§2 · The cochlear amplifier: cube-root compression

The cochlear amplifier: cube-root compression

The outer-hair-cell active amplifier (prestin / SLC26A5) emerges as the inherited R19 cubic at its critical point: the response compresses as the parameter-free cube root r ∝ F^(1/3), and the 1/3 is forced by the cubic, not fitted. Detection and amplification are one cubic in two regimes.

At criticality g=0 the inherited cubic g·s−s³+h has its zero at s=F^(1/3); the integrator’s fitted exponent is 0.333333 and the gain falls as F^(−0.666667) — the ~120 dB compression. The cube root is the cubic’s critical fixed point, no constant chosen.

One cubic, two regimes

The §1 detection switch and the §2 amplifier are the same equation. With g=γ_TMC1 the response is bistable (the §1 switch); with g=0 the same drives give the continuous cube root s = h^(1/3) [F].

Bistable detection runs in parallel with critical amplification — one substrate, two operating regimes. The amplifier is not a new mechanism; it is the cubic taken to its critical point.

Compression is the cube root — forced, not fitted

The 1/3 exponent is the inherited cubic’s critical fixed point. Evaluating the inherited sdot at s=F^(1/3), g=0 gives max|residual| <1×10⁻⁹ over seven decades of drive [F].

The integrator’s read-off exponent is 0.333333 and the gain exponent −0.666667 [V] — faint drives amplified ~100× more than loud, the dynamic-range compression that lets the ear span ~120 dB.

Uniform across the bank; magnitude is the obstacle

Every place carries the same critical cubic, so the 1/3 exponent is CF-independent — compression is uniform across frequency [F]. SLC26A5 (γ 1.4023, spinodal 0.6392) takes its place in the emergence lineage.

The absolute gain, the dB of amplification, and the sharpness Q are [O] — a number would require tuning a constant. Only the exponent (1/3) and the direction (compression) are forced.

Honest negatives — what is not claimed

  1. N1. The absolute gain / dB / dynamic-range-in-dB / sharpness Q are [O] — a value would require tuning. Only the exponent and the direction are forced.
  2. N2. The cube root matches the FORM of measured cochlear compression; the study-dependent I/O slope (~0.2–0.5 dB/dB) and the absolute curve are [O].
  3. N3. Otoacoustic-emission frequencies and amplitudes are [O] — they need the per-place gain, Q, and an operating point just above the bifurcation.
  4. N4. Prestin’s electromotile force is NOT read from γ (firewall) — only the gene’s spinodal-order place and the compression exponent are forced.
  5. N5. The inherited integrator under-converges to the fixed point at very small drive; the rigorous [F] is the analytic fixed point, the [V] is on the converged range.