Step 1: Recall the neutron density crossover effect.
In a gas bearing zone, the neutron log reads an apparent porosity lower than the true porosity because gas has a very low hydrogen index compared to oil or water. At the same depth the density log reads an apparent porosity higher than true porosity because gas is much lighter than the rock matrix. When both curves are plotted on compatible scales, the neutron curve moves toward lower porosity and the density curve moves toward higher porosity, so the two curves swing toward each other and cross. This is called the neutron density crossover, and it is the standard, well known gas indicator in log interpretation.
Step 2: Analyze option (A), neutron and density.
As shown in Step 1, this pair does show a clear crossover in gas zones. So option (A) is excluded from the answer because it DOES show crossover.
Step 3: Analyze option (B), caliper and resistivity.
The caliper log records the physical diameter of the borehole and detects washouts or mud cake buildup. The resistivity log records the electrical resistivity of the formation and fluids. These two logs respond to completely different physical properties and have no defined crossover relationship tied to the presence of gas. So option (B) does NOT exhibit a gas crossover.
Step 4: Analyze option (C), neutron and self-potential.
The self-potential or SP log responds to the electrochemical and electrokinetic potential difference between the borehole mud and the formation water, and is mainly used to pick shale versus permeable sand boundaries. It has no scale-based crossover behavior with the neutron log in gas zones. So option (C) does NOT exhibit a gas crossover.
Step 5: Analyze option (D), caliper and gamma-ray.
The gamma ray log measures natural radioactivity and is mainly a shale or clay indicator, while the caliper again measures hole diameter. Neither log carries a scale calibrated to show a gas crossover, so no crossover effect is defined between them. So option (D) does NOT exhibit a gas crossover.
Step 6: Final conclusion.
Only the neutron and density pair is designed to show a crossover in gas zones. The pairs that do NOT exhibit crossover are caliper-resistivity, neutron-self potential and caliper-gamma ray.
\[ \boxed{\text{(B), (C) and (D)}} \]