Geometric Deficit Invariants from Recursive Projective Closure

When a geometric framework enforces scale invariance through projective normalization, it pays a price at every step: some structural information gets discarded. This paper shows that this price is not arbitrary, it accumulates into a well-defined, non-vanishing, dimensionless quantity called the geometric deficit, denoted δ*, which is intrinsic to the closure operator and cannot be tuned away without altering the structure of the framework itself. The result is purely mathematical. Starting from the fixed-point program of recursive geometric closure on projective configuration spaces, we prove that the deficit exists uniquely at the fixed point, is independent of initial conditions, normalization conventions, and coarse-graining operations, and converges to its fixed-point value at the same universal exponential rate that governs the closure mechanism. We further show that this deficit is the foundational source of non-adjustable dimensionless content in recursive projective closure: any intrinsic scalar quantity the framework produces must originate from this geometric obstruction. No physical constants are identified and no phenomenological interpretation is assumed. This work completes the structural picture opened by the foundational fixed-point existence results, establishing the intermediate layer between convergence and the emergence of non-trivial invariants.