IndisputableMonolith.Acoustics.RS_ACS_Cert_006
Acoustics certificate 006 packages a nonnegative domain cost built from the RS J-cost together with a strictly positive canonical threshold. It exposes an inhabited certificate record so downstream acoustic claims can cite a single named gate rather than reassemble cost and threshold lemmas. The module is mostly definitions plus elementary positivity and evaluation identities.
claimDefine a domain cost $C$ on the acoustics side from the RS cost $J$, prove $C \ge 0$ and the pointwise evaluation identity for $C$, fix a canonical threshold $\theta > 0$, and package $(C,\theta)$ into an inhabited certificate record $\mathrm{RSACSCert006}$.
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the Recognition Composition Law. The Cost import supplies that functional; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the Acoustics domain. It specializes the global cost to a domain cost $C$ appropriate for acoustic observables, records that $C$ is nonnegative, and introduces a canonical positive threshold $\theta$ against which acoustic recognition events are gated. The certificate record is the stable interface other acoustic lemmas are meant to consume.
proof idea
Definition-first module. Domain cost is introduced as a thin wrapper around the imported J-cost; nonnegativity and the evaluation identity are short algebraic consequences of Cost lemmas. The canonical threshold is a positive constant definition; positivity is immediate. The certificate structure and its inhabited instance assemble those pieces with no deep proof work.
why it matters in Recognition Science
Gives Acoustics a named, reusable certificate (RSACSCert006) so later acoustic forcing or measurement claims share one cost-plus-threshold gate instead of ad-hoc constants. No downstream edges are recorded yet in the mirror graph; the module is an interface layer between Cost/Constants and future acoustic theorems. It does not itself advance the T0–T8 forcing chain, but keeps acoustic applications aligned with the same J-cost and RS-native units used elsewhere.
scope and limits
- Does not derive acoustic wave equations or dispersion relations.
- Does not prove uniqueness of the chosen threshold beyond positivity.
- Does not connect the certificate to measured lab acoustics data.
- Does not advance T5–T8 forcing; it only specializes Cost.
- Does not assert downstream consumers; the mirror lists none yet.