Vortex creep heating in neutron star cooling with direct Urca processes in heavy neutron stars

Old, thermally bright neutron stars imply internal heating at late times. Among candidate mechanisms, vortex creep heating (VCH) provides a robust link between spin-down and frictional dissipation in the pinned inner-crust superfluid, yet its interplay with fast DUrca cooling in massive stars remains insufficiently explored. We (i) implement VCH in our cooling code and validate it; (ii) identify the physically consistent domain where the steady-state form $L_{\text{h}}=J|\dot\Omega_\infty|$ applies; (iii) quantify how $(B,P_0)$ regulate observable VCH signatures under DUrca cooling; and (iv) introduce a 3D representation that resolves degeneracies hidden in standard 2D projections. Cooling is computed with BSk24 and APR EoS, standard pairing gaps, and iron/carbon envelopes. VCH is modeled with $J\simeq10^{42.9\text{--}43.8}$ erg s, and a quantum-creep coverage fraction $f_{\text{Q}}(t)$ diagnoses when steady-state heating is valid. We survey $B=10^{10\text{--}13}$ G and $P_0=10$--$570$ ms for $1.4$ and $2.0\,M_\odot$, and compare with a curated set of ordinary pulsars with measured $(P,\dot P)$. Results: (1) Our implementation reproduces published VCH bands. (2) The $(B,P_0)$ validity boundary follows magnetic-dipole spin-down, confirming consistency with $|\dot\Omega|$. (3) DUrca+VCH maintains $T_{\text{s}}^\infty\gtrsim10^5$ K for $B\gtrsim10^{11-12}$ G up to $P_0\sim10^2$ ms. (4) The 3D representation shows that sources appearing coincident in $(t,T_{\text{s}}^\infty)$ occupy distinct $B$-layers, removing degeneracies. VCH can substantially reshape late-time thermal states when spin-down power remains high; its observability depends chiefly on $(B,P_0)$ rather than on mass alone. We provide a practical $(B,P_0)$ validity map for $L_{\text{h}}=J|\dot\Omega_\infty|$ and advocate treating $B$ as a co-equal axis in cooling analyses. (Shortened due to the arXiv words limit.)