This preprint is the third part of the Constructive Tensor Theory (CTT) series, titled "Constructive Continuum Mechanics."It extends the geometric and ontological foundations established in Part I and Part II by reformulating the entirety of continuum mechanics using pullback-defined tensor structures. The manuscript introduces constructive definitions of stress, deformation, conservation, and constitutive laws. It reinterprets elasticity and viscosity as structurally distinct phenomena arising from commutative and non-commutative pullback paths. The classical Maxwell model is reconstructed as an observational approximation of dual geometric responses. Further chapters address entropy, chaos, and discontinuity as manifestations of degeneracy and breakdown in pullback coherence. Observation and consciousness are modeled as recursive pullback layers, culminating in a constructive epistemology and ontology, in which existence is defined by constructibility. E-mail: s183018[at]stn.nagaokaut.ac.jp This document is part of the Constructive Tensor Theory (CTT) project. For continuity, please also refer to: Part I: Mathematical Foundations [DOI:10.5281/zenodo.15505202, https://zenodo.org/records/15505202] Part II: Observation Geometry and Interpretations [DOI:10.5281/zenodo.15420549, https://zenodo.org/records/15522168] For more on applications of CTT to unresolved problems in AI semantic grounding and temporal modeling, visit:https://endohirokinut.github.io/CTT_Discovery_Trigger/