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Amit Sahai

Publications and source records attributed to Amit Sahai.

At least 19 recordsLinked to original sources
arXiv2026

Does Reasoning Improve Psychological Depth in Large Language Models?

Benchmark gains from reasoning-enhanced large language models encourage the assumption that these models should broadly perform better on complex generative tasks, but this remains underexplored in open-ended, creative domains shaped by nuances of reader experience. We tested this assumption through the lens of psychological depth in short stories. We compared DeepSeek-R1 vs. DeepSeek-V3, our closest available contrast for reasoning-oriented post-training, and GPT-5 vs. GPT-4o, through 60 blinded pairwise evaluations in which seven human annotators and an ensemble LLM-as-a-Judge rated prompt-matched story outputs. Human evaluations showed no universal reasoning advantage: GPT-5 was slightly preferred over GPT-4o (60.0-62.9% across two reasoning-effort settings), whereas DeepSeek-R1 fell behind DeepSeek-V3 (42.9%). Although this pattern was marked by low inter-annotator agreement (Krippendorff's $α= 0.070$), our annotation-quality checks suggested genuine evaluative subjectivity, plausibly reflecting distinct annotator decision schemes rather than annotation failure. In contrast, the LLM-as-a-Judge showed a strong pro-reasoning bias (98.3% of pairs vs. 55.2% of human votes), consistent across its five configurations and partly tracking sentence length. These findings suggest that, at the frontier, reasoning-related quality differences are small enough to be dominated by reader subjectivity, and that LLM-as-a-Judge evaluators can obscure this subjectivity in nuanced creative-generation tasks.

cs.CL↗
arXiv2026

From Solvers to Research: Large Language Model-Driven Formal Mathematics at the Research Frontier

Large language models have made rapid progress in generating machine-checked proofs for well-defined mathematical problems. Extending this progress to research mathematics requires more than solving isolated theorems. Research involves choosing questions, developing definitions and conjectures, connecting results, and combining formal proofs with expert judgment. This survey examines how LLM-driven systems can support these broader activities. We review formal-mathematics tasks, datasets, and autoformalization, and organize proof-generation methods into three overlapping areas: training strategies, inference-time search and adaptation, and agentic workflows. We then examine benchmark results, selected research projects, and platforms for collaborative mathematical work. A central distinction runs throughout the survey: checking a formal proof does not by itself establish that it captures the intended problem or makes a new mathematical contribution. We therefore distinguish theorem benchmarks, repository tasks, formalization projects, and research claims, and discuss the evidence needed to assess each. Finally, we identify five priorities for progress: better data and evaluation, representations of mathematical relationships, conjecture and concept generation, reliable tool integration, and human--AI collaboration. Together, these perspectives connect advances in proof generation to the practical demands of mathematical research.

cs.CL↗
arXiv2026

Lean Graph Search for Formal Theorem Proving

Retrieving relevant lemmas is crucial for formal theorem proving, enabling provers to build machine-checkable proofs from established mathematical results. Recent retrieval systems match library declarations to a target theorem or its intermediate proof steps, but make limited use of multi-hop dependencies among declarations. These dependencies can reveal supporting lemmas that are not necessarily lexically similar to the target theorem but recur in proofs of related problems. To this end, we propose LeanGraphSearch, a graph-based retrieval method for Lean 4 that constructs a graph from Mathlib's forward dependencies and traverses it to uncover supporting lemmas through multi-hop dependency chains. Specifically, LeanGraphSearch links each declaration to the premises it depends on, uses a graph-based ranking algorithm to propagate relevance from semantically matched declarations to their dependencies, and reranks the expanded candidates against the original query. Across two mathematical retrieval benchmarks, LeanGraphSearch improves recall by up to 12% over the strongest baseline, LeanSearch v2, on MathlibQR and increases coverage of complete annotated proof routes on MathlibMPR, covering up to 3.7% of extra groups. More importantly, we evaluate whether these retrieval gains translate into improved end-to-end theorem proving by integrating LeanGraphSearch into an iterative proving loop guided by Lean feedback. Across three theorem proving benchmarks, LeanGraphSearch improves pass rate at {8, 16, 32} reflection turns by as much as 40% compared to vanilla prover and at most 10% compared to LeanSearch v2, showing higher accuracy-cost efficiency. These findings highlight the value of dependency graph search for theorem proving and motivate further research on structure-aware search for proof discovery.

cs.LG↗
arXiv2026

Witness Encryption via Prime-Order Generic Groups

For every constant $c > 0$, we unconditionally construct witness encryption for NP in the classical generic cyclic group model with the following properties. For SAT instances of size $n$, encryption runs in time poly$(n)$, and any satisfying assignment can be used to decrypt in poly$(n)$ time with correctness error $2^{-n}$. If no satisfying assignment exists, every generic adversary making at most $n^{(\log n)^c}$ group queries has distinguishing advantage at most $n^{-(\log n)^c}$. Along the way, we give the first superconstant-factor NP-hardness of approximation result for homogeneous MinRank under randomized polynomial-time reductions. We obtain a gap of $(\log n)^c$ for any prescribed constant $c>0$, even when YES instances are required to have a rank-one matrix of the form $\mathbf u\mathbf v^T$ where $\mathbf v$ is a 0/1 vector. The main results of this paper were discovered by Codex / ChatGPT 6 Astra Ultra using harness components from the UCLA Moonshot Harness Project, in response to a general strategy for witness encryption suggested by the authors. See the end of the introduction for a discussion.

cs.CR↗
arXiv2026

Unconditional Unclonable Encryption

We give an unconditional construction of information-theoretically secure one-time private-key unclonable encryption for one-bit messages, with efficient encryption and decryption and exponentially small unclonable-indistinguishability advantage. Previous work by Bhattacharyya and Culf (Nature Physics, 2026) and Bhattacharyya, Broadbent, and Culf (arXiv:2603.08916) either only showed inverse-polynomial security loss or required inefficient encryption/decryption operations; we avoid both of these caveats. Our scheme's key is a uniformly random non-identity phase-free Pauli on $n$ qubits, and bit $a$ is encrypted as a random $(-1)^a$ eigenstate of that Pauli. We prove that the probability that both receivers recover the bit is at most $1/2 + O\left(2^{-n/2}\right).$ By a lower bound due to Broadbent, Culf, and Rochette (arXiv:2510.00903), this is the best probability bound achievable with $n$-qubit ciphertexts while ensuring perfect correctness (up to the constant hidden in the $O(\cdot)$).

quant-ph↗
arXiv2026

Continual Graph Memory for Mathematical Research Agents

Using frontier agent harnesses to tackle mathematical research problems has emerged as an effective means of advancing mathematics. However, solving frontier problems in mathematics may require a massive number of agents working in parallel for extended periods to construct proofs, thereby generating an enormous volume of intermediate proof results. Organizing these intermediate results throughout a long-horizon proof-search process and reusing knowledge gained from prior explorations remain major challenges. We present Ansatz, a mathematical research agent built around Continual Graph Memory, a graph-based, evolvable, cross-problem mathematical research memory system that explicitly organizes the entire proof search process and reuses information from exploration trajectories of previous problems. Specifically, we develop a unified graph memory that represents all intermediate exploration results, including facts, plans, and counterexamples, together with edges that explicitly represent the relationships among them; dependency-aware retrieval supplies precisely targeted local context; an evidence-sensitive curator updates the research frontier and distills lessons from prior attempts; and scoped recall surfaces earlier statements and negative findings for local re-proving rather than uncritical reuse. Experiments cover runs across all ten First Proof Second Batch problems, together with four component studies. Ansatz reports closure on all ten research tasks, demonstrating its ability to sustain and resume long-horizon mathematical search. Beyond these problems, Ansatz also produces solutions to the Jamison caterpillar conjecture and Erdős Problems 289, 348, and 488 without human intervention, and makes partial progress on several open problems, illustrating its strong ability to solve open mathematical research problems.

cs.AI↗
arXiv2026

Does Reasoning Improve Psychological Depth in Large Language Models? It Depends on Who's Judging

LLM-as-a-Judge evaluators are increasingly used to score open-ended generation, yet a judge's correlation with human ratings on its development set may not guarantee valid measurement when outputs are closely matched and human preferences are subjective. We study this failure mode through psychological depth in short stories. Seven human readers and an LLM-judge ensemble selected on the original scalar Psychological Depth Scale dataset ($ρ= 0.646$) evaluated 60 blinded, prompt-matched story pairs from GPT-5 vs.\ GPT-4o and DeepSeek-R1 vs.\ DeepSeek-V3. Human preferences showed no universal reasoning advantage: GPT-5 was modestly preferred over GPT-4o (60.0--62.9\%), whereas DeepSeek-R1 trailed V3 (42.9\%), and inter-reader agreement was near chance (Krippendorff's $α= 0.070$), with within-reader consistency and recurring weighting patterns suggesting structured heterogeneity rather than random responding. The judge, by contrast, favored reasoning outputs in 89.0\% of dimension-level comparisons and 59 of 60 pairs on aggregate PDS, uniformly across all five evaluator configurations, and its scores were associated with surface features such as sentence length and lexical diversity. These results suggest that development-set performance is insufficient evidence for deployment validity on a shifted distribution, and that point-estimate judges can obscure the heterogeneity in subjective human evaluation.

cs.LG↗
arXiv2026

Refuting a Conjecture of Umans and Wang on Arithmetic-Progression Divisor Covers

An \emph{$n$-divisor set} is a finite set of positive integers containing a multiple of every integer from $1$ through $n$. Umans and Wang proposed, as the arithmetic-progression version of their Strong $(α,β)$-Divisor Conjecture, an $n$-divisor arithmetic progression having at most $n^{2β}$ terms, each of magnitude at most $\exp(n^α)$. We prove unconditionally that an $n$-divisor arithmetic progression of height $H$ with $\log H=o(\sqrt n)$ must have length \[ L\ge \left(\sqrt{\frac{8}{27}}-o(1)\right) \frac{n^{3/4}}{\sqrt{\log n}}. \] Consequently, the arithmetic-progression version is false whenever $α<1/2$ and $β<3/8$. In particular, it is false at the proposed point $(α,β)=(1/3,1/3)$, even if both bounds are relaxed by $n^{o(1)}$ at the exponent level. The proof uses primes in a fixed band below $\sqrt n$ to turn semiprime divisibility into a finite incidence structure. An elementary bounded-degree linear-space estimate then gives the result. This theorem concerns the one-dimensional arithmetic-progression version only; it does not disprove the higher-rank Strong Divisor Conjecture.

math.NT↗
arXiv2026

List Recovery for Random Low-Rate Linear Codes

We prove a list recovery guarantee for random low-rate linear codes over sufficiently large prime fields. For fixed dimension $d$, error fraction $α$, and accuracy parameter $\varepsilon$, a random $d$-dimensional linear code $C \subseteq \mathbb{F}_p^n$ is, with high probability, $(α,\ell,\frac{1+\varepsilon}{1-α}\ell)$-list recoverable simultaneously for all input list sizes $\ell\le 2^{O_{α, \varepsilon, d}(n/\log n)}$. The proof is inspired by work of Matoušek, Př\'ıvětivý, and Škovroň on reconstructing point sets from their projections. It combines a deterministic graph-theoretic certificate, a nonvanishing determinant criterion, and the Schwartz--Zippel lemma. We also give a lower bound showing that any linear code $C \subseteq \mathbb{F}_p^n$ of dimension at least two cannot be $(α,\ell,\frac{1+\varepsilon}{1-α}\ell)$-list recoverable for feasible list sizes $\ell \geq 2^{Ω_{α, \varepsilon}(n)}$. In this sense, our result is nearly optimal.

cs.IT↗
arXiv2026

SLAM: Structural Linguistic Activation Marking for Language Models

LLM watermarks must be detectable without compromising text quality, yet most existing schemes bias the next-token distribution and pay for detection with measurable quality loss. We present SLAM (Structural Linguistic Activation Marking), a novel white-box watermarking scheme that sidesteps this cost by writing the mark into structural geometry rather than token frequencies: sparse autoencoders identify residual-stream directions encoding linguistic structure (e.g., voice, tense, clause order), and we causally steer those directions at generation time, leaving lexical sampling and semantics unconstrained. On Gemma-2 2B and 9B, SLAM achieves 100% detection accuracy with a quality cost of only 1-2 reward points - compared to 7.5-11.5 for KGW, EWD, and Unigram - with naturalness and diversity preserved at near-unwatermarked levels across both models. The trade-off is a complementary robustness profile: SLAM resists word-level edits but is vulnerable to paraphrase that restructures syntax (at a quality cost), the converse of token-distribution methods.

cs.CL↗
arXiv2026

Public Key Encryption from High-Corruption Constraint Satisfaction Problems

We give a public key encryption scheme with plausible quasi-exponential security based on the conjectured intractability of two constraint satisfaction problems (CSPs), both of which are instantiated with a corruption rate of $1 - o(1)$. First, we conjecture the hardness of a new large alphabet random predicate CSP (LARP-CSP) defined over an arbitrary but strongly expanding factor graph, where the vast majority of predicate outputs are replaced with random outputs. Second, we conjecture the hardness of the standard $k$XOR problem defined over a random factor graph, again where the vast majority of parity computations are replaced with random bits. In support of our hardness conjecture for LARP-CSPs, we give a variety of lower bounds, ruling out many natural attacks including all known attacks that exploit non-random factor graphs. Our public key encryption scheme is the first to leverage high corruption CSPs while simultaneously achieving a plausible security level far above quasi-polynomial. At the heart of our work is a new method for planting cryptographic trapdoors based on the label extended factor graph for a CSP. Along the way to achieving our result, we give the first uniform construction of an error-correcting code that has an expanding, low density generator matrix while simultaneously allowing for efficient decoding from a $1 - o(1)$ fraction of corruptions.

cs.CR↗
arXiv2026

Deterministic Hardness of Approximation For SVP in all Finite $\ell_p$ Norms

We show that, assuming NP $\not\subseteq$ $\cap_{δ> 0}$DTIME$\left(\exp{n^δ}\right)$, the shortest vector problem for lattices of rank $n$ in any finite $\ell_p$ norm is hard to approximate within a factor of $2^{(\log n)^{1 - o(1)}}$, via a deterministic reduction. Previously, for the Euclidean case $p=2$, even hardness of the exact shortest vector problem was not known under a deterministic reduction.

cs.CC↗
arXiv2026

SVP$_p$ is Deterministically NP-Hard for all $p > 2$, Even to Approximate Within a Factor of $2^{\log^{1-\varepsilon} n}$

We prove that SVP$_p$ is NP-hard to approximate within a factor of $2^{\log^{1 - \varepsilon} n}$, for all constants $\varepsilon > 0$ and $p > 2$, under standard deterministic Karp reductions. This result is also the first proof that \emph{exact} SVP$_p$ is NP-hard in a finite $\ell_p$ norm. Hardness for SVP$_p$ with $p$ finite was previously only known if NP $\not \subseteq$ RP, and under that assumption, hardness of approximation was only known for all constant factors. As a corollary to our main theorem, we show that under the Sliding Scale Conjecture, SVP$_p$ is NP-hard to approximate within a small polynomial factor, for all constants $p > 2$. Our proof techniques are surprisingly elementary; we reduce from a \emph{regularized} PCP instance directly to the shortest vector problem by using simple gadgets related to Vandermonde matrices and Hadamard matrices.

cs.CC↗
arXiv2026

TaoBench: Do Automated Theorem Prover LLMs Generalize Beyond MathLib?

Automated theorem proving (ATP) benchmarks largely consist of problems formalized in MathLib, so current ATP training and evaluation are heavily biased toward MathLib's definitional framework. However, frontier mathematics is often exploratory and prototype-heavy, relying on bespoke constructions that deviate from standard libraries. In this work, we evaluate the robustness of current ATP systems when applied to a novel definitional framework, specifically examining the performance gap between standard library problems and bespoke mathematical constructions. We introduce TaoBench, an undergraduate-level benchmark derived from Terence Tao's Analysis I, which formalizes analysis by constructing core mathematical concepts from scratch, without relying on standard Mathlib definitions, as well as by mixing from-scratch and MathLib constructions. For fair evaluation, we build an agentic pipeline that automatically extracts a compilable, self-contained local environment for each problem. To isolate the effect of definitional frameworks, we additionally translate every problem into a mathematically equivalent Mathlib formulation, yielding paired TaoBench-Mathlib statements for direct comparison. While state-of-the-art ATP models perform capably within the MathLib framework, performance drops by an average of roughly 26% on the definitionally equivalent Tao formulation. This indicates that the main bottleneck is limited generalization across definitional frameworks rather than task difficulty. TaoBench thus highlights a gap between benchmark performance and applicability, and provides a concrete foundation for developing and testing provers better aligned with research mathematics.

cs.LG↗
arXiv2026

Beyond What Seems Necessary: Hidden Gains from Scaling Training-Time Reasoning Length under Outcome Supervision

Training LLMs to think and reason for longer has become a key ingredient in building state-of-the-art models that can solve complex problems previously out of reach. Recent efforts pursue this in different ways, such as RL fine-tuning to elicit long CoT or scaling latent reasoning through architectural recurrence. This makes reasoning length an important scaling knob. In this work, we identify a novel phenomenon (both theoretically and experimentally): under outcome-only supervision, out-of-distribution (OOD) performance can continue improving as training-time reasoning length (e.g., the token budget in RL, or the loop count in looped Transformers) increases, even after in-distribution (ID) performance has saturated. This suggests that robustness may require a larger budget than ID validation alone would indicate. We provide theoretical explanations via two mechanisms: (i) self-iteration can induce a stronger inductive bias in the hypothesis class, reshaping ID-optimal solutions in ways that improve OOD generalization; and (ii) when shortcut solutions that work for ID samples but not for OOD samples persist in the hypothesis class, regularization can reduce the learned solution's reliance on these shortcuts as the number of self-iterations increases. We complement the theory with empirical evidence from two realizations of scaling training-time reasoning length: increasing the number of loops in looped Transformers on a synthetic task, and increasing token budgets during RL fine-tuning of LLMs on mathematical reasoning.

cs.LG↗
arXiv2025

Quantum Advantage via Solving Multivariate Polynomials

In this work, we propose a new way to (non-interactively, verifiably) demonstrate quantum advantage by solving the average-case $\mathsf{NP}$ search problem of finding a solution to a system of (underdetermined) constant degree multivariate equations over the finite field $\mathbb{F}_2$ drawn from a specified distribution. In particular, for any $d \geq 2$, we design a distribution of degree up to $d$ polynomials $\{p_i(x_1,\ldots,x_n)\}_{i\in [m]}$ for $m 2$, it is classically hard to find one based on a thorough review of existing classical cryptanalysis. Our work thus posits that degree three functions are enough to instantiate the random oracle to obtain non-relativized quantum advantage. Our approach begins with the breakthrough Yamakawa-Zhandry (FOCS 2022) quantum algorithmic framework. In our work, we demonstrate that this quantum algorithmic framework extends to the setting of multivariate polynomial systems. Our key technical contribution is a new analysis on the Fourier spectra of distributions induced by a general family of distributions over $\mathbb{F}_2$ multivariate polynomials -- those that satisfy $2$-wise independence and shift-invariance. This family of distributions includes the distribution of uniform random degree at most $d$ polynomials for any constant $d \geq 2$. Our analysis opens up potentially new directions for quantum cryptanalysis of other multivariate systems.

quant-ph↗
arXiv2025

Sandcastles in the Storm: Revisiting the (Im)possibility of Strong Watermarking

Watermarking AI-generated text is critical for combating misuse. Yet recent theoretical work argues that any watermark can be erased via random walk attacks that perturb text while preserving quality. However, such attacks rely on two key assumptions: (1) rapid mixing (watermarks dissolve quickly under perturbations) and (2) reliable quality preservation (automated quality oracles perfectly guide edits). Through large-scale experiments and human-validated assessments, we find mixing is slow: 100% of perturbed texts retain traces of their origin after hundreds of edits, defying rapid mixing. Oracles falter, as state-of-the-art quality detectors misjudge edits (77% accuracy), compounding errors during attacks. Ultimately, attacks underperform: automated walks remove watermarks just 26% of the time -- dropping to 10% under human quality review. These findings challenge the inevitability of watermark removal. Instead, practical barriers -- slow mixing and imperfect quality control -- reveal watermarking to be far more robust than theoretical models suggest. The gap between idealized attacks and real-world feasibility underscores the need for stronger watermarking methods and more realistic attack models.

cs.CR↗