rs_complexity_classes
Recognition Science complexity classes are summarized as a list of six descriptive entries covering ground-state verification, local dynamics, balance checks, J-cost optimization, phi-rung evaluation, and global configuration search. A physicist classifying the computational cost of RS-derived physics models would cite this definition to locate the framework within the complexity zoo. The definition is assembled directly from the convexity of J-cost and the eight-tick local update rule without additional lemmas.
claimThe RS complexity classes are the list: ground state verification at $x=1$ requires $O(1)$ time; local dynamics update in $O(1)$ per tick over an eight-tick period; balance verification requires a linear scan in system size; J-cost minimization admits polynomial-time gradient descent; rung computation on the phi-ladder requires exponential time; global RS state search is NP-hard.
background
In the module on computational complexity of physics from Recognition Science, the J-cost function is defined as $J(x) = (x + x^{-1})/2 - 1$, which is strictly convex with unique minimum at $x=1$. The fundamental time unit is the tick, with one octave comprising eight ticks. Upstream results establish the cost of recognition events as this J-cost and define rung levels for mass hierarchies in the phi-ladder.
proof idea
This definition is a direct list construction that enumerates complexity classes derived from the convexity of J-cost minimization and the structure of eight-tick local updates. No lemmas are applied; the entries follow immediately from the module summary of J-cost properties and phi-hierarchy growth.
why it matters in Recognition Science
This definition supplies the summary certificate for the complexity analysis in Recognition Science, linking to the eight-tick octave and the phi-ladder for mass rungs. It supports the claim that physics complexity is governed by J-cost minimization, feeding into broader questions on placement within the complexity zoo. It touches the open question of formalizing the NP-hard global optimization within the RS ledger.
scope and limits
- Does not establish rigorous big-O bounds or reductions to standard complexity classes.
- Does not prove convexity or uniqueness of the J-cost minimum.
- Does not address quantum complexity classes such as BQP.
- Does not provide algorithms for the listed classes.
formal statement (Lean)
219def rs_complexity_classes : List String := [
proof body
Definition body.
220 "Ground state (x=1): unique, 0 cost, O(1) to verify",
221 "Local dynamics: 8-tick update, O(1) per tick",
222 "Balance verification: O(N) linear scan",
223 "J-cost minimization: convex, polynomial gradient descent",
224 "φ-rung computation: EXPTIME (φⁿ grows without bound)",
225 "Global RS configuration: NP-hard analog (exponentially many states)"
226]
227
228/-! ## Summary Certificate -/
229