Syndrome
Syndrome records the boolean measurement outcomes that identify which error occurred in an 8-tick redundant encoding. Workers constructing fault-tolerant codes from Recognition Science phase structure cite it when building toric or surface codes. The declaration is a plain structure with a list of bits and a uniqueness flag defaulting to true.
claimA syndrome is a list of boolean bits together with a boolean flag (default true) that asserts distinct errors produce distinct measurement outcomes.
background
The module derives quantum error correction from the eight-tick octave, where the eight phases supply natural redundancy so that phase-shift errors can be detected by syndrome measurements without revealing logical information. Upstream, the module imports the active-edge count A from IntegrationGap, Masses.Anchor, and Modal.Actualization; these fix the per-tick coherence unit and the actualization operator that selects the realized configuration after each tick. The PeriodicTable.Block type is also imported to label chemical packing offsets that later anchor mass ladders in the same framework.
proof idea
The declaration is a structure definition that directly introduces the two fields bits and unique; no lemmas or tactics are applied.
why it matters in Recognition Science
Syndrome supplies the data type used by the downstream toricCode definition, which states that qubits lie on torus edges and logical information resides in homology classes. It realizes the module's core claim that 8-tick redundancy produces error syndromes from phase measurements, directly supporting the patent note for novel QEC codes based on Recognition Science. The structure therefore bridges the T7 eight-tick foundation to concrete error-correction bounds.
scope and limits
- Does not prescribe the length of the bit list.
- Does not define any error-to-syndrome mapping function.
- Does not enforce linearity or minimum-distance properties of the underlying code.
- Does not connect the bits to a specific physical error model or noise channel.
formal statement (Lean)
100structure Syndrome where
101 /-- Syndrome bits -/
102 bits : List Bool
103 /-- Syndrome uniquely identifies error -/
104 unique : Bool := true
proof body
Definition body.
105
106/-! ## Classical Code Foundation -/
107
108/-- A classical linear code [n, k, d].
109 - n: Block length
110 - k: Message length
111 - d: Minimum distance (corrects ⌊(d-1)/2⌋ errors) -/