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This is a spinoff of this earlier question of mine.

In particular, is there a description of which measures on $\omega_2^{L(\mathbb{R})}$ are potentially club? A reasonable guess is that the above argument with $Col(\omega,\omega_1)$ constructs such a measure; and under appropriate hypotheses (the nonstationary ideal on $\omega_1$ is $\omega_2$-saturated and $\mathcal{P}(\omega_1)^\sharp$ exists) this is verified by calculating the projective ordinal $\delta^1_2$ - since $L(\mathbb{R})$ computes $\omega_2$ correctly, $\omega_1^{L(\mathbb{R})[G]}=\omega_2^V=\omega_2^{L(\mathbb{R})}$. However, I don't know a snappy description for this measure. So we may ask:

This is a spinoff of this earlier question of mine.

In particular, is there a description of which measures on $\omega_2^{L(\mathbb{R})}$ are potentially club? A reasonable guess is that the above argument with $Col(\omega,\omega_1)$ constructs such a measure; and under appropriate hypotheses (the nonstationary ideal on $\omega_1$ is $\omega_2$-saturated and $\mathcal{P}(\omega_1)^\sharp$ exists) this is verified by calculating the projective ordinal $\delta^1_2$ - since $L(\mathbb{R})$ computes $\omega_2$ correctly, $\omega_1^{L(\mathbb{R})[G]}=\omega_2^V=\omega_2^{L(\mathbb{R})}$. However, I don't know a snappy description for this measure. So we may ask:

Long version:

In particular, is there a description of which measures on $\omega_2^{L(\mathbb{R})}$ are potentially club? A reasonable guess is that the above argument with $Col(\omega,\omega_2)$ constructs such a measure; and under appropriate hypotheses (the nonstationary ideal on $\omega_1$ is $\omega_2$-saturated and $\mathcal{P}(\omega_1)^\sharp$ exists) this is verified by calculating the projective ordinal $\delta^1_2$ - since $L(\mathbb{R})$ computes $\omega_2$ correctly, $\omega_1^{L(\mathbb{R})[G]}=\omega_2^V=\omega_2^{L(\mathbb{R})}$. However, I don't know a snappy description for this measure. So we may ask:

Long version: EDIT: Most of this is completely wrong, see the answer below.

In particular, is there a description of which measures on $\omega_2^{L(\mathbb{R})}$ are potentially club? A reasonable guess is that the above argument with $Col(\omega,\omega_1)$ constructs such a measure; and under appropriate hypotheses (the nonstationary ideal on $\omega_1$ is $\omega_2$-saturated and $\mathcal{P}(\omega_1)^\sharp$ exists) this is verified by calculating the projective ordinal $\delta^1_2$ - since $L(\mathbb{R})$ computes $\omega_2$ correctly, $\omega_1^{L(\mathbb{R})[G]}=\omega_2^V=\omega_2^{L(\mathbb{R})}$. However, I don't know a snappy description for this measure. So we may ask:

In particular, is there a description of which measures on $\omega_2^{L(\mathbb{R})}$ are potentially club? A reasonable guess is that the above argument with $Col(\omega,\omega_2)$ constructs such a measure; and this is verified by calculating the projective ordinal $\delta^1_2$ - since $L(\mathbb{R})$ computes $\omega_2$ correctly, $\omega_1^{L(\mathbb{R})[G]}=\omega_2^V=\omega_2^{L(\mathbb{R})}$. However, I don't know a snappy description for this measure. So we may ask:

In particular, is there a description of which measures on $\omega_2^{L(\mathbb{R})}$ are potentially club? A reasonable guess is that the above argument with $Col(\omega,\omega_2)$ constructs such a measure; and under appropriate hypotheses (the nonstationary ideal on $\omega_1$ is $\omega_2$-saturated and $\mathcal{P}(\omega_1)^\sharp$ exists) this is verified by calculating the projective ordinal $\delta^1_2$ - since $L(\mathbb{R})$ computes $\omega_2$ correctly, $\omega_1^{L(\mathbb{R})[G]}=\omega_2^V=\omega_2^{L(\mathbb{R})}$. However, I don't know a snappy description for this measure. So we may ask: