Mocanu · parametric, arity 1
The image of the unit disk under \(f\): concentric circles \(|z|=r\) and radial spokes mapped through \(f(z)=z+a_2z^2+\dots\); the bold curve is the boundary \(f(e^{i\theta})\). Rendered in your browser from the certified Taylor coefficients - nothing is computed server-side.
| Property | Kind | Params | Holds | Value / r | Status | Confidence |
|---|---|---|---|---|---|---|
| K | membership | {"alpha": 0.25} | ✗ | -8.1287 | unscreened | - |
| K | membership | {"alpha": 0.5} | ✗ | -1.2411 | unscreened | - |
| K | membership | {"alpha": 0.75} | ✓ | 0.0364 | unscreened | - |
| r* | radius | {"alpha": 0.25} | - | r ≤ 0.920 | unscreened | - |
| r* | radius | {"alpha": 0.5} | - | r ≤ 0.920 | unscreened | - |
| r* | radius | {"alpha": 0.75} | - | r ≤ 0.920 | unscreened | - |
| S* | membership | {"alpha": 0.35} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 0.4} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 0.45} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 0.55} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 0.6} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 0.65} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 0.7} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 0.8} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 0.85} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 0.9} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 0.95} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 1.0} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 0.1} | ✗ | r ≤ 0.800 | ✗ disproven | proven |
| S* | membership | {"alpha": 0.15} | ✗ | r ≤ 0.800 | ✗ disproven | proven |
| S* | membership | {"alpha": 0.2} | ✗ | r ≤ 0.800 | ✗ disproven | proven |
| S* | membership | {"alpha": 0.3} | ✗ | r ≤ 0.950 | ✗ disproven | proven |
| S* | membership | {"alpha": 0.0} | ✗ | r ≤ 0.600 | ✗ disproven | proven |
| S* | membership | {"alpha": 0.05} | ✗ | r ≤ 0.800 | ✗ disproven | proven |
| S* | membership | {"alpha": 0.25} | ✗ | r ≤ 0.900 | ✗ disproven | proven |
| S* | membership | {"alpha": 0.5} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S* | membership | {"alpha": 0.75} | ✓ | r ≤ 0.999 | ✓ proven | rigorous_verified |
| S | membership | {"alpha": 0.25} | ✗ | - | unscreened | - |
| S | membership | {"alpha": 0.5} | ✗ | - | unscreened | - |
| S | membership | {"alpha": 0.75} | ✗ | - | unscreened | - |
Because starlikeness is proven, these properties follow by classical implications (unconditional edges in the lattice):
| Verifier | Property | Direction | Outcome | Domain r | Witness | Engine |
|---|---|---|---|---|---|---|
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | disproves | pass | 0.800 | 0.9*exp(i*2.368466) | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | disproves | pass | 0.800 | 0.9*exp(i*2.396078) | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | disproves | pass | 0.950 | 0.99*exp(i*3.884039) | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | disproves | pass | 0.900 | 0.95*exp(i*2.399146) | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | proves | pass | 0.999 | - | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | disproves | pass | 0.600 | 0.8*exp(i*2.374602) | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | disproves | pass | 0.800 | 0.9*exp(i*3.951535) | boundary_v3.1 |
| boundary_scan_centered_v31 | starlike | disproves | pass | 0.800 | 0.9*exp(i*3.951535) | boundary_v3.1 |
| pointwise_starlike_iv | starlike | disproves | pass | 0.990 | [-0.7121464023352646, -0.6877117867544074] | gft-0.1.0-reaudit-20260610 |
| pointwise_starlike_iv | starlike | disproves | pass | 0.990 | [-0.7121464023352646, -0.6877117867544074] | gft-0.1.0-reaudit-20260610 |
| pointwise_starlike_iv | starlike | disproves | pass | 0.990 | [-0.7121464023352646, -0.6877117867544074] | gft-0.1.0-reaudit-20260610 |
| pointwise_starlike_iv | starlike | disproves | pass | 0.990 | [-0.7121464023352646, -0.6877117867544074] | gft-0.1.0-reaudit-20260610 |
| pointwise_starlike_iv | starlike | disproves | pass | 0.990 | [-0.7121464023352646, -0.6877117867544074] | gft-0.1.0-reaudit-20260610 |
| pointwise_starlike_iv | starlike | disproves | pass | 0.990 | [-0.7357133772226201, 0.6624393002952698] | gft-0.1.0-reaudit-20260610 |
| pointwise_starlike_iv | starlike | disproves | pass | 0.990 | [-0.7357133772226203, -0.6624393002952697] | gft-0.1.0-reaudit-20260610 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | disproves | pass | 0.800 | 0.9*exp(i*2.368466) | boundary_v3.0 |
| boundary_scan_polynomial | starlike | disproves | pass | 0.800 | 0.9*exp(i*2.386874) | boundary_v3.0 |
| boundary_scan_polynomial | starlike | disproves | pass | 0.800 | 0.9*exp(i*2.40068) | boundary_v3.0 |
| boundary_scan_polynomial | starlike | disproves | pass | 0.950 | 0.99*exp(i*2.40835) | boundary_v3.0 |
| boundary_scan_polynomial | starlike | disproves | pass | 0.600 | 0.8*exp(i*2.374602) | boundary_v3.0 |
| boundary_scan_polynomial | starlike | disproves | pass | 0.800 | 0.9*exp(i*2.380738) | boundary_v3.0 |
| boundary_scan_polynomial | starlike | disproves | pass | 0.900 | 0.95*exp(i*2.402214) | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |
| boundary_scan_polynomial | starlike | proves | pass | 0.999 | - | boundary_v3.0 |