At the heart of the Refined Quantum Geometric Physics framework sits a question that had never been formally answered: why do the proton and neutron have the specific internal structure they do, and where do the numerical constants that govern all RQGP calculations actually come from? RQGP Paper D provides that foundation. The starting point is a classical result from pure geometry. The Descartes circle theorem describes a simple packing problem: given three mutually tangent circles, there are exactly two circles that complete the arrangement — one small circle that fits in the central gap, and one large circle that encloses all three from outside. Paper D applies this theorem to the RQGP representation of a nucleon, where each of the three quarks occupies one of three spatial axes and is represented by a circle whose size reflects its structural extent within the confinement plane. The small inner circle becomes the void — the cavitation pocket at the nucleon's core. The large outer circle becomes the confinement boundary — the nucleon's outer shell. The central algebraic result of the paper is clean and striking. If you require that the gap between the confinement boundary and the void has a specific fixed size — the thickness criterion — then only two quark configurations satisfy this requirement exactly. The first is the symmetric configuration where all three quarks are the same size: this is the proton. The second is the asymmetric configuration where one quark is half the size of the other two: this is the neutron. Every other configuration produces a different thickness. The proton and neutron are not assumed — they are the unique solutions selected by a single geometric condition. This immediately raises the question of whether the geometry and the network physics of RQGP — two completely different mathematical languages — are telling the same story. The network layer assigns each quark axis a boundary condition that governs how much flux it sources or absorbs. For the neutron, this value on the defect axis has always been taken as minus one-half. Paper D shows that if you ask instead which boundary value makes the internal flux pressures as uniform as possible across all three nodes — a Kirchhoff equipartition condition — the answer is exactly minus one-half. The geometric calculation and the network calculation, starting from entirely different principles, land on the same number. The Thickness-Variance Compatibility Theorem is the formal statement of this agreement. From these results, the paper derives the internal piston resistance — the routing cost for flux travelling between quark axes within a single nucleon — directly from the geometry. It also provides the geometric motivation for the external piston resistance, the routing cost for flux crossing from one nucleon to another in composite systems, supported by four independent consistency checks. The internal resistance is fully derived; the external resistance is geometrically motivated and verified but not algebraically forced from the Descartes formula alone. The paper is explicit about this distinction throughout. Additional results include a third geometric threshold: at a quark size of one quarter, the enclosing confinement circle ceases to exist entirely, interpreted in RQGP as the geometric proxy for hadronic deconfinement. The paper also documents that the neutron's void pocket is displaced precisely in the direction of the defect quark — a direct algebraic consequence of the asymmetric packing — and that the distance from the confinement centre to the defect quark is exactly the golden ratio. These are exact results, not approximations. Paper D is the geometric foundation that all subsequent RQGP papers build on. It establishes what is derived, what is motivated, and what must be supplied as structural input from prior work, maintaining throughout a strict accounting of the epistemic status of every claim.