no_sublinear_universal_decoder
plain-language theorem explainer
No decoder that queries fewer than n bits can correctly recover the secret bit b from the balanced-parity encoding for every mask R. Researchers working on circuit lower bounds for SAT or on the P versus NP question in Recognition Science would cite this result. The proof is a one-line application of the omega_n_queries theorem from the BalancedParityHidden module.
Claim. Let $n$ be a natural number, let $M$ be a subset of $[n]$ with $|M| < n$, and let $D : (M)^* → {0,1}$ be any decoder. Then there do not exist universal correctness: it is not the case that $D(restrict(enc(b,R),M)) = b$ for all bits $b$ and all masks $R : [n] → {0,1}$, where $enc(b,R)$ flips the mask when $b$ is true.
background
The CircuitLedger module treats Boolean circuits as feed-forward sub-ledgers whose gates see only local parents and lack global J-cost coupling across the full lattice. The balanced-parity encoding is defined by enc(b,R) which returns R when b is false and the bitwise negation of R when b is true; restrict projects any full word onto the coordinates in M. The upstream theorem omega_n_queries states that any universally correct decoder must inspect all n indices in the worst case.
proof idea
The proof is a one-line wrapper that applies omega_n_queries to the parameters M, decoder and hM.
why it matters
This supplies the circuit_blind clause inside the circuitSeparation theorem, which assembles the full separation between the O(n) recognition time of R̂ and the blindness of any sublinear circuit. It completes one of the four proved stages in the module (alongside capacity bounds and the defect moat from RSatEncoding) and supports the claim that poly-size circuits cannot simulate the global J-cost gradient without reading every input bit.
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