A Loop-Inspired Polymerized Chronology Equation for Axisymmetric Chronology Surfaces

This Version 2.0 substantially revises the interpretation of the original preprint and introduces an explicit Taub / locally rotationally symmetric Bianchi IX benchmark. The work studies a loop-inspired spectral polymerization of the mixed extrinsic-curvature tensor, \[\mathsf P_\lambda(B)=\frac{\sin(\lambda B)}{\gamma\lambda},\] and derives a conditional bound on logarithmic metric contraction, \[\|D_\tau\ln q\|_{\mathrm{op}}\le\frac{3}{\gamma\lambda}.\] Within the proposed ansatz, this bound prevents a positive homogeneous metric eigenvalue from reaching zero in finite proper time. The construction is then tested against the classical Taub region of Taub-NUT spacetime, where the norm of the periodic fiber reaches zero at a Cauchy horizon in finite proper time while four-dimensional curvature invariants remain finite. A numerical evolution of a polymerized LRS Bianchi IX toy model is performed with \[m=\ell=1,\qquad\gamma=1,\qquad\lambda=0.1.\] The effective trajectory avoids the classical finite-time zero of the fiber norm and exhibits a transient bounce. However, the long-time evolution develops extreme anisotropy rather than approaching a stabilized regular configuration. The Taub benchmark also reveals an important limitation of the spectral polymerization: the modification becomes nonperturbative when the foliation-dependent extrinsic curvature becomes large, even though the four-dimensional invariant curvature remains small. At the point where \[\lambda |K_c|\sim1,\] the classical Kretschmann invariant gives \[\lambda^4 |\mathcal K_{\rm GR}|\simeq2.1\times10^{-5}.\] This demonstrates a foliation-dependent non-decoupling problem in the proposed ansatz. Version 2.0 therefore replaces the stronger chronology-protection interpretation of Version 1.0 with a more limited conclusion: bounded spectral polymerization kinematically prevents finite-time metric-eigenvalue collapse within the toy model, but its physical interpretation as a quantum-gravitational chronology-protection mechanism is not justified without a covariant, anomaly-free derivation. Version 2.0. Preprint, not peer reviewed.