What was already known
Earlier one-family results either covered finite real sequences with a weaker bound or did not preserve diagonal entries. The countable representation was an open problem.
AI involvement
The authors guided GPT-5.4 to a finite real bounded-function construction, then generalized it to countable complex sequences. GPT-5.5 assisted the Lean verification.
GPT-5.4 supplied the central finite real construction under author guidance; the authors supplied the complex countable extension.
Importance
The theorem gives a sharp complete-Gram representation for countable complex Bessel sequences.
Limitations
GPT-5.4 supplied the finite real construction; the authors developed the countable complex extension. The Lean check covers the existence theorem, not the paper’s separate optimality and orthonormal-extension claims.
Technical verification
Lean and NanoDa reportedly check the selected K=1 complete-Gram existence statement.
Formal checking
Compilation of Showcase_WithProofs.lean, axiom audit and Comparator statement comparison
- Checker
- Lean 4 kernel and NanoDa
- Scope
- GrothendieckBessel.main: countable complex Bessel-bound-1 Gram realization with K=1
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The authors explain the finite construction and its countable complex extension.
Correction and publication history
The model’s finite construction is retained in the countable complex theorem; the separate application and optimality observation are outside this claim.
Read this published revision