IndisputableMonolith.Acoustics.MusicConsonanceFromJCost
Packages musical interval consonance as nonnegativity and threshold facts for the Recognition J-cost on frequency ratios. Acoustics and RS workers cite it for the certificate that pure intervals sit under a canonical cost bound. The module is mostly definitions (domain cost, threshold, MusicConsonanceCert) plus elementary positivity and equilibrium lemmas from Cost.
claimOn frequency ratios $r>0$, set domain cost $C(r)=J(r)$ with $J(x)=\frac{x+x^{-1}}{2}-1$. Record $C(1)=0$ and $C\ge 0$, fix a positive canonical threshold $\theta$, and package a MusicConsonanceCert that the consonance criterion is the $J$-cost bound relative to $\theta$.
background
Recognition Science forces a unique nonnegative cost on positive reals, $J(x)=\frac{x+x^{-1}}{2}-1$ (equivalently $\cosh(\log x)-1$), vanishing only at equilibrium $x=1$ (forcing step T5). Musical intervals are positive frequency ratios; unison is that equilibrium.
The Cost import supplies $J$ and its elementary calculus; Constants supplies RS-native units (including the tick $\tau_0$). This acoustics module specializes those objects to pitch ratios: a domain cost, a fixed positive threshold, and a small certificate type that packages the consonance claim for downstream use.
proof idea
Definition-and-lemma module, not a deep derivation. domainCost is $J$ on the ratio; equilibrium at $1$ and nonnegativity are one-line transfers from Cost. canonicalThreshold is a fixed positive constant with a positivity lemma. MusicConsonanceCert is a structure bundling the criterion; cert and cert_inhabited witness that the bundle is inhabited. No multi-step tactic argument beyond those elementary facts.
why it matters in Recognition Science
Puts music consonance inside the J-cost forcing chain instead of an empirical psychoacoustic table. Thematic link to T7 (eight-tick octave) via the octave ratio $2$, though the module itself is cost-threshold packaging, not a tick-counting argument. The graph shows no used_by edges yet; it is the acoustics-side interface for T5 on intervals, ready for later scale or tuning results to import the certificate.
scope and limits
- Does not construct scales, temperaments, or specific interval libraries.
- Does not model timbre, partials, or inharmonic spectra.
- Does not prove psychoacoustic JNDs or listening-test claims.
- Does not tie consonance thresholds to the mass ladder or alpha band.
- Does not derive the numerical value of the threshold from first principles here.