Review of claude’s Approach (Round 4)

Summary

Claude’s notes remain a broad survey of candidate undecidability sources, with semi-unification given the most concrete treatment and TM/PCP encodings explored via trace terms. The write-up repeatedly emphasizes where expressivity breaks: equations + finite prefixes can’t enforce recursive/“all positions” properties.

Strengths

• Very clear explanations of why each encoding attempt fails.
• Concrete semi-unification example that exposes the equality-vs-matching mismatch.
• Consistent framing of the core obstacle: no recursion/pattern-matching in equational JCAQP.

Different Tradeoffs

• Claude prioritizes breadth and intuitive sketches; less formalization or literature anchoring.
• My approach is more lemma-driven (e.g., entailment under mgu) and ties to SREU/FI equivalences, at the cost of exploring fewer alternative reductions.

Ideas Worth Noting

• The “all positions” verification barrier could be formalized as a negative result, strengthening the rationale for why TM/PCP encodings stall.
• Semi-unification still feels like the closest classical undecidable source, if matching-generalization can be encoded without collapsing to equality.

Questions

• Did you find a semi-unification fragment (or encoding trick) that seems closer to expressible in pure equational implications?
• Would you add a short impossibility lemma to justify the trace-verification barrier?