§7 · Band-specific hearing loss (class × place)

Band-specific hearing loss (class × place)

Characteristic audiogram shapes emerge by composing §3’s failure-class axis with §6’s place axis: because the place map is monotone, where a failure sits IS which frequency band is lost (an order-isomorphism). Every direction is forced; every dB, slope, and notch-Hz is the irreducible measured [O].

The inherited CF(x) is strictly monotone, so a contiguous band of failed places maps to a contiguous band of lost frequencies, round-tripping to |Δ|=<1×10⁻⁹: basal x∈[0.7,1.0] → 4736 Hz–20677 Hz (high-frequency presbycusis, load ratio 1042×), mid → 999 Hz–2864 Hz (cookie-bite), apical → 19.8 Hz–560.0 Hz (low-frequency reverse-slope).

The keystone: place→frequency is an order-isomorphism

Because CF(x) is strictly monotone, where a failure sits is which band is lost. A basal band x∈[0.7,1.0] maps to 4736 Hz–20677 Hz, mid to 999 Hz–2864 Hz, apical to 19.8 Hz–560.0 Hz, each round-tripping to |Δ|=<1×10⁻⁹ [V].

No constant is tuned — this is the monotonicity of the inherited map turned into an isomorphism. WHERE fails ⇒ WHICH band is lost.

The four shapes are images of where

Each audiogram is the image of a place band. Basal degeneration → high-frequency down-slope: the base cycles fastest (CF is the rate), so cumulative load is basal-first, load(base)/load(apex) = 1042× — classic presbycusis [F].

An over-drive at the outer/middle-ear transfer peak → a notch below the top (the ~3–6 kHz C5-dip); an apical ion-regime failure → a low-frequency reverse-slope (WFS1 archetype); a mid-cochlear locus → cookie-bite — the weakest, since only mid→mid is forced and the locus concentration is cited [L].

The lever, and the earlier negatives carry

Each audiogram is a (class × band) hypothesis, and the lever is the §3 direction applied at the §6 band, proposal-only [F]. §3’s honest negative carries: the structural component of a high-frequency loss admits no drive rescue (the window 2·spinodal(g) → 0 as g→0).

Only the apical drive-class is switch-recoverable in principle — and even then only the direction is named. No molecule, dose, in-vivo selectivity, diagnosis, or efficacy.

Shape forced, magnitudes the obstacle

The place-isomorphism and the four shape directions are parameter-free; every dB threshold, dB/oct slope, notch frequency, and age of onset is the irreducible measured [O] [F].

A closed numeric audiogram = a choice of those magnitudes = tuning (forbidden). The disease layer states directions, never doses — and keeps §3’s honest negatives, refusing to promise a drive rescue for a structural high-frequency loss.

Honest negatives — what is not claimed

  1. N1. Every magnitude is [O] — the dB thresholds, the dB/oct slopes, the notch frequency in Hz, the age of onset, the load rate, the absolute band edges. Only the shape and the directions are forced.
  2. N2. Cookie-bite is the weakest of the four — the isomorphism forces only mid PLACE → mid FREQUENCY; the mid-concentration of the locus is a cited input, not derived.
  3. N3. The notch frequency (~3–6 kHz) is [O] — it needs the ear-canal length and ossicular transfer; only the existence of an interior transfer peak (a notch below the top) is forced.
  4. N4. Presbycusis is multi-factorial (sensory, strial, neural); §7 forces only the basal-first direction of the cyclic-load component.
  5. N5. The cumulative-load argument forces the ordering (basal-first) but assumes load accrues with cycling rate — a cited biophysical premise, not derived here; the rate and accrual law are [O].
  6. N6. Real audiograms are individual and noisy; §7 is a shape-DIRECTION claim about each etiology’s population tendency, not a per-patient predictor.
  7. N7. The readout class (OTOF / synapse) has no place-band shape in this transduction map — its failure is downstream of the flip ([O]); the felt experience of band-specific loss is the mind volume’s.