The Respect Principle: Viability-Based Invariance with Σ1-Certified Implementation

Multiple finite closures (FCs) inevitably interact when combined into a larger system.
In this paper we establish a \emph{Respect Principle} that characterises when such FCs can
coexist harmoniously.  Under mild regularity assumptions ensuring Carathéodory/Filippov
solutions, we derive uniform \emph{boundary-size bounds} for the monitors associated with each
FC.  These bounds lead to a simple budget inequality: a self–dissipation term
must dominate the interference influx from all other FCs.  We prove a forward–invariance
theorem showing this inequality is sufficient for the product closure to remain in a safe
region.  A converse shows the condition is also necessary: if it is violated, an
admissible coupling can force an FC to exit its safe region.  A harmony–equivalence
theorem identifies respect as the minimal and indispensable condition for global
invariance and yields a conservation/dissipation law on the boundary.  Beyond the core
analysis, we provide a Gershgorin-style interpretation, a quick–start checklist for
practitioners, and a $\SigOne$ verification procedure with explicit safety margins.
The framework provides a verifiable, patent-independent certificate for finite-closure invariance, publishable for universal public use.

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https://www.ghostdriftresearch.com/%E8%A4%87%E8%A3%BD-adic