Recursive Learning and the Archimedes Principle: Competence-Gated Tier Alternation as a Cost–Quality Mechanism for Autonomous Coding Agents

Autonomous coding agents achieve reliability by spending inference: they re-read, re-plan and re-verify until a task is carried out correctly. The dominant cost of such systems is therefore not capability but repetition — the same competence being re-purchased, at full price, on every task. This paper formalises and evaluates recursive learning and the Archimedes Principle as two coupled mechanisms for reducing that cost without an unbounded loss of quality. Recursive learning is defined as a four-stage operator over an episode store (execute, capture, consolidate, inject) that re-enters the agent's own history as context, and whose fixed point is analysed as a contraction under an explicit, falsifiable assumption. The Archimedes Principle is defined as a competence gate: a cheap local tier (the "displaced volume") attempts a task first, a per-task-pattern competence estimator decides whether that attempt is admissible, an independent verifier adjudicates the result, and a large tier is invoked only when the cheap attempt fails verification. We give a cost model, derive the break-even inequality q > c_S / c_L for the acceptance probability q, and prove that the gate's expected cost is strictly below always-large whenever that inequality holds. We further prove that the safest possible gate degrades quality only in proportion to the verifier's false-accept rate, and we prove — and then corroborate empirically — that a competence gate whose pattern keys are nearly unique is structurally unreachable. An audit of 1,225 real episodes from the Aura Code deployment (13 September–17 October 2026, 1.049 × 10^9 tokens) is reported in full, including a negative result: in the 153 logged alternation decisions, the cheap tier was attempted in 0 cases, and in the 143-task cost ledger 0 of 143 tasks ever satisfied the gate's minimum-evidence condition. We show that the theory predicts this outcome, identify the underlying defect as cold-start non-identifiability (attempts that do not happen are not recorded, so the estimator can never learn why they did not happen), and state the repair. We close with a pre-registered, powered evaluation protocol, an explicit list of what the audit cannot establish, and a delimitation of the principle against the claims it does not support.