Here are the issues to keep in mind.

1. You've done a pretty good job of addressing the 33% (1/3) vs 25% (1/4) arithmetic, which is one of the hardest to understand. That being for the case that the photons' hidden variables are exactly the same (or opposite, but let's ignore that for simplicity). So good job.

2. You are absolutely correct that in a hidden variables theory, the left and right photons potentially are not identical. In your example, that coincides with your (000,000), (000,001) (000,010) cases (and variations). In that case, you might be able to wiggle some changes to bring the hidden variables minimum down from 33% to closer to the actual of 25%. But...

3. There is also the requirement of "perfect" correlations with this scenario. Measuring both photons at the same angle settings - any same settings - always produces a matching (100%) result! That of course negates point 2. In your example, such cases as (000,001) or (000,010) can't/don't physically occur. You only see (000,000), (010,010), etc.

And in fact, it was the perfect correlations that were discovered first - in 1935, by Einstein and 2 others in the paper known as EPR. Bell discovered point 1 around 1964. They both must be met. So local realism cannot model the actual results of experiments.

A few more point. Around 1988, an important new paper was written introducing the GHZ Theorem. The also refutes local realism, and one of its authors won a Nobel for this and other work on entanglement. In GHZ, statistical probabilities are not involved. In every single GHZ run, Quantum Mechanics predicts the exact opposite of the local realistic assumption. Needless to say, experiment supports QM.

If you find this useful for anything, I also have a web page that loosely mirrors the video example you provided. It is called Bell's Theorem with Easy Math.

Here are the issues to keep in mind.

1. You've done a pretty good job of addressing the 33% (1/3) vs 25% (1/4) arithmetic, which is one of the hardest to understand. That being for the case that the photons' hidden variables are exactly the same (or opposite, but let's ignore that for simplicity). So good job.

2. You are absolutely correct that in a hidden variables theory, the left and right photons potentially are not identical. In your example, that coincides with your (000,000), (000,001) (000,010) cases (and variations). In that case, you might be able to wiggle some changes to bring the hidden variables minimum down from 33% to closer to the actual of 25%. But...

3. There is also the requirement of "perfect" correlations with this scenario. Measuring both photons at the same angle settings - any same settings - always produces a matching (100%) result! That of course negates point 2. In your example, such cases as (000,001) or (000,010) can't/don't physically occur. You only see (000,000), (010,010), etc.

Answering your question then: Yes, entangled photon pairs are always equal (or opposite).

And in fact, it was the perfect correlations that were discovered first - in 1935, by Einstein and 2 others in the paper known as EPR. Bell discovered point 1 around 1964. They both must be met. So local realism cannot model the actual results of experiments.

A few more point. Around 1988, an important new paper was written introducing the GHZ Theorem. The also refutes local realism, and one of its authors won a Nobel for this and other work on entanglement. In GHZ, statistical probabilities are not involved. In every single GHZ run, Quantum Mechanics predicts the exact opposite of the local realistic assumption. Needless to say, experiment supports QM.

If you find this useful for anything, I also have a web page that loosely mirrors the video example you provided. It is called Bell's Theorem with Easy Math.

Here are the issues to keep in mind.

1. You've done a pretty good job of addressing the 33% (1/3) vs 25% (1/4) arithmetic, which is one of the hardest to understand. That being for the case that the photons' hidden variables are exactly the same (or opposite, but let's ignore that for simplicity). So good job.

2. You are absolutely correct that in a hidden variables theory, the left and right photons potentially are not identical. In your example, that coincides with (000,000), (000,001) (000,010). In that case, you might be able to wiggle some changes to bring the hidden variables minimum down from 33% to closer to 25%. But...

3. There is also the requirement of "perfect" correlations with this scenario. Measuring both photons at the same angle settings - any same settings - always produces a matching (100%) result! That of course negates point 2. In your example, such cases as (000,001) or (000,010) can't/don't physically occur. You only see (000,000), (010,010), etc.

And in fact, it was the perfect correlations that were discovered first - in 1935, by Einstein and 2 others in the paper known as EPR. Bell discovered point 1 around 1964. They both must be met.

Here are the issues to keep in mind.

1. You've done a pretty good job of addressing the 33% (1/3) vs 25% (1/4) arithmetic, which is one of the hardest to understand. That being for the case that the photons' hidden variables are exactly the same (or opposite, but let's ignore that for simplicity). So good job.

2. You are absolutely correct that in a hidden variables theory, the left and right photons potentially are not identical. In your example, that coincides with your (000,000), (000,001) (000,010) cases (and variations). In that case, you might be able to wiggle some changes to bring the hidden variables minimum down from 33% to closer to the actual of 25%. But...

3. There is also the requirement of "perfect" correlations with this scenario. Measuring both photons at the same angle settings - any same settings - always produces a matching (100%) result! That of course negates point 2. In your example, such cases as (000,001) or (000,010) can't/don't physically occur. You only see (000,000), (010,010), etc.

And in fact, it was the perfect correlations that were discovered first - in 1935, by Einstein and 2 others in the paper known as EPR. Bell discovered point 1 around 1964. They both must be met. So local realism cannot model the actual results of experiments.

A few more point. Around 1988, an important new paper was written introducing the GHZ Theorem. The also refutes local realism, and one of its authors won a Nobel for this and other work on entanglement. In GHZ, statistical probabilities are not involved. In every single GHZ run, Quantum Mechanics predicts the exact opposite of the local realistic assumption. Needless to say, experiment supports QM.

If you find this useful for anything, I also have a web page that loosely mirrors the video example you provided. It is called Bell's Theorem with Easy Math.