Gulfstream-IV Captain reported erratic pressurization indications and checklist that did not match the situation during descent. Captain landed safely.

Narrative

On Day 0; at approximately XF23; I; Person A; while acting as PIC of a Gulfstream IV-SP; experienced a pressurization anomaly while on descent into the ZZZ airport. The crew has sufficient rest the day prior to the flight; and arrived at the aircraft at around XA30 to start pre-flighting the aircraft for a proposed 1H00 departure. All exterior and interior checks; as well as systems checks were completed per Gulfstream's Pilot Checklist. No anomalies were noted; and the pressurization checks passed with no defects noticed. The flight left the FBO ramp at around XF41 for departure on Runway XX at ZZZ1 airport. The aircraft then had climbed to altitude of FL410. The pressurization system operate normally and on schedule the whole time. At approximately XF13; we received a descent clearance to FL380. As selected a descent vertical speed of 1500 feet per minute; the autothrottles retarded the power and I felt a small pressure bump in my ears. I looked up and noticed a Bleed Manifold difference of 10 PSI from the left manifold of 24 PSI and the right manifold reading 14 PSI. I looked at the pressurization control panel and backed it up with the 'Triple-Needlel gauge and made the decision to expedite our descent. I informed the SIC to request a lower altitude to FL240 or lower. ATC had asked if we had any issues; and I stated that we are requesting the descent due to erratic pressurization readings and indications. We were then cleared to FL240. As I continued the descent to the lower altitude; the auto-throttles continued to regard the power to flight idle. This reduction in power resulted in less air coming into the cabin for sufficient pressurization. I then disconnected the autothrottles and pushed the power up to a point where I could see the bleed pressure increase. The pressurization system then started to climb to cabin at an increasing rate of up to 1700 feet per minute. I then tried to control the cabin rate through the Pressurization Control Panel Tapes. No change in rate. I then switched to ADC#2 and no change was noted. I then went to manual control to try and control the rate of pressurization and still no change. At this point we were descending through approximately FL290; and I then activated the right side anti-ice; in the attempts to increase bleed air by opening up the 10th stage air. The duct pressure increased to around 42 PSI on both sides; and the cabin started showing a slight descent in the cabin. The cabin pressure had rose to about 14100 and activated the cabin masks. We then received a Cabin Pressure Low annunciation; followed by Cabin Diff 9.8 (briefly flashed then extinguished); and a PAX CABIN OXYGEN on CAS (Crew Alerting System) message. As soon as we leveled off at FL240; the cabin had started a descent and was under 10;000 feet and continuing to descend. The previously stated CAS messages had all extinguished and were no longer displayed. We had now been level at FL240; for less than a minute when I then selected the Pressurization to 'Landing' mode and selected ADC#1 again and took it out of Manual mode. The system started operating normally at this point with the anti-ice selected on for increased air flow. No issues reoccurred until below 7;000' MSL; where the cabin started to climb and descend with no reason. We were then setting up for a visual approach to ZZZ airport when below 1000' MSL; the cabin started descending around 1200 feet per minute. Once we touched down; the cabin then started to show a climb again. The Cabin Altitude on the overhead pressurization panel showed 500'. Airport elevation was XXX' MSL. While taxiing in; I selected the pressurization to manual and opened the outflow valve full open. The cabin still showed a slight climb. I then deselected to Bleed Air to off; and it made no change to the climb. After approximately 10-15 seconds later; the cabin started to stabilize and finally fully equalized when the engines were shutdown. It of note to include as well; that while wewere at FL240; and had stabilized; the cabin descended to Sea Level; though showed a differential of 8.62. ATC had asked if we needed any assistance or if we were declaring an emergency. At this point; we had leveled off to a Sea Level altitude; aircraft had started acting normal; and oxygen was not needed. I informed ATC; that we are not declaring an emergency; that we were responding to an indication anomaly and needed no further assistance. I took this time to quickly go back into the cabin to check and ensure the passengers were OK and no one needed any medical assistance. Once I checked on each passenger; I informed the Lead and other passengers within earshot what had happened. Checklist was consulted and no checklist matched our situation except Loss Of Automatic Pressurization. We followed checklist and saw we had already complied with items.