Observer-quotient security studies interactive cryptographic systems whose security depends on what an admissible observer can distinguish across transcripts, leakage traces, and hidden implementation continuations. The paper defines observer-indexed experiments with session identifiers, adaptive schedulers, oracle forwarding, simulators, ideal quotient functionalities, and nonuniform environments, and proves a real/ideal emulation theorem in which sequential morphism defects add, parallel defects obey a product-TV bound, and adaptive observer choice is absorbed by an explicit wrapper construction. The resulting advantage bound is indexed by $δ_{\mathrm{obs},t}$, $δ_{K,t}$, $δ_{\mathrm{post},t}$, $δ_{\mathrm{sim},t}$, $η_t$, and the residual floor $ρ_T(\mathbb E Z_T)$. The framework is instantiated for IND-CPA encryption with timing leakage, deterministic encryption with entropy ledgers, and finite-state side-channel refinement under transcript, timing, cache, power, EM, and profiled observers. The optimization/control component identifies hidden continuations with observability kernels, treats sensor redesign as quotient refinement, and converts dissipativity, PL-type rates, and ISS residual bounds into concrete reductions in distinguishing advantage. Ancillary code and synthetic data reproduce the finite-state leakage audit and LTI observer-design benchmark.

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