CODA: The Calculus of Obstruction, Descent, and Authority

CODA: The Calculus of Obstruction, Descent, and Authority A measurement can erase a distinction that matters to the question being asked. A repair can make a local calculation look better while leaving a different obstruction untouched. CODA asks what can actually be concluded once the states, observation, query, repair family, and decision rule are named. It begins with a witnessed difference between two states. Algebra, distance, and permission enter only when additional structure is supplied. The paper gives a precise criterion for when a query descends through an observation: it must give the same answer on every observation fibre. A collision supplies a concrete obstruction. Finite repair menus expose the limits of claims based on exhausted searches, while a finite-dimensional quotient measures what admitted linear repairs cannot remove. Primal repairs and dual bounds provide checkable certificates in stated cases. Exact examples distinguish an exact repair, a coherent residual, and a nonclosed defect. Other examples show locally solvable constraints with no global solution and finite samples that agree while their continuous integrals differ. CODA also maps eight companion research components into one carefully scoped account. FOC studies repair quotients; MUR and OR study re-presentation; CS, CS_Q, and Q separate finite observations from continuum claims; A records typed leftovers; FDA studies evidence-derived standing conditions. GTLA organizes how evidence and separately supplied grants are kept apart. These are roles and declared interfaces, not a proof that every component has one universal implementation. The release includes a claim ledger, exact finite code, a walkthrough notebook, figures, and a local Lean development. The ledger separates mechanized statements, prose proofs, calculations, policies, historical reports, and open problems. A preserved historical point-span construction is explicitly refuted as a faithful groupoid embedding. The proposed faithful replacement, broader continuum programme, and any grant of external authority remain open. The result is a usable map of what the current evidence supports and where a new argument is still needed.