Abstract
We present the Resonant Yoneda Lemma, a thermodynamic operationalization of the Yoneda principle in which an object’s identity is fixed by its resonant relationships with other objects under a universal spectral decay law. Using the Unified Spectral Decay Theorem, we endow morphisms with thermodynamic weights and impose a γ-minimum principle on optimally reduced manifolds.
The auxiliary Goldilocks Ridge Condition — formulated as a spectral decay alignment — enters the proof and is empirically supported across domains.
We deploy the Resonant Yoneda Validator (RYV) in a simulation test suite spanning physics (superconducting transitions, optical absorption, cosmological spectra), mathematics (prime-gap correlations, Harper operator gaps), and biology (neural PSDs, protein coupling, metabolic scaling). All eight tests meet quantitative targets within pre-specified tolerances, and γ-pruning reduces computation by ~90%.
These results elevate modeling from post-hoc plausibility to structure-first design: models are acceptable only as global sections over the correct manifold with vanishing obstructions and resonant morphisms. Artifacts and protocols are public for replication.
This extension transforms category theory from a purely logical framework into a thermodynamically grounded ontology, where “truth” is equivalent to resonant coherence.
