Lean-certified four-point HRT results for three lattice points and one off-lattice point

Vignon Oussa

Lean-certified four-point HRT results for three lattice points and one off-lattice point

T0 review · 0 major / 2 minor · reviewed 2026-05-08 · grok-4.3

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Pith's one-line read If the symplectic area between a and b exceeds 1 and the fourth point lies off the lattice via irrational coefficients, then four time-frequency shifts of any nonzero L2 function remain linearly independent.

desk verdict This paper supplies a Lean-certified proof for one specific four-point HRT configuration with an off-lattice point, which is a clean verification artifact but limited in analytic reach. read the letter →

arxiv 2604.21228 v1 submitted 2026-04-23 math.FA

Vignon Oussa This is my paper · ORCID

classification math.FA

keywords HRTconjecturetime-frequencyanalysislinearindependenceLeancertificationsymplecticareaL2functionsGaborsystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper supplies a Lean-certified proof that in a four-point configuration with three points on a lattice and one off-lattice point, the corresponding time-frequency shifts of any nonzero square-integrable function stay linearly independent. This holds when the absolute value of the symplectic product of the lattice generators exceeds 1 and the coefficients that locate the off-lattice point, together with 1, are linearly independent over the rationals. A separate certified statement covers the case in which the coefficients are rational, reducing the problem to a known lattice result. The certification supplies machine-checkable confirmation of the analytic steps for this mixed configuration.

What carries the argument

The four-point set Lambda = {0, a, b, nu} together with the time-frequency shift operators pi(lambda) that map any f to its shifted version; the arithmetic conditions on a, b, r, s ensure the shifts produce independent vectors.

What would settle it

An explicit choice of a, b, r, s satisfying |symp(a,b)| > 1 and the irrationality condition, together with a nonzero f in L2(R) for which the four vectors are linearly dependent, would falsify the claim.

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Extended reading notes

Core claim

The principal result states that if |symp(a,b)| > 1 and 1, r, s are linearly independent over Q, then for every nonzero f in L2(R) the four vectors f, pi(a)f, pi(b)f, pi(nu)f are linearly independent, where nu = r a + s b lies outside the lattice generated by a and b. A companion theorem handles the rational-coordinate case by reduction to Linnell's theorem.

Load-bearing premise

The linear independence of 1, r, and s over the rationals, which places the fourth point off the lattice; if this fails the configuration collapses to a lattice case already covered by other theorems.

Editorial extensions

If this is right

The HRT conjecture holds for every such mixed three-lattice-plus-one-off-lattice configuration.
The Lean certification removes the possibility of hidden gaps in the analytic argument for this family of examples.
The rational subcase follows immediately once the off-lattice point is recognized as lying on a finer lattice.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Formal verification tools could be applied to additional partial cases of the HRT conjecture that remain open.
Relaxing the area or independence conditions might enlarge the set of configurations for which independence is known.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

This paper certifies in Lean that for the four-point set with three lattice points and one off-lattice point, under the conditions that the symplectic form exceeds 1 and the coordinates 1, r, s are linearly independent over the rationals, the corresponding time-frequency shifts are linearly independent for any nonzero f in L2. It also handles the rational case by direct appeal to Linnell's theorem. The main deliverable is the formal package itself, complete with an appendix that lists the exact Lean modules, analytic inputs, and certification ledger plus a download link. That explicit record is new and practical; prior informal arguments for this configuration did not come with machine-checked code against standard real-analysis axioms. The case split is handled cleanly and avoids any circular definitions or fitted parameters. The formalization therefore gives reproducible evidence for this fragment. The limitation is scope. The work does not derive a new analytic strategy or extend to larger sets; it verifies an expected statement for this exact setup. Anyone hoping to apply it more broadly will still need to bridge it to other configurations by hand. The paper stays in standard mathematical prose and does not overclaim. This is mainly for people already tracking the HRT conjecture or working with formal tools in analysis. A reader interested in verified results or in testing Lean on time-frequency problems will get concrete value from the ledger and the explicit inputs. I would bring it to a reading group if the group covers either formal verification or harmonic analysis. I would not cite it in my own papers unless I were doing similar formalization work. It deserves peer review because the certified package is a tangible, inspectable contribution that referees can evaluate directly against the stated axioms and imported theorems.

Referee Report

0 major / 2 minor

Summary. The paper records Lean-certified theorems for linear independence in the four-point Heil-Ramanathan-Topiwala configuration Λ = {0, a, b, ν} with ν = r a + s b off the lattice generated by a and b. The principal result states that if |symp(a,b)| > 1 and 1, r, s are linearly independent over ℚ, then f, π(a)f, π(b)f, π(ν)f are linearly independent in L²(ℝ) for any nonzero f; a separate theorem covers the rational case r, s ∈ ℚ via reduction to Linnell's theorem.

Significance. The result supplies machine-checked verification for a nontrivial special case of the HRT conjecture in time-frequency analysis. The explicit separation of the irrational case (direct Lean formalization against standard real-analysis axioms) from the rational case (via imported Linnell theorem), together with the appendix ledger of analytic inputs and certification details, constitutes a clear strength. This level of formal assurance is uncommon in the area and raises the reliability of the claimed independence statement.

minor comments (2)

The abstract states that 'a download link is provided' for the Lean package; the precise URL or repository reference should appear in the main text or appendix for immediate accessibility.
Notation for the symplectic form symp(a,b) and the time-frequency shift operator π is introduced without an explicit reminder of the standard definitions; a one-sentence recall in §1 would aid readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, for highlighting the value of the Lean certification in the irrational and rational cases, and for recommending acceptance.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; Lean formalization is self-contained against standard axioms

full rationale

The paper records a Lean-certified formal theorem establishing linear independence of the four time-frequency shifts f, π(a)f, π(b)f, π(ν)f under the stated conditions on the symplectic form and the Q-linear independence of 1, r, s. The development explicitly separates the irrational case (handled by direct formalization in Lean using standard real-analysis axioms) from the rational case (reduced to the externally known Linnell theorem). No quantity is defined in terms of another, no parameters are fitted to data, and the central claim does not reduce to its inputs by the paper's own equations or by a self-citation chain. The formal ledger and analytic inputs are recorded in the appendix, rendering the derivation self-contained and independently verifiable.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The claim rests on the standard axioms of Lebesgue integration and inner-product spaces together with the definition of the symplectic form; the only non-standard import is Linnell's theorem for the rational subcase. No free parameters or invented entities appear.

assumptions (2)

standard math Standard properties of L²(ℝ) as a Hilbert space and the definition of the time-frequency shift operator π
Invoked throughout the formal development to define the vectors whose independence is asserted.
domain assumption Linnell's theorem on linear independence for lattice configurations
Used without re-proof for the rational-coordinate subcase.

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Cite this review

Pith. "Pith review of Lean-certified four-point HRT results for three lattice points and one off-lattice point." pith.science (2026). https://pith.science/paper/2604.21228

@misc{pith2026260421228,
  author       = {Pith},
  title        = {Pith review of: Lean-certified four-point HRT results for three lattice points and one off-lattice point},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.21228}},
  note         = {Machine review of arXiv:2604.21228}
}

abstract

We record a Lean-certified theorem package for the four-point Heil--Ramanathan--Topiwala configuration \[ \Lambda=\{0,a,b,\nu\}\subset \R^2, \qquad \Lzero=\Z a+\Z b, \qquad \nu=r a+s b, \] with $a$ and $b$ linearly independent. The principal certified theorem states that if $|\symp(a,b)|>1$ and $1,r,s$ are linearly independent over $\Q$, then for every nonzero $f\in L^2(\R)$ the four vectors \[ f,\qquad \pi(a)f,\qquad \pi(b)f,\qquad \pi(\nu)f \] are linearly independent. A second certified theorem treats the rational-coordinate case $r,s\in \Q$, where the configuration lies in a finer full-rank lattice and linear independence follows from Linnell's theorem. The paper is written in standard mathematical prose. An appendix records the precise Lean certification ledger and the explicit analytic inputs used by the formal development and a download link is provided.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The Minimum Cardinality of a Dependent Finite Gabor System Is Four
math.FA 2026-08 conditional novelty 8.0 of 10

Four time-frequency shifts of a nonzero Schwartz function can be linearly dependent, and no smaller number can, so the minimum cardinality of a dependent Gabor system is exactly 4.
An intrinsically subcritical four-point counterexample
math.CA 2026-08 conditional novelty 7.0 of 10

Four distinct time-frequency shifts of a nonzero Schwartz function are linearly dependent while every symplectic triangle determinant is strictly below one.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · cited by 2 Pith papers

[1]

M. B. Bekka and P. Driutti,Restrictions of irreducible unitary representations of nilpotent Lie groups to lattices, J. Funct. Anal.168(1999), no. 2, 514–528

work page 1999
[2]

Gröchenig,Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, 2001

K. Gröchenig,Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, 2001

work page 2001
[3]

C. Heil, J. Ramanathan, and P. Topiwala,Linear independence of time-frequency translates, Proc. Amer. Math. Soc.124(1996), no. 9, 2787–2795

work page 1996
[4]

P. A. Linnell,Von Neumann algebras and linear independence of translates, Proc. Amer. Math. Soc.127 (1999), no. 11, 3269–3277

work page 1999
[5]

Ramanathan and T

J. Ramanathan and T. Steger,Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal. 2(1995), no. 2, 148–153

work page 1995

Pith tools

Reviewed May 8, 2026 · model on record in the stance chip above.

[1] [1]

M. B. Bekka and P. Driutti,Restrictions of irreducible unitary representations of nilpotent Lie groups to lattices, J. Funct. Anal.168(1999), no. 2, 514–528

work page 1999

[2] [2]

Gröchenig,Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, 2001

K. Gröchenig,Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, 2001

work page 2001

[3] [3]

C. Heil, J. Ramanathan, and P. Topiwala,Linear independence of time-frequency translates, Proc. Amer. Math. Soc.124(1996), no. 9, 2787–2795

work page 1996

[4] [4]

P. A. Linnell,Von Neumann algebras and linear independence of translates, Proc. Amer. Math. Soc.127 (1999), no. 11, 3269–3277

work page 1999

[5] [5]

Ramanathan and T

J. Ramanathan and T. Steger,Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal. 2(1995), no. 2, 148–153

work page 1995