pitchPerceptionFromPhiLadderCert
plain-language theorem explainer
The master certificate for pitch perception on the phi-ladder is inhabited by direct reference to eight pre-proved lemmas on the just-noticeable-difference function. Psychoacoustics researchers modeling auditory resolution via recognition lattices would cite this result to connect JND structure to the phi-self-similar lattice. The definition is a structure instantiation that bundles the positivity, strict decrease, geometric increment, and J-cost positivity properties of the JND sequence.
Claim. The pitch-perception-from-phi-ladder certificate is the structure asserting that the just-noticeable-difference function satisfies $1 < f(k)$ for all $k$, $f$ is strictly decreasing, the increment obeys $(f(k+1)-1)phi = f(k)-1$, the J-cost of each $f(k)$ is positive, and the rung-1 value equals $1 + phi^{-1}$ and lies in $(1.61,1.62)$.
background
In the Recognition Science framework, pitch perception is a recognition operation on the auditory-cortex phi-rung ladder. The just-noticeable-difference at rung $k$ is the smallest discriminable frequency ratio above unison, given by $f(k) = 1 + phi^{-k}$. Two pitches are distinct precisely when their J-cost separation exceeds the per-rung quantum $Jcost(f(k))$, with lower rungs corresponding to coarse perception and higher rungs to fine resolution in the sensitive band.
proof idea
This definition is a one-line wrapper that constructs the PitchPerceptionFromPhiLadderCert structure by supplying the eight sibling lemmas pitchJND_gt_one, pitchJND_pos, pitchJND_strict_anti, pitchJND_succ_lt, pitchJND_increment_geometric, pitchJND_jcost_pos, pitchJND_one, and pitchJND_one_band.
why it matters
This declaration inhabits the master certificate for Track L6, closing the derivation of auditory JND structure from the phi-ladder. It connects directly to J-uniqueness (T5) and the phi fixed point (T6) in the forcing chain, supplying a concrete falsifiable prediction for psychoacoustic data. The rung-1 value in (1.61,1.62) recovers the perfect-fifth ratio as the coarsest perceptual step.
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