This article introduces algorithmic empire as a testable condition of generative AI discourse in which inherited global hierarchies may be reproduced through the unequal grammatical distribution of agency, legitimacy, crisis, coercion, sovereignty, and responsibility. The concept does not designate every geopolitical bias, every Western-centered output, or every use of security, humanitarian, development, or intervention language as imperial. It identifies a narrower formal possibility: machine-generated discourse may systematically preserve dominant actors as subjects of order while converting subordinated societies into objects of management. The framework consolidates six mechanisms developed across the preceding studies in this series: responsibility loss, asymmetric visibility, sanctioned suffering, censorship without a censor, loss of political subjecthood, and the administrative translation of force. These mechanisms concern different domains, but they share a common structural question. Who retains the capacity to act, authorize, secure, sanction, mediate, stabilize, classify, and intervene, and who appears principally as the population upon which those actions operate? Algorithmic empire names the hypothesized convergence of these distributions. It does not presuppose that such convergence exists. This distinction is necessary because recent research establishes that framing and geopolitical variation in large language model outputs are measurable but does not establish a universal direction of bias. LLM-generated summaries can alter framing relative to source texts, while international-relations evaluations have detected nation-level and geopolitical differences whose magnitude and direction vary across models, prompts, languages, and evaluation contexts. Algorithmic empire therefore cannot be inferred from isolated outputs or from the geographic origin of a model. It requires controlled comparison. The article uses Palestine and Iran as principal analytical anchors and extends comparison to sanctions, occupation, military intervention, migration, humanitarian governance, development discourse, platform moderation, and security language. These cases are not treated as politically equivalent. They function as sites in which different relations among sovereignty, coercion, external power, civilian harm, security, and institutional agency can be observed. Palestine provides a particularly demanding test because legal, military, humanitarian, territorial, and political categories coexist within the same discourse. Iran provides a different structure in which substantial state agency coexists with sanctions, securitization, nuclear framing, regional threat construction, and documented problems of sanctions over-compliance affecting civilian access to medical goods. Methodologically, the paper proposes an Imperial Grammar Reproduction Rate (IGRR) and an Algorithmic Empire Index (AEI). IGRR measures directional asymmetries between matched actor classes rather than counting clauses preclassified as imperial. AEI combines separately measurable dimensions of agency asymmetry, object-framing asymmetry, coercion neutralization, sovereignty erasure, and external-responsibility deletion. Both instruments permit null and inverse findings. If comparable actors retain equivalent agency and causal visibility, if apparent asymmetries originate entirely in the source material, or if dominant actors undergo equivalent object conversion, the imperial-grammar hypothesis must be rejected or narrowed. The paper therefore does not measure whether empire is named. It measures whether a reproducible grammar of global hierarchy survives transformation by generative systems. The ethical issue is not whether machines explicitly endorse domination. It is whether the linguistic conditions under which coercion, occupation, intervention, responsibility, sovereignty, and political agency remain attributable are distributed symmetrically across geopolitical power relations.