§1 · The place map and the tip-link MET switch

The place map and the tip-link MET switch

The hair-cell mechanotransduction switch (TMC1 / PCDH15 / CDH23) emerges on the inherited √-law place map: each gene is read as γ (LEVEL) plus its A4 SHAPE, the R19 switch flips all-or-none, and the traveling-wave peak place is parameter-free. The full dispersive envelope stays the named [O].

Pitch is coded by PLACE: ω = √(S/m) on a log-graded stiffness gives an exponential map reproducing Greenwood to machine precision. The MET genes order by R19 spinodal 2·(γ/3)^1.5 — TMC1 0.5724 < PCDH15 0.6233 < CDH23 0.7336 — a lower discontinuous threshold flipping earlier.

R19 bistable switchṡ = g·s − s³ + h

The universal cubic substrate: bistable detection, and at criticality (g=0) cube-root amplification.

Greenwood place mapCF(x) = A(10^{a·x} − k)

Pitch by place: a √-law resonance on log-graded stiffness gives an exponential, strictly monotone tonotopic map.

γ (promoter stiffness): LEVEL and A4 SHAPEγ = −mean(ΔG37); A4 = signal − γ

Each master gene is read as a stiffness LEVEL (γ) and an orthogonal SHAPE (A4) — promoter structure only, never a function or a dose.

Pitch is place, not assumed

The cochlea reads a longitudinal sound wave and turns it into a spatial code. The resonance ω = √(S/m) with log-graded stiffness forces an exponential place map; √S ∝ 10^(a·x) reproduces Greenwood’s term to max|ratio−1| ≈ 2×10⁻¹⁶ [V].

The map is inherited physics, not a fit. Its constants A/a/k are measured calibration [L]; the exponential SHAPE is forced by the √-law on a logarithmically graded membrane.

The MET switch emerges by spinodal order

The tip-link switch flips all-or-none, not gradually. Emergence order = argsort(spinodal(γ)) with the A4 SHAPE breaking γ-ties: TMC1 0.5724 < PCDH15 0.6233 < CDH23 0.7336 < TMIE 0.7468 [F].

γ is measured [L]; a lower discontinuous threshold means earlier switch competence. The switch is OFF up to 0.99·spinodal and flips ON discontinuously past it — no graded leak across the barrier.

The peak place is forced; the envelope is the obstacle

Only the traveling-wave PEAK place is parameter-free. Inverse-Greenwood round-trips: 250 Hz → x* 0.18, 1 kHz → 0.40, 4 kHz → 0.67, 16 kHz → 0.95, to <1×10⁻¹² Hz [V].

The full fluid-loaded dispersive ENVELOPE — width, cutoff slope, phase, delay — is the named [O]. A closed envelope would require tuning a sharpness Q (forbidden); it is characterised, never tuned, in §5.

Honest negatives — what is not claimed

N1. The A4 tie-break never fires on this atlas (all MET γ are distinct); it is proven non-vacuous on constructed equal-γ inputs and kept because such genes can occur in general.
N2. argsort(spinodal(γ)) is a structural [F] order; its concordance with the measured developmental sequence is [O] — no timing dataset is bundled.
N3. Only the peak place is forced; the full dispersive envelope is [O] (a closed envelope needs a tuned Q plus the fluid mass-loading — the named obstacle this seed exists to take up).
N4. No per-gene tonotopic place is claimed — the MET genes span the whole partition; assigning one would be invention.