E8 Geometry Laboratory

An offline, deterministic browser laboratory for exploring the 240 roots of E8 as exact geometry and as a carefully bounded auditory display.

Launch the lab: https://qsolkcb.github.io/E8/

The Pages link becomes live after this change is merged and the repository's GitHub Pages source is set to GitHub Actions.

What the lab does

• Generates the standard 240-root E8 system from exact doubled-integer coordinates: 112 integer-family roots and 128 half-integer-family roots.
• Builds the complete inner-product root graph with 6,720 edges and 56 neighbours per root.
• Constructs a genuine order-30 Coxeter element and projects the roots onto its exponent-1 invariant plane, producing the characteristic eight rings of 30 roots.
• Adds optional display depth from an exponent-7 invariant plane without pretending the three-dimensional receiver is the native eight-dimensional object.
• Applies the explicit embedded D4 × D4 order-three triality used by QSOLKCB/SONIFICATION, giving exactly 12 fixed roots and 76 three-cycles.
• Reconstructs the normative ETQ-101 selector: two fixed roots plus the first 33 complete triality orbits, for 2 + 33×3 = 101 root-indexed states.
• Provides interactive filters, structural palettes, root inspection, graph edges, drag rotation, zoom, triality stepping, PNG capture and settings import/export.
• Plays and renders deterministic stereo WAV files from the ETQ symbolic note code while preserving the distinction between canonical mathematical identity and an authored audio receiver.

No server, package install, build step, CDN, web font, telemetry or network API is required at runtime. index.html can be opened directly from disk.

Visual receivers

Receiver	Exact input	Display choice
Coxeter plane	Exponent-1 invariant plane of an E8 Coxeter element	Screen scale, colour and rotation
Coxeter + depth	Same exact x/y plane	Exponent-7 plane supplies a declared z receiver
D4 × D4 blocks	The two four-coordinate blocks used by triality	Three linear combinations shown as x/y/z
Root graph	Inner product r·s = 1, tested as (2r)·(2s) = 4	Edge opacity and filtering

The exact two-dimensional Coxeter receiver is the least interpretive view. The other projections are useful ways to inspect relationships, not claims that E8 is intrinsically three-dimensional.

Sonification model

The canonical part comes directly from ETQ-101 v2:

basis j = 0, 1                       -> fixed singlets
basis j = 2 + 3m + q                -> m = 0..32, q = 0..2
M(s0) = 13
M(s1) = 113
M(m,q) = 14 + 33q + m

This gives an exact, reversible 101-note symbolic code:

State class	MIDI notes	ETQ meaning
Fixed singlet 0	13	Lower codebook boundary
q = 0	14–46	Authored low register lane; SCL +1
q = 1	47–79	Authored mid register lane; SCL −2
q = 2	80–112	Authored high register lane; SCL +1
Fixed singlet 1	113	Upper codebook boundary

MIDI note number is canonical symbolic identity, not canonical hertz. This lab adds a separately named receiver profile, e8-geometry-auditory-display-v1:

Geometry field	Receiver parameter	Why it is useful
Integer vs half-integer root family	Harmonic family	Distinguishes sparse and dense coordinate families
Selected-graph degree 22–55	Brightness and gain	Makes local connectivity contrast audible
Coxeter ring 1–8	Spectral colour	Preserves the eight-ring projection class
Qutrit label q	Register lane and stereo position	Retains the exact ETQ lane code
SCL value [1,−2,1]	Phase-modulation depth	Keeps curvature distinct from ordered pitch
ETQ-303 external fibre	Small stereo displacement	Exposes the independent three-state fibre

Tempo, step duration, A4 tuning, pan, gain, waveform, partials, sample rate, PCM and WAV serialization are all recorded receiver settings. They are not derived from E8.

The lab defaults to A4 = 432 Hz as an audible receiver preset because it is useful in QSOL music workflows. A4 = 440 Hz is one click away. Neither value is part of ETQ-101 structural identity, and every exported manifest says so explicitly.

Event traversals

• ETQ-101 basis order follows the canonical root-indexed basis.
• Triality orbit groups places the two singlets around 33 ordered q = 0,1,2 triples.
• Coxeter ring / angle makes the selected roots move through the exact eight-ring projection.
• Selected graph breadth-first follows deterministic connectivity, sorting newly discovered neighbours by degree and then basis index.
• ETQ-303 support traversal uses (j,a) → (j+1 mod 101, a+1 mod 3). Coprimality gives all 303 pairs exactly once before closure. The root note remains the ETQ-101 code; the independent fibre is a receiver field.

Run locally

Open index.html directly, or serve the directory with any static server:

python3 -m http.server 8000

Then open http://localhost:8000.

Run the dependency-free structural and audio smoke suite with Node.js 20 or newer:

npm test

or:

node tests/smoke.mjs

Deterministic checks

The smoke suite fixes:

• 240 unique doubled-coordinate roots, every norm squared 8;
• the 112 + 128 coordinate-family split;
• Coxeter element order 30 and eight 30-root projection rings;
• triality decomposition 12 + 76×3;
• full graph fixtures 240 / 6,720 / degree 56;
• ETQ graph fixtures 101 / 1,687 / degree 22–55;
• degree-potential receipts sum = 3,374, denominator 2,181;
• all 101 codebook notes, inverse mapping and triality lane shift;
• the 303-pair support traversal;
• deterministic stereo PCM and a valid 16-bit RIFF/WAVE container;
• local-only runtime resources and the Pages deployment workflow.

Repository layout

index.html                    Browser instrument
style.css                     Responsive offline interface
e8-core.js                    Exact E8, triality, graph and Coxeter construction
audio-core.js                 ETQ codebook receiver, schedules and WAV encoder
app.js                        Canvas, controls, Web Audio and local capture
docs/MATHEMATICAL_RECEIVERS.md Projection derivation and claim boundary
docs/SONIFICATION_PROFILE.md  Canonical mapping vs receiver contract
tests/smoke.mjs               Dependency-free regression suite
.github/workflows/            CI and GitHub Pages deployment

Scientific boundary

Supported descriptions include:

• “the standard 240 E8 roots in doubled integer coordinates”;
• “the E8 root graph under inner-product-one adjacency”;
• “a Coxeter-plane projection with eight 30-root rings”;
• “D4 triality embedded in the E8 root system”;
• “a deterministic, triality-closed 101-state root-indexed truncation”; and
• “an authored auditory receiver using the ETQ symbolic MIDI codebook.”

The lab does not claim:

• a 101-dimensional representation of E8;
• that the screen is the full eight-dimensional geometry;
• that low/mid/high is an intrinsic qutrit energy order;
• that MIDI or a pleasing WAV validates E8 or a physical theory;
• that E8 selects 432 Hz, 440 Hz, a waveform, a tempo or a musical scale; or
• that projection depth or sound is a measurement of a physical E8 system.

Lineage

• Product and offline-lab architecture: QSOLKCB/VORTEX
• Normative ETQ mathematics, selector and symbolic MIDI mapping: QSOLKCB/SONIFICATION

The source implementation is independently organized for this repository while preserving the cited ETQ formulas and claim boundaries.

Creator and licence

Created by Trent Slade / QSOL-IMC.

Source code and documentation are licensed under the Mozilla Public License 2.0. Preserve copyright and licence notices and identify modified files as required by the licence.