Cost-Amortised Reasoning via Self-Distilling Neuro-Symbolic Cascades: From Knowledge-Graph QA to Regulatory-Compliance Checking

Large language models make knowledge-graph question answering (KGQA) fluent, but every query pays the full price of an LLM call. We present TACET (Latin: it is silent), a self-distilling neuro-symbolic cascade that amortises that cost by progressively migrating a streamed workload off the LLM and onto cheap, checkable tiers, so that the expensive teacher falls silent over time. The cascade has three tiers: a sound forward-chaining Datalog rule engine that abstains when it cannot prove an answer (Tier 1), a confidence-calibrated ComplEx link predictor [60, 25] verified against a typed ontology (Tier 2), and an LLM teacher (Tier 3). The central mechanism is an online distillation loop. Whenever the teacher answers, TACET mines the answer into Datalog-checkable Horn rules and writes the facts back, so routing drifts toward the cheap tiers as the workload streams. A synthesised rule generalises to entities the teacher never saw; an answer cache cannot do that. On a controlled KGQA benchmark (8 seeds), the cascade answers at 0.981 accuracy while reducing blended cost by ∼3.9× relative to an LLM-only system under a simulated per-tier cost model. We then confirm this on real data with a real teacher. Streaming MetaQA (a 43k-entity movie KG) through a real Grok 4.3 teacher over 3 seeds in a controlled design (Tier-2 disabled and a single teacher answer shared across arms, so accuracy is matched by construction), TACET amortises measured LLM dollars by 2.8× on 1-hop and 5.1× on 2-hop in pooled cost; the per-seed cost ratio spans 1.5–6.3× on 1-hop because only non-empty teacher answers are cached. This saving is delivered by answer reuse, the cascade's caching tier. Under the real teacher, rule distillation adds no dollar advantage: full distillation ties the cache on every seed. The distillation-over-caching effect is itself teacher-quality-gated. With an oracle teacher the miner recovers a generalising composition rule and makes 87% fewer teacher calls than a cache, because a rule answers unseen heads and a cache cannot; under the noisy real LLM the miner recovers no installable rule and the cascade reduces to a cache. Sweeping the installation threshold across eleven 2026 frontier teachers refines this: at the paper's operating point none installs the rule, but at a lower threshold ten of eleven (favourable seed per model) install the world-precision-1.0 composition rule at 81–86% fewer calls, so the gate is the confidence threshold, not teacher identity. That regime installs a self-referential junk rule alongside the true one, which appears to demand gold labels to tell apart; the natural label-free substitutes (teacher agreement, self-consistency) are empirically inert. The apparent dependency is an artifact of candidate enumeration: excluding the mining target from its own rule body removes every junk rule while leaving the true rule and its savings untouched, so what still limits the regime is seed-dependence and the near-functional leakage of Section 7.1, not a need for gold (Section E.3). The one remaining structural foreclosure of that regime — genuinely recursive targets, excluded by the same blunt enumeration rule — is itself removable: a refined guard that bans only the pure self-loop body preserves the published result exactly on the real MetaQA grid (22/22 cells, by classification and generative replay) and, on a synthetic transitive-closure grid with oracle teachers, extends coverage to genuinely recursive targets (Appendix F). Tier-1 answers carry replayable Datalog proof trees, and we prove an ontology-preservation guarantee (Theorem 1) for the synthesised rules. The distillation-over-caching effect that is neutral under the noisy KGQA teacher turns positive on a second domain: streaming GDPR compliance cases from PrivaCI-Bench [37] past three real frontier teachers (gemini-3.5-flash, claude-sonnet-4.6, grok-4.3) over 3 seeds at matched verdict accuracy, online rule distillation amortises measured teacher cost by 1.5–2.1× versus an LLM-only system — against ∼1.1× for an exact-match cache and <1× for a faithful in-context-strategy baseline — with only a ∼10% repeat ceiling, so the saving is rule generalisation, not memoisation. We release the implementation, the benchmark generator, and the full experiment grid.