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A Sparse-Fock Proof of the Upper Edge for Fully Independent SparseStack

Diar Heidary

For a fixed n×d real matrix U with UᵀU=I, d≥1, 0<ε≤1 and 0<δ≤1, the fully independent signed-hash SparseStack construction gives failure probability at most δ with column sparsity below 1356 max(1,⌈log₂(d/δ)⌉)/ε and row count below 306456(d+max(1,⌈log₂(d/δ)⌉))/ε².

Sources for this record

What was already known

Nelson and Nguyen conjectured optimal row and sparsity scales. Earlier work addressed parts of the question under different constructions.

AI involvement AI-centered

Diar Heidary directed the project. ChatGPT 5.6 Sol and Claude Fable 5 proposed and challenged mathematical arguments used in the retained proof; Heidary reviewed the final report.

The disclosed model collaboration proposed mathematical arguments throughout the retained proof, while Heidary led the project and selected its scope.

Importance Notable

Achieves the target scales for this explicitly defined independent SparseStack law.

Limitations

The theorem is for a fixed matrix and the stated independent distribution, not every OSNAP law. A short-seed sampler and the paper’s separate refinement are outside the selected Lean result; independent human referee review is undocumented.

Technical verification Formal (Lean 4/Mathlib kernel, Comparator and NanoDa)

The source reports complete Lean proofs of the fixed-matrix theorem and the two scoped extensions, with no uninstantiated analytic hypotheses; kernel replay and axiom audit are reported. Human author review is not counted as a completed proof check.

Formal checking

The source reports complete Lean proofs of the fixed-matrix theorem and the two scoped extensions, with no uninstantiated analytic hypotheses; kernel replay and axiom audit are reported. Human author review is not counted as a completed proof check.

Checker
Lean 4 kernel, Palomar Comparator and NanoDa
Scope
Fully independent SparseStack fixed-matrix theorem and selected exact-s and 4q combined-selector transfer declarations
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Full

Heidary's technical report explains the retained fixed-matrix mechanism from convex-order transfer through moment bounds and rounding; author review is documented, although independent review is absent.

Correction and publication history