Where can I find my minimum tire pressure?

Year-round, you want to maintain your tires at the recommended tire pressure (COLD) for the stated vehicle load. For the most part, it is a fixed value to maintain (not a range from there to maximum) with the exception it can be increased (not to exceed maximum) if the load is increased.
This value is normally on a tag found inside the driver side door frame or in the owner's manual.
OEM values provided are set to a value to provide the most comfortable ride under normal conditions with optimized handling. The MAX PSI on the tire's sidewall is the PSI that the tire will carry the most weight.

Under-inflated tires can damage the tires, lose traction, lower fuel economy, and in wet weather increase chance of hydroplaning.
It has been shown in testing that a higher number of blowouts occur in under-inflated tires than over-inflated ones.

Based on these factors, lowering the pressure below the automobiles manufactures designed value cannot be approached without CALCULATIONS based on tire design, vehicle weights, atmospheric and road conditions and expected use.
Doing so will present possible unsafe conditions. Again, it is safer to go higher than to reduce pressure.

Tire footprint and sidewall pressure needs to be set at an optimum profile for the tire. A tire that becomes under-inflated will roll too much into cornering wearing heavy on the sidewall. Under-inflation will cause cupping of the tire, leading to outer edge wear, shortening the tire's life. It presents a smaller surface footprint on the ground, reducing traction.

You can go as low as the minimum recommended pressure. Lower than that will start to damage the tire and increase wear.

Mac Demere of Popular Mechanics states;

Without proper air pressure, the internal components of the tire—fabric, steel, rubber, and composites—flex beyond their designed limits.

Tire pressure is important to maintain the shape of the tire under load. With improper pressure, the tire can not only wear unevenly, it can cause loss of traction.

Why Tire Pressure Matters

Useful pressure info should be printed on the side of the tire.

The sticker on the drivers' door will give you the proper tire pressures. It's not a matter of minimum versus maximum, but a matter of contact patch.

For instance, you may think that lowering the tire pressure when it rains would help, but it's the opposite that happens an you

The tire pressure that is indicated on the door is based on the stock tire size, so if you replace your tires with an identical size, just follow the recommended pressures (to be measured when the tires are cold)

The basic "mechanical engineering principle" behind tire pressures is fairly simple.

Ignoring the stiffness of the side walls, the air pressure in the tires has to support the full weight of the vehicle. The weight of the vehicle has to be balanced by the tire pressure, multiplied by the contact area between the tire and the ground.

If the pressure is lower than recommended, the contact area will be larger (that becomes very obvious when you have a completely deflated tire!). The increased area will affect the sideways friction force on the tires when the vehicle is cornering, increase the amount of flexing of the tire tread as the wheels revolve (because of the "flat spot" in contact with the ground), and tend to increase the load through the side walls, which causes more flexing of the walls and potentially more damage to them.

On the other hand if the pressure is too high, the contact area with the road will be reduced, which makes it more likely that the tires will start to slide when cornering or braking.

The "minimum pressure" will probably correspond to an the design conditions for an unladen vehicle, and the pressures should be increased as the amount of load increases - that factor is obviously is more significant for a truck than for a compact passenger car.

Higher pressures increase the tension around the circumference of the tire, which resists the air pressure inside. The maximum pressure will correspond to the safe limit for that tension, before the structure of the tire is damaged - this is the same basic reason as the maximum pressure limit for a metal compressed air tank.

The tire pressure also affects the stiffness of the tire in resisting changes in vertical loads caused by bumps in the road. The stiffness will affect the behaviour of the vehicle suspension, and therefore affect the handling.

Here's my take on this. As @alephzero mentioned to support vehicle weight properly and not damage tires you need to maintain certain contact area which will guarantee correct tire profile. Let's consider Honda Civic which has a curb weight of 2800 lbs. Obviously manufacturer doesn't specify pressure as a function of load, so we can safely assume that recommended 30 PSI means full load, which for this car is 850 lbs, i.e. full weight is 3650 lbs. So if there are only two people in Civic, its weight would be around 2800 lbs + 400 lbs = 3200 lbs. Thus to maintain the same contact area we only need 30 * 3200 / 3650 = 26 PSI. Obviously weight may not be distributed 50/50, so 26 PSI is a bit optimistic, but in general if you only drive alone or with one passenger 26 PSI is enough (not that I recommend it though).

On the other hand if a tire is underinflated it will heat up fast because or extra flexing and pressure will rise, providing some negative feedback.