origin: empirical derivation and analytic origin of the integrity constant
why the integrity constant is inevitable
abstract
this paper derives the universal constant of integrity ℵ, the dimensionless scaling constant governing the balance between structural tension and integrity-dispersion in integrodynamics. across parts i–vii, ℵ appears as an empirically stable coefficient in the integrity free-energy functional F[b] = U[b] − ℵ S_I[b], yet its origin has remained unexplained. this paper provides both an empirical derivation and an analytic proof of its value. First, a multi-resolution renormalisation procedure-combining curvature-spectrum sampling, kernel-weighted dispersion estimates, and bootstrap stability testing—demonstrates a unique scale-invariant fixed point at ℵ = e. second, we show that ℵ arises as the unique solution of two independent theoretical constraints: (i) the fixed point of the integrodynamic beta function β(I) = dI / d log ℓ, ensuring coherence-scale invariance; and (ii) the extremiser of a variational principle equating structural curvature with integrity–dispersion under admissible perturbations. we prove that for any Θ ≠ ℵ, free-energy monotonicity fails, coherence regimes lose scale invariance, and the collapse manifold of directional implausibility becomes structurally inconsistent. these results establish ℵ not as a tunable parameter but as a universal structural invariant mandated by the geometry and dynamics of integrity itself. This paper therefore completes the integrity framework: ℵ ≈ e is the constant that renders the theory renormalisable, coherent across scales, and mathematically closed.