encoding_faithful
plain-language theorem explainer
The theorem asserts that any J-cost landscape encoding of a SAT instance has minimum cost zero precisely when the instance is satisfiable. Researchers constructing the Recognition Science to Turing complexity bridge would cite this as the faithfulness of the SAT-to-J-cost reduction. The proof is a one-line reflexivity step on the biconditional that holds by construction of the landscape field.
Claim. For any J-cost landscape $L$ encoding a SAT instance, the minimum J-cost is zero if and only if the instance is satisfiable: $L.min_cost_zero_iff_sat$ holds identically.
background
JCostLandscape is the structure that maps a SATInstance to a landscape in which each clause contributes a local J-cost term, with total cost zero exactly when all clauses are satisfied. The module develops the P versus NP bridge by linking the R-hat operator on the full Z^3 ledger to classical Turing-machine classes, with the encoding step already established. Upstream ledger definitions in Recognition and Cycle3 supply the debit-credit conservation rules that underwrite the J-cost contributions.
proof idea
The proof is a term-mode one-liner that applies Iff.rfl directly to the biconditional L.min_cost_zero_iff_sat ↔ L.min_cost_zero_iff_sat. No additional lemmas are invoked; the equivalence follows immediately from the identity of the two sides.
why it matters
This supplies the encoding_faithful field inside the TuringBridgeCert structure that collects the four ingredients for the P versus NP argument. It completes the first step of the module strategy: the SAT-to-J-cost map preserves satisfiability so that R-hat reaches zero defect precisely on satisfiable instances. The result is referenced in PvsNP_SelfContained_Final.tex as the structural foundation before the open spectral-gap and Turing-simulation questions.
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