IndisputableMonolith.Acoustics.RS_ACS_Cert_001
Certificate module packaging the first RS acoustics claim: a nonnegative domain cost on the J-cost scale together with a positive canonical threshold. Acoustics and metrology workers cite it when they need a Lean-checked witness that an acoustic observable sits above the RS recognition floor. The module is mostly definitions plus elementary nonnegativity and positivity lemmas imported from Cost and Constants.
claimDefine a domain cost $C$ on the RS $J$-cost scale, prove $C \ge 0$ and $C(1)=0$ at equilibrium, fix a canonical threshold $\theta > 0$, and package the inhabited certificate $\mathrm{RS\text{-}ACS\text{-}Cert\text{-}001}$ asserting that the acoustic observable meets the recognition threshold.
background
Recognition Science measures mismatch by the unique cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), forced by the Recognition Composition Law and the T5 uniqueness step of the forcing chain. The Cost import supplies that $J$ and its elementary calculus; Constants supplies the RS-native tick $\tau_0 = 1$.
Acoustics sits downstream of the eight-tick octave (T7) and the three-dimensional spatial forcing (T8). An acoustic certificate therefore needs a domain-specific cost pulled back to the $J$-scale, a proof that the cost vanishes only at equilibrium, and a positive numerical threshold against which the observable is compared.
This module introduces exactly those pieces: domainCost, its equilibrium and nonnegativity facts, canonicalThreshold with positivity, and the bundled certificate record RSACSCert001.
proof idea
Definition module with thin lemmas, not a deep derivation. domainCost is introduced as a pullback of the global $J$-cost to the acoustic domain; domainCost_at_eq and domainCost_nonneg are one-line appeals to the corresponding Cost facts. canonicalThreshold is a positive constant (positivity discharged by canonicalThreshold_pos). The record RSACSCert001 packages cost, threshold, and the comparison; cert and cert_inhabited simply inhabit that record. No multi-step tactic proof is required beyond nonnegativity and positivity.
why it matters in Recognition Science
Gives the first named, machine-checked acoustics certificate in the RS mirror (ACS-001). Downstream acoustics or metrology developments that need a Lean witness that an observable clears the recognition floor can import this inhabited cert rather than re-proving nonnegativity of $J$ and positivity of the threshold. It sits on the Cost and Constants foundation and does not yet feed a larger parent theorem in the current graph (used_by is empty), so it functions as a leaf certificate that later acoustic forcing or experimental-bound modules can cite. Framework landmarks touched: T5 $J$-uniqueness, the RCL, and the RS-native units carried by Constants.
scope and limits
- Does not derive a physical acoustic wave equation or dispersion relation.
- Does not prove uniqueness of the canonical threshold, only positivity.
- Does not connect the certificate to laboratory decibel or frequency data.
- Does not discharge any open forcing-chain step beyond Cost/Constants imports.
- Does not claim the certificate is used by a parent theorem in the current graph.