This monograph applies the Symmetric Core, Native Calculus, and cloning‑based algebra developed earlier in the series to a set of classical hard problems in mathematical physics, with the explicit aim of achieving maximal exact reduction rather than claiming solved status. The unifying framework is the Garycki Manifold, a coordinate system in which problems such as the Yang–Mills mass gap, the three‑body problem, and classical gravity are expressed using the same operational objects. The key algebraic element is the Apow Hermit unit τ, arising from the NC quadratic, whose horizontal cloning generates non‑flat interaction structure. For Yang–Mills theory, the monograph reformulates the root‑sector dynamics in Native Calculus coordinates, where the interaction potential becomes an affine Toda potential. Within this formulation, positivity of the mass gap is established structurally, and the remaining open step is identified as the decoupling of off‑diagonal fields. For the three‑body problem, the same unit τ resolves the triple‑collision branch point by making the Riemann‑sheet structure explicit. The shared algebraic core with Yang–Mills explains why both problems resist standard perturbative methods. The monograph also reformulates Newtonian gravity in the star sector of the Symmetric Core. In native coordinates u=lnru= ru=lnr, the gravitational potential becomes a pure exponential, and the Kepler problem is shown to be exactly equivalent to a Morse oscillator, yielding an integrable formulation. M16e isolates the single remaining obstruction in each problem, making further progress structurally well‑posed.
Paweł Łukasz Garycki (Fri,) studied this question.
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