When $c_{k,R}$ is $0^n$, $T_i$ for $i$ in $[n' + 1 + (n' / 2) + 1, 2 * n' + 1]$ is effectively the identity

[kayabaNerve]

As the first $n' / 2$ commitments to $t$ may be optimized out, them solely being the blinding factor $\tau$ with no actual value, the same is true for the first $n' / 2$ of the last $(n' / 2) + 1$ commitments. This does presumably remove the ability for a prover to prove their proof if their commitments contain such terms (either due to a completeness gap or an effective requirement no such terms exist), but as those terms cannot be used in constraints (due to the lack of a $W_{k, R}$), such terms are largely useless. Accordingly, the 25% reduction in commitments to the $t$ polynomial would be preferred.

[kayabaNerve]

Sorry. This idea is invalid as-is due to the $y \cdot s_R$ cross-products with $c_{k, L}$. One would need to remove the $s_R$ blinding terms which... may be fine? They mask indexes $(n' / 2) + n' + 1 \le i \le 2 (n' + 1)$ in the $t$ polynomial, while $s_L$ masks $n' + 1 \le i \le (n' / 2) + n' + 1$ (assuming $c_{k, R} = 0^n$ for $1 \le k \le n_c$). While that means $s_R$ is required to mask indexes $(n' / 2) + n' + 1 \lt i \le 2 (n' + 1)$, all of those indexes would be $0$ if $c_{k, R} = 0^n$ for $1 \le k \le n_c$ and $s_R = 0^n$, and therefore not actually need masking.

I'll leave this issue open in case there is a way to omit these higher-degree commitments, per this line of thought.

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To restate distinctly:

Can we introduce a completeness gap by constraining our prover to $c_{k, R} = 0^n$ for $1 \le k \le n_c$ and $s_R = 0^n$, allowing omitting 25% of the commitments to the $t$ polynomial, as already done for the first 25%? The question, as I see it, is if the resulting protocol would still be ZK to the witness. Even if not immediately, I believe we could have a non-ZK GBP protocol so long as it's composed with a ZK IPA argument, such as the one from BP+. While the BP+ IPA is two group elements and one scalar larger, we could omit $\mu$ from the GBP protocol (used to clear a term over $H$ while the BP+ IPA statement supports that without communicating the discrete logarithm, considering it part of the witness), closing the gap to just two group elements. This would be in exchange for omitting the high 25% of the commitments to the $t$ polynomial.

This theoretical GBP + BP+ IPA would communicate $A_I$ (blinded by $\alpha$), $A_O$ (blinded by $\beta$), $S$ (blinded by $\rho$), $T_i$ (blinded by $\tau_i$), $\hat{t}$ (blinded by $s_L$), and $\tau_x$. It is tricky to argue $\tau_x$ as still enabling zero-knowledge, it being the weighted sum of masks which could be used as a distinguisher, yet its final mask masks a term incorporating $s_L$. This means attempting to use $\tau_x$ as a distinguisher shouldn't work so long as $s_L$ is insufficiently short to allow forming a system of equations, such as when:

1. $s_L$ and $s_R$ simultaneously exist, each providing at least one masking term
2. $s_L$ alone exists, and $n \ge 2$

The requirement for at least two scalar masks (whether solely in $s_L$ or distributed between $s_L$ and $s_R$) is to simultaneously provide masking to $\hat{t}$ and not allow using $\tau_x$ as a distinguisher.

I think this would be feasible, and only a few hours of tweaking the code? So it may be worth considering.