IndisputableMonolith.Acoustics.Musical_Note_A4_Exact_RS
Exact Recognition Science treatment of concert pitch A4: a domain cost on frequency ratios, a canonical positive threshold, and a certified exact A4 value in RS-native units. Acoustics and constants workers cite it when locking musical pitch to the phi-ladder and J-cost. The module is mostly definitions plus nonnegativity and inhabitance certificates built from Constants and Cost.
claimIn RS-native units, fix a domain cost $C$ on frequency ratios (nonnegative, with an evaluation identity at the identity ratio), a canonical threshold $\theta>0$, and an exact A4 object together with an inhabited certificate that the A4 datum meets the cost/threshold specification.
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. Constants supplies the fundamental tick $\tau_0=1$ in RS-native time; frequencies are inverse durations on that tick lattice, and the golden ratio $\phi$ organizes self-similar rungs.
This acoustics module specializes that cost language to musical pitch. A domain cost scores how far a frequency ratio sits from the identity; the canonical threshold is the positive cutoff used to accept an exact pitch lock. A4 (concert A) is treated as an exact RS datum rather than an empirical 440 Hz convention, so downstream claims can quote a certified native value instead of a floating laboratory number.
Imports are deliberately thin: Mathlib for real arithmetic, Constants for $\tau_0$ and related RS units, and Cost for the underlying nonnegative cost infrastructure.
proof idea
Definition-heavy module, not a single deep theorem. It introduces the domain cost and proves an evaluation identity at the neutral ratio plus nonnegativity. The canonical threshold is defined and shown positive. The exact A4 object is packaged with a certificate type; inhabitance of that certificate is recorded so later developments can assume a witnessed exact pitch without re-deriving the cost bounds. Arguments are short algebraic or positivity appeals on top of Cost and Constants, not a long forcing chain.
why it matters in Recognition Science
Places concert pitch inside the same cost-and-threshold discipline used elsewhere in the monolith, so acoustics sits on RS-native units ($c=1$, tick $\tau_0$, $\phi$-rungs) rather than SI retrofit. No downstream edges are recorded yet; the natural consumers are further acoustics certificates, octave/eight-tick comparisons (T7), and any mass-or-frequency ladder statements that need a fixed audible reference rung. Sibling names (domain cost, threshold positivity, A4 exactness, certificate inhabitance) mark the closure surface for those extensions. It does not itself advance T5–T8 forcing; it applies the cost layer those steps justify.
scope and limits
- Does not derive A4 from the T0–T8 forcing chain or prove uniqueness of pitch standards.
- Does not equate the RS exact A4 to SI 440 Hz without an explicit unit bridge.
- Does not prove psychoacoustic or equal-temperament theorems beyond the cost/threshold package.
- Does not supply downstream consumers; used_by is currently empty.
- Does not redefine J-cost; it specializes Cost infrastructure to frequency ratios.