@cite{beltrama-schwarz-2024}: Empirical Data

@cite{beltrama-schwarz-2024}

Experimental data from @cite{beltrama-schwarz-2024}, showing that speaker persona stereotypes affect imprecision resolution for round numerals.

Experiments

1. Experiment 1 (Covered Screen Task): Participants choose which of two phone screens (COVERED vs VISIBLE) a speaker was looking at when making a round-numeral statement. Both Nerdy and Chill personae significantly affect choice rates.

2. Experiment 2 (Truth-Value Judgment Task): Participants judge whether a speaker's round-numeral statement is RIGHT or WRONG given a visible phone screen. Only the Chill persona significantly affects judgments.

Key empirical generalization

Speaker persona stereotypes modulate imprecision resolution bi-directionally in inference tasks (Exp 1) but asymmetrically in judgment tasks (Exp 2): Chill speakers increase tolerance for imprecision in both tasks, but Nerdy speakers only increase exactness demands in the inference task.

inductive BeltramaSchwarz2024.PersonaCondition : Type

Speaker persona condition (between-subjects). Participants read a short vignette describing the speaker (§4.1).

• nerdy : PersonaCondition
• chill : PersonaCondition
• noPersona : PersonaCondition

@[implicit_reducible]

instance BeltramaSchwarz2024.instDecidableEqPersonaCondition : DecidableEq PersonaCondition

Equations

• BeltramaSchwarz2024.instDecidableEqPersonaCondition x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance BeltramaSchwarz2024.instReprPersonaCondition : Repr PersonaCondition

Equations

• BeltramaSchwarz2024.instReprPersonaCondition = { reprPrec := BeltramaSchwarz2024.instReprPersonaCondition.repr }

def BeltramaSchwarz2024.instReprPersonaCondition.repr : PersonaCondition → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

inductive BeltramaSchwarz2024.ScreenFit : Type

Screen fit condition (within-subjects, §4.1). Determines the relationship between the speaker's round-numeral statement and the amount shown on the VISIBLE phone screen. In regression models, Match and Mismatch are collapsed into a binary "Control" factor, with Imprecise as the critical level (§4.5, §5.3).

• match_ : ScreenFit
• mismatch : ScreenFit
• imprecise : ScreenFit

@[implicit_reducible]

instance BeltramaSchwarz2024.instDecidableEqScreenFit : DecidableEq ScreenFit

Equations

• BeltramaSchwarz2024.instDecidableEqScreenFit x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[implicit_reducible]

instance BeltramaSchwarz2024.instReprScreenFit : Repr ScreenFit

Equations

• BeltramaSchwarz2024.instReprScreenFit = { reprPrec := BeltramaSchwarz2024.instReprScreenFit.repr }

def BeltramaSchwarz2024.instReprScreenFit.repr : ScreenFit → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

inductive BeltramaSchwarz2024.TaskType : Type

Experimental task type.

• coveredScreen : TaskType
• truthValueJudgment : TaskType

@[implicit_reducible]

instance BeltramaSchwarz2024.instDecidableEqTaskType : DecidableEq TaskType

Equations

• BeltramaSchwarz2024.instDecidableEqTaskType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def BeltramaSchwarz2024.instReprTaskType.repr : TaskType → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance BeltramaSchwarz2024.instReprTaskType : Repr TaskType

Equations

• BeltramaSchwarz2024.instReprTaskType = { reprPrec := BeltramaSchwarz2024.instReprTaskType.repr }

def BeltramaSchwarz2024.personaDescriptors : PersonaCondition → List String

The trait descriptors used for each persona condition (§4.1). Context sentences described the dialogue characters using these traits, e.g. "Arthur and Rachel, who have been described as Nerdy people, are looking for a plane ticket" (Figure 2).

Equations

• BeltramaSchwarz2024.personaDescriptors BeltramaSchwarz2024.PersonaCondition.nerdy = ["studious", "articulate", "introverted", "uptight"]
• BeltramaSchwarz2024.personaDescriptors BeltramaSchwarz2024.PersonaCondition.chill = ["laid-back", "sociable", "extroverted", "care-free"]
• BeltramaSchwarz2024.personaDescriptors BeltramaSchwarz2024.PersonaCondition.noPersona = []

structure BeltramaSchwarz2024.RegressionCoefficient : Type

A coefficient from a mixed-effects logistic regression model. Models fitted via lmer_alt from the afex package in R, which wraps lme4 (@cite{bates-etal-2015}; see footnote 6 of paper).

• predictor : String
• beta : ℚ

  Coefficient estimate (log-odds)

• se : ℚ

  Standard error

• significant : Bool

  Significant at p < 0.05?

@[implicit_reducible]

instance BeltramaSchwarz2024.instReprRegressionCoefficient : Repr RegressionCoefficient

Equations

• BeltramaSchwarz2024.instReprRegressionCoefficient = { reprPrec := BeltramaSchwarz2024.instReprRegressionCoefficient.repr }

def BeltramaSchwarz2024.instReprRegressionCoefficient.repr : RegressionCoefficient → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

def BeltramaSchwarz2024.exp1_n : ℕ

Experiment 1 participants (§4.4). 282 self-declared native English speakers recruited on Prolific.

Equations

• BeltramaSchwarz2024.exp1_n = 282

def BeltramaSchwarz2024.exp1_chill : RegressionCoefficient

Experiment 1 regression model (§4.5). DV: COVERED (1) vs VISIBLE (0). Persona: treatment-coded, NoPersona as reference. ScreenFit: binary (Control = Match+Mismatch collapsed vs Imprecise), sum-coded. Maximal converging mixed-effects logistic regression.

Equations

• BeltramaSchwarz2024.exp1_chill = { predictor := "Chill", beta := -67 / 100, se := 13 / 100, significant := true }

def BeltramaSchwarz2024.exp1_nerdy : RegressionCoefficient

Equations

• BeltramaSchwarz2024.exp1_nerdy = { predictor := "Nerdy", beta := 77 / 100, se := 14 / 100, significant := true }

def BeltramaSchwarz2024.exp1_screenFit : RegressionCoefficient

Equations

• BeltramaSchwarz2024.exp1_screenFit = { predictor := "ScreenFit", beta := 0, se := 10 / 100, significant := false }

def BeltramaSchwarz2024.exp1_chillScreenFit : RegressionCoefficient

Experiment 1 Persona × ScreenFit interactions (§4.5). Both are significant: persona effects are concentrated in the Imprecise condition. In Control conditions, no significant persona contrasts were found.

Equations

• BeltramaSchwarz2024.exp1_chillScreenFit = { predictor := "Chill:ScreenFit", beta := 66 / 100, se := 13 / 100, significant := true }

def BeltramaSchwarz2024.exp1_nerdyScreenFit : RegressionCoefficient

Equations

• BeltramaSchwarz2024.exp1_nerdyScreenFit = { predictor := "Nerdy:ScreenFit", beta := -78 / 100, se := 13 / 100, significant := true }

def BeltramaSchwarz2024.exp1_model : List RegressionCoefficient

Equations

• One or more equations did not get rendered due to their size.

def BeltramaSchwarz2024.exp2_n : ℕ

Experiment 2 participants (§5.2). 244 recruited on Prolific.

Equations

• BeltramaSchwarz2024.exp2_n = 244

def BeltramaSchwarz2024.exp2_chill : RegressionCoefficient

Experiment 2 regression model (§5.3). DV: acceptance rate. The paper describes fitting on "WRONG choices" but the positive Chill coefficient (β = 1.04) with the finding that Chill speakers had lower WRONG rates (§5.3, Figure 6) indicates the effective coding is acceptance (RIGHT = 1, WRONG = 0). Persona: treatment-coded, NoPersona as reference. ScreenFit: binary (Control vs Imprecise), sum-coded.

Equations

• BeltramaSchwarz2024.exp2_chill = { predictor := "Chill", beta := 104 / 100, se := 16 / 100, significant := true }

def BeltramaSchwarz2024.exp2_nerdy : RegressionCoefficient

Equations

• BeltramaSchwarz2024.exp2_nerdy = { predictor := "Nerdy", beta := 2 / 100, se := 15 / 100, significant := false }

def BeltramaSchwarz2024.exp2_chillScreenFit : RegressionCoefficient

Experiment 2 Persona × ScreenFit interactions (§5.3). Only Chill × ScreenFit is significant: the Chill tolerance effect is concentrated in the Imprecise condition.

Equations

• BeltramaSchwarz2024.exp2_chillScreenFit = { predictor := "Chill:ScreenFit", beta := 100 / 100, se := 16 / 100, significant := true }

def BeltramaSchwarz2024.exp2_nerdyScreenFit : RegressionCoefficient

Equations

• BeltramaSchwarz2024.exp2_nerdyScreenFit = { predictor := "Nerdy:ScreenFit", beta := 3 / 100, se := 18 / 100, significant := false }

def BeltramaSchwarz2024.exp2_model : List RegressionCoefficient

Equations

• BeltramaSchwarz2024.exp2_model = [BeltramaSchwarz2024.exp2_chill, BeltramaSchwarz2024.exp2_nerdy, BeltramaSchwarz2024.exp2_chillScreenFit, BeltramaSchwarz2024.exp2_nerdyScreenFit]

def BeltramaSchwarz2024.combined_nerdyTask : RegressionCoefficient

The critical Nerdy × Task interaction (§6): the Covered Screen Task features a higher rejection rate for Nerdy than No.Persona, but the Truth Value Judgment task does not. This is the formal statistical confirmation of the task asymmetry captured by task_asymmetry_derived.

Equations

• BeltramaSchwarz2024.combined_nerdyTask = { predictor := "Nerdy:Task", beta := 62 / 100, se := 31 / 100, significant := true }

def BeltramaSchwarz2024.exampleStatedAmount : ℕ

All critical items use round dollar amounts. Example from §2: "The ticket costs $200."

Equations

• BeltramaSchwarz2024.exampleStatedAmount = 200

def BeltramaSchwarz2024.exampleImpreciseAmount : ℕ

The imprecise screen shows a close-but-not-exact amount. Example from Figure 1: visible screen shows ~$207 when statement says $200.

Equations

• BeltramaSchwarz2024.exampleImpreciseAmount = 207

theorem BeltramaSchwarz2024.stimulus_is_round : Core.Roundness.roundnessGrade exampleStatedAmount = Core.Roundness.RoundnessGrade.high

Stimulus numerals are round: high roundness grade.

theorem BeltramaSchwarz2024.imprecise_value_is_nonround : Core.Roundness.roundnessGrade exampleImpreciseAmount = Core.Roundness.RoundnessGrade.none

The imprecise value is non-round: zero roundness.

theorem BeltramaSchwarz2024.roundness_gap : Core.Roundness.roundnessScore exampleStatedAmount = 6 ∧ Core.Roundness.roundnessScore exampleImpreciseAmount = 0

Roundness scores differ maximally: 6 vs 0.

theorem BeltramaSchwarz2024.exp1_both_personae_significant : exp1_chill.significant = true ∧ exp1_nerdy.significant = true

Core finding: both persona conditions significantly affect Exp 1 (inference task).

theorem BeltramaSchwarz2024.exp2_chill_only : exp2_chill.significant = true ∧ exp2_nerdy.significant = false

Key asymmetry: in Exp 2 (judgment task), only Chill is significant.

theorem BeltramaSchwarz2024.chill_reduces_exactness_exp1 : exp1_chill.beta < 0

Chill decreases COVERED rate (negative β): Chill persona leads listeners to favor the VISIBLE screen, treating the round numeral as imprecise (§4.5).

theorem BeltramaSchwarz2024.nerdy_increases_exactness_exp1 : exp1_nerdy.beta > 0

Nerdy increases COVERED rate (positive β): Nerdy persona leads listeners to favor the COVERED screen, treating the round numeral as exact (§4.5).

theorem BeltramaSchwarz2024.chill_increases_tolerance_exp2 : exp2_chill.beta > 0

Chill increases RIGHT rate (positive β on RIGHT DV), i.e., increases tolerance for imprecision in truth-value judgments (§5.3).

theorem BeltramaSchwarz2024.nerdy_null_in_exp2 : exp2_nerdy.significant = false

Nerdy has no significant effect on truth-value judgments (§5.3).

theorem BeltramaSchwarz2024.exp1_interactions_significant : exp1_chillScreenFit.significant = true ∧ exp1_nerdyScreenFit.significant = true

Significant Persona × ScreenFit interactions in Exp 1 (§4.5): persona effects are concentrated in the Imprecise condition. In Control conditions, no significant persona contrasts were found.

theorem BeltramaSchwarz2024.exp2_chill_interaction_only : exp2_chillScreenFit.significant = true ∧ exp2_nerdyScreenFit.significant = false

Exp 2 interaction asymmetry (§5.3): only Chill × ScreenFit is significant. Nerdy speakers show no interaction, consistent with the null main effect.

theorem BeltramaSchwarz2024.exp1_no_screenFit_main_effect : exp1_screenFit.significant = false

No main effect of ScreenFit in Exp 1 (§4.5): COVERED rates do not differ across Imprecise and Control conditions overall. The persona effect is entirely driven by the interaction.

theorem BeltramaSchwarz2024.chill_effect_larger_in_exp2 : exp2_chill.beta > -exp1_chill.beta

The Chill main effect is larger in Exp 2 (|β| = 1.04) than in Exp 1 (|β| = 0.67).

theorem BeltramaSchwarz2024.combined_nerdy_task_significant : combined_nerdyTask.significant = true

The combined analysis confirms the Nerdy × Task interaction is significant, validating the prejudiciality-based explanation.

theorem BeltramaSchwarz2024.combined_nerdy_task_positive : combined_nerdyTask.beta > 0

The combined analysis interaction sign is positive (§6): Nerdy speakers have higher rejection rates in CST than TVJ relative to baseline, consistent with rejection being blocked in TVJ.

def BeltramaSchwarz2024.precisionField : Phenomena.SocialMeaning.IndexicalField.IndexicalField Semantics.Numerals.Precision.PrecisionMode Sociolinguistics.SCM.SocialDimension

The indexical field for numeral precision, parameterized by PrecisionMode from Semantics.Numerals.Precision rather than a study-local type — the sociolinguistic variable whose social meaning is under study IS the semantic precision mode.

Equations

• One or more equations did not get rendered due to their size.

def BeltramaSchwarz2024.personaPrecision : PersonaCondition → Option Semantics.Numerals.Precision.PrecisionMode

The precision mode favored by a given persona condition.

Equations

• BeltramaSchwarz2024.personaPrecision BeltramaSchwarz2024.PersonaCondition.nerdy = some Semantics.Numerals.Precision.PrecisionMode.exact
• BeltramaSchwarz2024.personaPrecision BeltramaSchwarz2024.PersonaCondition.chill = some Semantics.Numerals.Precision.PrecisionMode.approximate
• BeltramaSchwarz2024.personaPrecision BeltramaSchwarz2024.PersonaCondition.noPersona = none

def BeltramaSchwarz2024.personaDimension : PersonaCondition → Option Sociolinguistics.SCM.SocialDimension

Persona conditions mapped to SCM social dimensions.

Equations

• BeltramaSchwarz2024.personaDimension BeltramaSchwarz2024.PersonaCondition.nerdy = some Sociolinguistics.SCM.SocialDimension.competence
• BeltramaSchwarz2024.personaDimension BeltramaSchwarz2024.PersonaCondition.chill = some Sociolinguistics.SCM.SocialDimension.warmth
• BeltramaSchwarz2024.personaDimension BeltramaSchwarz2024.PersonaCondition.noPersona = none

theorem BeltramaSchwarz2024.bidirectionality (p : PersonaCondition) (d : Sociolinguistics.SCM.SocialDimension) (v : Semantics.Numerals.Precision.PrecisionMode) (hd : personaDimension p = some d) (hv : personaPrecision p = some v) : precisionField.indexes v d

Bidirectionality theorem: production and comprehension mappings cohere.

theorem BeltramaSchwarz2024.scm_coherence_nerdy : precisionField.indexes Semantics.Numerals.Precision.PrecisionMode.exact Sociolinguistics.SCM.SocialDimension.competence

theorem BeltramaSchwarz2024.scm_coherence_chill : precisionField.indexes Semantics.Numerals.Precision.PrecisionMode.approximate Sociolinguistics.SCM.SocialDimension.warmth

theorem BeltramaSchwarz2024.opposite_directions (d : Sociolinguistics.SCM.SocialDimension) : precisionField.association Semantics.Numerals.Precision.PrecisionMode.exact d = -precisionField.association Semantics.Numerals.Precision.PrecisionMode.approximate d

Exact and approximate modes have opposite associations with every dimension.

inductive BeltramaSchwarz2024.ResponseShift : Type

• towardRejection : ResponseShift
• towardAcceptance : ResponseShift

@[implicit_reducible]

instance BeltramaSchwarz2024.instDecidableEqResponseShift : DecidableEq ResponseShift

Equations

• BeltramaSchwarz2024.instDecidableEqResponseShift x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def BeltramaSchwarz2024.instReprResponseShift.repr : ResponseShift → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance BeltramaSchwarz2024.instReprResponseShift : Repr ResponseShift

Equations

• BeltramaSchwarz2024.instReprResponseShift = { reprPrec := BeltramaSchwarz2024.instReprResponseShift.repr }

def BeltramaSchwarz2024.shiftDirection : Semantics.Numerals.Precision.PrecisionMode → ResponseShift

Equations

• BeltramaSchwarz2024.shiftDirection Semantics.Numerals.Precision.PrecisionMode.exact = BeltramaSchwarz2024.ResponseShift.towardRejection
• BeltramaSchwarz2024.shiftDirection Semantics.Numerals.Precision.PrecisionMode.approximate = BeltramaSchwarz2024.ResponseShift.towardAcceptance

def BeltramaSchwarz2024.rejectionIsPrejudicial : TaskType → Prop

Equations

• BeltramaSchwarz2024.rejectionIsPrejudicial BeltramaSchwarz2024.TaskType.coveredScreen = False
• BeltramaSchwarz2024.rejectionIsPrejudicial BeltramaSchwarz2024.TaskType.truthValueJudgment = True

@[implicit_reducible]

instance BeltramaSchwarz2024.instDecidableRejectionIsPrejudicial (t : TaskType) : Decidable (rejectionIsPrejudicial t)

Equations

• BeltramaSchwarz2024.instDecidableRejectionIsPrejudicial BeltramaSchwarz2024.TaskType.coveredScreen = isFalse BeltramaSchwarz2024.instDecidableRejectionIsPrejudicial._proof_1
• BeltramaSchwarz2024.instDecidableRejectionIsPrejudicial BeltramaSchwarz2024.TaskType.truthValueJudgment = isTrue trivial

def BeltramaSchwarz2024.shiftManifests (shift : ResponseShift) (task : TaskType) : Prop

Equations

• BeltramaSchwarz2024.shiftManifests BeltramaSchwarz2024.ResponseShift.towardAcceptance task = True
• BeltramaSchwarz2024.shiftManifests BeltramaSchwarz2024.ResponseShift.towardRejection task = ¬BeltramaSchwarz2024.rejectionIsPrejudicial task

def BeltramaSchwarz2024.personaEffectPredicted (persona : PersonaCondition) (task : TaskType) : Prop

Equations

• One or more equations did not get rendered due to their size.

theorem BeltramaSchwarz2024.acceptance_never_blocked (task : TaskType) : shiftManifests ResponseShift.towardAcceptance task

theorem BeltramaSchwarz2024.rejection_manifests_in_cst : shiftManifests ResponseShift.towardRejection TaskType.coveredScreen

theorem BeltramaSchwarz2024.rejection_blocked_in_tvjt : ¬shiftManifests ResponseShift.towardRejection TaskType.truthValueJudgment

theorem BeltramaSchwarz2024.cst_both_manifest : personaEffectPredicted PersonaCondition.nerdy TaskType.coveredScreen ∧ personaEffectPredicted PersonaCondition.chill TaskType.coveredScreen

Both personae's effects manifest in CST.

theorem BeltramaSchwarz2024.tvjt_chill_only : personaEffectPredicted PersonaCondition.chill TaskType.truthValueJudgment ∧ ¬personaEffectPredicted PersonaCondition.nerdy TaskType.truthValueJudgment

Only Chill's effect manifests in TVJT.

theorem BeltramaSchwarz2024.noPersona_no_effect (task : TaskType) : ¬personaEffectPredicted PersonaCondition.noPersona task

theorem BeltramaSchwarz2024.task_asymmetry_derived : (personaEffectPredicted PersonaCondition.nerdy TaskType.coveredScreen ∧ personaEffectPredicted PersonaCondition.chill TaskType.coveredScreen) ∧ (personaEffectPredicted PersonaCondition.chill TaskType.truthValueJudgment ∧ ¬personaEffectPredicted PersonaCondition.nerdy TaskType.truthValueJudgment) ∧ (exp1_nerdy.significant = true ∧ exp1_chill.significant = true) ∧ (exp2_chill.significant = true ∧ exp2_nerdy.significant = false) ∧ (exp1_chillScreenFit.significant = true ∧ exp1_nerdyScreenFit.significant = true) ∧ exp2_chillScreenFit.significant = true ∧ exp2_nerdyScreenFit.significant = false

Combined task asymmetry matches empirical significance pattern: theoretical prediction from prejudiciality aligns with both the main effects and the interaction structure across experiments.

def BeltramaSchwarz2024.impreciseReadingAvailable (n : ℕ) : Prop

Whether imprecise readings are available for a numeral.

Equations

• BeltramaSchwarz2024.impreciseReadingAvailable n = (Semantics.Numerals.Precision.inferPrecisionMode n = Semantics.Numerals.Precision.PrecisionMode.approximate)

@[implicit_reducible]

instance BeltramaSchwarz2024.instDecidableImpreciseReadingAvailable (n : ℕ) : Decidable (impreciseReadingAvailable n)

Equations

• BeltramaSchwarz2024.instDecidableImpreciseReadingAvailable n = BeltramaSchwarz2024.instDecidableImpreciseReadingAvailable._aux_1 n

theorem BeltramaSchwarz2024.roundness_gates_persona : impreciseReadingAvailable exampleStatedAmount ∧ ¬impreciseReadingAvailable exampleImpreciseAmount

Roundness gates persona effects: $200 supports imprecision, $207 does not.

theorem BeltramaSchwarz2024.div10_enables_imprecision (n : ℕ) (h10 : n % 10 = 0) : impreciseReadingAvailable n

Multiples of 10 always have imprecise readings.

The precision field can be lifted to a grounded field over the SCM property space via fromIndexicalField, connecting it to @cite{burnett-2019}'s persona-theoretic infrastructure.

def BeltramaSchwarz2024.precisionGroundedField : Sociolinguistics.EckertMontague.GroundedField Semantics.Numerals.Precision.PrecisionMode Sociolinguistics.SCM.scmSpace

The precision field converted to a grounded field over the SCM property space.

Equations

• BeltramaSchwarz2024.precisionGroundedField = Sociolinguistics.EckertMontague.fromIndexicalField BeltramaSchwarz2024.precisionField

theorem BeltramaSchwarz2024.exact_scmProperties : precisionGroundedField.indexedProperties Semantics.Numerals.Precision.PrecisionMode.exact = {Sociolinguistics.SCM.SCMProperty.competent, Sociolinguistics.SCM.SCMProperty.cold, Sociolinguistics.SCM.SCMProperty.antiSolidary}

Exact indexes {competent, cold, antiSolidary}.

theorem BeltramaSchwarz2024.approx_scmProperties : precisionGroundedField.indexedProperties Semantics.Numerals.Precision.PrecisionMode.approximate = {Sociolinguistics.SCM.SCMProperty.incompetent, Sociolinguistics.SCM.SCMProperty.warm, Sociolinguistics.SCM.SCMProperty.solidary}

Approximate indexes {incompetent, warm, solidary}.

@cite{beltrama-solt-burnett-2023}'s three-way variant contrast (precise/underspecified/approximate) refines B&S2024's binary distinction. The "underspecified" variant — a bare round numeral like "fifty" — is exactly the kind of stimulus B&S2024 studies: its precision is contextually determined.

theorem BeltramaSchwarz2024.bsb_stim_also_round : impreciseReadingAvailable BeltramaSoltBurnett2022.stimRound

BSB2022's round stimulus (50) has an imprecise reading available, just like B&S2024's stimulus (200). Both are round numbers whose precision resolution is the object of study.

The core finding of @cite{beltrama-schwarz-2024} — that numeral precision is jointly determined by roundness AND speaker identity — is captured by speakerModulatedHalo in the theory layer (Semantics.Numerals.Precision). Nerdy speakers get narrower halos (multiplier < 1), Chill speakers get wider ones (multiplier > 1).

theorem BeltramaSchwarz2024.wider_halo_of_larger_multiplier (m₁ m₂ : ℚ) (n : ℕ) (hm : m₁ < m₂) (hn : 0 < Semantics.Numerals.Precision.haloWidth n) : Semantics.Numerals.Precision.speakerModulatedHalo m₁ n < Semantics.Numerals.Precision.speakerModulatedHalo m₂ n

A larger multiplier produces a wider halo: the monotonicity that connects the competence/warmth ordering to tolerance width.

theorem BeltramaSchwarz2024.round_has_positive_halo : 0 < Semantics.Numerals.Precision.haloWidth exampleStatedAmount

Round numbers have positive halo width, so speaker modulation has a genuine effect.