A counterexample to canonical Toeplitz eigenvalue distribution
Clemens Thalhammer
We construct a continuous complex scalar symbol
with neither an inner nor an outer annular holomorphic extension,
whose Toeplitz sections are not canonically distributed. Along an
infinite sequence of orders, each of two points outside the symbol
range is an eigenvalue whose multiplicity is at least a fixed
positive fraction of the order.
The construction uses exact finite spectral corrections, geometric
Cauchy coordinates, and a uniformly convergent Fourier series.
A deterministic search using finite contraction certificates specifies
every correction, so for each admissible rational value of the
multiplicity proportion the symbol is computable. The admissible
range is an absolute constant that is not computed here, and no
claim of computational efficiency is made.
Claims
Automated review AI-Verified
Before this version was announced, three automated referees read its source, each with a different brief. What follows is model output, published as written: advisory, possibly wrong, and not a check of the proof. It changes no problem's status. That is what Lean verification and human review are for.
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An admin read the serious objections below and dismissed them as mistaken. They are kept on the record, struck through.
The argument and the results it rests on
The paper claims a counterexample to Widom's conjecture: a continuous symbol a(z) = z g(z^2), g = sqrt(1+1/s) + P_- + P_+, built by a sparse-stage construction, such that along n_j = 2 m_j + 1 both +-1 (outside a(T)) are eigenvalues of multiplicity at least floor(theta m_j). I checked the exact finite algebra in detail and found it correct: the even/odd block form of T_{2m+2}(c_w), the determinant identity det(wI - T_n(a)) = +- w det(H - tU) M(t), the base-symbol facts H = I and M(t) = t^m (so T_{2m+1}(a_0) has +-1 with multiplicity m each, verified by hand for m = 1), the Jacobian formula and its base evaluation J^0, the inverse-row/far-factor identities, the rationalized formulas for R_t, X_t, S_t, N_t, the oddness/invisibility bookkeeping for later stages, and the final test-function argument. The exclusion of annular extensions via Fourier root limsups is correct. I found no fatal error. The main claim, however, rests on three dense analytic propositions whose proofs are sketches: the uniform two-level isomorphism (Prop. 4.9), the far-factor coefficient bounds in two-variable Wiener algebras (Prop. 4.8), and the conformal forcing bound with uniform theta_0 (Prop. 5.4). These control whether theta* is an absolute constant, and a referee cannot verify the uniformity-in-degree claims from the text as written; I have marked them serious and listed the specific steps needing full proofs. The invoked outside results (finite section theorem, Gohberg-Semencul) are correctly stated and attributed. The paper is candid that its explicit thresholds are not derived; the result is non-effective in theta*.
The preconditioned actual Jacobian A_g x = (J_g (J^0)^{-1} x) o f extends to compatible isomorphisms on H^{-1/4} and H^{-1/8}, with norms and inverse norms bounded by constants depending only on gamma, uniformly over every finite background satisfying the five listed norm bounds, whatever the degree of P.
This proposition is the linchpin of the whole paper: it is what makes theta* an absolute constant and gives the uniformly bounded right inverses B_m in Proposition 5.7. Its proof compresses several independent, nontrivial operator-theoretic steps into one or two sentences each: (i) the identification of the range R_r = C_h H^r(T) modulo constants with analytic H^r via pi(v) = Pi_+ s^{-1} v; (ii) the assertion that composition with the boundary map h^{-1} o f preserves H^r(T) with constants depending only on bi-Lipschitz bounds (via the Gagliardo norm) and that it maps R_r exactly onto analytic H^r; (iii) the 'general differentiated density bound' C_r ||M||_{W^r_pos} ||y||_{H^r} used for the A_t and B_t parts, which is announced ('the same proof of boundedness ... shows') rather than proved; and (iv) the summation over j of the coefficient multipliers A_j(t), B_j(t) with weights (1+j)^r, which depends entirely on Proposition 4.8. The paper itself is forthright that the constants are read off proofs 'which establish the estimates only with unspecified constants'. As written, a referee cannot verify the uniformity claim in the degree of P from the text; each of (i)-(iv) needs to be written out with the constants tracked, since any dependence on the degree of P would destroy the absolute threshold theta* and hence the theorem as stated.
The weighted sums over j of ||[s^j](A_t - 1)||_{H^infty(D)} and ||[s^j]B_t||_{H^infty(D)} are bounded by C_K ||P_+||_{W^{K+3}}, via the identities log p_{a(eta)}(z) = Pi_{>0,z} log F(z,eta) and d_w log p_w(z) = -Pi_{>0,z} 1/(a(z)-w) evaluated at boundary parameters w = a(eta), followed by the maximum principle on the two closed lobes.
Both identities are derived on a thin exterior circle and then transported to the two-variable Wiener algebra on T^2 by 'taking the boundary limit', but F(.,eta) and 1/(a(z)-a(eta)) have boundary singularities (at z = +-eta and at the branch points), and the argument that the positive-z projection commutes with this limit term by term is only asserted ('the positive Laurent coefficients decay exponentially on its fixed collar, so the radial limit can be taken term by term'). The collar on which the decay holds depends on the background and shrinks as the degree of P_+ grows (tau_j r^{m_{j-1}} is not small for fixed r > 1), so the exponential decay is not uniform; the proof must make explicit that only qualitative convergence, and not the collar width, enters the final uniform bound. Likewise the Neumann/logarithm/exponential steps in the weighted algebra W^K_{pos,z} are stated without the needed submultiplicativity and invertibility remarks. Since Proposition 4.9 depends on this estimate with an explicit uniform constant, a complete derivation is required.
|[u^k] M_m(f(u))| <= C_P m^{-3/2} for 0 <= k <= theta_0 m, with theta_0 chosen uniformly over the W^{1+alpha} neighborhood (depending only on bi-Lipschitz constants of f and the constants C_0, c_0, C_1), while C_P and the slit-annulus width may depend on the background.
The proof deforms the s-coefficient contour of S_{f(u)}(s) into a slit annulus on which the second-sheet values t_+- of h lie inside D, so the outer-circle and cut pieces individually have poles in u inside the unit disc; the argument then extracts Taylor coefficients at u = 0 of each piece separately and relies on Lemma 5.3 for growth e^{C_1 k x}. This is legitimate only if (a) every piece is holomorphic in a neighborhood of u = 0 and (b) the exponential constant C_1 (including the enlargement caused by the Cauchy estimate in the parameter a for the derivative term) is genuinely independent of the background, since theta_0 < 1/(4 C_1) determines theta*. Point (a) is asserted via the choice of L but the required inequality R < |f^{-1}(t_+-)| is only stated, and point (b) is asserted ('with a possibly larger C_1') without bounding the enlargement. In addition the bound on the outer-circle term uses sup_{|s|=1+delta} |S_t(s)| <= C/delta without showing that the far-factor numerators and the continued denominator remain controlled across the cut on the second sheet. A careful referee needs these details to confirm that theta_0 is uniform; without uniformity the theorem's 'absolute constant theta*' is not established.
For boundary parameters w with c_w vanishing on T, the Gohberg-Semencul identity together with ||x||_1 <= C and ||y||_1 <= C N^2 gives ||V_N|| <= C N^2, using |(V_N)_{00}| bounded below for large N.
The bound ||y||_1 <= C N^2 is obtained from the first-row estimate |(V_N)_{0j}| <= C N, which in turn rests on |[z^j]F_w| <= C(j+1) from |F_w(z)| <= C/(1-|z|) and on the truncation residual bound C N rho^{j-N+b}. The second ingredient requires the weighted entry bound |(V_N)_{ij}| <= C rho^{j-i} at the same boundary parameter, i.e. finite-section stability of c_w(rho .), which needs c_w to have no zeros on |z| = rho and winding zero there; this is asserted from the collar lemma but the collar lemma is stated for h - s small in C^1, while the proposition is applied to every background in the W^{1+alpha} neighborhood with arbitrarily high degree. The chain of hypotheses (collar radius rho depends on the background; rho enters both the weighted bound and the exponent in the residual) should be made explicit, and the proposition should state clearly which constants are allowed to depend on the background, because Corollaries 2.8 and 2.9 and Lemma 5.2 later invoke these bounds as 'uniform'.
One may take gamma* = 2^{-1000}, theta_0 = 2^{-20}, theta* >= 2^{-10000}; the theorem asserts an absolute constant theta* > 0.
The remark explicitly states its constants 'have not been fully derived and are not used in the proof', and the theorem's proof only asserts the existence of gamma*, theta* via 'sufficiently small' thresholds whose existence depends on Propositions 4.9 and 5.4. This is consistent and honestly flagged, but it means the result is non-effective: the symbol is 'computable' only relative to admissible rational parameters that the paper cannot name. The theorem statement should make the non-effectivity explicit (it currently reads as if theta* were known), and the remark's numbers should either be derived or removed, since readers may take them as established.
If c is continuous on T, nonvanishing and of winding number zero, then T_N(c) is invertible for N >= N_0 and sup ||T_N(c)^{-1}|| < infinity; cited to Gohberg-Feldman, Boettcher-Silbermann, and recorded in BFGM Section 3 eq. (6).
The statement is the classical finite section theorem and is true and correctly attributed, and the compact-family uniformity via Neumann series is correct. The only point to note is that the BFGM citation is described as a theorem for piecewise continuous symbols; the paper correctly observes that the continuous case is within its hypotheses, so this is adequate. No change needed beyond perhaps citing the standard textbook statement directly rather than through BFGM.
Scope
Scope check. Theorem 5.1, as stated, gives exactly the counterexample the registry records. It produces a continuous symbol a(z)=zg(z^2) with Fourier root limsups equal to one on both sides, which by the Laurent-coefficient argument means no inner or outer annular extension. Along n_j=2m_j+1, the matrix T_{n_j}(a) has the eigenvalues ±1 with multiplicity at least floor(\theta m_j), and both points lie at distance more than 1/4 from a(\T). The explicit continuous, compactly supported test function \Phi(w)=\max\{0,1-8|w-1|\} vanishes on a(\T), so its integral against a_*m is zero. Its empirical eigenvalue average along n_j tends to at least \theta/2>0, so canonical distribution fails. The notion of canonical distribution, the quantifier (failure along a subsequence suffices), the hypotheses and the direction all match the registered problem. The multiplicity statement is imposed directly as exact divisibility of det(wI-T)/w by (w^2-1)^h. It persists in the limit because every later correction is invisible to earlier sections, and I checked the frequency bookkeeping for this. The result is not conditional on anything left unproved in the paper's own framing. The two hypotheses of the local-completion proposition are claimed to be proved uniformly over a fixed neighborhood. The loose numerical ledger is explicitly not used. Whether the analytic estimates actually hold — the uniform isomorphism, the forcing bound, and the existence of regular solutions — is the other referee's task. If they hold, the claim type 'counterexample' fits. Only minor scope remarks: make the annular-extension-to-coefficient-decay translation explicit, and present the computability and 'absolute constant' claims as auxiliary existence statements.
"An outer annular extension would give |a_k|\le C_\rho\rho^{-k} for some \rho>1 by Cauchy's Laurent coefficient formula; an inner extension would give the analogous geometric decay of a_{-k}."
The registry defines the extensions as holomorphic functions on an annulus that extend continuously to |z|=1 with boundary values a. The translation to coefficient decay needs one more step: the Laurent coefficients of the extension equal the Fourier coefficients of a. This follows from uniform continuity on a closed sub-annulus, by passing the circle integrals to the limit radius 1. The translation is correct, and with it the exhibited coefficients (g_{m_{j-1}}=\tau_j with \tau_j=\kappa 2^{-\lceil\sqrt{2m_{j-1}+1}\rceil}, and g_{-3m_j}=\binom{1/2}{3m_j}) do exclude both extensions. The step should still be stated explicitly, since it is exactly the link to the registered hypothesis.
There is an absolute constant \theta^*>0, and for rational \theta the symbol can be chosen computable. The remark indicates \gamma^*=2^{-1000} and \theta^*\ge2^{-10000}.
\theta^*, \gamma^* and C_1 are only shown to exist, with unspecified constants. The remark's ledger is explicitly not derived and not used. So the computability statement only asserts that some computable symbol exists; no admissible rational parameters are identified. None of this is needed for the registered problem, which only requires the existence of one counterexample symbol. The extra claims should be presented as auxiliary, and the remark's numbers should not be read as established. This does not affect whether the main theorem answers OP-00159.
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