The Setup
The 600-cell, viewed as the Cayley graph of the binary icosahedral group V600 = 2I, has 120 vertices. Inside 2I sits the binary tetrahedral group 2T, of order 24. The right cosets of 2T in 2I form five disjoint blocks of 24 vertices each — geometrically, five copies of the 24-cell whose union is the entire 600-cell.
600-cell, 120 vertices
Cayley graph of 2I with standard short-edge generating set; uniform degree 12 under H4 Coxeter transitivity.
Five disjoint 24-cells
Right cosets of 2T < 2I, each of size 24. The five cosets partition the 120 vertices of the 600-cell with no overlap.
The Lift Map
Each local 24-cell carries its own Laplacian L24. Eigenvectors of L24 are functions on 24 vertices. The natural lift extends such a function to all 120 vertices of the 600-cell by setting it to zero on the four other cosets. In general, a lifted local eigenvector is not an eigenvector of the full 600-cell Laplacian L600 — cross-coset coupling produces residuals on the off-coset edges.
The Selectivity Table
Exact-rational computation of the lift map at each local 24-cell eigenvalue gives:
| Local λ | Lift-image dim | dim E600(λ) | Intersection dim | Verdict |
|---|---|---|---|---|
| 0 | 5 | 1 | 1 | PARTIAL (kernel only) |
| 4 | 20 | 0 | 0 | E600(λ) = {0} |
| 8 | 45 | 0 | 0 | E600(λ) = {0} |
| 10 | 40 | 0 | 0 | E600(λ) = {0} |
| 12 | 10 | 25 | 10 | FULL — the bridge |
Only λ=12 sits in the spectrum of both the local 24-cell Laplacian and the global 600-cell Laplacian with a non-trivial lift intersection. The 2-dimensional local eigenspace on each coset, summed across the five cosets, contributes a 10-dimensional lifted subspace that sits inside the 25-dimensional global λ=12 eigenspace, exactly. λ=0 contributes only the trivial constant; λ ∈ {4, 8, 10} produce lift images that the global Laplacian does not share.
The Theorems
For each of the five 24-cell cosets, the 2-dimensional λ=12 eigenspace of the local Laplacian zero-extends to an eigenvector of the global 600-cell Laplacian at the same eigenvalue λ=12. Off-coset residuals vanish identically over ℚ.
The 10 = 5 × 2 lifted eigenvectors are linearly independent and span the lifted subspace Elifted ⊂ E600(12).
The non-trivial central element −1 ∈ 2I acts trivially on E600(12). Equivalently, the global λ=12 eigenspace descends to a representation of the icosahedral rotation group I ≅ A5.
The character values of A5 acting on E600(12) and on Elifted are exact integers, computed in ℚ(√5) arithmetic. No floating-point approximation enters the character calculation.
Under A5, the 25-dimensional global λ=12 eigenspace decomposes as:
Elifted ≅ 2 · Y5 Eresidual ≅ 3 · Y5
where Y5 is the 5-dimensional irreducible representation of A5. The lifted and residual sectors are both pure Y5-isotypic, with multiplicities 2 and 3 respectively. Total: 10 + 15 = 25 dimensions.
Why The Selectivity Is Non-Trivial
On a Cayley graph, a function supported on a single coset is generally not an eigenvector of the full Laplacian, because edges connecting that coset to other cosets contribute non-zero values at the boundary. The lift map respects the eigenvalue only when the local eigenvector lies in the kernel of the cross-coset coupling operator.
The selectivity table records that this annihilation happens exactly once across the five non-zero local eigenvalues — at λ=12. The other three non-zero local eigenvalues (4, 8, 10) are not in the global spectrum at all, so the lift cannot land. λ=0 sits in the global spectrum but with only the trivial constant in the global kernel, intersecting the 5-dimensional local-kernel lift in a single direction. Only λ=12 gives a full-rank match between the lift-image dimension and the global eigenspace dimension along the shared sector.
Mapping Numerics to Claims
Exact-rational claims
Selectivity table, λ=12 lift identity, 10+15 = 25 splitting, A5 character integers, 2·Y5 ⊕ 3·Y5 decomposition. All verified in pure ℚ arithmetic.
NumPy independence
Floating-point NumPy/SciPy is run only as an independent cross-check on the rational computation. Residual ≤ 6.2 × 10−15 across all certificates. Used for confidence, never as the source of truth.
Not claimed here
Standard Model derivation, particle masses, cosmology, black-hole physics, RH, consciousness. This note is narrow geometry. Downstream interpretation is a separate question.
Reproduction
The full result reproduces from a single Python entry point with no external network access required:
pip install -r requirements.txt
python closure_transform_engine/examples/run_wo008_keystone.py
pytest closure_transform_engine/tests/test_wo008_keystone_artifact.py
The repository vendors its own exact-arithmetic icosian quaternion package (vfd_v600); no other VFD repository is needed. Python 3.8+, NumPy, SciPy, SymPy. MIT licence.
What Is Explicitly Out of Scope
- Derivations of the Standard Model, particle masses, or any flavour physics
- Cosmology, expansion rates, CMB structure
- Black-hole thermodynamics or Hawking radiation
- Riemann hypothesis or any analytic number-theoretic claim
- Theories of consciousness
- Statements about whether the 600-cell is the unique substrate of any physical theory
- Continuum-limit physical interpretations of the λ=12 channel
One eigenvalue threads five 24-cells into the 600-cell. The selectivity is verified, the decomposition is exact, the arithmetic is rational.