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Open research questions in advanced mathematical theories

46 unresolved questions extracted from the limitations and future-work sections of 148 advanced mathematical theories papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • The central claim remains explicitly conditional. The paper does not assert an unconditional proof of BSD. The paper requires additional global assembly for a rational elliptic curve with bad reduction.

    A Spectral-Geometric Program of Birch and Swinnerton-Dyer Conjecture · 2026 · DOI

  • Future research should focus on verifying the hypotheses isolated in the paper. Future research should also focus on refining the congruent-number test suite. Future research should explore the potential applications of the paper's results in cryptography and coding theory.

    A Spectral-Geometric Program of Birch and Swinnerton-Dyer Conjecture · 2026 · DOI

  • The O(1) terms in the bounds need to be made explicit. The method cannot be extended to α = 2.

    Existence of a Sidon set for the distinct distance constant · 2026 · DOI

  • To improve the bounds for the distinct distance constant. To adapt the method to other constraint equations.

    Existence of a Sidon set for the distinct distance constant · 2026 · DOI

  • The paper identifies a gap in the existing literature regarding the analysis of threshold effects in a discrete quasi-1D scattering system.

    Topological Levinson’s Theorem and Corrections at Thresholds: the Full Picture in a Quasi-1D Example · 2026 · DOI

  • These open issues are not fatal; they are research frontiers. Page 169 Chapter 31 has provided a critical audit of the STC, celebrating its resolved tensions and honestly confronting its open problems. Finally, we confronted the open problems—the Z/Higgs degeneracy, the quantitative bridge, formalizing dynamics—and outlined a software toolkit, the Syntactic Reality Engine, to explore them.

    Quantum Laws of Form: A Syntactic Foundation for Physics: From The Calculus of Distinction to Ultrametric Cosmology · 2026 · DOI

  • To extend the results to other types of random functions. To apply the results to other areas of research. To further analyze the behavior of the random measures.

    A law of large numbers concerning the distribution of critical points of random Fourier series · 2026 · DOI

  • The current research on the subject has limitations. There is a gap in the understanding of the distribution of critical points of random Fourier series. The paper aims to fill this gap.

    A law of large numbers concerning the distribution of critical points of random Fourier series · 2026 · DOI

  • The paper identifies a gap in the derivation of the two-term formula for the inverse fine-structure constant. The paper identifies a gap in the study of the almost-Mathieu operator and its application to the boundary Hamiltonian.

    Golden-Ratio Torus Holography: Cantor Spectrum, AdS₂ Geometry, and a Two-Term Formula for the Inverse Fine-Structure Constant · 2026 · DOI

  • P1 (ζ(3) correction). Derive Δ₂ = 2Tʜ·ζ(3)/F₉² from the Bogoliubov integral. Note: ζ(3) is odd and cannot arise from the Laurent-Bernoulli expansion. The mechanism is genuinely open. P2 (Central charge). Derive c = 6 from (1.1)–(1.2). Three motivated paths; none complete. P3 (Boundary value). Derive 20φ⁴ from a single argument. Three contributing factors, not unified. P4 (Coupling). Derive λ = 2 from R/r = 4. P5 (3+1 dimensions). Extend to AdS₄/CFT₃. Formula (3.4) gives R₃ = −12(lnφ)² for AdS₄. The 3D boundary theory is conjectured to relate to the icosahedral quasi-periodic Schrödinger operator, consistent with Shechtman quasi-crystals and the Fibonacci quasi-crystal literature. Not derived from (1.1)–(1.2). Structural parallel (not a claim): the Aubry-André self-duality (λ→4/λ) has fixed point λ = 2 producing dimₕ = 1/2; the Riemann functional equation (s→1−s) is symmetric about Re(s) = 1/2. Both are Fourier self-dualities at the geometric centre of their respective domains. See Connes and Berry-Keating.

    Golden-Ratio Torus Holography: Cantor Spectrum, AdS₂ Geometry, and a Two-Term Formula for the Inverse Fine-Structure Constant · 2026 · DOI

  • Developing new algorithms and techniques for evaluating Γ_p(N) efficiently. Exploring the function's connections to other areas of number theory, such as elliptic curves and modular forms. Analyzing the function's potential applications in cryptography and coding theory.

    The Morita p-Adic Gamma Function: Computation, Adelic Structure, and the Factoring Question · 2026 · DOI

  • The lack of a factoring algorithm based on the Morita p-adic Gamma function. The high computational cost of evaluating Γ_p(N). The limited analysis of the function's properties and connections to other areas of number theory.

    The Morita p-Adic Gamma Function: Computation, Adelic Structure, and the Factoring Question · 2026 · DOI

  • The paper identifies a gap in the literature on the derivation of interface evolution laws from microscopic models. The paper aims to connect mechanisms at a microscopic level to macroscopic models of image processing.

    Horizontal Mean Curvature Flow as a Scaling Limit of a Mean Field Equation in the Heisenberg Group · 2026 · DOI

  • The paper suggests that future research should empirically test the five-regime phase diagram and falsifiable predictions. The paper proposes that future research should explore the implications of the ultrametric pattern for the study of hierarchical organization in various domains. The paper suggests that future research should investigate the limitations and potential biases of ultrametric models.

    The Tree and Its Echo: From Pre-Numeric Ground to a Unified Account of Hierarchical Organization, Convergence, and the Independence Fallacy · 2026 · DOI

  • The paper identifies a gap in understanding where ultrametric organization appears and where it fails. The paper notes that prior work has argued that ultrametricity is the signature of a deep generative logic operating across physics, biology, and cognition, but has not fully explored the limitations and potential biases of ultrametric models.

    The Tree and Its Echo: From Pre-Numeric Ground to a Unified Account of Hierarchical Organization, Convergence, and the Independence Fallacy · 2026 · DOI

  • Establishing a precise correspondence between the analytic zero set Z(G) and the observed modular stabilization phenomena. Investigating the behavior of exponential equations in different regimes. Applying the results to number theory and algebraic geometry.

    Valuation Thresholds and Analytic Finiteness in Exponential p-Adic Lifting · 2026 · DOI

  • The lack of understanding of the analytic and valuation-theoretic structure underlying the phenomenon of congruence aliasing. The need for a rigorous correspondence between the finite analytic zero set Z(G) and the observed modular stabilization phenomena.

    Valuation Thresholds and Analytic Finiteness in Exponential p-Adic Lifting · 2026 · DOI

  • To further develop the DKKC framework and its applications. To compare the framework with existing frameworks and methods.

    The Mathematics of Infinity · 2026 · DOI

  • The existing frameworks may not provide a nuanced understanding of mathematical processes. There is a need for a framework that preserves code and standard values.

    The Mathematics of Infinity · 2026 · DOI

  • Public blockchains preserve artifacts indefinitely, creating a “record now, break later” risk profile for protocols whose security relies on assumptions that may be weakened by future capabilities. There is a need for a reference implementation for fully on-chain verification of ZK-STARK proofs and post-quantum signatures on Solana.

    solana-pqzk-fullchain: Full on-chain verification of ZK-STARK proofs and post-quantum signatures on Solana · 2026 · DOI

  • Future research can focus on constructing new families of association schemes using the characterization of ℓ-form plateaued functions. The authors suggest exploring the applications of the paper's findings in coding theory, cryptography, and combinatorial design.

    Characterization of $$\ell $$-form plateaued functions via association schemes · 2026 · DOI

  • The paper identifies a gap in the characterization of ℓ-form plateaued functions via association schemes. There is a need to provide a necessary and sufficient condition for the p-ary plateaued functions of ℓ-form to yield symmetric association schemes.

    Characterization of $$\ell $$-form plateaued functions via association schemes · 2026 · DOI

  • To apply the result to other reductions of the KP hierarchy. To use the result to compute connected bosonic N-point functions for the BKP hierarchy. To study the Witten-Kontsevich tau-function and the Witten Conjecture/Kontsevich Theorem.

    A formula for connected bosonic N-point functions for BKP hierarchy · 2026 · DOI

  • The lack of a formula for connected bosonic N-point functions for the BKP hierarchy. The need for a new method for computing connected bosonic N-point functions for the BKP hierarchy.

    A formula for connected bosonic N-point functions for BKP hierarchy · 2026 · DOI

  • The paper does not provide a comprehensive survey of all related work. The study is limited to the specific context of the quantitative identity of the formalization remainder.

    The Quantitative Identity of the Remainder: From ρ≠∅ to Euler's Formula / 余项的定量身份:从ρ≠∅到欧拉公式 · 2026 · DOI

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46 open questions have been extracted from the limitations and future-work passages of 148 advanced mathematical theories papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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