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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.19103944 |
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Table of 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>