Artemev’s conjecture

Artemev’s Conjecture Resonance Transformations for the (2,2p+1) Minimal String and the x–y Swap in Topological Recursion DR.ARIE-ARIADNE DEWATSON Research manuscript — September 2026 A rigorous manuscript based on the supplied 2026 project dossier, incorporating the published theorem, the strengthened pure-singularity question, the parity reductions, and the finite-difference research program. Status statement: the published resonance/x–y-swap correspondence is proved; the stronger pure-singularity assertion is treated here as a partially reduced open strengthening, not as a completed theorem. Abstract Artemev’s conjecture proposes a precise relation between the resonance transformations required in the (2,2p+1) minimal string and the transformation induced by interchanging the two coordinates x and y in topological recursion. The conjecture is motivated by the observation that the x–y swap implements, at the level of the spectral curve and its recursion data, a duality closely related to the p–q exchange of minimal-string descriptions. The 2026 work of Dekinga, Shadrin and Verlinde proves the resonance-transformation/x–y-swap correspondence for every integer p≥1, including stabilization, the exceptional (g,n)=(0,1) sector, and an explicit polynomial correction in u². The original stronger formulation contains an additional assertion: the dual coefficients A_n^(g),∨ should be purely singular in u². The supplied project dossier develops a rigorous status analysis of this strengthening. It shows that positive genus is automatically singular by homogeneity, reducing the unresolved part to genus zero; residue selection yields a strong parity constraint; several low-complexity sectors vanish rigorously; and the remaining obstruction is a positive-even genus-zero sector involving the genus-zero topological-recursion kernel. The dossier further identifies the finite-difference mechanism behind Lemma 4.6 and proposes a route toward a new regular-sector vanishing theorem. The central conclusion is deliberately conservative: the published theorem is established, while the general pure-singularity statement remains an open strengthening unless the remaining finite-difference/topological-recursion cancellation is independently proved. Keywords minimal string; (2,2p+1) model; resonance transformation; topological recursion; x–y swap; Chebyshev spectral curve; pure singularity; Pochhammer symbols; finite differences; genus-zero amplitudes.

1. Introduction Minimal string theory admits several complementary descriptions, including worldsheet, matrix-model and spectral-curve formulations. A recurring issue is that the natural matrix-model couplings do not initially reproduce the desired worldsheet amplitudes term by term. The required nonlinear change of couplings is the resonance transformation. For the (2,2p+1) family, a striking alternative description was proposed by Artemev: instead of introducing the resonance transformation directly, one may interchange the spectral-curve coordinates x and y and use topological recursion on the swapped curve. The 2026 paper by Dekinga, Shadrin and Verlinde gives a proof of this correspondence. Its main theorem establishes the equality after the appropriate stabilization, normalization and an explicit polynomial correction. The theorem is therefore the correct basis for calling the resonance/x–y-swap correspondence solved. At the same time, the authors explicitly distinguish their result from the stronger original formulation: they control the polynomial terms in u² but do not derive the assertion that the dual coefficients are purely singular in u². This manuscript reorganizes the supplied project dossier into a journal-style research article. Its purpose is twofold: first, to give a self-contained mathematical account of the published correspondence and, second, to isolate precisely what remains to be proved for the stronger statement. The distinction is essential. A plausible cancellation mechanism is not itself a theorem, and the remaining genus-zero sector should not be declared solved without a new identity.
2. The Minimal-String Spectral Curve Let p≥1. The spectral curve for the (2,2p+1) minimal string is represented on CP¹ by Chebyshev polynomials. Using the global coordinate w on the double cover defined by z=(w+w⁻¹)/2, one may write x = 2u T₂(z) = u(w² + w⁻²), y = 2u^(p+1/2) T_(p+1)(z) = u^(p+1/2)(w^(2p+1) + w^(-(2p+1))). (2.1) The standard Bergman kernel is B(z₁,z₂)=dz₁dz₂/(z₁−z₂)². Topological recursion produces symmetric differentials ω_n^(g). At the initial levels, ω_1^(0)=y dx and ω_2^(0)=B; in the stable range the differentials have poles only at the branch points of x.
3. The x–y Swap The x–y swap interchanges the two functions while leaving the underlying Riemann surface and Bergman kernel fixed. Thus the dual curve is x^∨ = y = u^(p+1/2)(w^(2p+1)+w^(-(2p+1))), y^∨ = x = u(w²+w⁻²). (3.1) Let ω_n^(g),∨ denote the amplitudes obtained from topological recursion on the swapped curve. The basic conceptual question is whether these dual amplitudes reproduce the resonance-transformed minimal-string amplitudes. The x–y swap is a general construction in topological recursion and is connected to broader structures such as KP integrability and p–q duality. For the present family, the swap changes which Chebyshev polynomial plays the role of x, and hence changes the branch-point structure governing recursion.
4. Resonance Transformations and Stabilization The supplied project dossier follows the notation of the 2026 proof. The resonance side is expressed in terms of stabilized amplitudes. The stabilization is essential because the unstable sectors (0,1) and (0,2) behave differently from the stable topological-recursion range. ω̄_1^(0) = 0, ω̄_2^(0) = ω_2^(0) − B(w₁,w₂) = dw₁dw₂/(1−w₁w₂)², ω̄_n^(g) = ω_n^(g) when 2g−2+n>0. (4.1) With this convention the resonance-transformation expansion can be expressed uniformly through residues at w=0. The project dossier emphasizes that the stabilized differentials are holomorphic at w=0, which is the source of the residue-selection rule used later.
5. The Published Correspondence Theorem The central result of Dekinga–Shadrin–Verlinde identifies the stabilized resonance-transformation coefficients with the coefficients generated by the x–y swap, after accounting for the exceptional (0,1) term and an explicit generalized-topological-recursion correction. In schematic form, for n≥1 and k_i∈{1,…,p}, the theorem has the structure A_n^(g),st/(2^n(2p+1)^n) + δ_(g,0)δ_(n,1)·u^(2p+1)/(p−1/2) = A_n^(g),∨ − δ_(g,0)δ_(n≥2)·C_n(p,k;u)·ω̃_n^(0). (5.1) Here C_n is an explicitly determined polynomial in u² for n≥2, and ω̃_n^(0) denotes the relevant generalized-topological-recursion differential. The exact expression is obtained by matching the resonance-side calculation with the universal x–y-swap formula. The theorem therefore settles the principal correspondence for every p≥1. A key point for the present manuscript is that the published theorem is more precise than a bare equality of correlators: it explicitly describes the polynomial sector generated by the swap. However, the theorem is also weaker than the original strongest formulation because it does not establish that A_n^(g),∨ contains no nonnegative even powers of u.
6. The Strong Formulation The strong form can be expressed by separating the singular and regular sectors in the u² expansion. Following the supplied dossier, define Π_reg to project onto the nonnegative even powers of u. The desired stronger assertion is Π_reg A_n^(g),∨ = 0. (6.1) Equivalently, A_n^(g),∨ should contain only negative powers of u and odd positive powers, with the precise meaning of “singular in u²” inherited from Artemev’s formulation. The project dossier stresses that the published paper itself does not prove (6.1) in complete generality.
7. Homogeneity: Why Positive Genus Is Not the Obstruction The coefficient A_n^(g)(k₁,…,k_n) is homogeneous in u. The degree recorded in the project dossier is d = (2−2g)p − (3g−3+n) − Σ_i k_i. (7.1) For g>0 the degree is negative throughout the relevant range. Consequently, the dual coefficient cannot contain a nonnegative even power of u at positive genus. This reduces the genuinely unresolved strong statement to genus zero. For g=0: d = 2p + 3 − n − Σ_i k_i. (7.2) Thus the strong problem becomes a finite coefficient problem in genus zero: determine whether the positive-even part of the dual coefficient vanishes.
8. Residue Selection and the Parity Constraint The stabilized differentials entering the resonance-side residue formula are holomorphic at w=0. Consequently, a residue is nonzero only when the exponent of w equals −1. For a block I of a partition, the dossier derives a corresponding selection condition. After rewriting that condition in terms of the block index and the block sum k_I, the surviving residue index is constrained to have a fixed parity. This parity rule is not merely cosmetic. Summing the block conditions over a partition links the parity of the total degree to the parity of the number of blocks. In the positive regular sector, this produces a global selection rule that eliminates all parity-forbidden configurations. In particular, when the regular degree is odd, the parity constraints are incompatible and the corresponding coefficient vanishes. For positive even degree, only a narrower odd-block sector can survive.
9. Rigorous Partial Results for the Strong Form The project dossier separates the strong statement into what is already forced by degree, what is eliminated by residue parity, and what requires a new cancellation identity. The following conclusions are supported by the supplied analysis. • All positive-genus contributions are singular by homogeneity. • The parity-forbidden genus-zero regular sectors vanish because no admissible residue configuration exists. • Several low-complexity partition sectors are eliminated directly by the singleton residue condition and positivity constraints. • The first genuinely nontrivial obstruction occurs in the positive-even genus-zero sector. • The generalized-topological-recursion correction can contribute degree-zero terms, so degree zero must be treated separately. These statements sharpen the original open problem substantially. They do not, however, amount to a proof of the entire strong form.
10. The Exact Partition Formula The resonance-side coefficient can be organized as a sum over set partitions. For a partition I₁⊔⋯⊔I_ℓ={1,…,n}, each block contributes a residue factor involving a Pochhammer symbol. Suppressing the common normalization, the structure is A_n^(g),st = u^(−Σk_i−n) Σ_{ℓ=1}^n [(-2)^ℓ(2p+1)^ℓ u^((p+3/2)ℓ)/ℓ!] × Σ_{I₁⊔⋯⊔I_ℓ} ∏{a=1}^ℓ Res{w_a=0}[ block(I_a,w_a) ] · ω̄_ℓ^(g). (10.1) The explicit block is the one appearing in Proposition 3.1 of the published proof and contains the Pochhammer factor, the block sum k_I, the block size |I| and the summation variable s_I. The important structural point for the stronger problem is that the regular projection turns the apparently infinite residue expansion into a finite sum: the residue condition fixes the allowed s_I, while Pochhammer zeros eliminate sufficiently large values.
11. Lemma 4.6 and the Finite-Difference Mechanism Lemma 4.6 is one of the central combinatorial identities in the published proof. It converts a single-block Pochhammer expression into a sum over partitions. Its proof uses a generating function, a shift of variables, and a finite-difference identity. The supplied dossier correctly identifies a crucial limitation. Lemma 4.6 proves the combinatorial equivalence required to match the resonance expansion with the x–y-swap expansion; it does not by itself imply that the regular projection of the dual coefficient is zero. This distinction prevents a common logical error: a finite-difference identity that reproduces the published partition formula cannot automatically be promoted to a new vanishing theorem. A further identity must connect the regular projection to a finite difference of sufficiently high order. 11.1. Universal finite-difference identity The mechanism can be understood through the backward difference operator Δ_i f(x_i)=f(x_i)−f(x_i−1). For a polynomial of degree d, every difference of order greater than d vanishes. The generating operator used in the published argument packages repeated differences into an exponential series. exp(−Σ_i Δ_i) = ∏_i exp(−Δ_i), (11.1) The resulting coefficients can be organized by complete homogeneous symmetric polynomials in the difference operators. This explains the finite truncation appearing in the auxiliary identity used in the proof of Lemma 4.6. For the stronger conjecture, the desired strategy is therefore to show that every surviving positive-even regular contribution contains a finite-difference operator whose order exceeds the effective polynomial degree of the relevant block expression.
12. The Genus-Zero Obstruction The remaining difficulty is not a generic failure of the correspondence. The published theorem already matches the two constructions. The issue is whether the polynomial sector left visible on the dual side is identically zero after the appropriate projection. For g=0, the positive-even sector is constrained by the global parity rule. When the number of blocks is odd, certain sectors survive the elementary parity test. The simplest such surviving configuration leads to a convolution involving the genus-zero topological-recursion differential. At this point the Pochhammer factors simplify strongly, and the nontrivial dependence migrates into the topological-recursion kernel. This is precisely the point at which a new theorem is required. The dossier proposes to expand the relevant genus-zero differential at w=0, insert the expansion into the finite partition convolution, and determine whether the resulting coefficient is an exact finite difference.
13. A Proposed Regular-Sector Vanishing Theorem The natural research target is the following. Regular-Sector Vanishing Conjecture. For every p≥1, n≥1 and k_i∈{1,…,p}, the positive-even genus-zero coefficient produced by the stabilized x–y-swap expansion vanishes after imposing the residue constraints. Π_reg A_n^(0),∨ = 0. (13.1) A proof of (13.1), together with the already established degree and parity reductions, would upgrade the published correspondence to the full pure-singularity formulation. The important methodological point is that (13.1) should be proved as a new identity. It should not be presented as a consequence of Lemma 4.6 unless every intermediate implication is explicitly established.
14. Low-Complexity Tests and the First Nontrivial Sector The supplied project analysis studies the first few block configurations. Singleton blocks obey a particularly restrictive residue rule, and multiple singleton blocks rapidly consume the available total degree. This eliminates several small cases by contradiction. The first configuration that survives the elementary parity and positivity constraints is a genus-zero odd-block sector in which the block weights simplify and the remaining sum is controlled by the genus-zero topological-recursion kernel. This sector is therefore the correct laboratory for the first genuinely new cancellation identity. The analysis also shows why the problem should not be attacked by indiscriminate graph-by-graph expansion. The block decomposition, residue constraints and finite-difference structure already reduce the problem to a finite algebraic object. The remaining task is to identify the exact kernel identity satisfied by that object.
15. General Research Program
16. Define the regular projection explicitly and isolate the positive-even coefficient before any asymptotic or graph expansion.
17. Use the residue condition to replace every residue by a finite sum and record the associated parity constraint.
18. Exploit Pochhammer zeros to obtain finite support in the block variables.
19. Normalize the block weights in terms of block invariants rather than individual indices.
20. Express the resulting partition sum through generating functions or Bell-polynomial-type expansions.
21. Insert the explicit genus-zero topological-recursion differential and determine its Taylor coefficients at w=0.
22. Rewrite the surviving convolution as a finite-difference expression.
23. Prove that the finite-difference order exceeds the relevant polynomial degree, or identify an exact cancellation with the generalized-recursion correction.
24. Treat degree-zero terms separately, because the generalized-topological-recursion correction may contribute there.
25. Only after these steps state the pure-singularity theorem in full generality.
26. Computational Verification The supplied project dossier also describes a reproducible symbolic verification protocol. Its purpose is verification of algebraic ingredients, not replacement of proof by numerical evidence. • Pochhammer identities are checked symbolically. • The degree formula is recomputed independently. • The regular-sector projector retains only nonnegative even powers. • Positive-genus regular projections vanish by degree. • Genus-zero parity exclusions are checked by exact residue conditions. • The remaining positive-even genus-zero sector is reported as unresolved unless an independent symbolic identity proves its cancellation. A useful baseline sweep is 1≤p≤5, 0≤g≤3, 1≤n≤4, with all k_i∈{1,…,p}. Such computations can identify patterns and test candidate identities, but an exact finite range cannot by itself establish the theorem for arbitrary p and n.
27. Conceptual Interpretation The resonance transformation and the x–y swap can be viewed as two ways of implementing the same duality between descriptions of the minimal string. The resonance transformation removes non-universal polynomial terms in the coupling parametrization, while the x–y swap changes the polarization of the topological-recursion spectral data. The published theorem demonstrates that these mechanisms agree at the level required for the physical amplitudes, after stabilization and the explicit polynomial correction. The stronger pure-singularity statement asks for more: it asks that the dual coefficient itself already lie entirely in the singular sector. From this perspective, the missing result is naturally a projection/annihilation theorem. One wants the dual residue functional to annihilate the regular polynomial sector. The parity selection rule explains why this is plausible, while the finite-difference structure suggests how such annihilation could arise algebraically.
28. Main Results of This Manuscript The manuscript establishes the following status theorem, directly reflecting the supplied project dossier and the published 2026 result. Theorem A — Published resonance/x–y-swap correspondence For every integer p≥1, the resonance transformations of the (2,2p+1) minimal string coincide with the transformations induced by the x↔y swap in topological recursion, after the stated stabilization, normalization, exceptional (0,1) contribution and explicit polynomial correction. This is the theorem proved by Dekinga, Shadrin and Verlinde and published in Letters in Mathematical Physics in 2026. Theorem B — Rigorous partial strengthening The additional assertion that A_n^(g),∨ is purely singular in u² is automatic at positive genus by the degree formula and is rigorously established in the parity-forbidden genus-zero sectors isolated in the supplied analysis. The general positive-even genus-zero sector remains the outstanding strengthening addressed by the regular-sector vanishing problem.
29. What Is Not Claimed For mathematical clarity, this manuscript does not claim a complete proof of the strong pure-singularity assertion. In particular, it does not identify Lemma 4.6 as sufficient by itself, and it does not convert the proposed finite-difference mechanism into a theorem without proving the missing connection to the genus-zero topological-recursion kernel. This qualification is not a weakness of the project. It is precisely what makes the research program reproducible: the published theorem is separated from the new statement, the already proved reductions are isolated, and the remaining identity is explicitly formulated.
30. Conclusion Artemev’s conjecture has a clear two-level mathematical status. At the first level, the central correspondence between resonance transformations and the x–y swap for the (2,2p+1) minimal string is proved for every p≥1. The proof includes stabilization, the exceptional unstable sector and the explicit polynomial correction. At the second level, the stronger original claim that the dual coefficients are purely singular in u² remains a sharper statement. The supplied project dossier reduces this strengthening substantially: positive genus is settled by homogeneity; residue holomorphicity produces a parity selection rule; several low-complexity sectors vanish; and the remaining obstruction is a positive-even genus-zero partition sum governed by the genus-zero topological-recursion kernel. The natural next theorem is therefore a regular-sector annihilation identity. The finite-difference machinery behind Lemma 4.6 provides the correct algebraic language, but the decisive step is to prove that the surviving genus-zero topological-recursion convolution is exactly an annihilated finite difference or otherwise vanishes identically. Accordingly, the strongest scientifically defensible conclusion at present is: the published Artemev correspondence is proved; the pure-singularity strengthening has been reduced to a concrete finite genus-zero problem; and the remaining cancellation is a well-defined target for further research. Appendix A. Compact Formula Sheet z = (w+w⁻¹)/2. x = u(w²+w⁻²), y = u^(p+1/2)(w^(2p+1)+w^(-(2p+1))). x^∨ = y, y^∨ = x. d = (2−2g)p − (3g−3+n) − Σ_i k_i. g=0: d = 2p+3−n−Σ_i k_i. Strong form: Π_reg A_n^(g),∨ = 0. Positive genus: d<0 ⇒ regular sector absent. Remaining target: positive-even genus-zero regular coefficient = 0. Appendix B. Research Checklist • Verify every residue-selection exponent directly from the published formula. • Separate the unstable (0,1) and (0,2) sectors before applying stable formulas. • Keep the generalized-topological-recursion correction explicit. • Derive the first nontrivial positive-even genus-zero convolution completely. • Identify the exact finite-difference operator acting on the genus-zero kernel coefficients. • Prove the corresponding annihilation identity for arbitrary p. • Check degree-zero terms independently. • Only then state the strong pure-singularity theorem. References [1] K. Dekinga, S. Shadrin, E. Verlinde, “Resonance transformations for the (2,2p+1) minimal string via x−y swap: a proof of Artemev’s conjecture,” Letters in Mathematical Physics 116, 114 (2026). DOI: 10.1007/s11005-026-02140-1. [2] A. Artemev, “x−y swap for a (2,2p+1) minimal string,” 2025 preprint. [3] A. Alexandrov, B. Bychkov, P. Dunin-Barkowski, M. Kazarian, S. Shadrin, “A universal formula for the x−y swap in topological recursion,” arXiv:2212.00320 (2022). [4] A. Alexandrov, B. Bychkov, P. Dunin-Barkowski, M. Kazarian, S. Shadrin, “Degenerate and irregular topological recursion,” Communications in Mathematical Physics 406, 94 (2025). [5] A. Alexandrov, B. Bychkov, P. Dunin-Barkowski, M. Kazarian, S. Shadrin, “KP integrability through the x−y swap relation,” Selecta Mathematica 31, 42 (2025). [6] Supplied project dossier: “Artemev_Conjecture_Project_Completion_2026,” containing the detailed status analysis, parity reductions, finite-difference program, and symbolic-verification protocol. Source and Status Note This manuscript was prepared from the supplied project file and checked against the published 2026 Springer record for the primary theorem. The manuscript intentionally distinguishes published results, deductions recorded in the project dossier, and the remaining proposed research step. The strong pure-singularity assertion is therefore not presented as proved.

Artemev’s Conjecture Resonance Transformations for the (2,2p+1) Minimal String and the x–y Swap in Topological Recursion DR.ARIE-ARIADNE DEWATSON Research manuscript — September 2026 A rigorous manuscript based on the supplied 2026 project dossier, incorporating the published theorem, the strengthened pure-singularity question, the parity reductions, and the finite-difference research program. Status statement: the published resonance/x–y-swap correspondence is proved; the stronger pure-singularity assertion is treated here as a partially reduced open strengthening, not as a completed theorem. Abstract Artemev’s conjecture proposes a precise relation between the resonance transformations required in the (2,2p+1) minimal string and the transformation induced by interchanging the two coordinates x and y in topological recursion. The conjecture is motivated by the observation that the x–y swap implements, at the level of the spectral curve and its recursion data, a duality closely related to the p–q exchange of minimal-string descriptions. The 2026 work of Dekinga, Shadrin and Verlinde proves the resonance-transformation/x–y-swap correspondence for every integer p≥1, including stabilization, the exceptional (g,n)=(0,1) sector, and an explicit polynomial correction in u². The original stronger formulation contains an additional assertion: the dual coefficients A_n^(g),∨ should be purely singular in u². The supplied project dossier develops a rigorous status analysis of this strengthening. It shows that positive genus is automatically singular by homogeneity, reducing the unresolved part to genus zero; residue selection yields a strong parity constraint; several low-complexity sectors vanish rigorously; and the remaining obstruction is a positive-even genus-zero sector involving the genus-zero topological-recursion kernel. The dossier further identifies the finite-difference mechanism behind Lemma 4.6 and proposes a route toward a new regular-sector vanishing theorem. The central conclusion is deliberately conservative: the published theorem is established, while the general pure-singularity statement remains an open strengthening unless the remaining finite-difference/topological-recursion cancellation is independently proved. Keywords minimal string; (2,2p+1) model; resonance transformation; topological recursion; x–y swap; Chebyshev spectral curve; pure singularity; Pochhammer symbols; finite differences; genus-zero amplitudes.

Solutions is attached.