V01.05 — Mirror Mathematics II: Minimal Observer Thresholds

This paper forms part of Mirror Programme, Volume I: Observerhood. Mirror Mathematics II develops the formal threshold layer of the Mirror Mathematics arc. Building on Mirror Mathematics I, it asks when a self-model becomes observer-relevant rather than merely predictive, self-referential or descriptive. The paper introduces a minimal Mirror observer predicate grounded in self-model reliability, action-guidance, cost-sensitive repair and positive viability coupling under perturbation. The central condition is expressed through a salience functional measuring whether the expected viability benefit of reliability-guided repair exceeds the representational and action costs of maintaining and using that reliability structure. The paper synthesises the computational evidence from Mirror Observerhood Labs I-V. The Labs show that self-model reliability can help under self-relevant perturbation, that decomposed reliability alone is insufficient, that repair must be actionable and cost-sensitive, that reliability has a bounded positive threshold region, and that recursive reliability is useful only when lower-order reliability can itself fail in viability-relevant ways. The result is a conservative mathematical predicate for minimal recursive observerhood. It is not a consciousness claim, not a moral-status claim and not a claim that all artificial agents are observers. It defines a testable threshold: a system becomes minimally observer-like only where self-model reliability is represented, action-guiding, cost-sensitive and positively coupled to viability under perturbation.