What was already known
Hladík studied extrema of symmetric tridiagonal interval matrices; its outermost-eigenvalue endpoint theorem does not directly give this interior gap. Standard bidiagonal singular-value results supply ingredients, but the project reports no located one-switch optimizer or sharp arbitrary-channel theorem.
AI involvement
JD Jones directed the project; its disclosure credits GPT-6 Pro with the initial one-switch and sharp-channel arguments and GPT-6 Astra with extension and Lean proof work.
JD Jones set scope; GPT-6 Pro developed the one-switch and sharp-channel arguments and GPT-6 Astra reconstructed, extended and formalized them.
Importance
Reduces a continuous uncertainty problem to finitely many ordered configurations and extends the sharp bound to noncommuting channel matrices.
Limitations
The one-switch value theorem includes zero lower bounds, but the project’s point-or-edge classification of all minimizers requires strictly positive weak lower bounds. Cyclic and infinite-volume systems are outside scope. The bounded prior-art search leaves global priority unresolved; no independent human specialist review is reported.
Technical verification
The source reports formal proof and statement comparison for one-switch value and arbitrary-channel bound, with standard-axiom audit. This applies to the selected claim and is not an independent reproduction by AIProofWatch.
Formal checking
The source reports Lean proofs of the listed Comparator declarations, axiom audit and exact-commit Linux replay through Lean, NanoDa and con-ron; supplementary boundary-threshold and determinant results are outside that contract.
- Checker
- Lean 4 kernel and documented Comparator or independent-kernel replay run by repository authors
- Scope
- For a finite open chain with positive strong-bond lower bounds βᵢ, boundary parameter s≥0 and weak-to-strong ratios qᵢ∈[ℓᵢ,ρᵢ], the minimum scalar squared gap minq λmin(DβQ(q)ᵀQ(q)Dβ) is positive and equals the minimum over the ordered one-switch endpoint vectors q⁽ᵏ⁾, with lower endpoints before k and upper endpoints after k. The same value sharply bounds the squared nonzero gap for arbitrary finite-dimensional complex matrix channels satisfying the stated singular-value constraints.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Only internal agent review is documented, and no human specialist explanation or check of the one-switch optimizer is reported. 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