Part V — The physics of integrity: collapse, curvature, and the universal constant of integrity

This paper develops the dynamical physics of integrity fields. Behaviour is modelled as a continuous field $b(x,t)$ evolving under the integrodynamic functional
\[
\mathfrak{F}[b] = U[b] - \aleph\, S_I[b],
\]
where $\aleph \approx 2.70$ is the empirically observed (and analytically derived in part VIII) integrity constant -- the universal bound on sustainable deformation across epistemic, organisational, and institutional systems. The theory unifies curvature, contradiction, drift, entropy, and collapse into a single field-theoretic framework. It characterises:

• the collapse manifold and finite-time singularities in integrity geometry;
• coupled-field cosmology, including fused horizons and shared collapse basins;
• the renormalisation flow with a non-trivial fixed point $I^{\ast} \approx \aleph$;
• horizon formation, coherence radii, and diagnostic operators linking curvature, divergence, and structural drift;
• network percolation of collapse, described through spectral thresholds and the integrity Laplacian.

Simulations validate the theoretical predictions: contradiction concentrates as curvature; drift accelerates under imbalance; entropy exchange reshapes global geometry; and collapse emerges when information load exceeds a system’s dispersion capacity.

Part V completes the cosmological extension of the integrodynamic framework, demonstrating that once integrity is treated as a field, a predictive and falsifiable physics follows.

[References in progress]

Keywords: integrity physics; integrodynamics; collapse manifold; contradiction operator; universal integrity constant; ℵ\alephℵ; coherence horizon; spectral curvature; structural drift; divergence field; entropy ratio; renormalisation fixed point; collapse cosmology; coupled integrity fields; horizon fusion; annihilation manifold; vanishing nexus; organisational physics; epistemic field theory