summary
This definition assembles a five-item list that captures the central claims of error correction bounds derived from the eight-tick phase structure. Information theorists and quantum information researchers would cite the list when summarizing how Recognition Science supplies natural redundancy mechanisms. The definition is realized by direct enumeration of the five strings.
claimThe summary is the list containing the statements ``Channel capacity limits reliable transmission'', ``8-tick phases provide natural redundancy'', ``Majority voting corrects 3/8 errors'', ``8-tick syndromes for error detection'', and ``QEC from 8-tick phase coherence''.
background
The module derives error correction bounds from the eight-tick structure. Upstream, tick denotes the fundamental RS time quantum with value 1, while phase(k) for k in Fin 8 returns kπ/4, giving the eight discrete phases that repeat every 2π. The module doc states that Shannon's channel capacity theorem supplies the maximum reliable transmission rate, and that eight-tick phases furnish natural redundancy by encoding each bit across the phases with error correction arising from phase correlations.
proof idea
The definition is a direct construction that enumerates the five strings in order.
why it matters in Recognition Science
This definition supplies the high-level summary for the INFO-005 module on error correction bounds from the eight-tick octave (T7). It connects the channel-capacity correction factor from QuantumChannelCapacityFromPhi to the concrete redundancy claims that later siblings such as hamming_bound_8tick and eight_tick_corrects_3 will quantify. The list therefore serves as the entry point for any derivation that begins from the eight-tick phase coherence.
scope and limits
- Does not derive numerical values for channel capacity or correction factors.
- Does not contain proofs of the five listed statements.
- Does not specify the functional form of the Hamming or singleton bounds.
- Does not address continuous-time limits or non-eight-tick encodings.
formal statement (Lean)
217def summary : List String := [
proof body
Definition body.
218 "Channel capacity limits reliable transmission",
219 "8-tick phases provide natural redundancy",
220 "Majority voting corrects 3/8 errors",
221 "8-tick syndromes for error detection",
222 "QEC from 8-tick phase coherence"
223]
224
225/-! ## Falsification Criteria -/
226
227/-- The derivation would be falsified if:
228 1. Error correction exceeds Shannon limit
229 2. 8-tick doesn't support redundancy
230 3. Majority voting doesn't work in quantum regime -/