A projective-line quotient of the Rogalski–Sierra algebra

Abstract

For the exact six-relation presentation R(alpha,beta) printed in the 2010 Oberwolfach report and 2012 Rogalski–Sierra article, the linear ideal I=(x3-x1,x4-x2) is nonzero, homogeneous and completely prime, strictly below the augmentation ideal. The exact quotient is the projective-line twisted homogeneous coordinate ring B(P1,O(1),sigma), with sigma[s:t]=[s+beta*t:beta*s+t]. The unital graded surjection has Hilbert series(1-z)^(-2), positive-dimensional projective target and GK-dimension 2. This decisively refutes both literal printed assertions for all prescribed algebraically independent parameters: uniqueness of the nontrivial graded prime and absence of nontrivial maps to any THCR. No classification, embedding, common graded quotient ring or surface-dimensional image is claimed. The self-contained presentation theorem holds over every field for arbitrary alpha and beta^2!=1. In characteristic two it concerns the displayed algebra; the journal's fractional parameter identification with its geometric noetherian family is used only in characteristic different from two. Singular beta=+/-1 is excluded and has linear zero divisors. Classical quantum/Jordan-plane, degree-reduction, graded-twist and THCR machinery is credited. Historical priority remains unresolved. Two independently positioned bounded audits verified no full matching predecessor in inspected primary passages. Nevins–Sierra arXiv:1206.0760v1, page 25, and v2, pages 28–29 repeat the contrary no-map assertion; their canonical-factor/moduli theorems require extra hypotheses. The inspected arXiv versions are distinguished from the unread journal body, and the v2 arXiv revision 14 April 2014 from its internal 15 November 2018 date. Campbell's baseline infinitesimal deformations and function skew fields of another family are not this exact generic algebra's homomorphic curve quotient. Older papers/books, unread versions, incomplete citation coverage and nonindexed work remain material gaps. No firstness, certified novelty, exhaustive absence or current worldwide-open claim is made. The companion archive preserves all 58 immutable candidate files, including three original archives, exact selected-record and statement identities, two independent mathematical families, two independent priority families and standard-library Python checks. Finite diagnostics support the universal written proof and do not establish all-parameter conclusions by sampling or provide proof-assistant certification. Optional primary-body custody is distinguished from authored-source verification; full third-party PDFs, extracted texts and page images are excluded. AI tools were used extensively in mathematical development, literature investigation, drafting, computation, reproduction and adversarial review. This is an unrefereed preprint without external human peer review or formal proof-assistant verification. The note and authored companion materials are licensed CC BY 4.0.