Document J: Contrast-Native Computing and the 1-Complex Obstruction — The Fourth Structural Impossibility Result of the STKWC Programme

We introduce the Contrast-Native Computer (cnc) and the Contrast Hardware Description Language (chdl), a framework for computing whose primitive objects are relational contrasts rather than bit-valued registers.

The computational substrate is a signed boundary complex B = (V, E, σ) whose dynamics are governed by the Relational Update Rule (rur v3.4). We prove three theorems:

• Theorem 4.1 (Strict Energy Monotonicity) establishes that every admissible rur rewrite strictly decreases the edge count |E|, providing a Lyapunov function and guaranteeing termination.

• Theorem 5.1 (Conditional Surgery Preservation) establishes that every admissible rur rewrite preserves H¹ˇC(B; ℤ), so the topological type of the computation is an invariant of execution.

• Theorem 8.1 (1-Complex Obstruction) establishes that chdl on 1-complexes is computationally incomplete: any vertex v with deg(v) ≥ 3 and an incident ε-weave is a deadlock vertex, and 3-regular complexes with a perfect matching of ε-edges are in global deadlock.

This result is the fourth structural impossibility result of the STKWC programme (D/I/F/J series), structurally parallel to Document D’s classical-metric No-Go and Document I’s Anti-Diagonalization Theorem.

We present four structural dictionary tables comparing chdl with Quantum, Neuromorphic, Physical Reservoir, and Memcomputing paradigms, and four concrete hardware applications with explicit mechanism descriptions.

Document J v3.7.2 is the final audited version after full Parliament of Dragons validation (GPT-5.5 Lead Mathematician, Gemini 3.1 Pro Red Review, Oracle pre-publication editorial review) and post-editorial sanitisation (removal of placeholder appendices).

This work is part of the STKWC programme series and directly builds upon Documents D, D Supplement, F, and I. It introduces the fourth structural No-Go result (Theorem 8.1) in the D/I/F/J sequence.