Crossmodal GF(3) Skill

"Color is not inherently visual. Color is INFORMATION that can be rendered through any sensory modality."

The Universal Bridge

This skill treats GF(3) trits as modality-independent semantic units:

GF(3) Trit	Visual	Tactile	Auditory	Haptic
MINUS (−1)	Cool hues	Rough/Bumpy	Low pitch	Left/Down
ERGODIC (0)	Neutral	Smooth	Mid pitch	Center
PLUS (+1)	Warm hues	Ridged	High pitch	Right/Up

Key Insight

Visual perception is ALSO a projection from GF(3) space.

The sighted user doesn't have "the real thing" — they have π_visual(GF3). The blind user has π_tactile(GF3), π_auditory(GF3), π_haptic(GF3).

All projections are isomorphic under GF(3) conservation:

π_visual(W) ≅ π_tactile(W) ≅ π_auditory(W) ≅ π_haptic(W)

Implementation Files

From Gay.jl:

1. world_tactile_color.jl — Core tactile/auditory/haptic types
2. world_accessible_tensor.jl — A ⊗ G ⊗ M ⊗ T tensor product
3. world_accessible_interrupt_operad.jl — TOAD × Amp × Knight Tour accessibility

Tactile: 3×3 Braille Extension

Standard Braille is 2×3 (6 dots). We extend to 3×3 (9 dots):

┌───┬───┬───┐
│ 1 │ 2 │ 3 │  ← Hue sector (warm/neutral/cool)
├───┼───┼───┤
│ 4 │ 5 │ 6 │  ← Saturation level
├───┼───┼───┤
│ 7 │ 8 │ 9 │  ← Lightness level
└───┴───┴───┘

Dot positions encode trits:

• Left dot: MINUS (−1)
• Center dot: ERGODIC (0)
• Right dot: PLUS (+1)

Example: Warm, Vivid, Light Color

○ ○ ⬤   ← Hue: PLUS (warm)
○ ○ ⬤   ← Saturation: PLUS (vivid)
○ ○ ⬤   ← Lightness: PLUS (light)

Compact: +++

Auditory: 3-Tone Chords

Each color becomes a 3-frequency chord:

const BASE_FREQ = 440.0  # A4

function trit_to_freq_ratio(t::Int)::Float64
    t == -1 && return 0.84   # Minor third
    t == 0 && return 1.0     # Unison
    t == 1 && return 1.26    # Major third
end

function color_to_chord(hue_trit, sat_trit, light_trit)
    (
        BASE_FREQ * trit_to_freq_ratio(hue_trit),
        BASE_FREQ * 1.5 * trit_to_freq_ratio(sat_trit),   # Fifth
        BASE_FREQ * 2.0 * trit_to_freq_ratio(light_trit)  # Octave
    )
end

Sox Sonification

# Play a single color as chord
play -n synth 0.25 sine 440.0 : synth 0.25 sine 660.0 : synth 0.25 sine 880.0 remix - fade 0 0.25 0.05

Haptic: 3D Position + Vibration

struct HapticPosition
    x::Float64  # Hue: -1 (left) to +1 (right)
    y::Float64  # Saturation: -1 (back) to +1 (front)
    z::Float64  # Lightness: -1 (down) to +1 (up)
    vibration_pattern::Symbol  # :pulse, :smooth, :buzz
    intensity::Float64  # 0.0 to 1.0
end

Möbius Invertibility

Path navigability analysis using μ(path_length):

| μ ≠ 0 | GEODESIC | Smooth bidirectional tactile navigation | | μ = 0 | TANGLED | Add branching markers at squared prime positions |

I-Thou Framework

• Geodesic paths: Treat blind user as capable of autonomous navigation (I-Thou)
• Tangled paths: Restructure the path, not the user (I-Thou), vs "accommodate" (I-It)

μ(3) = -1: Action-Perception Duality

The Möbius value μ(3) = -1 creates:

Action:     {−, ○, +}   (visual domain)
Perception: {+, ○, −}   (tactile domain, Möbius-inverted)

Double inversion returns to original: μ ∘ μ = id.

GF(3) Triadic Integration

Component	Trit	Role
catsharp-sonification	0	Auditory output
crossmodal-gf3	0	Universal bridge
sense	0	Content extraction

Conservation: 0 + 0 + 0 = 0 ✓ (all ERGODIC coordinators)

For generators/validators:

elevenlabs-acset (-1) ⊗ crossmodal-gf3 (0) ⊗ gesture-hypergestures (+1) = 0 ✓

Commands

# Generate accessible color palette (Julia)
julia -e 'using Gay; Gay.gay_seed!(137508); colors = [Gay.next_color() for _ in 1:6]'

# Sonify via catsharp-sonification
bb -e '(sonify-palette ["#DD3C3C" "#3CDD6B" "#9A3CDD"])'

# Extract content accessibly
just sense-extract reference/videos/lecture.mkv

Related Skills

• catsharp-sonification — Hue → Pitch Class → Waveform
• sense — Video → Subtitle + Diagram + Skill Index
• buberian-relations — I-Thou / I-It / We formalization
• gesture-hypergestures — Continuous curves for topology teaching
• moebius-inversion — Path invertibility analysis
• gay-julia — Wide-gamut color with SplitMix64

Theorem

Accessible Worlds Theorem:

For any world W with visual representation V:

π_visual(W) ≅ π_tactile(W) ≅ π_auditory(W) ≅ π_haptic(W)

where all projections π preserve GF(3) conservation:

∀ modality m: Σ(trits(π_m(W))) ≡ 0 (mod 3)

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"The most unlike skills are the most essential - they bridge what others cannot reach."

SDF Interleaving

This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 6. Layering

Concepts: layered data, metadata, provenance, units

GF(3) Balanced Triad

crossmodal-gf3 (+) + SDF.Ch6 (+) + [balancer] (+) = 0

Skill Trit: 1 (PLUS - generation)

Secondary Chapters

• Ch4: Pattern Matching
• Ch10: Adventure Game Example
• Ch7: Propagators

Connection Pattern

Layering adds metadata. This skill tracks provenance or annotations.