Molecular trust is frequently equated with physical unclonable functions (PUFs) that treat fabrication variability as a source of binary authentication. While powerful as security primitives, such models capture only one dimension of trust when computation becomes material. In molecular and biological systems, variability is stochastic, measurement may be invasive, artefacts can be re-synthesised, and verification unfolds within institutional workflows. Unclonability and challenge–response authentication are therefore only part of how the material memory sits within a larger ecology of trust. This paper introduces Trusted Molecular Memory (TMM) as a conceptual framework for reasoning about molecular trust in hybrid bio-digital systems. TMM does not propose a new authentication mechanism. Instead, it examines how molecular traces participate within systems of documentation, interpretation, accountability, and governance, surfacing hidden assumptions in prevailing molecular verification models. Drawing on analysis of existing PUF-style systems, lab practice, and scenario-based reasoning across lifecycle of molecular artefacts, we distinguish unclonability-based authentication from stochastic, process-bound trust. From this analysis, we derive a five-pillar architecture operationalised through a Molecular Chain of Custody (M-CoC) framework. Three stress-test scenarios demonstrate how trust accumulates, degrades, and remains contestable across regeneration, maintenance, and institutional change. We argue that molecular trust is not only object-bound but also procedural and institutionally mediated. Molecular traces contribute to trust when interpreted within documented transitions, attributable decisions, and recognised governance structures.