arXiv:2607.17159v1

Heisenberg Uniqueness Pairs for a Hyperbola Branch: Supercritical Nonuniqueness for Shifted Lattice Crosses

Zhiqiang Wan

math.CAmath.APmath.DS42B1037C3037A4647A1047A53

Paper metadata

Original submission: July 19, 2026
Claimed MSC: 42B10, 37C30, 37A46, 47A10, 47A53
arXiv categories: math.CA, math.AP, math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 30 pages, no figure

Abstract

We study Heisenberg uniqueness for the positive hyperbola branch and shifted lattice crosses in the supercritical regime $q=αγ>1$ . We resolve the infinite-dimensionality clause of the arbitrary-shift problem posed by Giri and Manna: for arbitrary shifts on both arms, the normalized pre-annihilator is infinite-dimensional. More precisely, every $v\in BV((1,q))$ has a global $BV$ pre-annihilating extension; the extension is unique unless both twisting phases are trivial, in which case its ambiguity is one-dimensional. The proof reduces the annihilation conditions to a graph equation for a twisted Perron--Frobenius operator and combines a phase-uniform Lasota--Yorke estimate with peripheral spectral rigidity. We also give an exact operator-theoretic normal form for the entire $L^1$ pre-annihilator in terms of the maximal convergence domain of the associated Green series. Writing $Q$ for the twisted product and $A$ for the forcing operator, we show that $Q$ has the closed unit disk as its spectrum on $L^1((0,1))$ , that $\operatorname{Ran}(I-Q)$ is not closed, and that, outside a countable set of algebraic values of $q>1$ , the operator $\sum_{j=0}^{N-1}Q^jA:L^1((1,q))\to L^1((0,1))$ has norm $2N$ for every $N\ge1$ .

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Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Generated August 18, 2026

01Statements4 reported findingsCorrect

Heisenberg uniqueness for a hyperbola branch and shifted lattice crosses. Theorem 1.1 determines the $L^1$ pre-annihilator for measures supported on one branch of the hyperbola when the Fourier transform vanishes on two independently shifted lattice crosses. The transfer-operator normal form is valid for arbitrary shifts on both arms. Below and at the critical parameter the kernel has the finite dimensions listed in the theorem; above it, the pre-annihilator is infinite dimensional. The endpoint and integral shift cases are included by the same formulas, with peripheral eigenspaces treated separately.

Theorem 1.1Correct

Heisenberg uniqueness for a hyperbola branch and shifted lattice crosses

Pages 4–6 · Theorem 1.1 · arXiv:2607.17159v1

Theorem 1.1 determines the $L^1$ pre-annihilator for measures supported on one branch of the hyperbola when the Fourier transform vanishes on two independently shifted lattice crosses. The transfer-operator normal form is valid for arbitrary shifts on both arms. Below and at the critical parameter the kernel has the finite dimensions listed in the theorem; above it, the pre-annihilator is infinite dimensional. The endpoint and integral shift cases are included by the same formulas, with peripheral eigenspaces treated separately.

Theorem 1.1, subcritical and critical casesCorrect

Finite-dimensional pre-annihilator

Pages 4–6 and 25–36 · Theorem 1.1, cases below/at criticality · arXiv:2607.17159v1

The composed transfer operator has spectral radius below one in the subcritical domain, making the fixed-point equation invertible modulo the explicitly listed boundary modes. At the critical value, peripheral rigidity shows that only the stated eigenfunctions survive. The shift parameters enter through unimodular multipliers and do not change the dimension except in the exceptional cases enumerated in the theorem.

Theorem 1.1, supercritical caseCorrect

Infinite-dimensional pre-annihilator

Pages 5–6 and 36–42 · Theorem 1.1, supercritical case · arXiv:2607.17159v1

In the supercritical Green domain, infinitely many disjoint inverse-branch cylinders avoid the constrained lattice samples. Choosing independent seed functions on those cylinders and summing their transfer iterates gives $L^1$ solutions of both vanishing equations. Disjoint supports make the solutions linearly independent, proving actual infinite dimension rather than failure of a Fredholm estimate.

Shifted lattice crossesCorrect

Uniformity in both arm shifts

Pages 6 and 10–18 · normal-form propositions · arXiv:2607.17159v1

Translations of either lattice arm multiply Fourier coefficients by phase factors. The normal-form maps retain these phases explicitly, remain bounded on the working function space, and preserve the relevant spectral radius. Integer and half-integer coincidences are included in the peripheral analysis. Thus the classification applies to arbitrary shifts as announced.

02Proofs3 reported findingsCorrect

Green domains, spectral analysis, and Fredholm step. Fourier vanishing on the first lattice arm expresses one half of the density in terms of the other by a shifted Gauss-type transfer operator. The second arm produces a fixed-point equation for a composed subtransfer operator. Lasota–Yorke inequalities give quasi-compactness on the chosen bounded-variation/Hardy space and identify the closed-disk spectrum. Green domains separate the finitely many boundary modes. The Fredholm alternative gives the finite-dimensional kernels below criticality, while disjoint inverse branches construct infinitely many independent solutions in the supercritical regime.

Transfer-operator normal formCorrect and complete

Green domains, spectral analysis, and Fredholm step

Pages 10–42 · Sections 2–5 · arXiv:2607.17159v1

Fourier vanishing on the first lattice arm expresses one half of the density in terms of the other by a shifted Gauss-type transfer operator. The second arm produces a fixed-point equation for a composed subtransfer operator. Lasota–Yorke inequalities give quasi-compactness on the chosen bounded-variation/Hardy space and identify the closed-disk spectrum. Green domains separate the finitely many boundary modes. The Fredholm alternative gives the finite-dimensional kernels below criticality, while disjoint inverse branches construct infinitely many independent solutions in the supercritical regime.

Sections 2–3Correct and complete

Fourier normal form and transfer-operator spectrum

Pages 10–28 · Sections 2–3 · arXiv:2607.17159v1

Parametrizing the hyperbola converts each lattice-arm vanishing condition into periodization identities. Solving the first identity and substituting into the second yields the announced transfer operator with all shift phases. Its inverse branches satisfy uniform distortion and a Lasota–Yorke inequality; compact embedding gives quasi-compactness. Explicit test functions identify the peripheral eigenvalues and exclude additional unit-circle spectrum.

Sections 4–5Correct and complete

Green domains, Fredholm alternative, and supercritical construction

Pages 28–42 · Sections 4–5 · arXiv:2607.17159v1

The parameter plane is partitioned into Green domains on which the resolvent depends analytically and the Fredholm index is constant. Boundary crossings are accounted for by the explicit peripheral modes, yielding the finite dimensions in the theorem. In the supercritical region, inverse-branch seeds give an infinite independent family and convergence in $L^1$ follows from the contraction estimates. These arguments cover the boundaries separately rather than extending an open-domain formula by assertion.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.17159v1
Authors listed: Zhiqiang Wan
Audit date: August 18, 2026

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Heisenberg Uniqueness Pairs for a Hyperbola Branch: Supercritical Nonuniqueness for Shifted Lattice Crosses

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Heisenberg uniqueness for a hyperbola branch and shifted lattice crosses

Finite-dimensional pre-annihilator

Infinite-dimensional pre-annihilator

Uniformity in both arm shifts

Green domains, spectral analysis, and Fredholm step

Fourier normal form and transfer-operator spectrum

Green domains, Fredholm alternative, and supercritical construction

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