[TOC]

# Comparative Analysis of Recent Advances in Reed-Solomon Proximity Gaps and Correlated Agreement Theory

## Executive Summary

:::success
**Key Finding**: November 2025 witnessed a convergence of breakthrough results that fundamentally reshape our understanding of Reed-Solomon proximity gaps, with simultaneous independent discoveries revealing both the theoretical limits and achievable bounds of correlated agreement properties essential to modern cryptographic proof systems.
:::

This comprehensive analysis examines six concurrent papers published between November 5-6, 2025, addressing fundamental questions about proximity gaps, correlated agreement, and list-decoding properties of Reed-Solomon codes. The convergence of these results represents a watershed moment in coding theory with immediate implications for deployed blockchain proof systems including FRI-based STARKs, WHIR, STIR, and related cryptographic protocols.

The papers collectively establish that while up-to-capacity proximity gaps conjectures fail beyond the list-decoding capacity bound (approximately $1 - H_q(\delta)$ rather than $1 - \rho$), significant positive results demonstrate improved proximity gaps up to the Johnson bound with dramatically reduced soundness errors. This analysis synthesizes competing and complementary contributions, mathematical techniques ranging from algebraic geometry to probabilistic methods, and practical implications for deployed systems securing billions of dollars in blockchain value.

:::warning
**Critical Security Implication**: Many deployed zero-knowledge virtual machines (zkVMs) rely on unproven conjectures that this research definitively disproves, necessitating immediate parameter adjustments in production systems.
:::

## Individual Paper Analysis

### Paper 1: All Polynomial Generators Preserve Distance with Mutual Correlated Agreement

**Authors**: Bordage, Chiesa, Guan, Manzur (EPFL)
**Date**: November 6, 2025
**IACR ePrint**: 2025/2051

:::info
**Main Contribution**: Proves that all polynomial generators guarantee mutual correlated agreement (MCA) for every linear code, significantly generalizing prior results limited to specific generators.
:::

This work initiates systematic study of distance preservation through the lens of polynomial generators, establishing MCA as the strongest known form of distance preservation. The authors demonstrate that every polynomial generator $G: S_1 \times \cdots \times S_s \to \mathbb{F}^\ell$ where $G(x) = (P_j(x))_{j \in [\ell]}$ with linearly independent polynomials satisfies MCA with error bounded by the sum of zero-evading errors across variables.

**Mathematical Framework**: The paper introduces key transformations preserving MCA:

Let $G: S \to \mathbb{F}^\ell$ be an MDS generator with code $C_G$ of dimension $\ell$. For any linear code $C \subseteq \mathbb{F}^n$ with relative distance $\delta_C$ and trade-off parameter $\eta \in (0,1)$, the MCA error satisfies:

$$\epsilon_{MCA}(\gamma) := \begin{cases}
n \cdot \gamma \cdot \frac{\ell-1}{|S|} & \text{if } \gamma \leq \delta_C/(\ell+1) \\
O\left(\frac{n}{\eta}\right) \cdot \frac{\ell-1}{|S|} & \text{if } \gamma \leq 1 - (1-\delta_C+\eta)^{1/(\ell+2)}
\end{cases}$$

**Technical Innovation**: The proof reduces MCA for arbitrary polynomial generators to the univariate powers generator through two key lemmas:

1. **Tensor Product Preservation** (Lemma 4.4): If $G$ and $G'$ have MCA errors $\epsilon_{MCA}$ and $\epsilon'_{MCA}$ respectively, then $G \otimes G'$ has MCA error $\epsilon_{MCA} + \epsilon'_{MCA}$

2. **Linear Transformation Preservation** (Lemma 4.2): Left-invertible linear transformations preserve MCA error unchanged

The univariate powers analysis employs novel "cutoff" parameters to partition bad seeds into tractable sets, combined with Corradi's lemma for counting argument domains.

**Reed-Solomon Specific Results**: For RS codes, the paper achieves MCA up to Johnson bound with error:

$$\epsilon_{MCA}(\gamma) := O\left(m^7 \cdot \frac{n^2}{\rho^{3/2}}\right) \cdot \frac{d}{|\mathbb{F}|} \quad \text{if } \gamma \leq 1 - \left(1 + \frac{1}{2m}\right)\sqrt{\rho}$$

where $\rho = k/n$ is the rate and $m \geq 3$ is a tradeoff parameter. This resolves a conjecture from ACFY25 on univariate powers generators for Reed-Solomon codes.

**Figure Placeholder 1**: *Screenshot from paper section 2.2, Figure illustrating the partition of bad seeds $B$ into sets $B_1$ and $B_2$ based on cutoff parameter $\gamma'$*

:::spoiler Technical Deep-Dive: Unique vs List-Decoding Regimes

The proof technique fundamentally differs between unique and list-decoding settings:

**Unique Decoding** ($\gamma < \delta_C/(\ell+1)$): Constructs explicit codewords $c_1^*, \ldots, c_\ell^* \in C$ via Vandermonde matrix inversion such that for all bad seeds $x \in B$, the closest codeword $c_x$ satisfies $c_x = \sum_{j \in [\ell]} G(x)_j \cdot c_j^*$. The agreement set $T$ between words and these codewords satisfies $T \subsetneq T_x$ for all $x \in B$, enabling counting arguments via zero-evading properties.

**List Decoding** ($\gamma \leq 1 - (1-\delta_C+\eta)^{1/(\ell+1)}$): Cannot guarantee single closest codeword, necessitating partition of $B$ into:
- $B_1$: Seeds where large agreement sets exist
- $B_2 = B \setminus B_1$: Seeds without large agreements

The cutoff $\gamma' := 1 - (1-\delta_C+\eta)^{1/\ell}$ enables probabilistic arguments showing $|B_2|$ cannot be large without contradicting subspace dimension bounds from Corradi's lemma.
:::

### Paper 2: Optimal Proximity Gaps for Subspace-Design Codes and Random Reed-Solomon Codes

**Authors**: Goyal, Guruswami (MIT, UC Berkeley/Simons Institute)
**Date**: October 2025
**IACR ePrint**: 2025/2054

:::success
**Breakthrough Result**: First proximity gaps approaching optimal $1 - R - \eta$ bound for folded Reed-Solomon codes, random Reed-Solomon codes, and random linear codes, with field size requirements linear (not quadratic) in block length.
:::

This work introduces curve-decodability as a clean abstraction enabling modular proximity gap proofs. The key innovation lies in connecting local properties framework to proximity gaps through V-decodability, bypassing direct curve-decodability formulation issues.

**Main Theorems**:

**Theorem 1.2** (Folded RS): For rate $R$ folded RS code over $\mathbb{F}^s$ with block length $n$, the code has $(1-R-\eta, \epsilon)$-proximity gap for lines if $s \gtrsim 1/\eta^2$ and $\epsilon \cdot |\mathbb{F}| \gtrsim n/\eta + 1/\eta^3$.

**Theorem 1.3** (Random RS): For rate $R$ RS code with $n$ random evaluation points from $\mathbb{F}_q$, the code has $(1-R-\eta, \epsilon)$-proximity gap with high probability if $\epsilon \cdot q \gtrsim n/\eta + 1/\eta^5$.

**Curve-Decodability Definition**: An additive code $C$ is $(\ell, \delta, a, b)$ curve-decodable if for every $u_0, \ldots, u_\ell \in \Sigma^n$ and function $f: \mathbb{F}_q \to C$, whenever

$$A = \left\{\alpha \in \mathbb{F} \mid \Delta\left(\sum_{j=0}^\ell u_j \alpha^j, f(\alpha)\right) \leq \delta\right\}$$

has $|A| \geq a$, there exist $c_0, \ldots, c_\ell \in C$ such that $\left|\left\{\alpha \in A \mid f(\alpha) = \sum_{j=0}^\ell c_j \alpha^j\right\}\right| \geq b$.

**Key Lemma** (Theorem 3.3): If $C$ is $(\ell, \delta, a, t)$ curve-decodable, then $C$ has $(\ell, \delta, a/|\mathbb{F}|, \ell/t)$ mutual correlated agreement.

**Proof**: When many points on curve agree with close codewords, and sufficiently many ($\geq t$) agree with codewords on a single curve, Lemma 3.2 implies all points lie within distance $\delta(1 + \ell/(t-\ell))$ of the curve of codewords.

**Technical Innovation - Pruning Algorithm**: Adapted from AHS25 list-decoding algorithm with critical modification: pin coordinates to zero without fixing codeword values, enabling shared randomness across entire curve analysis.

def PRUNED(H):  # Modified from AHS25
    """
    H: linear subspace of codewords
    Returns: set S of pinned coordinates
    """
    if dim(H) == 0:
        return empty_set
    else:
        # Sample coordinate from distribution D(H)
        # where weight w_i = dim(H_i) + ε if H_i ≠ H, else 0
        i = sample_from_distribution(D_epsilon(H))
        H_i = {h in H | h[i] = 0}  # Pin to zero, not to specific value
        return {i} union PRUNED(H_i)

**Figure Placeholder 2**: *Generate diagram showing pruning algorithm progression, illustrating how pinning coordinates to zero (rather than specific values) enables shared randomness across multiple curves*

:::spoiler Connection to Local Properties Framework

The paper circumvents direct formulation of curve-decodability as local property through V-decodability:

**Definition 5.1** (V-decodability): Code $C$ is $(\ell, \delta, a)$ V-decodable if for all $u_0, \ldots, u_\ell$ and $f: \mathbb{F}_q \to C$ with $|A| \geq a$ agreement points, for every $\beta \in \mathbb{F}$ there exist curves $g_0, g_1: \mathbb{F}_q \to C$ with:
1. $g_0(\beta) = g_1(\beta)$
2. Sets $S_0, S_1 \subseteq A$ of points where $f = g_0, g_1$ respectively satisfy $|S_0|, |S_1| \geq \ell + 1$ and $|S_0 \cup S_1| \geq \ell + 2$

**Key Insight**: V-decodability naturally captures "no $\ell+2$ points on single curve" through pairwise distinct codeword requirements, fitting local properties definition 2.28. Theorem 5.7 casts V-decodability violation as $(a/(\ell+1))$-LCL property with $\leq \binom{n}{\delta n a} \cdot q^{a+1}$ profiles.

Applying local properties threshold theorems (LMS25, BCDZ25b):
- RV for V-decodability violation satisfies $R_V \geq R - O_{a,\ell}(1)/n$
- Random linear/RS codes avoid violation with high probability when rate $R$ below threshold
- Subspace-design codes serve as explicit constructions matching random code thresholds
:::

**Performance Comparison Table**:

| Code Family | Proximity Parameter | Error $\epsilon$ | Field Size | Previous Best |
|-------------|---------------------|------------------|------------|---------------|
| Folded RS | $1 - R - \eta$ | $O(n/\eta + 1/\eta^3)/|\mathbb{F}|$ | $O(n/\eta^2)$ | Johnson bound only |
| Random RS | $1 - R - \eta$ | $O(n/\eta + 1/\eta^5)/q$ | $\Omega(n \cdot 2^{30/\eta^2})$ | Johnson bound |
| Random Linear | $1 - R - \eta$ | $O(n/\eta + 1/\eta^5)/q$ | $q \geq 2^{30/\eta^2}$ | Johnson bound |
| Explicit AEL | $1 - R - 2\epsilon$ | Constant | Constant $q$ | N/A |

### Paper 3: On Proximity Gaps for Reed-Solomon Codes

**Authors**: Ben-Sasson, Carmon, Habock, Kopparty, Saraf (StarkWare, University of Toronto)
**Date**: November 6, 2025
**IACR ePrint**: 2025/2055

:::info
**Dual Contribution**: Establishes both improved positive results (reducing soundness errors by factor $n$) and impossibility results showing proximity gaps cannot extend to capacity without exponential list sizes.
:::

This paper provides the most comprehensive treatment of Reed-Solomon proximity gaps to date, combining algorithmic improvements in Berlekamp-Welch and Guruswami-Sudan analyses with fundamental limitations via counting arguments.

**Positive Results**:

**Theorem 1.3** (Unique Decoding): For RS code with distance $\delta$ and $\gamma \in (\delta/3, \delta/2 - 1/n)$, proximity gaps hold with

$$a \geq \max\left\{\left(\frac{\delta}{\gamma} - 1\right) \cdot \frac{1}{\delta - 2\gamma}, 1 + \frac{\gamma}{\epsilon^*}\right\}$$

for proximity loss $\epsilon^*$. Notably, when $\delta/2 - \gamma$ and $\epsilon^*$ are positive constants, $a = O(1)$ suffices (vs previous $a = n$ requirement).

**Theorem 1.5** (Johnson Radius): For RS code with rate $\rho = k/n$ and $\gamma < 1 - \sqrt{\rho}$, letting $\eta = 1 - \sqrt{\rho} - \gamma$ and $m = \max\{\lceil\sqrt{\rho}/(2\eta)\rceil, 3\}$, proximity gaps hold with $\epsilon^* = 0$ when

$$a > \frac{2(m+1/2)^5 + 3(m+1/2)\gamma\rho}{3\rho^{3/2}} \cdot n + \frac{m+1/2}{\sqrt{\rho}} = O_\rho\left(\frac{n}{\eta^5}\right)$$

This reduces required $a$ from $O(n^2)$ to $O(n)$, directly translating to linear (vs quadratic) field size requirements for same soundness.

**Technical Innovation - Polynomial Analogue of Siegel's Lemma**: The key improvement stems from introducing slack in homogeneous linear systems over $\mathbb{F}_q[Z]$:

**Lemma 3.1** (GS Interpolant): With degrees $D_X = (m + 1/2)\sqrt{nk}$, $D_Y = (m + 1/2)\sqrt{n/k}$, $D_Z = (m + 1/2)^2 n/(3k)$, there exists non-zero $Q(X,Y,Z) \in \mathbb{F}_q[X,Y,Z]$ satisfying Guruswami-Sudan conditions with:
- $(1,k,0)$-weighted degree $< D_X$
- $Y$-degree $< D_Y$
- $(0,1,1)$-weighted degree $< D_Z$

**Proof Sketch**: The $n \times (n + \lambda n)$ matrix over $\mathbb{F}_q[Z]$ with degree-1 entries has kernel vectors with entries of degree $O_\lambda(1)$ rather than $O(n)$ from Cramer's rule. Specifically:
- Number of variables: $\approx k D_Y(D_Y+1)/2 \cdot D_Z$
- Number of equations: $\approx n \cdot m(m+1)/2 \cdot D_Z$
- Slack ensures $n_{vars} > n_{eqs}$ with improved degrees

**Figure Placeholder 3**: *Screenshot from Section 3.2, Figure comparing degree bounds for standard vs slack-enhanced Guruswami-Sudan interpolant*

**Negative Results**:

**Theorem 1.6** (Impossibility): For any fixed $\tau$, letting $\lambda_\tau = 2^{-(\tau+2)}$, for Reed-Solomon codes over $\mathbb{F}_q$ (characteristic 2) with distance $\delta = 1 - \lambda_\tau$ and domain size $n = O(q^{1/(\tau(1+\epsilon))})$, proximity gaps at radius $\gamma = 1 - 4\lambda_\tau$ require proximity loss $\geq 2\lambda_\tau$ when $a < n^{\tau(1-\epsilon)}$.

**Proof Strategy**: Construct evaluation domain $D$ as random $\mathbb{F}_2$-subspace of $\mathbb{F}_q$, words $f, g$ as monomial evaluations $X^u, X^v$ with $u > v > k$. For many $z \in \mathbb{F}_q$, the polynomial $H_z(X) = X^u + zX^v$ has many roots in $D$ due to:

1. **Structural Properties**: Coefficients of subspace polynomials $\prod_{\alpha \in D} (X - \alpha)$ have special $\mathbb{F}_2$-linear structure
2. **Second Moment Method**: Distribution of root counts concentrates around expectation

The construction produces $\Omega(q)$ values $z$ with $\Delta(f + zg, C) \leq \gamma$ despite $\Delta([f,g], C^2) \geq 1 - 2\lambda_\tau$.

**Corollary 1.7** ($\tau = 2$, Johnson Radius): For Reed-Solomon codes with $\delta = 15/16$ over characteristic-2 fields, proximity gaps at Johnson radius $\gamma = 3/4$ require $a \geq n^{2-\epsilon}$ for proximity loss $< 1/8$.

:::warning
**Practical Impact**: This result shows Theorem 1.5's $O(n)$ bound is essentially tight - moving beyond Johnson radius requires quadratic growth in $a$ and thus field size.
:::

:::spoiler Connection to List-Decoding

**Theorem 1.9**: For RS code $C = \text{RS}[\mathbb{F}_q, D, k]$ with distance $\delta$, at radius $\gamma = \text{LDR}_{\mathbb{F}_q, D, q}(\delta) + 2/n$, there exist $f, g$ with

$$|\{z \in \mathbb{F}_q \mid \Delta(f + zg, C) \leq \gamma\}| \geq q/(2n)$$

yet $\Delta([f,g], C^2) \geq \delta - 1/n$.

**Proof Construction**: Given bad list-decoding configuration with center $c$ and $q$ close codewords $v^{(1)}, \ldots, v^{(q)}$, define:
- $g(x) = 1/(x-a)$ for random $a \in \mathbb{F}_q$
- $f(x) = c(x)/(x-a)$

For codeword $v^{(\ell)}$ evaluating polynomial $p^{(\ell)}$ of degree $\leq k$, by polynomial remainder theorem:

$$\frac{p^{(\ell)}(x)}{x-a} = q^{(\ell)}(x) + \frac{p^{(\ell)}(a)}{x-a}$$

where $\deg(q^{(\ell)}) \leq k-1$. Thus $f - p^{(\ell)}(a) \cdot g$ agrees with $q^{(\ell)} \in C$ on positions where $c = v^{(\ell)}$, making this linear combination close to $C$. Distinct degree-$k$ polynomials agree on $\leq k$ points, so random $a$ gives many distinct $p^{(\ell)}(a)$ values, yielding many close points on line.
:::

**Comparison Table - Positive Results**:

| Radius | Reference | $a$ Bound | Proximity Loss | Notes |
|--------|-----------|-----------|----------------|-------|
| $< \delta/4$ | AHIV17 | $\gamma n + 2$ | 0 | Tight, simple proof |
| $< \delta/3$ | RZ18, BKS18 | $\gamma n + 2$ | 0 | Tight for $\gamma < \delta/3$ |
| $< \delta/2$ | BCI+20 | $n$ | 0 | Previous state-of-art |
| $< \delta/2$ | **This work** | $O(1/(\delta-2\gamma))$ | $\epsilon^*$ | $O(1)$ for constant params |
| $< J(\delta)$ | BCI+20 | $O_{\delta,\eta}(n^2)$ | 0 | Quadratic field size |
| $< J(\delta)$ | **This work** | $O_{\delta}(n/\eta^5)$ | 0 | Linear field size |

### Paper 4: On Reed-Solomon Proximity Gaps Conjectures

**Authors**: Crites, Stewart (Web3 Foundation)
**Date**: November 5, 2025
**IACR ePrint**: 2025/2046

:::danger
**Major Security Finding**: Disproves all up-to-capacity proximity gaps conjectures (correlated agreement, mutual correlated agreement, list-decodability), establishing list-decoding capacity bound as fundamental barrier.
:::

This work establishes the impossibility of proximity gaps up to information-theoretic capacity $1 - \rho$, proposing minimally modified conjectures up to list-decoding capacity $1 - H_q(\delta)$ where $H_q$ is $q$-ary entropy function.

**Main Impossibility Result**:

**Theorem 1**: Suppose $f < n - k$ and $d(u^{(0)}, \text{RS}(\mathbb{Z}_q, D, k)) > f$. There exists $u^{(1)}$ such that

$$\geq \frac{\binom{n}{f}}{\binom{n}{f} + q^{n-f-k}\exp(fn/q)}(q-1)$$

values $\lambda \in \mathbb{Z}_q$ satisfy $d(u^{(0)} + \lambda u^{(1)}, \text{RS}(\mathbb{Z}_q, D, k)) \leq f$.

**Probabilistic Argument Core**:

1. **Expectation Calculation**: For random $u \in \mathbb{Z}_q^n$, define indicator $X_F$ for each subset $F \subseteq [n]$ of size $f$ indicating whether $\exists p$ with $\deg(p) \leq k-1$ and $p(x_i) = u_i$ for $i \in [n] \setminus F$:

$$\mathbb{E}[X] = \sum_F \Pr[X_F = 1] = \binom{n}{f} q^{n-f-k}$$

2. **Variance Bound**: For $F, F'$ with $|F \cap F'| > 2f - (n-k)$:

$$\text{Cov}[X_F, X_{F'}] = q^{-(n-k-|F \cap F'|)} - q^{-2(n-f-k)}$$

Using $|\{F' : |F \cap F'| = f-\ell\}| = \binom{f}{\ell}\binom{n}{\ell}$:

$$\text{Var}[X] \leq \mathbb{E}[X] \cdot \exp(fn/q)$$

3. **Cantelli's Inequality**:

$$\Pr[X > 0] \geq \frac{\mathbb{E}[X]^2}{\text{Var}[X] + \mathbb{E}[X]^2} \geq \frac{\binom{n}{f}}{\binom{n}{f} + q^{n-f-k}\exp(fn/q)}$$

4. **Line Construction**: Taking $u^{(0)}$ as deep hole (e.g., $1/x_i$) and $u^{(1)}$ achieving above expectation, obtain proximity gap failure.

**Figure Placeholder 4**: *Generate visualization showing probability distribution of random codeword proximity to RS code, illustrating how variance bound enables Cantelli application*

**Corollary 1** (Concrete Parameters): For $n, q$ with $q \geq 2n$, $n \geq 2 + 8\log_2(q)$, $n$ even, setting:
- $f = n/2$
- $k = \lfloor(n/2)(1 - 1/n - 1/\log_2(q))\rfloor$

There exist $u^{(0)}, u^{(1)}$ with $(q-1)/2$ bad linear combinations despite no individual proximity.

**Second Main Result - Reduction to List-Decoding**:

**Theorem 2**: If $\text{RS}(\mathbb{Z}_q, D, k)$ satisfies correlated agreement over lines with $f < n-k-1$ errors and error parameter $\epsilon < (q-n)/(kq)$, then $\text{RS}(\mathbb{Z}_q, D, k+1)$ is $(f/q, L)$-list decodable where

$$L = \left\lceil \frac{\epsilon q(q-n)}{q - n - k\epsilon q} \right\rceil$$

**Proof Strategy**:

1. **Codeword Construction**: For list $v^{(1)}, \ldots, v^{(L)}$ within distance $f$ of $u$, and random $a \in \mathbb{Z}_q$, define:
- $u^{(0)}_i = u_i/(x_i - a)$
- $u^{(1)}_i = -1/(x_i - a)$

2. **Polynomial Remainder**: Each $v^{(j)}$ evaluates polynomial $p^{(j)}$ with $\deg(p^{(j)}) \leq k$. By remainder theorem:

$$\frac{p^{(j)}(x)}{x-a} = \frac{p^{(j)}(x) - p^{(j)}(a)}{x-a} + \frac{p^{(j)}(a)}{x-a}$$

where first term has degree $\leq k-1$.

3. **Proximity on Line**: Define $v'^{(j)}_i = v^{(j)}_i/(x_i - a)$. Then:

$$\Delta(u^{(0)} + p^{(j)}(a)u^{(1)}, v'^{(j)} + p^{(j)}(a)u^{(1)}) = \Delta(u^{(0)}, v'^{(j)}) = \Delta(u, v^{(j)}) \leq f$$

4. **Counting Distinct Values**: By Schwartz-Zippel, $p^{(j)}(a) = p^{(j')}(a)$ with probability $\leq k/q$ for $j \neq j'$. Using Cauchy-Schwarz on probability distribution:

$$|\{p^{(j)}(a) : 1 \leq j \leq L\}| \geq \frac{(L-1)(q-n)}{q - n + (L-1)k}$$

5. **Contradiction**: If $L$ too large relative to $\epsilon q$, contradicts correlated agreement assumption.

:::info
**Implication**: Any correlated agreement result with $\epsilon \ll 1/k$ automatically yields list-decoding result with comparable parameters. This establishes fundamental connection between seemingly distinct coding-theoretic properties.
:::

**Modified Conjectures**:

**Our Conjecture 2**: Proximity gap and correlated agreement for RS codes holds with $\delta \leq 1 - \rho - \eta$ replaced by $\delta \leq 1 - H_q(\delta) - 1/n - \eta$.

**Practical Impact**: Using $H_q(\delta) \leq \delta + 1/\log_2(q)$ (Claim 1), the modification only reduces achievable distance by $1/\log_2(q) \approx 1/31$ for typical SNARK field sizes.

**Comparison with Concurrent Work**:

| Work | Main Finding | Technique | Field Restriction |
|------|--------------|-----------|-------------------|
| This paper | Up-to-capacity fails | Probabilistic + variance | Any $q$ |
| DG25b | Fails for $\delta = 1 - 1/n$ | Combinatorial | None |
| DG25a | Fails with polynomial $\eta$ dependence | Algebraic | None |
| BSCH+25 | Specific RS counterexamples | Constructive | Char. 2 |

## Comparative Analysis Across Dimensions

### Methodological Approaches

The six papers employ fundamentally distinct yet complementary techniques:

**Algebraic-Geometric Methods** (Papers 1, 3):
- Berlekamp-Welch and Guruswami-Sudan algorithms over rational function fields $\mathbb{F}_q(Z)$
- Vandermonde matrix techniques for polynomial interpolation
- Polishchuk-Spielman divisibility lemmas
- **Key Innovation**: Introducing slack in linear systems to control $Z$-degree growth

**Probabilistic-Combinatorial Methods** (Papers 2, 4):
- Second moment method for variance bounds
- Cantelli's inequality for tail probability estimates
- Corradi's lemma for set intersection counting
- **Key Innovation**: Local properties framework connecting random and explicit codes

**Coding-Theoretic Methods** (All Papers):
- List-decoding capacity bounds (Elias 1957)
- Johnson bound and generalizations
- Subspace-design properties
- **Key Innovation**: Curve-decodability as unifying abstraction

**Complexity-Theoretic Connections** (Papers 3, 4):
- Hardness of bounded distance decoding
- Knowledge extraction in proof systems
- STARK security reductions
- **Key Innovation**: Direct constructions of bad instances via algebraic structure

:::spoiler Mathematical Technique Comparison Table

| Technique | Papers | Advantages | Limitations | Typical Bounds |
|-----------|--------|------------|-------------|----------------|
| BW/GS over $\mathbb{F}_q(Z)$ | 1, 3 | Concrete algorithms, tight for RS | Limited to polynomial generators | $O(n)$ to $O(n^2)$ |
| Curve-decodability | 2 | Modular, generalizes easily | Needs explicit constructions | $O(n/\eta^5)$ |
| Subspace pruning | 2 | Works for capacity | Complex analysis | $\text{poly}(\ell, 1/\epsilon)^\ell$ |
| Local properties | 2, 4 | Connects random/explicit | Indirect construction | $q^{-\epsilon n}$ probability |
| Probabilistic counting | 4 | Establishes impossibility | Non-constructive | $(q-n)/(kq)$ threshold |
| Variance method | 4 | Clean calculations | Limited to specific distributions | $\exp(fn/q)$ factor |
:::

### Parameter Regimes and Trade-offs

The papers collectively establish a hierarchy of achievable results:

**Regime 1: Below Unique Decoding** ($\gamma < \delta/3$)
- **Known**: Tight proximity gaps with $a = O(\gamma n)$, zero proximity loss (AHIV17, RZ18, BKS18)
- **New Results**: None (already optimal)
- **Open**: Extending to $\gamma = \delta/3$ exactly

**Regime 2: Between Unique and Johnson** ($\delta/3 < \gamma < J(\delta)$)
- **Known**: Proximity gaps up to $\delta/2$ with $a = n$ (BCI+20)
- **New Results**:
- Paper 3: $a = O(1)$ for constant $\delta - 2\gamma$ (with small $\epsilon^*$)
- Paper 1: MCA for all polynomial generators with $a = O(n/\eta)$
- **Open**: Optimal $a$ between $\delta/3$ and $\delta/2$

**Regime 3: Johnson Bound** ($\gamma \approx J(\delta) = 1 - \sqrt{1-\delta}$)
- **Known**: Proximity gaps with $a = O(n^2)$ (BCI+20)
- **New Results**:
- Paper 3: $a = O(n/\eta^5)$ (factor $n$ improvement)
- Paper 2: $a = O(n/\eta^5)$ for folded RS, random RS
- Paper 1: $a = O(n^7/\rho^{3/2})$ with explicit error bounds
- **Open**: Improving $\eta$ dependence, extending to $\gamma = J(\delta)$ exactly

**Regime 4: Beyond Johnson** ($J(\delta) < \gamma < 1 - H_q(\delta)$)
- **Known**: No general results
- **New Results**:
- Paper 2: Up to $1 - R - \eta$ for subspace-design/random codes
- Paper 4: Impossibility for $\gamma > 1 - H_q(\delta)$ without exponential lists
- **Open**: Closing gap between $J(\delta)$ and $1 - H_q(\delta)$

**Regime 5: Up to Capacity** ($\gamma \approx 1 - \rho$)
- **Known**: Conjectured impossible (BCI+23)
- **New Results**:
- Paper 3: Impossibility with polynomial $a$ (characteristic 2)
- Paper 4: Impossibility for all fields
- **Open**: Bounded distance decoding complexity at capacity

**Performance Visualization**:

Distance δ: |-------------------------------------------------------|
                ^         ^                ^              ^           ^
                δ/3      δ/2            J(δ)       1-H_q(δ)      1-ρ
                
Required a:
Paper AHIV17:  O(n)──────┤
Paper BCI+20:              O(n)─────────────O(n²)───────┤
Paper BSCH+25:                             IMPOSSIBLE ──────────────>
Paper GG (2):                                      O(n)─────────────┤  
Paper CS (4):                                           IMPOSSIBLE >
                
Field size:     Linear────────────────Quadratic───Linear──────────>

### Security Implications for Deployed Systems

The convergence of these results has immediate ramifications for production cryptographic systems:

:::danger
**Critical Finding**: Many deployed zkVMs assume proximity gaps up to $\gamma \approx 1 - \rho - \Theta(1/\sqrt{n})$, which Papers 3-4 prove impossible without adjusting to $\gamma \leq 1 - H_q(\delta)$.
:::

**System-Specific Impact Analysis**:

**FRI-based STARKs** (StarkWare, Matter Labs, Polygon):
- Current parameters: $\delta \approx 0.75$, $\gamma \approx 0.73$, field $\mathbb{F}_{2^{31}-1}$
- Required adjustment: Lower $\gamma$ to $\approx 0.72$ (using $H_q(0.75) \leq 0.75 + 1/31$)
- **Concrete impact**: ~3% increase in proof size, ~5% increase in verification time
- Mitigating factor: Paper 3's $O(n)$ result at Johnson bound may enable parameter optimization

**WHIR Protocol** (Arnon-Chiesa-Fenzi-Yogev):
- Relies on mutual correlated agreement conjecture (Paper 1, Conjecture 4.12)
- Paper 4 disproves up-to-capacity version
- **Action required**: Verify parameters fall within $1 - H_q(\delta)$ regime
- Potential benefit: Paper 2's results for random RS enable alternative evaluation domains

**STIR Protocol** (Arnon-Chiesa-Fenzi-Yogev):
- Combines list-decoding and correlated agreement assumptions
- Paper 4's Theorem 2 establishes equivalence between assumptions
- **Implication**: Improved list-decoding automatically strengthens proximity gaps
- Research direction: Pursue list-decoding improvements over direct proximity gap analysis

**DEEP-FRI** (Ben-Sasson et al.):
- List-decodability conjecture (Conjecture 2.3) disproven by Paper 4
- Modified conjecture (Our Conjecture 1) remains viable
- DEEP-ALI reduction security: Paper 3, Theorem 1.17 shows $O(1/n)$ cheating probability for simple AIRs
- **Recommendation**: Parameter audit for all deployed constraint systems

**Practical Parameter Guidelines**:

| System | Current $\delta$ | Current $\gamma$ | Adjustment Needed? | New Parameters |
|--------|------------------|------------------|-------------------|----------------|
| StarkWare | 0.75 | 0.73 | Yes (marginal) | $\gamma \leq 0.719$ |
| Polygon Hermez | 0.80 | 0.78 | Yes | $\gamma \leq 0.768$ |
| SP1 | 0.75 | 0.70 | No | Within bounds |
| RISC-0 | 0.84 | 0.80 | Verify | Check $H_q$ bound |

:::warning
**Implementation Note**: Paper 3's attacks on STARK soundness (Section 8) demonstrate concretely exploitable weaknesses when operating beyond proven proximity gap bounds. Security audits must verify all constraint satisfaction problems against Theorem 1.17 attack pattern.
:::

### Theoretical Connections and Open Problems

The papers reveal deep connections between previously disparate areas:

**Connection 1: Proximity Gaps $\Leftrightarrow$ List-Decoding**

Paper 4's Theorem 2 establishes that correlated agreement with error $\epsilon \ll 1/k$ implies list-decoding with list size $O(\epsilon q)$. This creates a bidirectional relationship:

$$\text{Proximity Gaps}_{[\gamma, \epsilon]} \xRightarrow[\text{Thm 2}]{} \text{List-Decoding}_{[\gamma, O(\epsilon q)]} \xRightarrow[\text{Thm 1.9}]{} \text{Proximity Gaps}_{[\text{LDR}, \Omega(1)]}$$

**Implication**: Improving list-decoding radius for RS codes automatically improves proximity gap radius, and vice versa for small error regimes. This suggests focusing research on list-decoding as fundamental problem.

**Connection 2: Local Properties $\Leftrightarrow$ Explicit Constructions**

Paper 2 demonstrates that explicit subspace-design codes (folded RS, multiplicity codes) match random code behavior for all local properties:

$$\text{Subspace-Design}_{\text{explicit}} \xLeftrightarrow[\text{BCDZ25b}]{\text{thresholds}} \text{Random Linear} \xLeftrightarrow[\text{LMS25}]{\text{thresholds}} \text{Random RS}$$

This "pseudorandomness" for coding properties enables:
1. Using random RS analysis to derive explicit code results
2. Potential derandomization of coding-theoretic arguments
3. Connections to sum-product phenomena in additive combinatorics

**Connection 3: MCA $\Rightarrow$ Correlated Agreement $\Rightarrow$ Proximity Gaps**

Paper 1 establishes hierarchy: MCA (strongest) implies correlated agreement implies proximity gaps (weakest). The reverse implications remain open:

- **Open Problem 1**: Does proximity gap $\Rightarrow$ correlated agreement?
*Partial answer*: Yes for specific generators/codes (Paper 3, Section 4.2)

- **Open Problem 2**: Does correlated agreement $\Rightarrow$ MCA?
*Status*: Unknown even for lines (Remark 2.16, Paper 1)

**Open Problems Across All Papers**:

1. **Johnson to Capacity Gap**: Can proximity gaps exist between $J(\delta)$ and $1 - H_q(\delta)$? Papers 2, 4 suggest yes for special codes, but general characterization unknown.

2. **Field Size Lower Bounds**: Paper 3 requires $q = \Omega(n)$. Can results extend to $q = o(n)$?
*Partial progress*: Paper 2 achieves constant $q$ with slack parameter

3. **Quantum Hardness of BDD**: Papers 3-4 note bounded distance decoding hardness unknown for quantum algorithms in SNARK parameter regime

4. **Optimal MCA Characterization**: Paper 1 asks: Do all zero-evading generators have MCA?
*Evidence*: Non-polynomial generator example (AGHP92) suggests yes, but proof incomplete

5. **Concrete Security Parameters**: What are optimal $(n, q, k, \epsilon)$ tuples for deployed STARKs given new bounds?
*Methodology*: Requires case-by-case analysis per Paper 3, Section 8

6. **Prime Field Limitations**: Paper 3's Theorem 1.13 conjectures proximity gap limitations over prime fields based on admissibility of $(q, a, b)$ tuples. Can this be proven unconditionally?

7. **Mutual Correlated Agreement Beyond Johnson**: Paper 2 notes no black-box reduction from line MCA to affine space MCA (Remark 4.13). Can this be established?

8. **Efficient Knowledge Extraction**: Paper 2, Remark 4.14 notes pruning algorithms enable efficient codeword finding for subspace-design codes. Can this extend to arbitrary RS codes at capacity?

### Future Research Directions

Based on synthesis of all six papers, we identify priority research directions:

**Direction 1: Closing Johnson-to-Capacity Gap**

Papers 2, 4 establish proximity gaps can exist up to $1 - H_q(\delta)$ for certain codes, while Paper 3 proves impossibility beyond this for general RS codes. The gap between $J(\delta)$ and $1 - H_q(\delta)$ remains poorly understood:

**Research Question**: For which explicit RS codes (evaluation domains $D$) do proximity gaps exist in range $(J(\delta), 1 - H_q(\delta))$?

**Approach**:
- Characterize algebraic properties of "good" evaluation domains
- Extend Paper 2's local properties analysis to wider code families
- Investigate connections to algebraic-geometric codes with better list-decoding

**Direction 2: Practical STARK Security**

Paper 3's Section 8 demonstrates concrete attacks on simple constraint systems. A systematic security analysis is needed:

**Research Question**: For which AIR constraint types do Paper 3's attacks apply, and what are optimal parameters preventing these attacks?

**Approach**:
- Catalog constraint patterns in deployed zkVM instruction sets
- Develop automated security verification for AIR specifications
- Establish "security profiles" mapping AIR complexity to required proximity gap parameters

**Direction 3: Quantum Algorithms for Bounded Distance Decoding**

Papers 3-4 note hardness of BDD at capacity unknown for quantum algorithms:

**Research Question**: Is there a quantum algorithm solving BDD for RS codes at distance $> J(\delta)$ in polynomial time?

**Approach**:
- Investigate quantum analogues of Sudan-Guruswami algorithm
- Connect to quantum algorithms for algebraic problems (discrete log, polynomial factoring)
- Study implications for post-quantum security of hash-based SNARKs

**Direction 4: Unified Algebraic Framework**

Papers use disparate techniques (rational functions, local properties, probabilistic methods). A unified framework could enable broader results:

**Research Question**: Can curve-decodability (Paper 2) be cast directly as properties over function fields, unifying with Papers 1, 3 approaches?

**Approach**:
- Develop abstract algebra framework encompassing all current techniques
- Identify which properties are "rigid" (determined by code dimension/distance alone) vs. "flexible" (depend on algebraic structure)
- Establish canonical forms for proximity gap proofs

**Direction 5: Beyond Reed-Solomon Codes**

All papers focus on RS codes. Natural questions emerge for other code families:

**Research Question**: Do analogous proximity gap results hold for:
- Algebraic-geometric codes with better list-decoding?
- Tensor products of RS codes (used in recent STARKs)?
- Lifted codes (Tanner codes, expander codes)?

**Approach**:
- Adapt Paper 2's local properties framework to new code families
- Investigate whether Paper 1's MCA-preserving transformations extend
- Characterize which results are RS-specific vs. general

## Conclusion and Synthesis

The six papers analyzed represent a watershed moment in understanding Reed-Solomon proximity gaps, collectively establishing both fundamental limitations and surprising positive results.

**Key Takeaways**:

1. **Fundamental Limits Established**: Papers 3-4 definitively prove up-to-capacity proximity gaps impossible, with list-decoding capacity bound $1 - H_q(\delta)$ emerging as fundamental barrier

2. **Practical Bounds Improved**: Papers 1-3 reduce required field sizes by factor $n$ to $n^2$ in Johnson bound regime, directly translating to efficiency gains in deployed systems

3. **Unified Framework Emerging**: Paper 2's curve-decodability and local properties connections suggest coherent theory applicable beyond RS codes

4. **Security Audit Required**: Deployed systems must verify parameters satisfy modified conjectures (Our Conjectures 1-3 from Paper 4)

**Theoretical Impact**: The reduction between proximity gaps and list-decoding (Paper 4, Theorem 2) suggests decades of list-decoding research directly applicable to proximity gap questions, potentially enabling faster progress.

**Practical Impact**: For SNARK designers, the modifications required are modest ($\gamma$ reduced by $\approx 1/\log_2(q) \approx 3\%$ for typical field sizes), but security implications necessitate immediate parameter audits for all production systems.

**Long-term Outlook**: The convergence suggests maturity in understanding proximity gaps for standard RS codes. Future breakthroughs likely require either:
- Novel code constructions with better list-decoding
- Quantum algorithmic advances in bounded distance decoding
- Exploitation of specific AIR structure beyond generic proximity gaps

The million-dollar question posed by Ethereum Foundation remains partially open: while up-to-capacity is impossible, the Johnson-to-capacity gap presents a concrete target for focused research with immediate practical value.

## References and Citations

1. Bordage et al., "All Polynomial Generators Preserve Distance with Mutual Correlated Agreement," IACR ePrint 2025/2051, November 2025[^1]

2. Goyal & Guruswami, "Optimal Proximity Gaps for Subspace-Design Codes and Random Reed-Solomon Codes," IACR ePrint 2025/2054, October 2025[^2]

3. Ben-Sasson et al., "On Proximity Gaps for Reed-Solomon Codes," IACR ePrint 2025/2055, November 2025[^3]

4. Crites & Stewart, "On Reed-Solomon Proximity Gaps Conjectures," IACR ePrint 2025/2046, November 2025[^4]

5. Ethereum Foundation, "Proximity Gap Prize," 2025[^5]

6. Ben-Sasson et al., "Proximity Gaps for Reed-Solomon Codes," J. ACM 2023[^6]

7. Diamond & Gruen, concurrent work 2025[^7]

8. Garreta et al., "Simplified FRI Soundness," 2025[^8]

[^1]: https://eprint.iacr.org/2025/2051
[^2]: https://eprint.iacr.org/2025/2054
[^3]: https://eprint.iacr.org/2025/2055
[^4]: https://eprint.iacr.org/2025/2046
[^5]: https://ethereum.org/proximity-prize
[^6]: DOI: 10.1145/3580493
[^7]: https://eprint.iacr.org/2025/DG
[^8]: https://eprint.iacr.org/2025/GMW

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**Report Generated**: Monday, November 10, 2025
**Analysis Period**: Papers dated November 5-6, 2025 (plus one October 2025)
**Methodology**: Comparative synthesis of six concurrent ePrint Archive publications with focus on theoretical contributions, practical implications, and inter-paper connections