Volume or Coupling? A Scale-Dependent Dissociation in Constraint Recovery of Language-Model Loops

Recent work on self-referential large-language-model (LLM) loops suggests that isolated recursive loops degrade toward closed attractor states and that exchange with an independent process supports recovery. We ask what the operative ingredient of such coupling is, using a minimal, fully reproducible perturbation-recovery protocol: an instruction-tuned model maintains an explicit output constraint (all-capital letters), a conflicting instruction is injected, and constraint recovery is tracked over eight exchanges under fixed criteria, across four conditions (coupled dual-model, single, volume-matched single, and a four-fold-volume condition; 30 opener-paired runs each). At 1.5B parameters, the coupled loop's advantage over a single model (43\% vs. 17\%, exact McNemar \(p = .039\)) is fully reproduced by a volume-matched single model (37\%; C vs. X \(p = .774\)): the apparent benefit of coupling is a context-volume effect. Extending the identical protocol across scale reverses this picture. At 3B, all conditions are at or near ceiling (90–100\%): the perturbation fails to capture. At 7B, the perturbation captures single-model conditions completely (0/30 for both single and volume-matched; 1/30 at four-fold volume), while the coupled loop alone recovers (8/30; C vs. X, \(b = 8\), \(c = 0\), \(p = .0078\))—slowly (delays 2–7) and durably in five of eight cases. Deterministic trajectory replays localize the mechanism: the agent that directly receives the perturbation suffers deep, recurrent capture, whereas its partner—which never receives the instruction—is captured only shallowly by mimicry and rapidly re-anchors, supplying constraint-consistent evidence that pulls the system back. We propose a unified account: recovery depends on the availability of constraint-consistent generation that is not bound by the perturbing instruction—supplied by raw volume at small scale, and only by structural shielding at larger scale. The deflationary volume account is thus itself scale-bounded. All code, raw outcomes, and figures are released for exact reproduction on free- and low-cost-tier hardware.