quantumSpeedups
Quantum speedups lists three canonical examples of quantum algorithms that outperform classical counterparts: Shor's for factoring, Grover's for search, and quantum simulation. Researchers working on physical computation models would cite this when connecting RS ledger structure to quantum advantage. The definition is a direct enumeration of strings together with a comment that attributes the advantage to parallel 8-tick paths.
claimRecognition Science enumerates quantum computational advantages as the list of strings: factoring $N$ in $O( (log N)^3 )$ time, search in $O( sqrt N )$ time, and efficient simulation of quantum systems. These illustrate how the 8-tick ledger structure permits superposition through simultaneous exploration of multiple computation branches.
background
The module derives the Church-Turing thesis from ledger universality: any effectively computable function reduces to a sequence of ledger updates, and the 8-tick structure supplies a universal gate set. Upstream, tick is defined as the fundamental RS time quantum with one octave equal to eight ticks. The Measurement structure records name, value, and error for hardcoded data. PrimitiveDistinction and the integers-from-logic constructions supply the underlying distinction and counting primitives used throughout the ledger model.
proof idea
The definition directly constructs a static list of three strings. It is accompanied by an attached comment block that states quantum parallelism arises from 8-tick superposition, permitting multiple phases, path interference, and measurement collapse to a single outcome. No lemmas or tactics are invoked; the body is a literal enumeration.
why it matters in Recognition Science
This definition supports the module claim that the Church-Turing thesis follows from ledger universality, as referenced in the paper proposition on the physical basis of universal computation. It ties directly to the eight-tick octave (T7) in the forcing chain, where the period-8 structure provides the universal gate set. No downstream theorems reference it, leaving open its integration into formal universality proofs.
scope and limits
- Does not derive the listed time complexities from RS axioms.
- Does not prove realizability of superposition inside the physical ledger.
- Does not connect the speedups to specific RS constants such as phi or J-cost.
- Does not address uncomputable functions or halting problems.
formal statement (Lean)
202def quantumSpeedups : List String := [
proof body
Definition body.
203 "Shor's algorithm: Factor N in O((log N)³)",
204 "Grover's algorithm: Search in O(√N)",
205 "Quantum simulation: Efficient for quantum systems"
206]
207
208/-- **THEOREM**: Quantum parallelism from 8-tick superposition.
209
210 The 8-tick structure allows:
211 - Multiple phases simultaneously
212 - Interference between paths
213 - Measurement collapses to one outcome -/