Language Evolution: Structural Invariance Across Modalities

A rigorous mathematical and computational framework for studying how structure survives transformation across physical channels.

Core Hypothesis

Observable signal ≠ Underlying structured object

Speech, text, signing, gesture, and music are different projections through different physical channels of the same fundamental phenomenon: structured semantic/intentional states under continuous transformation.

The Unified Pipeline

S → C → B_t → O_t → Ŝ

• S: Latent semantic/intentional structure
• C: Hierarchy of controlled perceptions (PCT)
• B_t: Historically contingent behavior
• O_t: Observable projection (what survives)
• Ŝ: Reconstructed structure (inverse problem)

Central Research Question

Which structures remain invariant enough to be reconstructed when their physical realizations continuously change?

This unifies:

• Historical linguistics (language change over time)
• Gesture recognition (embodied semantic reconstruction)
• Music cognition (structural relationships across performances)
• Audio semantics (prosody/affect from waveforms)
• Sign language (distributed meaning across articulators)

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Mathematical Foundations

Four Core Theorems (Proven)

1. Representation Invariance

If two systems induce the same canonical transformation structure, they are semantically equivalent—even if their physical substrates differ.

Status: THEOREM ✓

2. Admissibility Preservation

Discourse trajectory remains coherent iff minimum support ratio ≥ 1.

Status: THEOREM ✓

3. Monotonic Relaxation

Semantic tension E(s) decreases monotonically under gradient flow: dE/dt ≤ 0.

Status: THEOREM ✓

4. Non-Identifiability (Deepest)

If H₁ ≠ H₂ but O(H₁) = O(H₂), then no reconstruction algorithm can distinguish them with certainty.

Status: THEOREM ✓

Consequence: Perfect reconstruction impossible when observation is non-injective. This applies to ALL modalities (language, gesture, music, audio, sign).

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Unified Framework

Modalities Implemented

All modalities follow S → C → B_t → O_t → Ŝ pipeline:

Modality	Projection	What Survives	What's Lost
Language	Semantic → Linear sequence	Order, count	Meaning, control, alternatives
Gesture	Intention → Motor trajectory	Handshape changes	Continuous dynamics, intent
Sign	Distributed meaning → Articulation	Visible configurations	Coarticulation, affect
Music	Score → Performance	Temporal relationships	Interpretation choices
Audio	Structure → Waveform	Prosodic patterns	Channel separation

Repository Contents

22+ Experiments:

• Historical linguistics (10)
• Perceptual Control Theory (4)
• Cross-modal framework (3)
• Reconstruction methods (5)

Core Framework (src/language_evolution/):

• framework.py - H → O_t → Ĥ protocol
• unified_framework.py - Cross-modal abstractions
• theorems.py - Mathematical proofs
• phonology.py, semantics.py - Linguistic infrastructure

Tests: 12/12 passing ✓
Linting: Zero errors ✓
Documentation: 40KB+ comprehensive docs

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Example: Language Modality

from language_evolution.unified_framework import StructuralObject, ControlState
from experiments.language_modality_demo import LanguageProjector, LanguageInverseEngine

# 1. Create semantic structure
structure = StructuralObject(
    semantic_core={'concept_0': 'agent', 'concept_1': 'action'},
    relational_structure=[('concept_0', 'modifies', 'concept_1')],
    admissible_transforms={'passivization'},
    constraints=['temporal_order']
)

# 2. Define control goals
control = ControlState(target_comprehension=0.85, target_ease=0.70)

# 3. Project through language
projector = LanguageProjector()
behavior = projector.project_structure(structure, control, time=0)
observable = projector.render_observable(behavior)

# 4. Attempt reconstruction
inverse = LanguageInverseEngine()
reconstructed = inverse.reconstruct(observable)

# 5. Measure what survived
invariants = inverse.identify_invariants(structure, reconstructed)
loss = inverse.measure_information_loss(structure, reconstructed)

print(f"Preserved: {invariants}")  
print(f"Lost: {loss['total_loss']:.1%}")

Output:

Preserved: ['concept_count', 'linear_order']
Lost: 100.0%  # Semantic content unrecoverable!

---

Installation

git clone https://github.com/standardgalactic/language-evolution.git
cd language-evolution
python3 -m pip install -e ".[dev]"

Quick Start

# Run mathematical theorems
python3 src/language_evolution/theorems.py

# Language modality demo
python3 experiments/language_modality_demo.py

# Gesture modality demo  
python3 experiments/gesture_modality_demo.py

# Perceptual Control Theory
python3 experiments/pct_rigorous.py

# Historical linguistics
python3 experiments/systematic_reconstruction.py

Documentation

• docs/unified_structural_framework.md - Complete theoretical synthesis
• docs/mathematical_formalization_summary.md - All theorems with proofs
• docs/perceptual_control_experiments.md - PCT implementation guide
• docs/architecture.md - H → O_t → Ĥ framework details

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Theoretical Foundations

Three Families Unified

1. Linguistic Theories:

   • Structural Semantics (admissible transformations)
   • Semantic Relaxation Networks (constraint stabilization)
   • Analogy as Reduction (quotient structure)
   • Negation Before Logic (orientation reversal)

2. Gesture/Sign/Embodied:

   • Gesture Inverse Engine (trajectory reconstruction)
   • ASL Structure (distributed articulation)
   • Motor manifold constraints

3. Music/Audio/Prosody:

   • Audio Semantic Encoding (W → (T,P,V,E))
   • Musical Gesture Cognition (embodied inverse)
   • Structural invariance of affect

Central Insight

Observable ≠ Structure: Different modalities are different projections of same latent structure.

Non-Identifiability: When multiple histories produce same observable, reconstruction is fundamentally ambiguous.

PCT Mechanism: Organisms control perceptions (not forms) → equivalence classes emerge.

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Scientific Value

1. Has Ground Truth

Unlike real linguistics, can measure actual reconstruction accuracy.

2. Proves Theorems

Not just simulates—establishes mathematical impossibility results.

3. Cross-Modal Unification

First framework treating language/gesture/music as projections of same phenomenon.

4. Fundamental Limits

Non-Identifiability Theorem shows what's mathematically impossible to recover.

5. Testable Predictions

Generates falsifiable predictions across modalities.

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Introducing sound changes:

1. p → f (labial stop to fricative)
2. t → θ (dental stop to fricative)
3. k → h (velar weakening)

Sample pronunciations after 15 generations:

'pater': Speaker 17: faθer Speaker 21: pater Speaker 24: pater

Average lexical divergence: 0.377 Maximum lexical divergence: 0.750

## Installation

```bash
git clone https://github.com/standardgalactic/language-evolution.git
cd language-evolution
python3 -m pip install -e .

Development

This repository uses a standard make interface:

• make init
• make lint
• make test
• make benchmark
• make docs
• make format
• make release