IndisputableMonolith.Information.Quantum_Error_Rate_RS
Defines the Recognition Science quantum error-correction threshold from the J-cost on a discrete domain, together with nonnegativity and positivity lemmas and a certificate type. Information theorists and QEC analysts working in RS units would cite the exact threshold identity. The module is mostly definitional, with short algebraic proofs of sign and evaluation facts.
claimOn a discrete recognition domain one defines a nonnegative domain cost $C$ built from the RS $J$-cost, a positive canonical QEC threshold $\theta_{\mathrm{can}}$, and an exact RS quantum-error-rate threshold identity relating the two; a certificate type packages the numerical claim.
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ forced by the Recognition Composition Law. The Cost import supplies that $J$-calculus; Constants supplies the RS tick $\tau_0=1$ used to normalize rates.
This Information-domain module lifts $J$ to a domain-level cost on discrete configurations relevant to quantum error correction. Sibling objects include the domain cost, its evaluation identity and nonnegativity, a canonical positive threshold, the exact RS QEC threshold statement, and an inhabited certificate type that packages the claim for downstream checking.
The setting is RS-native units ($c=1$, rates in ticks), not laboratory SI. The module does not re-derive $J$-uniqueness (T5) or the eight-tick octave (T7); it consumes them via Cost and Constants.
proof idea
Definition-heavy module. Domain cost is introduced as a $J$-based functional on the discrete domain; evaluation and nonnegativity are short algebraic or positivity wrappers off Cost. The canonical threshold is a positive constant expression; positivity is immediate from the closed form. The exact QEC threshold identity equates the RS error-rate bound to that threshold. The certificate is a structure inhabited by a trivial constructor once the identity is in hand.
why it matters in Recognition Science
Places a concrete quantum-error-rate threshold inside the RS information stack, tying QEC performance to the same $J$-cost that forces $\phi$, the eight-tick period, and $D=3$ in the T0–T8 chain. No downstream consumers are recorded yet in the mirror graph, so the module currently stands as a self-contained certificate source rather than a lemma feeder.
It gives RS a sharp, checkable number for the error-rate boundary instead of an external phenomenological fit. Open follow-ons would connect the threshold to the Berry creation scale $\phi^{-1}$ or to mass-ladder gap structure; those links are outside this file.
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law; imports Cost.
- Does not prove hardware-specific fault-tolerance thresholds for named codes (surface, color, etc.).
- Does not convert the RS-native threshold into SI error rates without external unit maps.
- Does not claim experimental QEC measurements; only the RS exact identity and certificate.
- Does not feed recorded downstream theorems in the current mirror graph.