# Exact source and accepted theorem

The frozen row is AMR-022-1032, Hayman--Lingham Problem 1.32. It asks whether the length-zero exceptional directions in Gross's star theorem can be refined for a fixed regular inverse germ of a meromorphic function on the plane. The original 2018 source records no reported progress.

The accepted local theorem is the q>1 iterated-logarithmic construction in main.tex. It is a complete theorem, not a finite experiment or a pending outside-review candidate. It is NOT counted as a new closure of the historical capacity question: Sina Nadi's 2609.29797v1 already answers that negatively, and Eremenko's current index attributes dimension one to Nadi.

Excluded: q=1; equality of the whole exceptional set with the constructed compact set; realization of every null set; finite-order entire functions; every conceivable refinement or classification of exceptional sets. The historical record receives zero new original or exact-question closure increments from this manuscript.


# Gross exceptional sets: working proof of an iterated-logarithmic refinement

Date: 2026-10-28 (Europe/Istanbul). Worker: primary. Source target: AMR-022-1032.
This is an actual constructive proof attempt, not a theorem inferred from a finite computation.
The baseline construction, the crosscut estimate, and the starlike Arakelian-set lemma are due to Sina Nadi, arXiv:2609.29797v1. The original capacity question must be credited to that work. Eremenko's live problem index already describes a dimension-one example; dimension one alone is not claimed as new here.

## Proposed precise theorem

For every q>1, put h_q(t)=t[log log(e^e/t)]^q for all sufficiently small t>0 (extend monotonically away from zero). There exist a transcendental entire f, a simple zero z_0, the corresponding inverse germ phi at 0, and a compact set K of angles, such that

* 0 < H^{h_q}({exp(i theta): theta in K}) < infinity;
* K has length zero;
* for every theta in K, the SAME germ phi continues along [0,exp(i theta)), its continuation tends to infinity as t tends to 1, and it cannot continue holomorphically through exp(i theta).

Consequently the full exceptional direction set has dimension one and length zero. The compact subset has infinite H^{g_epsilon} measure for every epsilon>0, where g_epsilon(t)=t[log(e/t)]^epsilon, because h_q/g_epsilon tends to zero. No exact equality E=exp(iK), universal characterization of all exceptional sets, endpoint q=1 result, or novelty for dimension one alone is asserted.

## 1. Angular intervals and their exact gauge

Fix ell_0=1/8, gamma=(q-1)/2>0, and for n>=0 set

ell_n = ell_0 2^{-n} [log(3)/log(n+3)]^q,
r_n = 1 - (1/2)[log(3)/log(n+3)]^gamma,
h_n = r_{n+1}-r_n.

Thus r_0=1/2, r_n increases to 1, and
h_n asymp 1/[(n+3)(log(n+3))^{gamma+1}].
The constants may depend on q. Put a_n=ell_{n+1}/ell_n, so 0<a_n<1/2 and inf a_n>0.

Starting from the centered interval J_empty of length ell_0, give a level-n interval centered at c_v two children of length ell_{n+1}, centered at c_v +/- ell_n/4. They are disjoint, strictly inside the parent. Define
tau_0=5/8,
tau_n=1/2+(1-2a_{n-1})/8 for n>=1,
I_v=(c_v-tau_n ell_n,c_v+tau_n ell_n), |v|=n.
The identity
1/4-a(1/2+(1-2a)/8)=(1-2a)(2-a)/8>0
shows that every closed child I lies strictly inside its parent J. In particular the I intervals at each level have disjoint closures. Let U_n be their union and K=intersection_n union_{|v|=n} J_v.

The Bernoulli probability nu on K gives each J_v mass 2^{-n}. Since 2^n ell_n tends to zero, K has length zero. The exact scale relation is
2^n h_q(ell_n) -> ell_0 (log 3)^q in (0,infinity).
For ell_n<=t<ell_{n-1}, an interval of length t meets at most t/ell_n+2 level-n intervals. The ratios ell_{n-1}/ell_n are uniformly bounded. Hence nu(L)<=C 2^{-n}<=C' h_q(t) for small intervals L. The mass-distribution argument gives positive H^{h_q}(K); the level-n covers give finite H^{h_q}(K). Exponentiation on J_empty is bi-Lipschitz, and h_q(ct)/h_q(t)->c for each fixed c>0, so the same two conclusions hold on the unit circle.

## 2. Thin radial corridors and small disk shadows

Let D=Delta(0,1/2) union union_{n>=0}{re^{i theta}:0<r<r_{n+1}, theta in U_n}. This is an open bounded starlike domain. At r_n<=r<r_{n+1} its angular section is exactly U_n. Every theta in K reaches the boundary point exp(i theta) radially.

For |v|=n, set x_n=r_n+h_n/4, y_n=r_n+3h_n/4 and take the crosscut Gamma_v={y_n exp(i theta):theta in I_v}. Let D_v^+ be its component away from 0. All curves from a fixed central disk to Gamma_v cross the logarithmic rectangle (x_n,y_n) x I_v. Therefore their extremal distance is at least
log(y_n/x_n)/(2 tau_n ell_n) >= c h_n/ell_n
>= c' 2^n (log(n+3))^gamma/(n+3)
for all sufficiently large n.

Let chi:unit disk -> D be conformal, chi(0)=0. Its inverse extends across each radial boundary segment containing an endpoint of Gamma_v. The preimage of Gamma_v is a disk crosscut. Nadi's Lemma 2.2 (or the classical planar continuum-modulus estimate used there) gives, for the component V_v away from the central disk,
diam closure(V_v) <= delta_n,
delta_n=C exp[-c 2^n (log(n+3))^gamma/(n+3)].
After increasing the starting level, delta_n is decreasing and tends to zero.

The nested crosscuts along theta have impressions {exp(i theta)}: diam D_v^+<=1-y_n+2 tau_n ell_n tends to zero. The disk closures intersect in one boundary point beta(theta). Distinct theta give distinct prime ends, hence distinct beta. If two theta share their level-n interval, their beta images are within delta_n. Thus beta:K->unit circle is a continuous injection. Put B=beta(K), B_v=beta(K intersect J_v). These are disjoint compact partitions of B, and diam B_v<=delta_n. Moreover V_v and B_v lie in the same closure of diameter delta_n.

## 3. A nonatomic measure and a multiscale potential estimate

Use mu=beta_*nu, NOT a level-weighted atomic approximation. Then mu(B_v)=2^{-n} exactly. Define on the disk
A(zeta)=-integral_B Log(1-conj(xi) zeta) dmu(xi),
with the branch at 1. It is holomorphic, Re A>=-log 2, and |Im A|<pi/2: each 1-conj(xi)zeta lies in the open right half-plane.

If zeta is in V_v for a word of length n, let v_j be its ancestors. For every j<n the disjoint shell B_{v_j} minus B_{v_{j+1}} has mass 2^{-j-1}, and its points are within delta_j of zeta. All other contributions are bounded below by -log 2. Consequently, starting at one fixed large j_0,
Re A(zeta) >= sum_{j=j_0}^{n-1} 2^{-j-1} log(1/delta_j) - C
>= c sum_{j=j_0}^{n-1} (log(j+3))^gamma/(j+3) - C
>= c' (log(n+3))^{gamma+1} - C'.

The same lower bound, with adjusted constants, holds whenever dist(zeta,B)<delta_n: choose beta in B within delta_n, use its ancestral shells, and bound distances by delta_j+delta_n<=2 delta_j. Thus Re A tends uniformly to infinity as dist(zeta,B) tends to zero inside the disk.

Set Q(zeta)=zeta exp(2A(zeta)). Since mu is a probability,
zeta Q'(zeta)/Q(zeta)=integral (1+conj(xi)zeta)/(1-conj(xi)zeta) dmu,
whose real part is positive. The analytic starlikeness criterion makes Q conformal onto a starlike domain Omega. Omega is proper (otherwise the inverse would be bounded and entire). Also |Q(zeta)| tends uniformly to infinity as dist(zeta,B) tends to zero. Put Phi=Q chi^{-1}:D->Omega and g=Phi^{-1}.

## 4. The necessary amplification of the approximation weight

For theta in K and 0<=t<1, put rho_theta(t)=epsilon dist(te^{i theta},boundary D) and T_theta,t=closed disk centered at te^{i theta} of that radius. Choose one fixed sufficiently small epsilon>0.

For r_n<=t<r_{n+1}, the angular margin of theta in I_{v_{n+1}} is at least
(tau_{n+1}-1/2)ell_{n+1} asymp ell_{n+1}/[(n+4)log(n+4)].
A disk of a sufficiently small constant times this radius changes the argument by at most half that margin and crosses at most one radial level: the adjacent radial gaps are only polynomially small, whereas ell_n decreases exponentially. Nesting places the entire disk in D. The radial boundary of I_{v_n} supplies an upper bound. Thus
c ell_{n+1}/[(n+4)log(n+4)] <= dist(te^{i theta},boundary D) <= C ell_n,
and log(2/rho_theta(t))<=C_q(n+1).

Since ell_n/h_{n-1} tends to zero, decrease epsilon once to put T_theta,t beyond the level-(n-1) crosscut whenever n is sufficiently large. Therefore chi^{-1}(T_theta,t) lies in V_{v_{n-1}}, and throughout this disk
Re A(chi^{-1}(xi)) >= c(log(n+2))^{gamma+1}-C.
This tends to infinity; also |chi^{-1}(xi)| tends uniformly to 1. Hence Phi(T_theta,t) escapes uniformly to infinity as t tends to 1.

Crucially, a linear weight P=A composed with Q^{-1}+constant would NOT dominate the O(n) error demand: (log n)^{gamma+1}=o(n). Use instead
P_0(w)=exp[(1/2) A(Q^{-1}(w))].
The bound |Im A|<pi/2 gives
Re P_0(w) >= 2^{-1/2} exp[(1/2) Re A(Q^{-1}(w))].
Since gamma+1>1,
exp[c(log n)^{gamma+1}]/n -> infinity.
Thus there is a real C_* such that for P=P_0+C_*,
Re P(w)>=log(2/rho_theta(t)) on Phi(T_theta,t), simultaneously for all theta and t.
For t in a compact subinterval of [0,1), the disks form a compact subset of D and their radii have a positive lower bound, so a single additive constant handles those t. No uniformity in q is asserted.

## 5. Entire approximation and preservation of one inverse germ

Let S be the union of all Phi(T_theta,t). It is closed in the plane: a bounded convergent sequence cannot have t tending to 1 by uniform escape; compactness of K and continuity of the disk centers/radii handle the remaining case. S is a subset of Omega.

By Nadi's starlike-envelope Lemma 3.2, there is an Arakelian closed set F with S subset F subset Omega. This lemma is used as an attributed input. Its proof opens tapered radial tubes around every ray in the complement of Omega; every point of a tube escapes with nondecreasing modulus, giving connected spherical complement and local connectivity at infinity.

Apply the classical Arakelian uniform approximation theorem first to -P on F, obtaining entire H with |H+P|<log 2. Apply it a second time to g exp(-H), obtaining entire G with |G-g exp(-H)|<1/8. Then f=exp(H)G is entire and
|f-g| < (1/4)exp(-Re P) <= rho_theta(t)/8 on Phi(T_theta,t).

On the boundary of the Jordan domain Phi(interior T_theta,t), g(w)-te^{i theta} has modulus rho_theta(t) and exactly one zero inside. Rouche gives exactly one zero counted with multiplicity for f(w)-te^{i theta}. It is simple. At t=0 all disks and domains are the same, so all zeros are the same simple zero z_0. For fixed theta, the zero depends continuously on t (compactness, strict boundary inequality, and uniqueness), and the inverse function theorem patches these local zeros into the continuation of the SAME inverse germ along the ray. The zeros belong to Phi(T_theta,t) and therefore tend uniformly to infinity as t tends to 1. They cannot continue holomorphically through the endpoint. f is nonconstant and cannot be a polynomial, since it remains bounded along this escaping curve.

This proves the stated q>1 construction, subject to the listed standard/theorem inputs; none of the infinite-dimensional claims is inferred from finite numerical checks.

## Credit, scope, and unresolved publication questions

* Nadi supplies the original positive-capacity counterexample and its construction architecture. That result is not ours.
* Eremenko's public index already attributes dimension one to Nadi. Do not claim first dimension-one example.
* The explicit contribution attempted here is the h_q(t)=t(log log(e^e/t))^q gauge for every q>1, obtained by variable angular/radial scales, the full Bernoulli shell mass, and the exponential holomorphic error weight.
* No sharpness at q=1, no prescribed arbitrary null set, no equality with the entire exceptional set.
* The original source question is open-ended. This is a complete explicit refinement theorem, not a new claim to settle every interpretation of Problem 1.32.
* A bounded current literature search must be completed and the argument self-audited before manuscript/publication acceptance.
* A checker may verify scale algebra, finite geometry and intentionally false controls; it cannot prove the conformal mapping, Arakelian approximation, or universal quantified result.

## Rejected or superseded routes

1. Taking a limit of Nadi's separate f_s examples does not establish one f of dimension one; no normal-family argument is available.
2. A constant binary ratio a=1/2 destroys strict interval separation and yields positive length, incompatible with Gross's theorem.
3. A one-level atomic lower bound throws away the ancestral shell contributions and cannot justify the stated slowly varying scales.
4. Retaining the source's linear error weight fails because the available lower bound is polylogarithmic in n while the required precision is exponential in n.
5. The endpoint q=1 sets gamma=0 and makes r_n identically 1/2. It is explicitly excluded; taking q down to 1 is not a proof at the endpoint.
6. The inherited near-basepoint obstruction is false for a FIXED inverse germ: its initial analytic disk supplies a positive uniform lower bound on all stopping radii. Entire functions of finite lower order do have finitely many finite asymptotic values by Eremenko 2110.06134v1, p.6; this does not extend to meromorphic functions.



# Detailed self-audit of the iterated-logarithmic construction

Acceptance: COMPLETE_PROOF of the explicitly stated q>1 gauge theorem. All listed obligations are discharged in the written proof; this is originating-researcher acceptance, without an independent reviewer or proof assistant.

1. Quantifiers. One q is fixed, one f and one simple zero are constructed, and all theta in K use the same disk at t=0. No limit of separate Nadi examples is used. The theorem's every-epsilon consequence holds for that single example.
2. Angular geometry. Exact identity (1-2a)(2-a)/8>0 gives closed child enlargement inside the parent core. The level intervals are disjoint and contraction ratios have a strictly positive lower bound depending on q.
3. Hausdorff gauge. The level-cover sum tends to ell_0(log 3)^q. An arbitrary short interval meets a uniformly bounded number of level intervals at the matching scale. Bernoulli mass bounds give a positive lower measure and the covers a finite upper measure. The bi-Lipschitz circle map preserves these bounds because h(ct)/h(t) tends to c.
4. Radial geometry. gamma=(q-1)/2>0 makes r_n increase from 1/2 to 1. Its gaps have the stated derivative asymptotic. Thus h_n/ell_n grows exponentially, while r_n still tends to a finite endpoint.
5. Crosscut estimate. A logarithmic annular rectangle lies in every crossing path. Nadi Lemma 2.2 applies with one fixed central disk. Analytic radial boundary pieces allow inverse reflection. Crosscut impressions have diameters tending to zero and distinct endpoint values, so prime ends give an injective continuous beta and disjoint B_v.
6. Measure and shell estimate. The pushforward is a probability and has exact B_v mass 2^{-n}. An ancestral shell has mass 2^{-j-1}. These shells are disjoint; their distance bounds give the sum of (log j)^gamma/j. Negative kernel contributions outside the selected shells are bounded below by minus log 2. The same argument works uniformly near B with a factor of 2 in distance.
7. Complex phase. All 1-conjugate(xi)zeta lie in the right half-plane, so |Im A|<pi/2. This is essential: exp(A/2) has real part at least exp(Re A/2)/sqrt(2); arbitrary exp(2A) does not.
8. Conformality. The Herglotz expression for zeta Q'/Q has positive real part and Q'(0)=1. The standard starlike criterion applies. The image cannot be the whole plane, by the bounded inverse and Liouville. Uniform escape follows from Re A tending to infinity and |zeta| tending to one.
9. Disk scale. The angular margin is comparable to ell_{n+1}/[(n+4)log(n+4)]. A small disk at this scale crosses at most one radial boundary and remains in the nested enlarged interval. This gives the lower distance bound. Radial sides give the upper bound and disks lie beyond the prior crosscut for all large n.
10. Weight. Re A is bounded below by c(log n)^{gamma+1}-C, which alone is too small for the linear-in-n demand. The sector-preserving exponential produces exp(c(log n)^{gamma+1}), and gamma+1>1 makes this dominate n for every positive fixed c. One additive real constant handles the compact remaining parameters.
11. Closed approximation set. Uniform image escape excludes t tending to 1 in a bounded convergent sequence. For bounded-away-from-1 t, compactness and continuous disk radii prove closedness. Nadi Lemma 3.2 applies in the proper starlike domain. Its radial tubes also verify the standard connectedness and local-connectedness-at-infinity criterion.
12. Entire approximation. Both inputs are holomorphic on the domain containing the closed approximation set. Two uniform Arakelian approximations produce the exact error less than rho/8. The entire approximants are existential; no finite truncation is represented as them.
13. Rouché and coherence. Every bounded Jordan image disk contains exactly one zero counted with multiplicity, hence a simple zero. At t=0 it is the same zero for every theta. Continuous dependence and the inverse function theorem patch the same germ, not unrelated branches.
14. Endpoint. The inverse values escape uniformly while f stays finite, so f is transcendental and the branch fails holomorphically at the unit endpoint.
15. Scope. The compact set has positive finite h_q measure. Its larger power-log gauges are infinite. The whole E has dimension one but still length zero by Gross. Nothing asserts its exact set or its finite h_q measure.
16. Novelty and evidence. The architecture and original counterexample are Nadi's. Dimension one is already attributed on Eremenko's index. Bounded searches did not locate the precise q>1 iterated-log gauge. No absolute-priority certificate is claimed.
17. Checks. 16,878 mixed exact/symbolic and 100-digit numerical regression checks agree in normal and optimized Python. Four false controls are rejected. The numerical samples are not rigorous interval enclosures and do not establish the analytic existence theorem.

Known mathematical gaps in the accepted q>1 statement: none identified in this self-audit.


# Primary-source and novelty audit

Cutoff: 28 October 2026, Europe/Istanbul. Search records and original bytes are retained in the problem dossier; third-party full texts are excluded from the public source ZIP.

- Hayman--Lingham, arXiv:1809.07200v2, printed p.16: Problem 1.32 and the old no-progress update exactly match the frozen target.
- Eremenko, gross.pdf, 4 April 2015, p.1: asks for a smaller exceptional set and describes the old Volkovyskii capacity-zero example.
- Sina Nadi, arXiv:2609.29797v1, 24 September 2026, 10 pages: Theorem 1.1 gives positive capacity and positive H^s for each fixed s<1; Lemmas 2.2 and 3.2 supply the geometric inputs used here. The inspected version has no iterated-logarithmic gauge statement.
- Eremenko's live uns1.html, archived with TLS and SHA-256, explicitly attributes a dimension-one example to Nadi. We preserve that attribution and claim no priority for dimension one. The literal versioned Nadi theorem has the quantifier for each s<1 there exists f_s; we do not silently strengthen that theorem based only on the website sentence.
- Eremenko, Singularities of inverse functions, arXiv:2110.06134v1, pp.1 and 6: an obstructed inverse path ends at a critical point or escapes along an asymptotic curve; finite-lower-order ENTIRE functions have finitely many finite asymptotic values. The finite-order meromorphic analogue is false in general.
- Classical Arakelian uniform approximation and the analytic starlikeness criterion are external mathematical inputs. The novel quantitative part attempted here is the explicit gauge h_q(t)=t[log log(e^e/t)]^q, q>1, using the full nonatomic shell measure and the sector-preserving exponential error weight.

Bounded web searches for Gross exceptional sets, Nadi, Hausdorff gauges and iterated logarithms found no matching statement beyond the cited work. This is not exhaustive and does not certify priority. Nadi's separate implicit-functions paper 2609.37451 concerns an entire relation in two variables and is not the inverse-function target.

Inherited upstream audit: the capacity-zero and dimension-zero proposals are outdated after Nadi. The finite-order entire countability consequence is supported by the primary survey. The claimed uncontrolled near-basepoint stopping obstruction is false for a fixed germ, whose initial analytic disk supplies a positive lower bound on all finite stopping radii. For the singular-value reduction, use actual critical/asymptotic values rather than automatically replacing their set by its possibly larger closure. These corrections do not become new discovery counts.