The Geometric Identity of the Zero-Square-Sum Constraint under Scale Invariance: Isotropic Cone, Projective Quadric, and Intrinsic Quantum Structure — An Expository Note

This expository note contains no new theorems; its contribution is the synthesis of known mathematical results and their placement onto one object. Consider a system defined by exactly two postulates: (1) states are complex vectors x in C^N minus the origin satisfying the zero-square-sum constraint x_1^2 + ... + x_N^2 = 0, and (2) only ratios are physical, i.e., x and lambda x (lambda a nonzero complex number) represent the same state. The note identifies the mathematical identity of the resulting state space by citation alone. Five conclusions: (i) the solution set of the constraint is the complex isotropic (null) cone, and projectivization by postulate 2 makes the state space a smooth projective quadric Q_{N-2} in CP^{N-1} - a compact Kaehler manifold; (ii) by the general theory of geometric quantization (Kostant, Souriau; explicit compact example in Berezin), a system with compact phase space has, with no room for choice, finite-dimensional state spaces and discrete spectra: quantum discreteness is not an additional hypothesis but follows automatically from the two postulates - the system is not merely quantizable, it is born quantized; (iii) writing x_n = q_n + i p_n, the real part of the constraint is equipartition (sum q^2 = sum p^2) and the imaginary part is the vanishing of the dilatation generator (sum q_n p_n = 0), so scale invariance may already be contained in the imaginary part of the constraint; (iv) the function space on the cone is exactly the space of harmonic polynomials (Stein-Weiss), i.e., spherical harmonics - integer-labelled multiplets - so the 'harmonics' structure is a theorem of the state space, not a metaphor; (v) line bundles on compact projective manifolds are classified by integers (Chern classes), so the exact integrality of winding-type quantities is the integrality of cohomology itself; for N = 3 the projectivized cone is a conic isomorphic to the Riemann sphere CP^1, null vectors are squares of spinors (Cartan's construction, given explicitly), and the minimal nontrivial state space is isomorphic to that of a qubit (projective state space of spin 1/2). The note claims only kinematic identifications: no correspondence with any particular dynamics or physical system is asserted. Adjacent known structures (Penrose's twistor algebra) are noted.