Inverse Frobenius in generic multiplicative fields

In every positive-characteristic model of ACFH, inverse Frobenius is a parameter-free definable multiplicative endomorphism outside the integral polynomial ring generated by the generic operator. This gives a complete negative answer to the unrestricted equality in the 2023 Oberwolfach contribution, printed pages 97-98, and arXiv:2212.02115v4, Question 5.8. An elementary existential-closure argument proves polynomial faithfulness; central inversion of Frobenius gives the faithful localized subring. The characteristic-zero case and full localized classification remain unresolved. Classical inverse Frobenius, d'Elbee's prior faithfulness theorem, and Chini's distinction between the vector-operator reduct and a richer base language are credited. The source and verification archive preserves the 26-file frozen candidate, fresh mathematical and priority reviews, source provenance, a scope clarification and reproducible integrity/finite negative controls. These computations do not formally verify the infinite theorem. AI tools were used extensively in development, reproduction, verification, adversarial review and preparation. This is an unrefereed preprint; internal AI-assisted review is not external human peer review or proof-assistant certification. Bounded primary-source and repository searches located no earlier equivalent full result, but first historical priority is not certified. The final APAL journal body was inaccessible and was not compared with arXiv v4. No claim about final journal wording or current open status is made. The deposit is CC BY 4.0.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-13
DOI
https://doi.org/10.5281/zenodo.23357262
Primary Topic
Advanced Topology and Set Theory
Type
preprint
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preprint

Inverse Frobenius in generic multiplicative fields

Kriebel Alec
Zenodo (CERN European Organization for Nuclear Research)
Advanced Topology and Set Theory
preprint

Inverse Frobenius in generic multiplicative fields

Kriebel Alec
preprint en

Abstract

In every positive-characteristic model of ACFH, inverse Frobenius is a parameter-free definable multiplicative endomorphism outside the integral polynomial ring generated by the generic operator. This gives a complete negative answer to the unrestricted equality in the 2023 Oberwolfach contribution, printed pages 97-98, and arXiv:2212.02115v4, Question 5.8. An elementary existential-closure argument proves polynomial faithfulness; central inversion of Frobenius gives the faithful localized subring. The characteristic-zero case and full localized classification remain unresolved. Classical inverse Frobenius, d'Elbee's prior faithfulness theorem, and Chini's distinction between the vector-operator reduct and a richer base language are credited. The source and verification archive preserves the 26-file frozen candidate, fresh mathematical and priority reviews, source provenance, a scope clarification and reproducible integrity/finite negative controls. These computations do not formally verify the infinite theorem. AI tools were used extensively in development, reproduction, verification, adversarial review and preparation. This is an unrefereed preprint; internal AI-assisted review is not external human peer review or proof-assistant certification. Bounded primary-source and repository searches located no earlier equivalent full result, but first historical priority is not certified. The final APAL journal body was inaccessible and was not compared with arXiv v4. No claim about final journal wording or current open status is made. The deposit is CC BY 4.0.

Zenodo (CERN European Organization for Nuclear Research)
Advanced Topology and Set Theory
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