Free High-Order Cognition: Zero-Training, Zero-Parameter Structural Emergence in the UHGP Symbolic System

We report a cognitive-science study of a symbolic system, the UHGP (Unified Hyperoperation Gauge Program), in which 30 high-order cognitive capabilities — deduction, long-chain stability, abduction, counterfactual reasoning, analogy, induction with remote extrapolation, multi-step planning, self-falsification, normative self-awareness, isomorphism detection, semantic/syntactic distinction, cross-domain transfer, hyper-compression, one-shot learning, condition-number self-diagnosis, operator polymorphism, the spontaneous emergence of a positional counting system, order isomorphism, algorithmic shortcuts, lossless round trips, closed-form sensitivity, the generation of novel formal systems, implicit-axiom attribution, meta-creation, meta-meta-creation, and dual-path self-proof — are measured, item by item, with independent ground truth. The central finding is that every one of these capabilities is a structural corollary of a single closed-form flow law P_λ⁽ʰ⁾ = f_h⁻¹(λ f_h(x)), not a trained skill: the system has zero training, zero learned parameters, zero gradients, and no external proof assistant. A self-contained, single-file, numpy-only script (uhgp_quick30.py, included) reproduces all 30 capabilities in under one second (30/30, deterministic, 0.4–0.5 s). The companion full-scale suite scores 108.2/120 on a 20-item cognitive battery, with the honest weak points (error-correction self-check, verification) disclosed. We also state the system's unstructured-capability boundary in its own terms: the input must be expressible as an element of some group; torsion structures are detected as out of scope by the system itself (blind test 5/5), in agreement with independently proved theorems. This work demonstrates that high-order cognition can be a structural property of a mathematical system rather than a skill acquired through training. All experimental results are deterministic and fully reproducible by running a single Python script.