A piece of circuit reverse engineered around the buzzer will help, or at least a board photo. The buzzer may be passive, driven by some pulse series from the chip, or active DC driven with internal oscillator. In the case of passive buzzer, some simple peak rectifier circuit needs to be hacked to keep MOSFET or solid state relay or whatever input steady.

Do you have an oscilloscope?

So a while ago, I posted on fluorescent under cabinet lights for my laundry room. But now that I’m ready to buy the fixtures, I have a question. There is a 33 inch model, and a 42 inch, but if I bought two 33 inch I would fill basically the entire length underneath. But if I bought one 42 inch, I would have maybe 6 to 8 inch’s of unlit space on each side. Is it better to get two smaller fixtures, or one bigger one?

For the clock control chip it would either be some microcontroller with a proprietary firmware, so datasheet won't help you to get the functional description.
Or it would be a some clone of one of the common LED clock chip like LM8560 or similar (they differ in display configuration - the most recent and most commonly used I remember the LM8560 has a two common cathode "duplex" display, the earlier chips I have seen jut in datasheet used single common cathode)
What backup battery it uses? 3V or 9V?
What display it uses? LED or LCD? Is it multiplexed?

As far as I remember, (at least for the EU market) the "4100K" standard flew out of the window with the introduction of triphosphors in the 80's, when the rated CCT became 4000K.

And I don't think it makes that much an effect, mainly compare to what the uneven room furniture/paint color scheme does - I had a pair of 4000K tubes in a kitchen, but the cabinet door were painted red. It looked like there were different tubes in the fixture, just what the reflected red from the cabinet did when reaching the fixture. And I was even able to even that out by using one tube of 3500K and the second of 4000K.
And needed to be said, in fact I was the only one who was bothered by it at all ( tell me something about the lighting bug  ) and it was not my kitchen, so I put back the originals, after all the lamps were not the culprit and it was not that bad, I just noticed it...

So I don't think 4100 vs 4000K would be that much of a problem, I would guess a bit of dust collected on/in the fixtures and the unevenness effects would be way worse even with tubes exactly the same...

In other words it is the time needed for the filaments to warm up to a temperature enough to even initiate the ionization.
And as mentioned, it is not just about the ballasts alone, it is really a combination of the ballast OCV (so combination of ballast step up ratio and the actual mains voltage), the lamp properties, temperature, lamp surface contamination, the exact design and even tolerances of the fixture and the lamp position (the fixture metal work is acting as an external capacitively coupled electrode for the lamp), whether the fixturfe is properly grounded,...

Rapid start doesn't mean instant start though so keep that in mind. 

There is a S-like curve between lamp ignition voltage and cathodes temperature (which depends on preheat voltage and time). Lamp starts to glow when OCV is above this curve. Curve shape depends on a lamp type, exact lamp sample, grounded objects around and even glass surface resistance. And of course different ballasts will have different OCVs. Later ignition is beneficial on the lifetime as the lamp spends less time in glow discharge, but the risk of overall ignition fail increases.

We all know that fluorescent lamps sort of fade on while starting up on a rapid-start ballast. Most of the time, the lamps illuminate dimly as soon as they are turned on, and then they get gradually brighter or snap to full brightness. However, I have noticed that on some ballasts, there can be a split-second delay between when you turn the fixture on and when the lamps start glowing dimly. It doesn’t seem to have any adverse effects on the functionality of the fixture, just a weird effect.

One example of a ballast that behaves this way is the Universal 446-LR-TC-P ballast from 1985 that I installed in my bathroom light recently. I have noticed that after it has been switched off for a while, when you go to turn it on again, there is a ~0.5-1 second delay before the lamps begin starting up. If the lamps are already warm when starting up, the delay goes away.

Does anyone know why this delay occurs? I have noticed it on a handful of ballasts over the years, but the vast majority don’t seem to have this effect. Is it something that caused by a specific pairing of lamps and ballast, usage conditions, or just something intrinsic to the ballast?

Sorry if this might come as annoying but what has the better value?

A single GTE era Sylvania 400W deluxe white clear-bander mercury vapor bulb for $29.19 USD.

A lot of four GTE era Sylvania 250W deluxe white mercury vapor lamps; one of the lamps is a Safeline. They all have a standard fully coated top. The cost is $36.30.

Which listing has the best overall value?

I want to covert an old Dana Lighting Aerolite alarm clock-radio with built in folding halogen lamp to a deaf alarm clock. My plan includes sticking a relay across the lamp switch and tapping the alarm signal at the buzzer position on the switch to trigger the relay coil through a MOSFET.
Can you hunt up the datasheet for the MSM5096B alarm clock chip? I've been unable to find it.