I've thought about the possibility that the tranny drive gear may be worn out. Will that cause the speed to read further off the faster you drive? I found today that the rebuilt speedo head is around 5 mph off when you're going 10 MPH, the GPS ground speed is 15 MPH. 25 MPH on the gauge is 35 on the GPS, and going 58 on GPS, the gauge was reading ~45 MPH.

I think rather than seeing the speed being off you'd see intermittent or no speedometer operation if the gear was broken.

It was a fiasco, but I got the speedometer back and it's now working to within 2 MPH of the right speed. This is probably tire error. It's good enough for me. They sent my cluster back and it wasn't even completely together. I sent it back once again and they set it straight.

That's very close to road speed and better than many of those old speedos. If you read road tests from back in the day where speedometer error was checked it was common for them to be off by 5 mph or more at higher speeds, usually reading lower than actual. (The error is a percentage, so less at low speeds.)

I noticed that about mine, before they did the recalibration. The error increases as your speed increases. I couldn't find much about how they're set up inside, but I was able to find out they're magnetic. The cable spins a magnet, which pulls on another magnet, attached to a shaft and spring. The spring compresses as the spinning magnet tugs the stationary magnet and tries to move it. Kinetic energy (or rotational energy) converts to potential energy in the spring. You can measure it on a scale if you know how far it compressed at a known speed. Higher rational energy compresses the spring further. I realized they were pretty geniusly designed devices when I figured that out. It's just a conversion of energy, but it took some extensive physics knowledge to actually realize it. There's quite a bit of energy loss to friction as the speed gets higher, which explains how they deviate at high speed.

I haven't done that yet. I bought a new cable, in case it's worn out. My friend having his fluid leakage issue with his '77 Impala's speedometer cable has me a little skittish of messing with the cable if I don't need to. He hasn't been able to find someone who knows how to fix the problem. It seems fine on the speedometer head end, and there were certain speeds where the reading would stabilize, before the speedometer head was serviced.

The problem that the original cable is running in dried-up lube that is over 40 years old. Really best to take it out, clean it, lubricate it, and reinstall. I guess though you could try checking it the way you describe though you probably wouldn't be able to see small variations in the cable's rotation. It would be safest if you could position it so someone else can drive while you watch the cable. (You wouldn't want a kid or even a dog darting in front of you while eyes are on the cable.)

I took it for a quick drive in the country cul-de-sac across the road that's roughly 600 feet long and drove it slowly. The cable seemed to twist well enough to feel comfortable hooking it to the cluster. I did the same thing, then drove down the 55 MPH country road. The needle seems to move smoothly, but it's around 10 MPH slower than I'm going once it hits 35-40 MPH. Pull the cable and check it out, or do you think this a calibration issue? I put WD-40 in the sheath last summer to soften up old grease and try to lubricate it in case that was the issue.

It sure seems like a calibration issue. If it was the cable binding I'd expect a shaky speedometer needle and you're saying it's smooth. (If anything was restricting cable motion it would bind up and release, bind up and release, repeatedly like a spring.)

I'm assuming nothing was done to the car over the last 43 years to change gearing, like changing the rear axle ratio or installing a trans with different gearing or a big difference from original tire size. (On most cars of that era a different gear would be used at the transmission end of the cable for different rear axle ratios to make the speedometer read correctly.)

I don't think it was. The new tires I bought last year were the specified size I'm pretty sure. I'll upload a video I took while I was in the cul-de-sac.

Here's a quick recording I did on the straight stretch of the cul-de-sac. https://www.youtube.com/watch?v=1ofyG2Hdc44

Needle looks quite steady so it I'd say the cable is working OK. I'd check with the speedometer rebuilder about the calibration being off.

I'll send them an email and see what they say. When I accelerate in the video, I noticed there seems to be a very slight wiggle to the needle, but it stays in place at a steady pace. It's mechanical, and not a computer readout like any new car, so I would expect that to happen.

If the cable were cleaned and lubed that small amount of wiggle would probably go away but it's not bad enough to cause a bad speedometer reading.

I live in an odd, partly suburbanized rural area where 2-lane roads are extremely busy during the day and completely empty at night. I took the Catalina out for a good 8 mile drive after midnight so I could compare the speed to GPS and just monitor the needle at different speeds without irritating people or causing accidents. Haha. It's definitely working, but poorly calibrated. It feels like a new car, not hearing those annoying clunks, dings and taps coming from that gauge. The rebuilder said they will recalibrate it under warranty.

@chucktobais I pulled the cable and housing out of the car this afternoon. It's still in pretty good shape. I ran some hot water through the housing to get any dirt and bad grease out, a fair amount came out. I chased the water with a bit of WD-40 to make sure it all came out, then I sprayed the cable with a light coating of white lithium grease. I put it all back together and drove it a bit. Needle is bouncier than before when going slow. Any idea on why it would do that? If I had to guess, the lithium grease may not be the best lubricant for the cable.

Lithium grease is really not suitable. There are specific lubes for speedometer cables and most are graphite-based like the stuff you use in locks.

That's what I thought. I routed the tube as close as possible to how it was originally. I'll pull it apart again and clean it out, then use the cable grease. Any suggestions on how to get the white lithium grease out of the tube if I pull that out again as well?

Probably a solvent that won't leave residue, like brake parts cleaner, would do the trick. Then dry it with compressed air to make sure all the solvent is gone.

This is probably the closest I can find to a graphite based dry lube.
https://www.oreillyauto.com/detail/c/b-laster/b-laster-dry-lubricant/pbb0/8gs?q=Dry+lubricant&pos=0

They also carry a $3 squeezeable tube of graphite lube that looks to be for locks.
https://www.oreillyauto.com/detail/c/versachem/oil--chemicals---fluids/grease---lube/graphite-lubricant/1efe28ee2d3e/versachem-0-21-ounce-dry-graphite/ver0/10250

Will either of those work?

You can use the stuff that's used in locks or you can use purpose-made cable lube.

https://www.amazon.com/Ags-Kable-Ease-Cable-Lubricant-Tube/dp/B000ZO0EA2

Yeah, they'll always do that, bigger tires cover more ground for every revolution because the circumference is bigger. The deviation won't be noticeable if the change is really small. I found out today that the tires the car called for originally were FR 78-15s. I've never heard of that before. Haha. I put 205 75/R15s on last year, which I read is the closest modern tire to that size. Research says there's only 0.1" difference in the diameter of the, which is less than a percentage point difference.

D&M Restoration in South Carolina did the work. Was recommended by a few members in the Pontiac Catalina group on Facebook.

That gear looks to be in perfect shape. (They made 'em good back then!) I do believe different gears were used depending on the rear axle ratio but it doesn't seem likely that either would have been messed around with on that car.

I agree, I don't think that gear has seen daylight since March of 1979. Haha. I figured it was worth a look. I can't really tell if it was white or yellow when it was new. I lubricated the shaft and o-ring with with new transmission fluid like an oil filter before putting it back in. I'm starting to think it's a speedo head problem. I don't really want to send the thing back to the place on my dollar and have them check it again, then send it back for weeks and that ending up not the issue either. I'm actually a little concerned I may have inadvertently damaged the unit, the speedo cable sprang up a bit when I was checking it out last week and I heard it smack the input shaft. It made the same clunk noise as it did when it was broken from the impact. Hasn't affected the general error, but it's noticeably bouncier than before when going slow, both in the car and when using a drill. 

I think some time back you said you had purchased a new cable so you could use that (though it's probably inferior to the 1979-vintage part). However if the speedometer needle is bouncing around powered from a drill - assuming that's constant speed - then it does suggest there's still a problem in that speedometer head, and certainly the calibration is off.

I did buy a new cable. It's around a foot too long. I'm not sure how to trim it down to make it fit, so it's sitting around. I just used it to test the speedo again. I put the drill on the highest speed and ran it, I neglected that part before. I was using it on the lowest speed clutch setting and it hardly moved the needle on my last try. No real bounce on the re-test. 

I'd stick with the original cable if it's serviceable but if you need to use the replacement maybe you can give it a gentle curve underneath the car to take up the slack.

That's exactly what I think happened. They called me late Friday and it was at UPS on Monday, haha.