Acoustics Speech Intelligibility From Jcost Sr Cost Reciprocal Symm

The symmetry of recognition cost

Speech intelligibility depends on the signal-to-noise ratio (SNR), the power of a speech signal divided by the power of background noise. When the signal and noise have equal power, the ratio is 1, and recognition is easiest. As the ratio moves away from 1 in either direction, recognition becomes harder. The declaration srCost_reciprocal_symm establishes a precise symmetry in this difficulty: the cost of recognizing speech at a signal-to-noise ratio r is exactly the same as the cost at the reciprocal ratio 1/r. For example, a signal twice as strong as noise (r = 2) carries the same recognition cost as a signal half as strong as noise (r = 1/2).

This symmetry is not an assumption but a proved consequence of the framework's core cost function. In Recognition Science, the cost of recognition is forced by five plain conditions, and the resulting function is J(x) = (x + 1/x)/2 - 1. The reciprocal symmetry follows directly: J(2) = J(1/2) = 0.25, and the same holds for every positive ratio. The declaration states this as a theorem, proved in the framework's machine-checked library of formal theorems, with no unproved assumptions. It also carries a companion result: the cost is zero exactly when the ratio is 1, and positive for every other positive ratio.

The symmetry has a practical consequence for hearing. The speech-reception threshold (SRT) is the SNR at which a listener recognizes 50 percent of words; healthy adults sit near -7 dB, where the ratio r is about 0.2. Hearing impairment shifts the threshold by +5 to +15 dB, which the framework models as one to three steps of a penalty ladder based on the golden ratio. Because of the reciprocal symmetry, the penalty for a degraded SNR is the same whether the loss is expressed as a negative decibel shift or as the corresponding positive ratio. The framework's library proves the penalty is zero at the starting rung and nonnegative at every rung.

What the declaration does not claim is equally important. It does not say that human speech recognition actually follows this cost function in every listener; that is a modeling choice, not a proved fact. It does not assert that the reciprocal symmetry holds at r = 0, where the ratio is undefined, or for negative ratios, which carry no physical meaning as power ratios. It does not claim that the framework derives the fine-structure constant or any other coupling from this acoustic result. The declaration is a theorem about a defined cost function, not an empirical claim about hearing.

theorem srCost_reciprocal_symm {r : ℝ} (hr : 0 < r) :
    srCost r = srCost r⁻¹ := Cost.Jcost_symm hr

theorem srCost_zero_at_threshold : srCost 1 = 0 := Cost.Jcost_unit0

theorem srCost_pos_off_threshold {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < srCost r := Cost.Jcost_pos_of_ne_one r hr hne

/-- The hearing-loss penalty at zero rungs is zero. -/
theorem hearingLossPenalty_zero : hearingLossPenalty 0 = 0 := by
  unfold hearingLossPenalty
  simp
  exact Cost.Jcost_unit0

/-- The penalty is nonnegative at every rung. -/
theorem hearingLossPenalty_nonneg (k : ℕ) : 0 ≤ hearingLossPenalty k := by
  unfold hearingLossPenalty
  apply Cost.Jcost_nonneg
  exact zpow_pos Constants.phi_pos _

What this page does not claim

This does not claim that human speech recognition empirically follows this cost function in every listener. This does not claim the symmetry holds at r = 0 or for negative ratios. This does not claim the framework derives the fine-structure constant from this acoustic result.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Acoustics/SpeechIntelligibilityFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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