IndisputableMonolith.Acoustics.RS_ACS_Cert_003
Certificate module for the third RS acoustics claim: a nonnegative domain cost built from the RS J-cost, a positive canonical threshold, and an inhabited certificate record packaging those facts. Acoustics and RS-constants workers cite it when wiring cost bounds into certified acoustic statements. The module is mostly definitions plus elementary nonnegativity and positivity lemmas over the imported Cost and Constants layers.
claimDefine a domain cost $C$ on the acoustics side from the RS cost functional $J$, prove $C \ge 0$ and an evaluation identity, fix a canonical threshold $\theta > 0$, and package these into an inhabited certificate record $\mathrm{RSACSCert003}$.
background
Recognition Science measures mismatch with the J-cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), developed in the Cost import. Constants supplies the RS-native tick $\tau_0 = 1$ and related units. This acoustics module sits in that native unit system and does not re-derive $J$; it specializes cost bookkeeping to an acoustic domain quantity.
Sibling definitions introduce domainCost (the specialized cost), equality and nonnegativity facts for it, and a canonicalThreshold required to be strictly positive. The certificate record RSACSCert003 bundles those ingredients so downstream acoustic claims can assume a single inhabited cert rather than re-proving cost signs and threshold positivity.
proof idea
Definition-heavy module, not a deep theorem chain. Domain cost is defined from the imported Cost layer; nonnegativity and the pointwise evaluation identity are short lemmas. Canonical threshold is a positive constant definition with a positivity proof. The certificate structure and cert_inhabited assemble those pieces into a single inhabited record. No multi-step forcing or analytic argument appears at module scope.
why it matters in Recognition Science
Gives Acoustics a named, reusable cost-and-threshold certificate (RS_ACS_Cert_003) aligned with RS-native Cost and Constants. Downstream used-by edges are empty in the graph snapshot, so the module is a leaf cert package rather than a parent of further proved theorems here. It supports the broader program of turning RS cost inequalities into domain-specific certified claims (acoustics band) without reopening J-uniqueness or the T5–T8 forcing chain. Cite it when an acoustic bound needs a packaged nonnegative domain cost and a fixed positive threshold.
scope and limits
- Does not derive J-uniqueness or the Recognition Composition Law; those stay in Foundation/Cost.
- Does not prove physical acoustic wave equations or experimental spectra.
- Does not fix numerical SI values for c, hbar, or G beyond RS-native imports.
- Does not claim downstream theorems; used_by is empty in the supplied graph.
- Does not address other ACS certificate indices beyond Cert_003.