@cite{ackerman-malouf-2013}: Bridge Theorems @cite{ackerman-malouf-2013}

@cite{carstairs-mccarthy-2010}

Verification theorems connecting the cross-linguistic typological data to the LCEC predictions. Each theorem proves that a language's reported I-complexity falls below the LCEC threshold.

Structure

• §1: Per-language LCEC verification (all 10 languages)
• §2: E-complexity / I-complexity dissociation
• §3: Mazatec case study (observed vs. random baseline)

Each language's reported I-complexity is below the 1-bit threshold. These are "per-datum verification theorems" in linglib's sense: changing a language's avgCondEntropy breaks exactly the corresponding theorem.

theorem Phenomena.Morphology.AckermanMalouf2013.fur_lcec : Typology.fur.avgCondEntropy ≤ Typology.lcecThreshold

theorem Phenomena.Morphology.AckermanMalouf2013.ngiti_lcec : Typology.ngiti.avgCondEntropy ≤ Typology.lcecThreshold

theorem Phenomena.Morphology.AckermanMalouf2013.nuer_lcec : Typology.nuer.avgCondEntropy ≤ Typology.lcecThreshold

theorem Phenomena.Morphology.AckermanMalouf2013.kwerba_lcec : Typology.kwerba.avgCondEntropy ≤ Typology.lcecThreshold

theorem Phenomena.Morphology.AckermanMalouf2013.chinantec_lcec : Typology.chinantec.avgCondEntropy ≤ Typology.lcecThreshold

theorem Phenomena.Morphology.AckermanMalouf2013.mazatec_lcec : Typology.mazatec.avgCondEntropy ≤ Typology.lcecThreshold

theorem Phenomena.Morphology.AckermanMalouf2013.finnish_lcec : Typology.finnish.avgCondEntropy ≤ Typology.lcecThreshold

theorem Phenomena.Morphology.AckermanMalouf2013.german_lcec : Typology.german.avgCondEntropy ≤ Typology.lcecThreshold

theorem Phenomena.Morphology.AckermanMalouf2013.russian_lcec : Typology.russian.avgCondEntropy ≤ Typology.lcecThreshold

theorem Phenomena.Morphology.AckermanMalouf2013.spanish_lcec : Typology.spanish.avgCondEntropy ≤ Typology.lcecThreshold

theorem Phenomena.Morphology.AckermanMalouf2013.all_satisfy_lcec (l : Typology.LanguageData) : l ∈ Typology.ackermanMalouf2013 → l.avgCondEntropy ≤ Typology.lcecThreshold

All 10 languages satisfy the LCEC.

The LCEC's key prediction: E-complexity and I-complexity are dissociated. A language can have enormous E-complexity but low I-complexity.

theorem Phenomena.Morphology.AckermanMalouf2013.mazatec_max_eComplexity (l : Typology.LanguageData) : l ∈ Typology.ackermanMalouf2013 → l.numClasses ≤ Typology.mazatec.numClasses

Mazatec has maximal E-complexity in the sample (109 classes).

theorem Phenomena.Morphology.AckermanMalouf2013.mazatec_high_e_low_i : Typology.mazatec.numClasses = 109 ∧ Typology.mazatec.avgCondEntropy ≤ 1

Mazatec's I-complexity is still below 1 bit despite 109 classes.

theorem Phenomena.Morphology.AckermanMalouf2013.eComplexity_doesnt_predict_iComplexity : Typology.kwerba.numClasses < Typology.german.numClasses ∧ Typology.german.avgCondEntropy < Typology.kwerba.avgCondEntropy

Kwerba has minimal E-complexity (2 classes) but its I-complexity is not the lowest — German (7 classes) has lower I-complexity. This shows E-complexity doesn't predict I-complexity in either direction.

theorem Phenomena.Morphology.AckermanMalouf2013.spanish_minimal_iComplexity (l : Typology.LanguageData) : l ∈ Typology.ackermanMalouf2013 → Typology.spanish.avgCondEntropy ≤ l.avgCondEntropy

Spanish has only 3 classes but 57 cells — yet its I-complexity is the lowest in the sample (0.003 bits). More cells with fewer classes means more implicative structure.

The Mazatec case study (§4 of the paper) demonstrates that the observed I-complexity is far below what random assignment of inflection-class patterns would produce.

theorem Phenomena.Morphology.AckermanMalouf2013.mazatec_well_below_random : Typology.mazatec.avgCondEntropy < Typology.mazatecRandomBaseline

Mazatec's observed I-complexity is far below the random baseline. Observed: 0.709 bits. Random permutation baseline: ~5.25 bits. The observed value is less than 14% of the random baseline.

theorem Phenomena.Morphology.AckermanMalouf2013.mazatec_ratio_to_random : Typology.mazatec.avgCondEntropy * 7 < Typology.mazatecRandomBaseline

The ratio of observed to random I-complexity is less than 1/7. (0.709 / 5.25 ≈ 0.135, i.e., ~13.5% of random)

theorem Phenomena.Morphology.AckermanMalouf2013.mazatec_violates_synonymyAvoidance : Typology.mazatec.avgCondEntropy > 0

Mazatec has nonzero I-complexity: it violates @cite{carstairs-mccarthy-2010}'s synonymy avoidance but satisfies the LCEC. This witnesses that the LCEC is strictly weaker than synonymy avoidance.