# Verification report — OWR-4531-016 (Dyakonov: Fermat's last theorem for model spaces K^p_θ, Problem 2)

Verification date: 2026-10-17.

**Verdict.** The note answers Problem 2 of Dyakonov **negatively for n = 3 and every p ∈ [1, ∞]**; every statement it
makes is proved. The record is answered **in part**. Three AI-assisted verification runs examined the work: runs A and
B the first written version, run 2 the final text of the note (it found no error in a proof and no earlier answer;
its required corrections, all of presentation or precision, were made).
- **Settled.** There are an explicit inner function θ_0 = zBB̃ (B, B̃ Blaschke products whose zeros accumulate on the
  circle exactly at −1 and 1, so that the boundary spectrum is {−1, 1}) and F, G, H ∈ K^∞_{θ_0} ⊂ K^p_{θ_0} (all p)
  with F³ + G³ = H³ and dim span{F, G, H} = 3; moreover F + iH = B and the inner factors of F, G, H are pairwise
  coprime Blaschke products (Theorem 1.1). Consequences: the hypothesis "at most one boundary spectral point" in
  Garcia's cubic theorem (arXiv:2610.17180, Theorem 1.1(a)) cannot be relaxed to two points (Corollary 1.2); every
  closed E ⊂ T with at least two points is the boundary spectrum of a Blaschke product θ for which K^∞_θ contains a
  nontrivial cubic Fermat triple (Corollary 1.3).
- **Known results, credited.** The question and the finite-Blaschke case are Dyakonov's (Oberwolfach Reports 7 (2010);
  IEOT 78 (2014)). The positive results for n ≥ 4 with boundary spectrum of logarithmic capacity zero and for n = 3 with
  at most one boundary spectral point are Garcia's (arXiv:2608.11653; arXiv:2610.17180). Dixon's elliptic functions and
  the use of elliptic functions for the cubic Fermat equation are classical.
- **Not addressed.** The case n ≥ 4; whether every inner function with two boundary spectral points admits a
  nontrivial cubic triple; Problem 1 of the record (extreme points of the unit ball of K^∞_θ).

The note is unrefereed.

## Statement checked
- **Primary source.** K. M. Dyakonov, "Two problems on star-invariant subspaces", in: Operator Theory and Harmonic
  Analysis (organisers A. Borichev, R. Mortini, N. Nikolski, K. Seip), Oberwolfach Report No. 49/2010, Oberwolfach
  Reports 7 (2010), no. 4, pp. 2813–2875, doi:10.4171/OWR/2010/49; list of open problems, item 3, p. 2870 (PDF page
  58 of the open file on EMS Press; read in this stage and rendered as an image). Definition
  K^p_θ := H^p ∩ θ·conj(H^p_0), 1 ≤ p ≤ ∞ (the bar is printed); Problem 2: prove or disprove Fermat's Last Theorem for
  K^p_θ, i.e. no solutions with n ≥ 3 unless the three functions lie in a one-dimensional subspace.
- **Published version.** K. M. Dyakonov, "Two problems on coinvariant subspaces of the shift operator", Integral
  Equations Operator Theory 78 (2014), no. 2, 151–154, doi:10.1007/s00020-013-2110-0; read as arXiv:1401.8031v1 (4
  pages). Problem 2 (p. 3): whenever θ is inner, p ∈ [1, ∞] and n ∈ {3, 4, 5, ...}, no nontrivial triple f, g, h ∈ K^p_θ
  satisfies fⁿ + gⁿ = hⁿ; nontrivial = not in a single one-dimensional subspace. Also p. 2: K²_θ = H² ⊖ θH², K^p_θ is the
  kernel of T_θ̄, and K^∞_θ contains inner functions iff θ(0) = 0, these being the divisors of θ/z.
- **Corpus record.** ulamai/UnsolvedMath, OWR-4531-016 (dataset version 1.6.0; upstream status open). Its statement
  bundles Problem 1 and Problem 2 and reads "K^p_θ = H^p ∩ θH^p_0": the bar over H^p_0 has been lost. Read literally
  this space is θH^p_0, and the question would be trivial (footnote 1 of the note gives the triple
  (θz·z/2, θz(1 − zⁿ/2ⁿ)^{1/n}, θz)). The note uses the definition of the sources and says so in a footnote.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Problem 2 as stated (universal over θ, p ∈ [1, ∞], n ≥ 3) | false: n = 3, θ_0, every p | Theorem 1.1 |
| "nontrivial" = not in a one-dimensional subspace (the source's definition) | dim span{F, G, H} = 3 | Theorem 1.1(c) |
| stronger readings: pairwise non-proportional; no common inner divisor (analogue of coprime solutions) | also satisfied: inner factors pairwise coprime | Theorem 1.1(d), Lemma 5.3 |
| p = 1 and p = ∞ | the triple is in K^∞_{θ_0} ⊂ K^p_{θ_0} for all p | Lemma 2.1(a), Lemma 5.2 |
| θ(0) ≠ 0 required? | not a hypothesis of the source; θ_0(0) = 0 is needed for this triple (B = F + iH is inner) | Section 5.3 |
| refined question by boundary spectrum, n = 3 | for closed E ⊂ T: every θ with σ(θ) ∩ T = E admits no nontrivial cubic triple iff #E ≤ 1 (with Garcia's theorem) | Corollaries 1.2, 1.3 |
| n ≥ 4 | not addressed | Scope and priority |
| corpus text read literally (bar lost) | trivial; not the question | footnote 1 |

## Results in the paper
- **Lemma 2.1** (criterion): f ∈ K^p_θ iff θ z̄ f̄ = g a.e. for some g ∈ H^p; consequences (a) K^∞_θ ⊂ K^p_θ, (b) passage
  to multiples θθ_1, (c) 1 ∈ K^p_θ iff θ(0) = 0 (stated as a consequence of the definition).
- **Lemma 2.2** (B_Λ ∘ φ = c B_{φ^{-1}(Λ)} for automorphisms φ; pseudo-hyperbolic invariance). **Lemma 2.3** (an H^∞
  function that continues analytically across an arc has no singular inner mass there).
- **Lemma 3.1** (Dixon's functions sm, cm from the differential system sm' = cm², cm' = −sm²: meromorphy, sm³ + cm³ = 1,
  conjugation and rotation symmetries, the period lattice, the real period p_0 and the period ωp_0 = −p_0/2 + ih),
  proved from scratch via the holomorphic vector field y²∂_x − x²∂_y on the projective Fermat cubic. **Corollary 3.2**:
  reality on Im w = 0 and Im w = h, simple zeros of sm ∓ i, the six quotients (sm, cm, 1)/(sm ± i) and their conjugation
  on the lines, their zero sets.
- **Lemma 4.1** (strip map ψ, deck map γ(z) = (z + τ)/(1 + τz), τ = tanh(π/√3)); **Lemma 4.2** (zero sets: finite modulo
  p_0, nonempty, off the lines; Blaschke condition by 1 − |z|² ≤ C_0 e^{−|σ|}); **Proposition 4.3** (θ_0, boundary
  spectrum {−1, 1}, |B∘γ| = |B|).
- **Section 5**: holomorphy and the identities F³ + G³ = H³, F + iH = B; **Lemma 5.1** boundedness (maximum principle);
  **Lemma 5.2** membership (θ_0 z̄ F̄ = F̃ := B̃(q̃∘ψ) ∈ H^∞); linear independence (coefficient system with constant term
  c³/b³ = −1); **Lemma 5.3** inner factors (no singular part: Lemma 2.3 on T \ {±1}, invariance under γ at ±1);
  **Remark 5.4** sup norms equal to 1.
- **Section 6**: proof of Corollary 1.2; **Proposition 6.1** (transport by any automorphism φ with θ_φ = z(B∘φ)(B̃∘φ));
  proof of Corollary 1.3 (prescribed closed E); **Remark 6.2** (θ_0∘φ works only if φ(0)F̃(φ(0)) = 0; rotations give only
  antipodal pairs); **Remark 6.3** (why Garcia's one-point argument does not apply: the glued map lives on C \ {0}).
- **Section 7**: numerical checks (not proofs), Table 1 (the zeros of sm ± i per period), the Verification and the Scope
  and priority paragraphs.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program of the note** (`check_note.py`; mpmath, numpy, sympy; about 6 s; TOTAL failures: 0). Exact identities of
  the proofs (sympy); Dixon's functions to 40 digits from the Weierstrass function with g2 = 0, g3 = 1/27 (lattice
  tested, formulas tested against the differential system, an Abel-integral check that does not use ℘): p_0 = B(1/3,1/3)
  = 5.29991625085634987194..., exactly three zeros of sm + i and of sm − i per period (agreeing to 7e-25 with the first
  computation), centroid p_0/3 + 2ih/3, three poles per period including 2p_0/3; on the disc in double precision:
  |<f, θ_0 z^k>| ≤ 2.2e-16 for f = F, G, H, k = 0..8; wrong inputs rejected (no z: 0.10; no B̃: 0.28; height 0.97h:
  7.9e-3; B̃ in place of B: negative Fourier coefficients 0.12); Proposition 6.1 for a non-antipodal automorphism
  (≤ 2.6e-16), and |<F∘φ, θ_0∘φ>| = 0.0572858 = |φ(0)F̃(φ(0))| as predicted by Remark 6.2; Fermat identity and
  F + iH = B to 6.4e-16 at 4000 points; deck invariance; suprema 0.999998, 0.999998, 0.999978; rank 3.
- **First computation** (`original/`): Taylor-series integration of the system with a chart at the poles (30 digits),
  zeros to 25 digits, Abel-integral cross-check of the zero set, disc computations, pseudocontinuation check.
- **Verification run A** (`verification_run_A/`): ℘ by Laurent series (45 digits); membership by orthogonality with
  controls (B̃ omitted; height 0.97h); the corrected Corollary for a non-antipodal automorphism.
- **Verification run B** (`verification_run_B/`): ℘ by Jacobi theta functions (40 digits); membership by Fourier
  coefficients with five wrong inputs; FFT discretisation; the corrected Corollary for the pair {1, i}; wrong lattices
  and perturbed zeros as negative controls.
- **Verification run 2** (`independent_run_2/`, on the final text): Dixon's functions by Taylor integration of the
  system, by Abel integrals and through Jacobi's cn (40 digits); the zeros of Table 1 from Abel integrals and the
  order-three automorphism u ↦ ω(u − p_0/3) + p_0/3 of the cubic (agreement with Table 1 to 4.6e-16, the rounding, and
  with the recorded 22-digit values to 4.6e-22); the poles; sm(ih/2) = −0.62996… + 0.42916…i ≠ ±i; on the disc, in the
  variable z with ordinary Blaschke factors and graded Gauss–Legendre rules on T: boundary identity at 7776 nodes
  (9e-24), |<f, θ_0 z^k>| ≤ 4.9e-19, wrong inputs (θ = BB̃ gives exactly conj f̃(0) at k = 0 and 0 for k ≥ 1, with
  |F̃(0)|, |G̃(0)|, |H̃(0)| = 0.0774, 0.1016, 0.1016; θ = zB: 0.2819; zeros of B̃ moved: 1.2e-3; B̃ in place of B: 0.1195),
  Proposition 6.1 and Remark 6.2 for another non-antipodal automorphism (7.2e-19; prediction confirmed), the value
  |φ(0)F̃(φ(0))| = 0.0572858 for the automorphism of Section 7; 65 checks, 0 failures.
- **Re-run.** On 2026-10-17 all programs of the package were run again from a copy; all recorded outputs were
  reproduced (two outputs as first saved also contained warning lines from the error stream, which are not part of
  the recorded outputs). Verification run 2 extracted `source.zip` and ran all 15 programs again: all 17 output files
  identical byte for byte. The programs of run 2 were then run from the final archive, with identical outputs.
  Details in `reproducibility/README.md`.

## Independent verification runs
The result was first obtained, with proofs and the programs in `original/`, in a first written version. Two
independent verification runs, both AI-assisted, examined that version on 2026-10-17; each wrote its own programs and
did not see the other's report. The changes made when writing the note are listed below; verification run 2 then
examined the final text (last subsection).

| Item | Run A | Run B |
|---|---|---|
| Statement, definition (bar), "nontrivial", readings | CONFIRMED (sources read; literal corpus reading trivial) | CONFIRMED |
| Lemma 3.1 (Dixon's functions from scratch) | CONFIRMED (one notation clash) | CONFIRMED |
| Strip map, zero sets, Blaschke condition, boundary spectrum {−1, 1} | CONFIRMED (also: exactly three zeros per period) | CONFIRMED |
| Boundedness of F, G, H | CONFIRMED (norms equal to 1) | CONFIRMED (norms equal to 1) |
| Membership in K^p_{θ_0}, p = 1, p = ∞, all p | CONFIRMED (re-derived; orthogonality test with two failing controls) | CONFIRMED (Fourier-coefficient test, five wrong inputs) |
| Fermat identity, dim span = 3 | CONFIRMED (sign slip c³/b³) | CONFIRMED (two cosmetic slips) |
| Primitive triple (F + iH = B, pairwise coprime inner factors) | PROVED by run A | — |
| Corollary B (sharpness of Garcia's Theorem 1.1(a)) | CONFIRMED | CONFIRMED |
| Corollary C (any two points; any closed E with ≥ 2 points) | WRONG AS WRITTEN (rotation), CONFIRMED after repair | ERROR in the justification, CONFIRMED after repair |
| Numerics of the first computation | REPRODUCED (own method; re-runs) | REPRODUCED (own method; re-runs identical) |
| Novelty | not contradicted; arXiv API unavailable | supported, not proved |

**Corrections required by run A** (all applied; details in `audit/fixes_applied.md`): (1) Corollary C with a disc
automorphism and the factor z; (2) a symbol used for two points in the proof of Lemma 3.1 (now O = (0, 1) and Q_k);
(3) the misprint conj(f) = conj(θ) z g in the proof of the criterion; (4) c³/b³ = −1; (5) the St. Petersburg Math. J.
paper of Dyakonov is 34 (2023), no. 3, 453–462; (6) the criterion for constants as a consequence of the definition;
(7) the poles of sm lie in [0, π_3) × [0, h), and the triangle of the listed zeros has centroid p_0/3 + 2ih/3; (8) for
n ≥ 4 there are no nonconstant meromorphic solutions in C, and solutions in H^∞ without the model-space condition
exist for every n; (9) "not addressed here" instead of "open"; (10) novelty wording "to our knowledge", final arXiv
check, no n ≥ 4 sketch in public text; (11) the lost bar in the corpus statement, and the partial status of the record.

**Corrections required by run B** (all applied): (1) Corollary C as above; (2) q_H does vanish (at the poles of sm),
only H ≢ 0 is needed; (3) c³/b³ = −1; (4) bibliographic data of the St. Petersburg Math. J. paper, and the DOI of
Dyakonov 2014; (5) the centroid p_0/3 + 2ih/3; (6) the docstring of check3_abel.py had (1+i)^{1/3} and (1−i)^{1/3}
interchanged.

**Changes made when writing the note, after the two runs.** Lemma 2.2 (an elementary proof that |B∘γ| = |B|, replacing
an argument with harmonic majorants), Lemma 2.3 and the present proof of Lemma 5.3 (run A had proved the statement with
a different argument for the singular measure on T \ {±1}); Remark 6.2 written out (the condition φ(0)F̃(φ(0)) = 0);
the program `check_note.py`. The n ≥ 4 sketch of the first written version is not in the note (one sentence says that
n ≥ 4 is not addressed).

**Verification run 2 (on the final text, 17 October 2026).** A further AI-assisted run examined the note as written:
first the parts that runs A and B had not seen (Lemma 2.2, Lemma 2.3, the proof of Lemma 5.3, Remark 6.2 and
`check_note.py`), then every other proof, line by line. It read the sources again (the problem page, rendered, with the
bar over H^p_0; Dyakonov 2014, Garcia I and II, Dyakonov's two papers of 2023 and Section 1 of Conrad–Flajolet, all but
the problem page in their arXiv versions), checked the DOIs with Crossref, wrote its own programs (`independent_run_2/`),
re-ran the whole package from the source archive and repeated the literature search. Its report:

| Item | Run 2 |
|---|---|
| Statement, definition (bar), "nontrivial", scope | CONFIRMED (sources read again) |
| Lemma 2.2 (B_Λ∘φ = c B_{φ^{-1}(Λ)}) | CONFIRMED, proof complete |
| Lemma 2.3 (no singular mass on an arc of continuation) | CONFIRMED, proof complete |
| Lemma 5.3 (inner factors, pairwise coprime; new proof) | CONFIRMED, proof complete |
| Remark 6.2 ("if and only if") | CONFIRMED |
| All other proofs (Sections 2–6) | CONFIRMED |
| Cited results (Dyakonov, Garcia I and II, Conrad–Flajolet; GMR as quoted by Garcia) | CONFIRMED; page numbers of Dyakonov 2014 are those of the arXiv version |
| `check_note.py` | CHECKED (each check computes what it says; all numbers of Section 7 and Table 1 in its output) |
| Own programs | 65 checks, 0 failures |
| Re-run of the package | 17 output files identical |
| Novelty | no earlier answer found (searches below) |

**Corrections required by run 2** (all applied): (1) the page numbers cited for Dyakonov 2014 are those of the arXiv
version, which the bibliography now says; (2) Section 5.3 now gives the reason why the factor z cannot be omitted
(B̃(0) ≠ 0, i.e. ih/2 is not a zero of sm − i; numerical, Table 1); (3) the symbol τ, used for tanh(π/√3) and for
Im s, is now β in the proof of Lemma 4.2(b); (4) the digits of p_0 and h in Section 7 are now truncated, not rounded
before an ellipsis; (5) Dyakonov's papers of 2023 are described as papers on extreme points that mention Problem 1 and
not Problem 2; (6)–(8) the Verification and Scope paragraphs and Section 7 record this run, its programs and its
searches; (9) this report, the README files and the archive were updated. The PDF now has 14 pages.

## Relation to the literature, novelty and scope
- **Searches.** By the first computation and the two runs on 2026-10-17: arXiv (abstract pages, PDFs, Garcia's author
  listing, the recent listings of math.CV and math.FA), Semantic Scholar (works citing Dyakonov 2014: Garcia I and II,
  Garcia–Mashreghi–Ross 2016, Dyakonov's two papers on extreme points of 2023, one unrelated paper), zbMATH Open,
  Crossref, OpenAlex (mostly refused), and four web searches. The arXiv API refused keyword queries (HTTP 429), so no
  keyword search over all of arXiv was made. At the writing stage, on 17 October 2026: Garcia's arXiv author feed
  (newest entry arXiv:2610.17180 of 12 October 2026), the math.CV and math.FA listings announced from 9 to 16 October
  2026 (the only hit for Fermat or star-invariant subspaces is arXiv:2610.17180), Semantic Scholar citations of Garcia I
  (only Garcia II) and Garcia II (none), one more web search (only known items), and Crossref for all DOIs; the arXiv
  API again answered 429, also on the retry after 60 s. By verification run 2, on 17 October 2026 (UTC): Garcia's
  arXiv feed at 04:29 (unchanged: newest arXiv:2610.17180v1; Garcia I and II still in version 1), the math.CV and math.FA
  "new" and "recent" listings at 04:35 (newest announcement Friday 16 October 2026; the only Fermat or star-invariant
  item is arXiv:2610.17180), Crossref (two bibliographic queries; nothing relevant), zbMATH Open (the query "Fermat
  star-invariant" returns exactly Garcia I, Garcia II and Dyakonov 2014), Semantic Scholar (the same six works citing
  Dyakonov 2014; none citing Garcia II), one web search (only known items); the arXiv API (three keyword queries) and
  OpenAlex (citation filter, searches) were refused (429/503). Six web searches in all.
  - No answer to Problem 2 was found. The closest work is Garcia II (12 October 2026), which proves the complementary
    positive statement for at most one boundary spectral point and says nothing about two points.
- **What was read.** Dyakonov's problem page (OWR, p. 2870) and the arXiv version of Dyakonov 2014, Garcia I and
  Garcia II in full; Section 1 of Conrad–Flajolet (arXiv version); by run 2 also the arXiv versions of Dyakonov's papers
  of 2023 (St. Petersburg Math. J. 34 (2023); IEOT 95 (2023)), which mention Problem 1 and not Problem 2. Dixon 1890,
  the papers of Gross and Baker, and the books
  of Garcia–Mashreghi–Ross and Garnett were not consulted; what the note says about them is taken from Conrad–Flajolet
  and Garcia, or is standard.
- **Caveats.** Garcia I is cited by Garcia II as in press in Proc. Amer. Math. Soc.; no journal record was found. This
  negative search is not a proof of priority, and the note claims none ("to our knowledge").
- **Scope.** The note answers the Fermat part (Problem 2) of the record negatively for n = 3 and all p. The case n ≥ 4 and
  Problem 1 are not addressed. Suggested record status: partially solved.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/