Fixing the Structural Efficiency Constant by Invariant Normalisation on ℂℙ¹

Geometric normalisation companion to Unified Dynamics from Time Generation. This work determines the structural efficiency constant κ by identifying the physical time yield of a recording event with the invariant measure of quantum distinguishability resolved by that event. Under a single physical postulate—the Yield Principle—vacuum symmetry selects the Fubini–Study measure on the pure state space of a binary recording degree of freedom, ℂℙ¹. The invariant volume Vol_FS(ℂℙ¹) = π then fixes the constant uniquely: κ = π The result eliminates the final undetermined dimensionless parameter in the gravitational scalar sector of the generative time framework. With κ fixed, the bridge law, Yukawa coupling, species identity, and cosmological constant relation become closed-form expressions involving π and experimentally measurable quantities. Main contributions • Derivation of κ as an invariant normalisation from the geometry of projective Hilbert space.• Identification of binary record formation with ℂℙ¹ state space via vacuum symmetry.• Independent consistency check through the species–coupling identity, predicting Nₑff ≈ 3π³ and implying three generations with Majorana neutrinos as a testable consequence.• Emergence of a single remaining scale ratio 𝓡 = Mₚ / mφ (“record hierarchy”) governing the separation between quantum and gravitational scales.• Closed-form scalar-sector predictions including ε = 1/(6π²) and ρΛ ∝ π⁻⁸. Within the broader programme, this paper provides the geometric closure of the bridge law and establishes the invariant normalisation underlying the statistical and cosmological extensions developed in companion works.