Abdullah Scaling Equations: From the Sphere Volume Equation to General Shape Relations

Core Equation: By computing V, V′, V″ by hand and combining them algebraically, r²V″ + 2rV′ − 12V = 0 was constructed. This is a Cauchy–Euler equation; its roots are m=3 and m=−4; the general solution is V = C₁r³ + C₂/r⁴, with the sphere volume being the C₂=0 branch.Universality: The same equation returns zero for a rectangular prism, cylinder, cone, octahedron, ellipsoid, torus, and arbitrary degree-3 homogeneous polynomials (verified with SymPy) — it is not shape-specific but universal.Operator Form: Using the Euler scaling operator D̂ = x∂x+y∂y+z∂z, the equation factors as Â_xyz ≡ D̂²−D̂−6 = (D̂−3)(D̂+2); its fully expanded Cartesian form is x²∂x²+y²∂y²+z²∂z²+2xy∂x∂y+2yz∂y∂z+2zx∂z∂x−6. Coordinate-Free Form: In the single-variable case, 3VV″ = 2(V′)²; its multivariable generalization is 2∇VᵀH⁻¹∇V = 3V (H: the Hessian matrix) — all position variables are eliminated. General Degree-k Form: (k−1)∇VᵀH⁻¹∇V = kV — valid for length (k=1), area (k=2), volume (k=3), and 4D hypervolume (k=4). Physical Connection: The equation's second solution branch (C₂/r⁴) coincides exactly with the radial part of Laplace's equation for l=3 (octupole potential), since l(l+1)=12 — the r³ branch corresponds to the interior solution, and 1/r⁴ to the exterior (multipole) falloff.Circumference–Surface Area–Volume Relation: For the sphere, eliminating π and r algebraically gives S² = 6CV; the general shape-dependent form is S² = κCV, where κ is a shape-specific constant (analogous to an isoperimetric ratio).Honesty Note: Euler's homogeneity theorem and its second-order extension are established, published results; the specific combinations presented here (the operator, the coordinate-free identity, the degree-k generalization, the three-way C–S–V relation) were not found in this closed form during a literature search, and are presented as an original construction and application of known tools rather than as new theorems. ----- This work presents a series of scaling equations constructed starting from the sphere volume formula V(r) = (4/3)πr³. All ideas, questions, and the direction of discovery belong to Abdullah Baran (ORCID: 0000-0003-2935-1835, x.com/realABaran, [e-mail]); Claude (Anthropic) served only as a scribe, providing typesetting and mathematical verification.