CoxeterViewer5D is an offline-capable viewer for Coxeter groups, finite cover complexes, wall systems, and local Morse data. It combines five related views:
- Davis: a finite Cayley ball with visible Davis cells;
- hat X (
\hat X): a finite cover of the standard Coxeter presentation 2-complex; - bar X (
\bar X): the compression of\hat Xused by Jankiewicz--Wise; - Gamma (
\Gamma): the defining graph of the Coxeter system; - Projection: chamber barycenters drawn from supplied reflection data.
The app is a research and teaching instrument, not a theorem prover. Incidence computed from validated finite data can be exact while the 3D placement remains a drawing. The interface labels those two claims separately.
The main research path is Covers + Walls:
- Choose a Coxeter system.
- Ask an exact backend to find a finite-index torsion-free subgroup
Hand its coset action. - Verify torsion-freeness by testing prime-order torsion from the spherical special subgroups.
- Build
\hat Xfrom the lifted presentation cells. - Compress it to
\bar X. - Find and coorient the walls of
\bar X. - Extract the induced homomorphism
H -> Z. - Inspect lawful cells and the ascending and descending links used in the Morse-theoretic argument.
This follows the setup in Kasia Jankiewicz and Daniel T. Wise, Incoherent Coxeter Groups. The viewer can also search over wall coorientations to retain many lawful cells. That search is an application feature, not a theorem from the paper.
Automatic cover discovery is the intended primary backend path. The current
source already validates finite actions and carries out the cover,
compression, wall, lawfulness, and finite-link calculations. The controlled
desktop job now runs a bounded strategy ladder: exact Sage congruence images,
matrix and table-of-marks screening, reusable partial modules and their
Everitt-style diagonal products, then a small GAP low-index fallback. An exact
reduction can certify a large normal torsion-free kernel without pretending
that its cover has been built. The browser constructs \hat X only when a
manageable generator action is also present and passes an independent check.
The fibering step then
writes a deterministic Reidemeister--Schreier presentation, evaluates the wall
map on every Schreier generator and relator, normalizes its period gcd, and
checks every stated PL Morse hypothesis. Manual finite-action import stays
available as the advanced fallback.
See
Automatic torsion-free cover discovery
for the algorithm and the exact status language.
For a first pass:
- Load
I2(5)and open Davis to see its decagonal rank-two cell. - Load Ideal 3-cube, all m=3 (S4 cover) for the smallest bundled example that connects certified hyperbolic reflection data to a nontrivial torsion-free cover. Open Gamma to see its octahedral finite-relation graph, then compare hat X and bar X.
- Open Gamma to read the defining generators and finite relations.
- In Covers + Walls, press Find torsion-free cover. The desktop app
runs the bounded automatic strategy ladder; the bundled
I2(5)action remains ready as a quick example. - Compare hat X with bar X and inspect the compression fibers.
- Open Walls, select one wall, and then flip its coorientation.
- Open Lawful cells to see which polygons have one source and one sink.
- Press Run lawful-first certification; use Check full Davis quotient for the all-cell fallback. Inspect or export the resulting certificate.
The Start Here panel names these paths directly:
- Explore a Coxeter example
- Find a torsion-free cover
- Find walls in bar X
- Coorient walls
- Inspect exactness and data status
The Focus Inspector answers three questions throughout the app:
- What is selected?
- Why is it here?
- Is it exact data, a browser check, or a drawing?
Davis shows the Cayley graph and cells associated to spherical special subgroups. Its finite-radius boundary may clip cells.
hat X shows the lifted Coxeter presentation complex before compression. Its directed generator lifts and lifted 2-cells come from a discovered or imported finite action. Calling this data a torsion-free cover requires a complete prime-order fixed-point certificate or equivalent subgroup evidence; a permutation action by itself does not prove torsion-freeness.
bar X shows the compressed even-sided 2-complex. The two lifted generator
bigons based at opposite ends of an s_i orbit collapse with their two
directed boundary edges to one geometric edge. The 2m_ij relation lifts in
one finite-dihedral orbit become one 2m_ij-gon. Walls and lawful cells are
computed here.
Gamma shows the defining Coxeter graph. The app can include m = 2 edges
when a full finite-relation graph is useful. Pairs with m = inf are omitted
because they do not define a finite rank-two relation.
Projection applies supplied reflection data to chamber barycenters and projects the result to three dimensions. A Klein, Poincare, axes, or PCA view is still a projection unless the displayed certificate states a narrower verified claim.
The one-vertex complex and state/move legal-system reader from earlier releases are not part of the current model switch. The wall-coorientation workflow is the general object used by the current source tree.
The app uses four deliberately different status levels:
- Certified source data: a stored artifact and hashes support a stated transcription, Gram, geometry-interval, or external-checker scope.
- Exact incidence: finite combinatorial data pass the in-repo validators. Examples include signed attaching maps, compression fibers, wall classes, and lawful-cell tests.
- Browser diagnostic: a deterministic computation has passed, but it is not an external theorem certificate. Wall pathology checks and a completed small exact coorientation search normally belong here.
- Drawing: coordinates, spacing, clipping, transparency, and camera choices used to make the same incidence data legible.
A result called maximum must come from a completed exhaustive or branch-and-bound search with matching bounds. A timed or heuristic search is reported as best found, together with its lower bound and any available upper bound.
The app never treats a convenient 3D placement as part of the cell complex. Node coordinates, force relaxation, parallel-rail offsets, wall arcs through a polygon, transparency, clipping, and camera choices are drawings. The objects they refer to can still be exact: a wall arc, for example, connects the exact pair of opposite boundary occurrences recorded for that relation cell.
Projection mode is also a drawing. Even when interval certificates support the reflection data or bound selected coordinates, the final three-dimensional axes or PCA view need not preserve hyperbolic distances, angles, or intersections.
The wall workflow does not by itself prove incoherence or a fibering theorem. The Jankiewicz--Wise argument uses additional hypotheses, including an appropriate finite torsion-free cover, globally coorientable two-sided walls, an aspherical affine 2-complex, and nonempty connected ascending and descending links. Embeddedness and absence of self-osculation support the paper's random orientation estimates; they are not extra gates once one concrete coorientation and all of its cells and links are checked directly. Their incoherence result adds further group-theoretic and Euler-characteristic input.
The intended virtual algebraic-fibering output is nevertheless concrete: a
finite-index subgroup H, an explicit primitive homomorphism H -> Z, checked
cell-boundary sums, and the relevant Morse links. The app should call this a
verified algebraic fibration only when the finite-index, torsion-free,
surjectivity, affine/aspherical, and finitely-generated-kernel hypotheses all
carry matching evidence.
This matters for the compact hyperbolic 5-dimensional examples. Their
rank-two compression is useful for finding and coorienting walls, but it is not
the complex on which the final five-dimensional Morse links are checked. The
full certificate reconstructs every spherical Coxeter cell of
K = H\Sigma, gives those cells one compatible pulling subdivision, builds an
exact rational height, and checks both directed links at every quotient vertex
orbit. Until a complete torsion-free action and every later stage pass, the UI
reports an incomplete calculation rather than a fibering claim.
There are therefore two independently replayed certification tracks:
- the lawful-subcomplex-first track,
which checks the actual retained polygonal complex and transfers finite
generation through
pi_1(Y) -> H; - the full Davis-quotient track,
which constructs an explicit PL Morse model on every Coxeter cell of
K = H\Sigma.
The first track also records an optional generalized lawful subcomplex: remove every unlawful 2-cell and all of its higher cofaces. That rule produces a genuine maximal subcomplex, but closure alone does not prove its asphericity or extend the Morse map across retained higher cells. Those are separate gates.
The full-Davis note defines the complete cell poset, quotient walls, Reidemeister--Schreier generators, gcd normalization, pulling triangulation, quotient-periodic tie breakers, full ascending and descending links, and the optional collapsibility check. It also states exactly why a passing algebraic fibration is not automatically a locally trivial topological bundle.
The end-to-end command-line path now accepts a complete transitive right coset action for any bundled Coxeter system. The input is the action, not merely a list of subgroup generators:
{
"id": "i2-5-regular-action",
"index": 10,
"generatorImages": [
[1, 0, 3, 2, 5, 4, 7, 6, 9, 8],
[9, 2, 1, 4, 3, 6, 5, 8, 7, 0]
]
}Each generator row must contain exactly index zero-based images; the number
and order of rows must match the bundled Coxeter generators. Run:
corepack pnpm cover:fiber:user-action -- \
--example I2_5 \
--action my-action.json \
--output promotion.jsonThe command distrusts any torsion-free label, rechecks the Coxeter relations and every spherical-special-subgroup orbit, constructs the quotient, and runs the bounded lawful-first/full-Davis wall-character search. A passing result is a replayed virtual algebraic fibration. A failed or incomplete result is only about the recorded search family and bounds. For an infinite compact Coxeter group, subgroup words alone are not yet converted to a finite coset action by the generic exporters.
This is not yet a full integral H^1 or smooth-fibering orchestrator. The
scalable generic integral H^1 backend can prepare and certify large action
matrices, but its emitted kernel witness is not wired into the all-character
Morse/link search. The materialized path searches wall characters. Smooth
fibering is not certified: the repository has no source-bound
manifold/PL/smoothing verifier, and caller-supplied IMM-style booleans are
ignored.
Current theorem-facing results are:
- the JNW rank-eight control passes virtual algebraic fibering but is a two-dimensional Davis complex, not a compact hyperbolic 5-manifold;
- the imported compact-cube action has
H^1 = Z^19, but its recorded Track-B height complex obstructs every nonzero integral character and its bounded rescue found no passing original-vertex link system; - the P0/P1 and Tumarkin compact portfolio has certified source plans but no materialized torsion-free action, hence no fibering result.
See the research portfolio, the generic external job contract, and the scalable integral-H1 protocol.
Install Node.js with Corepack enabled, then run:
corepack enable
corepack pnpm install
corepack pnpm devVite prints a local address, usually http://127.0.0.1:5173/. After the
dependencies are installed, ordinary use of the viewer is offline. Sage, GAP,
KBMAG, and CoxIter are optional external research tools, not browser runtime
dependencies.
For a production-style build:
corepack pnpm build
corepack pnpm previewThe static build is written to dist/.
The desktop application is a Tauri v2 wrapper around the same viewer. Desktop development also requires Rust and the Tauri prerequisites for your operating system.
corepack pnpm desktop:devBuild an unsigned local bundle with:
corepack pnpm desktop:buildPlatform packages are written below src-tauri/target/release/bundle/.
The v0.2.0 research preview contains the previously published web and desktop artifacts. Those binaries may predate the cover-compression rewrite described by the current source tree. Windows artifacts are unsigned and macOS artifacts are not notarized, so the operating system may show a first-launch warning.
The repository includes small finite examples, generated Sage/GAP fixtures, the certified regular ideal hyperbolic 3-cube, certified compact 5-cube and compact 5-prism-family data, and the compact eight-facet catalogue transcribed from Tumarkin's classification. Each certificate has a limited scope. A passed Gram/signature check, for example, does not certify the browser's 3D placement.
The ideal 3-cube is the golden cover example. Its six facet generators are the
transpositions t12, t13, t14, t23, t24, and t34. Two generators
have m = 3 when the transpositions share a letter, so Gamma is an octahedron;
the three disjoint pairs have m = inf. Sending tij to (ij) gives a
surjection onto S4. The bundled regular action has 24 points, and the app
checks every spherical A1 and I2(3) restriction before calling its kernel
torsion-free. The cube is finite-volume and ideal, not compact: each vertex
link is the Euclidean triangle (3,3,3).
Open Choose Example -> Certified eight-facet catalogue to reach all 16 eight-facet cases without expanding the first-use interface.
Finite cover construction needs more than a Coxeter matrix, but users should not normally have to write the missing action by hand. The primary backend enumerates prime-order torsion in spherical special subgroups, constructs exact congruence images, and rejects impossible action degrees from matrix-group and fixed-point-mark data before it constructs a coset action. Compatible partial actions are cached and may be combined on diagonal orbits. Generic GAP low-index enumeration remains a bounded fallback. A selected action receives a second, independent spherical-action certificate in the app.
The status panel separates torsion-free finite-index kernel, exact index certified, and usable finite cover materialized. Exact matrices over a finite field plus the complete spherical-injectivity checks can establish the first before a structural computation determines the image order. The wall and fibering pipeline requires the third.
The browser accepts the generated artifact or a complete manually supplied generator action on a finite vertex set. Manual import is an advanced compatibility path. When evidence is absent, the app may validate and display the incidence, but it does not call the action torsion-free.
External tools follow the same rule. A missing tool produces a skipped or blocked artifact, never a silent downgrade to a stronger in-repo claim.
Run the ordinary release checks from the repository root:
corepack pnpm format
corepack pnpm lint
corepack pnpm test
corepack pnpm build
corepack pnpm exec playwright test
corepack pnpm bench:timed:check
corepack pnpm workflow:validate
corepack pnpm validate:research-gradeUseful research checks include:
corepack pnpm compare:backends
corepack pnpm compare:quotient-backends
corepack pnpm validate:virtual-fibering
corepack pnpm registry:validate
corepack pnpm session:validate
corepack pnpm certify:compact-5-cube
corepack pnpm certify:compact-5-prism
corepack pnpm check:independentCommands that invoke Sage, GAP, KBMAG, or CoxIter require those tools to be installed or available through the documented container/WSL path.
- Mathematical conventions:
X,\hat X,\bar X, walls, lawful cells, links, and theorem boundaries. - Automatic torsion-free cover discovery:
spherical torsion witnesses, bounded GAP search, exact Sage congruence
kernels, Everitt-style composite actions, and the route from a subgroup to
H -> Z. - Coordinated compact-cube cover search:
the finite-target, composite-module, and geometry-informed search tracks,
including the exact reason every cover degree is divisible by
5,760. - Virtual algebraic fibering certificate: Schreier generators, wall periods, primitivity, lawful cells, and the exact PL Morse checklist.
- Data format: import contracts, signed attaching maps, compression certificates, wall results, and export status.
- Viewer design: model switch, Covers + Walls workflow, drawing rules, performance, and interaction design.
- Walkthroughs: short guided readings of Davis cells, cover compression, walls, lawful cells, Gamma, and projections.
- References: sources and the exact claims each source supports.
- Tooling: external backends, containers, desktop builds, and release commands.
Unless otherwise noted, CoxeterViewer5D source code, scripts, bundled JSON examples, and documentation are released under the Apache License 2.0. Source references cited in the data remain the property of their authors and publishers; this project licenses only its own transcriptions, code, and generated artifacts.
Academic citation metadata is in CITATION.cff.