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{\it If} the total number of punctures($N$) of a quantum isolated horizon isconsidered to be a macroscopic parameter alongside the Chern-Simons level($k$)or equivalently classical area$(A_{cl})$ a strict analysis of the {\itmicrocanonical} ensemble reveals that the {\it microcanonical} entropy has theform $S_{MC}=A_{cl}/4\ell_p^2+N\sigma(\gamma)$, only for values of theBarbero-Immirzi(BI) parameter$(\gamma)$ greater than a certain number. It isargued that the term $N\sigma(\gamma)$ must be negative definite, which leadsto the bound on the BI parameter.

The Microcanonical Entropy of Quantum Isolated Horizon, `quantum hair' $N$ and the Barbero-Immirzi parameter fixation