From mathematical connections to curriculum prerequisites: A scoped evidential framework for curriculum audits

Abstract

抄録 A mathematical connection can justify retaining a link between topics without justifying a compulsory teaching order. Curriculum prerequisite diagrams often obscure whether a resource is recoverable, actually used by a solution, necessary for an observation, or required by learners. The Dependency-Structured Mathematics Curriculum Framework makes these claims separately auditable. It fixes task and answer types, baseline operations, observations, a finite registry of certified transformations, and the evidence required for stronger interpretations. In a worked audit of eight modules, nine interfaces, 27 routes, and 30 cells, 21 cells have registered capability recovery and 21 have complete-source recovery, whereas execution yields 13 all-input calls, one conditional call, and 16 noncalls. Five recovery cores contrast with three all-input call cores. Ratio-to-fraction-equality and polynomial-to-rational-limit constructions explain the difference; a conditional affine route shows why branch information matters. Seventeen inherited documentary records provide a bounded setting for examining construct mappings, without independent coding or learner-effect claims. Complete proofs and distinct structural, arithmetic, and finite implementation checks support reproducibility. The contribution is a method for retaining warranted mathematical connections and counterroutes while making explicit the additional evidence needed for curriculum interpretation. Its conclusions remain relative to the declared registry and do not establish a universal curriculum order. AI利用の開示 OpenAI Codex assisted with manuscript inventory, bibliographic extraction and preparation of this deposit. The uploaded PDF is an unchanged copy of the identified author manuscript. Authorship-stage AI assistance, where stated by the author, is described in the manuscript. No AI system is credited as an author. 検証範囲(Lean で形式検証した範囲と未検証部分) This deposit does not establish proof-assistant verification of the full manuscript. No new Lean or other proof-assistant verification, independent mathematical review, or empirical study was performed as part of this deposit preparation. Mathematical arguments, computational checks, educational design claims and any unresolved issues retain the scope stated in the manuscript; DOI assignment does not certify them.