Value Headroom Moderates Whether Specification Beats Style in LLM Negotiation

When two language-model agents negotiate, does it matter more how they are styled (warm, dominant) or how their objective is specified (ranked priorities, a reservation value, a concession rule)? A large autonomous-negotiation study reported that interpersonal warmth, not structure, predicted success. This note offers an independent test – a reconstruction from that study's public materials, not a one-to-one reproduction – and identifies value headroom (the joint surplus a naively cooperative pair leaves on the table) as a first-order moderator. Across seven arms spanning three model families (4,920 dyads), specification-first prompting ties interpersonal style on near-ceiling scenarios (a preregistered null) but yields strictly higher joint value once headroom is present (Cohen d = .314, p = .009 on a mid-capability model; d = .569, p < .001 at the frontier – the advantage grows with capability), with a graded design confirming the headroom slope. An ablation shows the advantage comes from teaching logrolling, not leaking the payoff matrix; it survives prompt paraphrase and replicates across model families. Specification moves the objective value frontier while remaining affectively neutral; style games the agreement proxy. The apparent "style beats structure" result is recovered as the zero-headroom boundary case of a more general law. Includes paper.yaml (Paper Spec v0.1.0) – a machine-readable specification of the paper's claims, assumptions, and dependencies. See https://github.com/spectralbranding/paper-spec for the standard.