What was already known
Taylor, Hendrickx and Glineur posed the smooth-convex final-gradient conjecture and supplied sharp examples. Chikhi proved the balancing-step case and a factorized multiplier shape, leaving the full constant-step curve open in the cited version.
AI involvement
JD Jones is responsible human author. The disclosure credits GPT-6 Astra with mathematical arguments, uniform construction and proximal extensions, Lean formalization and checking, with collaborating agents reviewing the work.
JD Jones is the responsible author; the disclosure credits GPT-6 Astra with the uniform multiplier construction, mathematical arguments and proximal extensions central to the selected result.
Importance
Completes the uniform last-gradient rate across all nonnegative step sizes and finite horizons, with exact witnesses and equality structure.
Limitations
The cited earlier conjecture also covers positive strong convexity; that regime is not solved here. Proximal bounds assume supplied actual minimizers and count a terminal evaluation. Written quantitative stability outside the ten compared declarations is not included in the formal check. No independent human specialist review is claimed.
Technical verification
The source reports ten completed lean declarations, challenge/solution statement comparison, standard-axiom audit and independent proof replay. This applies to the selected claim and is not an independent reproduction by AIProofWatch.
Formal checking
The source reports completed Lean proofs of ten principal declarations, exact statement comparison, axiom audit and fresh Linux Lean/NanoDa/con-ron replay tied to the checked commit.
- Checker
- Lean 4 kernel and documented Comparator or independent-kernel replay run by repository authors
- Scope
- For every N≥0 and h≥0, constant-step gradient descent xᵢ₊₁=xᵢ−(h/L)∇f(xᵢ) on a differentiable convex function with L-Lipschitz gradient in a real Hilbert space satisfies ‖∇f(xN)‖≤L‖x0−x*‖max{1/(1+Nh),|1−h|^N}. One-dimensional Huber and quadratic examples attain the two branches. The compared Lean contract also classifies sampled equality trajectories and proves the same sharp factor for the terminal residual of relaxed proximal minimization with actual minimizers.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The repository documents agent review of the analytic transfers and formal statements but no independent human explanation of the central uniform construction. Few is the evidence default.
Correction and publication history
Claim-level primary-source review of exact theorem, prior baseline, AI contribution and documented checking scope.
Read this published revision