An Orientation-Locking Kernel in the Continuum–Crystalline Correspondence: Unoriented bordism versus admissible spatial backgrounds in (2 + 1) dimensions

Abstract

We examine the compatibility between the unoriented-bordism formulation of the continuum–crystalline correspondence and the background fields admitted by its spatial-symmetry definition. For a finite point group G ≤ O(2), an admissible spatial background identifies the orientation character of the G-bundle with the orientation double cover of spacetime. The formal bordism classification used in the continuum-limit theorem instead allows arbitrary maps to BG. In two spatial dimensions, the resulting formal continuum group is isomorphic to (Z/2Z)^3. We give three explicit characteristic-number generators and show that restricting them to orientation-locked backgrounds annihilates two while the third remains detectable on an admissible Z2 × Z2 reflection background over RP2 × S1. The physical restriction map therefore has the exact kernel (Z/2Z)^2. This result does not contradict the formal split epimorphism proved by Chari and Hughes; rather, it prevents that algebraic statement from being interpreted as an injective, response-level encoding of formal continuum classes by physically admissible crystalline probes. We formulate two consistent repairs: replace ordinary BG-bordism by a mixed tangential structure that incorporates orientation locking, or quotient the formal target by the classes invisible on admissible backgrounds. AI-use disclosure. OpenAI’s ChatGPT, using GPT-5.6 Sol, was used extensively to assist with structural exploration, candidate argument development, mathematical stress-testing. The author reviewed and revised the resulting arguments. A public snapshot of the relevant interaction is available here: https://chatgpt.com/share/6a83ce19-c494-83ee-93eb-741f63054b8f