Frequency-Scaled Weights

@cite{coetzee-pater-2008} @cite{coetzee-kawahara-2013}

The "frequency lives in the grammar, continuously" theory: constraint weights themselves vary as a linear function of an item's log-frequency. Unlike IndexedConstraints (discontinuous, two strata), scaled-weights produce a gradient response to frequency within a single grammar.

The schema:

effectiveWeight(c, item) = baseWeight(c) + slope(c) · tokenLogFreq(item)

Different alternations / constraints can have different slopes: positive slope means "high-frequency items are more strongly subject to the constraint" (the empirical pattern in @cite{coetzee-kawahara-2013} for Japanese geminate devoicing); negative slope means the opposite (high-frequency items more lenient — closer to representation-strength predictions).

Empirical signature

Scaled-weights theories predict a continuous monotonic dependence of violation force on log-frequency. Where the empirical distribution shows two distinct distributions and no gradience, IndexedConstraints under-fits less. Where the distribution is continuous and monotonic but always in the same direction across constraints sharing the same target, scaled-weights is the better fit. Where direction flips across constraints (e.g., compound-frequency raises nasalisation but N2-frequency lowers it, as in Phenomena/Phonology/Studies/BreissKatsudaKawahara2026.lean), this theory needs separate slopes per constraint — which it has, and so remains a viable account.

def Phonology.ItemSpecificity.Scaled.scaledWeight {α : Type} [HasTokenFreq α] (baseWeight slope : ℝ) (a : α) : ℝ

The effective weight of a constraint at a given item: a linear function of the item's log-frequency.

Equations

• Phonology.ItemSpecificity.Scaled.scaledWeight baseWeight slope a = baseWeight + slope * Phonology.ItemSpecificity.tokenLogFreq a

structure Phonology.ItemSpecificity.Scaled.ScaledConstraint (α : Type) [HasTokenFreq α] : Type

Build a frequency-scaled NamedConstraint. The eval field returns a Nat, so we cannot directly store a scaled ℝ weight there. Instead, this constructor returns a weighted constraint representation — see Theories/Phonology/OptimalityTheory/Constraints.lean for the weighted-constraint type. The implementation here is the abstract scaled-weight rule; the OT-side wiring is supplied by the consumer.

• base : Core.Constraint.OT.NamedConstraint α
• baseWeight : ℝ
• slope : ℝ

def Phonology.ItemSpecificity.Scaled.ScaledConstraint.harmony {α : Type} [HasTokenFreq α] (s : ScaledConstraint α) (a : α) : ℝ

The harmony contribution of a scaled constraint at a candidate: scaledWeight × violations. The harmony is the negative of the cost; consumers minimising cost equivalently maximise harmony.

Equations

• s.harmony a = Phonology.ItemSpecificity.Scaled.scaledWeight s.baseWeight s.slope a * ↑(s.base.eval a)

theorem Phonology.ItemSpecificity.Scaled.scaledWeight_monotone_in_freq {α : Type} [HasTokenFreq α] (baseWeight slope : ℝ) (h : 0 ≤ slope) (a b : α) (hab : tokenLogFreq a ≤ tokenLogFreq b) : scaledWeight baseWeight slope a ≤ scaledWeight baseWeight slope b

Linear monotonicity: when the slope is positive, the scaled weight is monotonically non-decreasing in token log-frequency. The contrast with indexed constraints' constant-within-stratum is direct.