V01.04 — Mirror Mathematics I: A Constraint Calculus for Recursive Observerhood

This paper opens the Mathematics Arc of Mirror Programme, Volume I: Observerhood by introducing a constraint calculus for recursive observerhood. It formalises the core vocabulary developed across the Theory Arc: constraint regimes, viability functionals, bounded local modelling, self-models, reliability variables and salience gain. The purpose is not to replace existing mathematical frameworks, modify physics or prove consciousness, but to provide a disciplined formal grammar for asking when observerhood becomes a viable organisational predicate. The central contribution is a set of narrow mathematical structures for distinguishing persistence, prediction, self-modelling and observer-relevant reliability. A system is not treated as an observer merely because it computes, stores information or predicts its environment. In the Mirror framework, observerhood requires bounded modelling, viability-relevant self-modelling and positive reliability salience after cost. This release begins the Mathematics Arc following V01.01 — Mirror Theory I, V01.02 — Mirror Theory II and V01.03 — Mirror Theory III.