What was already known
The textbook rate is O(T⁻¹), while silver schedules reach about O(T⁻¹·²⁷¹⁵). The general first-order Ω(T⁻²) benchmark did not separate prescribed-step gradient descent from accelerated methods.
AI involvement
GPT-5.6 Sol Pro developed the main proof after Ma and Chen supplied the objective and high-level resisting-oracle plan. They reviewed, verified and revised it; Codex later helped formalize it.
The model developed the substantive proof after the authors supplied the objective and high-level strategy. That human strategic input prevents an autonomous label, while the final proof is model-led.
Importance
Rules out a uniform O(T⁻²) last-iterate rate from predetermined nonnegative stepsizes alone.
Limitations
The theorem does not cover adaptive or signed schedules, best iterates, or the endpoint exponent. The authors report a completed Lean build and their own proof review; the manuscript-to-Lean mapping was assessed through their traceability report.
Technical verification
The authors explicitly report reviewing and verifying the generated proof. Their Lean project reports a completed main theorem with matching finite-horizon quantifiers, a successful clean build, no placeholders or project axioms, and only ordinary mathlib axioms. Both checks are source-reported.
Formal checking
Full Lean/mathlib build, traceability map and axiom/placeholder audit
- Checker
- Lean 4 kernel, reported by Ma and Chen
- Scope
- GDLowerBound.mainTheorem: finite-horizon last-iterate Ω((T+1)^−p) for every prescribed nonnegative schedule and p>sqrt(2+sqrt(3))
- Basis
- Reported by the linked checker
Formal checking
Reviewed, verified and revised the generated argument
- Checker
- Jianhao Ma and Yuxin Chen, the human authors
- Scope
- The generated main lower-bound proof in Theorem 2.1
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Ma and Chen give a detailed roadmap and explanations of the geometric trajectory, order-free matching and rank-growth mechanisms after revising the model proof.
Correction and publication history
Rules out a uniform O(T⁻²) last-iterate rate from predetermined nonnegative stepsizes alone.
Read this published revision