A Pre‑Geometric Origin for Quantum Dynamics: Adjoint Memory, Temperley-Lieb Projection, and a Consistency Locked r=10

We introduce and develop the role of the adjoint left distributive monoid within a generative wholeness approach as a natural dynamical extension of quantum mechanics. By projecting the adjoint structure onto the Temperley-Lieb algebra, we obtain a dual product structure: sequential time--like evolution (composition) and generative formative acts (application). This structure preserves both the operator stacking of Temperley-Lieb and an``adjoint memory'' trace that retains information about formative origins even after projection.

We show that this adjoint memory provides a natural origin for the statistical weights of the Born rule, which emerge as an algebraic trace, rather than requiring it as an independent postulate. The resulting dynamics is unitary in the Temperley-Lieb sector but also carries dissipative components from the adjoint shadow, yielding a new, internally consistent equation of motion that provides a unified algebraic mechanism whose projection reproduces Born weights and yields a unitary–dissipative selection law.

Finally, we derive an essentially unique reader constraint—unique up to a trace‑preserving monoidal equivalence—corresponding to the tenth‑root loop value. This truncation aligns adjoint memory with physical fusion cutoffs. We conclude that the adjoint monoid structure not only addresses foundational issues but also establishes the groundwork for extensions beyond the Standard Model, to be developed in subsequent papers in this series.