractonic system that conserves both total charge and dipole moment, the local continuity equation requires a higher derivative tensor current form, given by $\partial_{t}\rho+\partial_{i}\partial_{j}J^{ij}=0$ .Using a coframe-based dipole gauge formulation, the study identifies a mixed axial-dipole term in the anomaly polynomial. This polynomial takes the form

I_{6}=\frac{1}{(2\pi)^{3}}F_{5}\wedge[\frac{k_{VV}}{2}\mathcal{F}\wedge\mathcal{F}+\frac{k_{DD}}{2}\delta_{ab}F^{a}\wedge F^{b}+\frac{k_{R}}{24}tr(R\wedge R)]

. The anomalous Ward identities generated by these descent equations reveal that axial charge converts directly into dipole and subdimensional transport channels rather than unrestricted vector currents. Equilibrium generating functionals and Kubo formulas then yield anomaly-induced constitutive relations. A notable result is the fractonic chiral magnetic effect, which produces a nondissipative tensor response

J_{anom}^{ij}=\frac{k_{DD}}{2\pi^{2}}\mu_{5}\mathcal{B}^{ij}

governed by a higher-rank magnetic field. The work ultimately classifies boundary and hinge inflow channels and confirms the theoretical predictions through numerical simulations of spectral flow, nonequilibrium dynamics, and hydrodynamic modes.