Graded realization
The associated graded category of ind(mod A) for the radical filtration is the mesh category k(ΓA) (Proposition 2.6), and evaluation at the projective objects is an equivalence onto mod A0 (Proposition 2.10).
Graded realizationMathematical navigation
The proof in three parts
The paper first builds the standard form A0 from the radical grading of the module category, then proves Theorem B outside characteristic two, and finally realizes A as a binary fibre of a family of deformations of A0 in characteristic two.
Part I: the standard form and characteristic not two
Rays and scalar normalization
Rays give a basis compatible with all two-sided ideals. Lifting to the universal cover of the mesh complex and descending along the free deck group gives a normed multiplicative basis of A0 (Proposition 4.14).
Scalar normalizationMesh deformations
Outside characteristic two the linearized mesh derivative is surjective (Proposition 5.3), so higher error terms in the mesh relations can be removed degree by degree (Proposition 5.4). This proves Theorem B.
Theorem BPart II: deformations in characteristic two
Positive deformations
Internal-degree bounds for Ext lift the indecomposable modules of A0 to every positive filtered deformation (Lemma 6.3, Proposition 6.5).
Module liftingWeights and parameters
Integer ray weights and residual characters control the positive Hochschild cohomology (Propositions 7.2–7.6). A family whose fibres keep the ray basis recovers A as a binary fibre (Theorem 7.7, Proposition 7.9).
Family criterionPart III: multiplicative bases in characteristic two
Local ray geometry
Bimodule retractions onto representation-infinite incidence and bound-quiver algebras restrict how rays factor (Section 8, Appendix C).
Proper productsIntegral lifting
A Bockstein map for coefficients modulo four shows that zero-supported cocycles span the positive Hochschild cohomology (Theorem 9.1, Corollary 10.5).
Bockstein argumentBinary families
Zero-supported cocycles form an associative family over a polynomial ring (Proposition 11.3). A is a binary fibre, where the basis of A0 stays normed and multiplicative (Theorem 11.7).
Final basisFurther results
Section 12 of the paper, also formalized, gives completeness of the lifted modules, a normal form for the filtered mesh relations, flatness of polynomial mesh perturbations, the full Hochschild comparison and the polynomial mesh comparison.
Paper correspondence
Every numbered statement of the October 17, 2026 revision is listed with its Lean declarations. Some statements are proved in an equivalent or more general form, and some proofs take a shorter route than the paper.