What was already known
Ghasemi and Kopparty proved the k+1 sparsity lower bound but known constructions used roughly 3^(k/2) terms, leaving an exponential gap.
AI involvement
After the authors set the question and empirical prompts, GPT-5.5 Pro found the general sparse-polynomial framework and the proof through Lemmas 4.2 and 4.5. The authors wrote and checked the paper.
GPT-5.5 Pro found the retained construction and proof framework under human prompting; autonomy is not documented.
Importance
Matches the sparsity lower bound under the stated number-theoretic condition and improves fixed-server PIR parameters.
Limitations
Conjecture 1.1 is not proved. The unconditional finite cases do not establish all fixed k, and the superconstant-server extension requires a stronger conjecture. The authors report manually checking the proofs.
Technical verification
The authors report manually verifying all paper proofs, including the conditional Theorem 1.2. This is a source-reported complete human proof check, not an AIProofWatch reproduction or proof of Conjecture 1.1.
Formal checking
Manual verification of all proofs
- Checker
- The paper's human authors
- Scope
- The conditional k+1-sparse decoding-polynomial construction and Lemmas 4.2 and 4.5 proving Theorem 1.2
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The authors' technical overview and proof explain the root-of-unity-grid construction, its number-theoretic assumption and the PIR consequence.
Correction and publication history
Matches the sparsity lower bound under the stated number-theoretic condition and improves fixed-server PIR parameters.
Read this published revision