singleton_bound_8tick
plain-language theorem explainer
Verification of the Singleton bound for the 8-tick code shows that minimum distance 8 meets d ≤ 8-1+1 with equality, confirming MDS status. Information theorists applying Recognition Science to error correction cite this when checking redundancy from the phase structure. The proof executes as a direct numerical check.
Claim. For a code of length $n=8$ and dimension $k=1$ with minimum distance $d=8$, the Singleton bound $d ≤ n - k + 1$ holds with equality.
background
Recognition Science encodes information across the eight phases kπ/4 for k=0 to 7, where each phase is a multiple of the fundamental tick τ₀=1. The module derives error-correction bounds from this periodicity, noting that phase correlations allow recovery after single-phase errors via majority voting and yield an effective rate-1/8 code. Upstream constants supply the tick definition as the RS-native time quantum and the phase function as periodic with period 2π.
proof idea
The proof is a one-line wrapper that applies norm_num to the arithmetic statement 8 ≤ 8-1+1.
why it matters
This places the Singleton bound inside the 8-tick error-correction setting of Recognition Science, directly supporting the eight-tick octave (T7) that supplies natural 8-fold redundancy. It fills the INFO-005 target of deriving fundamental bounds from the phase structure and connects to the module's discussion of Shannon capacity in RS-native units. No downstream theorems are recorded, leaving open how the bound interacts with the Recognition Composition Law in later information-theoretic results.
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