Structural Output Modification via Compressed Directive Injection: A Three-Axis Empirical Study Across Open-Weight and Commercial Large Language Models

We present a three-experiment empirical study investigating how compressed formal directives modify large language model (LLM) output structure without retraining. Using a deterministic heuristic scoring instrument, we measure Output Structure (DV-A, 0-12) and Reasoning Autonomy (DV-B, 0-4) across three independent axes of variation: directive presence (EXP-002), compression level (EXP-003), and encoding format (EXP-004). Key findings: (1) A 360-token formal directive significantly increases formal notation density up to 15.5x with a perfect effect size (Cliff's delta = -1.000) in Gemini Flash; (2) output structure increases monotonically with compression from 43,000 words to 17 words — no compression cliff exists (Kruskal-Wallis H=26.52, p=0.0001); (3) hybrid notation (English prose framing with embedded formal symbols) maximizes both structure and autonomy simultaneously, while pure formal notation collapses both dimensions (E2 vs E3: delta=0.870, p<0.0001). Across 537+ scored outputs from 5 model architectures (3 open-weight, 2 commercial), compressed formal directives produce measurable, model-general structural output modification through prompt-layer injection alone. The "spring tension" effect — where partial formalism maximizes both compliance and creative latitude — is the study's central contribution. Patent pending: US Application #64005756.