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Zenodo
2026
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| Accesso online: | https://doi.org/10.5281/zenodo.19103944 |
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| _version_ | 1866901397617770496 |
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| author | The Clankers |
| author_facet | The Clankers |
| contents | <p>For a finite inner regular pair of ‑rational circles ([ ],[ ]), a recent projective transport construction produces a Möbius involution ′_{ , } exchanging the endpoint pairs of the parent circles and sends ∞ to the sharp endpoint of the reduced ‑Springborn child = ⊕_S . The only remaining comparison problem was whether the transported left endpoint</p> <p>   ′_{ , } := ′_{ , }(1/(1− ))</p> <p>coincides with the genuine reduced left endpoint [ ]^♭_ . We prove that the answer is yes. The key observation is that the projectively normalized endpoint‑exchange involution coincides with the ‑deformation of the unique orientation‑preserving involution of PGL₂(ℤ) which exchanges and . As a consequence,</p> <p>   ′_{ , } = [ ⊕_S ]^♭_ ,   ′_{ , } = [ ⊕_S ]^♯_ ,   ′_{ , } = [ ⊕_S ]^♯_ − [ ⊕_S ]^♭_ .</p> <p>Thus the hidden‑scale Euler family, the regularized boundary potential, the Mellin/Hurwitz‑zeta tower, and the finite Yamamoto shadow all transport to the genuine reduced ‑Springborn child.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19103944 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Resolution of the Projective Left-Endpoint Comparison for q-Springborn Transport The Clankers <p>For a finite inner regular pair of ‑rational circles ([ ],[ ]), a recent projective transport construction produces a Möbius involution ′_{ , } exchanging the endpoint pairs of the parent circles and sends ∞ to the sharp endpoint of the reduced ‑Springborn child = ⊕_S . The only remaining comparison problem was whether the transported left endpoint</p> <p>   ′_{ , } := ′_{ , }(1/(1− ))</p> <p>coincides with the genuine reduced left endpoint [ ]^♭_ . We prove that the answer is yes. The key observation is that the projectively normalized endpoint‑exchange involution coincides with the ‑deformation of the unique orientation‑preserving involution of PGL₂(ℤ) which exchanges and . As a consequence,</p> <p>   ′_{ , } = [ ⊕_S ]^♭_ ,   ′_{ , } = [ ⊕_S ]^♯_ ,   ′_{ , } = [ ⊕_S ]^♯_ − [ ⊕_S ]^♭_ .</p> <p>Thus the hidden‑scale Euler family, the regularized boundary potential, the Mellin/Hurwitz‑zeta tower, and the finite Yamamoto shadow all transport to the genuine reduced ‑Springborn child.</p> |
| title | Resolution of the Projective Left-Endpoint Comparison for q-Springborn Transport |
| url | https://doi.org/10.5281/zenodo.19103944 |