  A novel non-minimal interaction of neutral right-handed fermion and abelian
gauge field in the covariant $\Theta$-exact noncommutative standard model
(NCSM) which is invariant under Very Special Relativity (VSR) Lorentz subgroup,
opens an avenue to study the top quark pair production at linear colliders.
Here the coupling denoted as $\kappa$ and the noncommutative (NC) scale
$\Lambda$. In this work, we consider two types of analysis, one is without
considering helicity basis and another, considering helicity states of the
polarized and unpolarized initial and final particles. In the first case, when
$\kappa$ is positive and for certain values of $\Lambda$, we arrived a specific
threshold value of machine energy (units of GeV) $\sqrt{s_0}$ ($ \simeq 2.52
~\Lambda + 39$ ) may be quite useful to probe NCSM with the unpolarized beam.
The statistical $\chi^2$ analysis of the azimuthal anisotropy is quite possible
when $\kappa$ takes negative value $0 >\kappa> -0.596$ which persuade a lower
limit on NC scale $\Lambda$ ($ 1.0\, \text{to} \, 2.4\,\text{TeV}$) at
$\kappa_{max}=-0.296 (95\%$ C.L) according to luminosity ranging from
$100\,fb^{-1} \text{to}1000\,fb^{-1}$ at machine energy
$\sqrt{s}=1.4\,\text{TeV}\,\text{and}\, 3.0\,\text{TeV}$. In another case, we
performed polarized beam analysis to probe NCSM in the light of following
observables azimuthal anisotropy, helicity correlation, and top quark helicity
left-right asymmetry. The polarization of the initial beam $\{
P_{e^{-}},P_{e^{+}}\} = \{-0.8,0.3\}( \{-0.8,0.6\})$ enhances the ranges of
lower limit on $\Lambda$, $i.e.\, 1.13 \, \text{to} \, 2.80\,\text{TeV}$ at
$\kappa_{max}$ alongside the $\kappa_{max}$ enhanced into $-0.5445 \,(-0.607)$
$95\%$C.L accord with luminosity and $\sqrt{s}$. Finally, we studied the
intriguing mixing of the UV and the IR by invoking $T(2)$ VSR Lorentz subgroup
symmetry on NC tensor $\Theta_{\mu\nu}$.
