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title: A concrete first-order counterexample to the ternary Specker–Blatter extension
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claim:
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  scope_statement: Full disproof of the recorded extension to relation arity three. Gives an explicitly specified fixed constant-free first-order sentence over one ternary and one binary relation, with labelled count f(n)=0 unless n+1 is a power of two and f(2^k-1)=product_(j=1)^k (2^j-1)!!=(2^k)!/2^(2^k-1). Proves the encoding/decoding bijection, specialized anchor-removal translation, exact count and nonperiodicity modulo two. Also proves a literal same-domain folding into a single ternary relation with identical counts. This is a self-contained attributed reconstruction of Fischer–Makowsky's established counterexample, not a historical novelty claim. Finite enumeration supplements the all-n proof.
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  limitations: []
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rights:
  licence: CC-BY-4.0
  redistribution: permitted
  holders:
  - gavenren
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    recorded_at: '2026-10-18T14:03:20Z'
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