Beyond the Monochord: From Pythagorean Harmony to the Transport-Retention Adjunction

Seven intellectual traditions (the Pythagorean school, Plato, Euclid, Wittgenstein, Patanjali's Yoga Sutras, early Buddhism, and the Upanishads) are read as addressing complementary fragments of a single structural problem: under what formal conditions can validated knowledge survive transport across representation contexts? The formal core constructs verification spaces, whose open sets are the positively verifiable properties of a context, and the transport-retention adjunction, whose unit and counit locate round-trip epistemic loss as explicit subsets and whose fixed-point core is the claimwise lossless content. The results at this level are the classical image-preimage and frame adjunctions; the contribution is their verification-semantic reading, together with a comparative rational reconstruction of the traditions. The adjunction is then coupled to resource-constrained dynamics. Governor compliance preserves and restores the budget-accessible verification structure with explicit re-admission times; sustained violation contracts it along an equilibrium capacity curve whose marginal rate worsens without bound at a fold; collapse and recovery are hysteretic with closed-form thresholds; and a maintenance semantics carries the degradation into the executable topology itself, breaking continuity, openness and presentability of the lossless core by explicit criteria. The underlying scalar fold and hysteresis are proved here, including well-posedness across the nonsmooth clipping boundaries. Left open, as an explicit conjecture: the two-dimensional coherence of the capacity-indexed family, that is whether coarsening commutes with chained transport. A counterexample shows that the category of verification spaces and continuous open maps does not inherit pullbacks from Top, so any indexed formulation must name its base category. 52 pages. MSC classes: 18A40, 37G10, 00A30. Categories: math.CT, math.HO, math.DS.