The Prime-Hyperoctahedron Witt Tower: Self-Similar Spectra from Primorial Residues

The Prime-Hyperoctahedron Witt Tower: Self-Similar Spectra from Primorial Residues (Version 3) The eight units modulo 30, {1, 7, 11, 13, 17, 19, 23, 29}, form the vertex set of the 16-cell, the four-dimensional cross-polytope β₄. The antipode v ↦ −v is realised as r ↦ 30 − r, and the four antipodal axes are {1, 29}, {7, 23}, {11, 19} and {13, 17}. This paper lifts this graph through the primorial chain N₁ = 30, N₂ = 210, N₃ = 2310, …, N_T = p₁p₂⋯p_{T+2} (with p₁ = 2) by a single rule: two units modulo N_T are adjacent if and only if their reductions modulo N_{T−1} are distinct and adjacent. The resulting family of graphs G_T is the prime-hyperoctahedron Witt tower. Main results (proved) • Tensor form: A_T = A₁ ⊗ J_{f₂} ⊗ ⋯ ⊗ J_{f_T}, with f_j = p_{j+2} − 1 and J_f the all-ones matrix. • Spectral self-similarity: spec(A_T) = {6Π_T (once), −2Π_T (three times), 0 (φ(N_T) − 4 times)}, where Π_T = ∏_{j=2}^{T} (p_{j+2} − 1) = φ(N_T)/8. Hence Π₂ = 6, Π₃ = 60, Π₄ = 720. The rescaled matrices A_T/Π_T carry the 16-cell spectrum {6, −2, −2, −2} unchanged to every level • Axis rule: two units r, s modulo N_T are adjacent if and only if r ≢ ±s (mod 30). • Cayley form: G_T is the complete four-partite Cayley graph K_{m,m,m,m} with m = 2Π_T on the unit group (ℤ/N_Tℤ)^×. Its parts are the preimages of the four antipodal axes modulo 30. • Characters: the Dirichlet characters modulo N_T form an orthogonal eigenbasis. Only the lifts of the four even characters modulo 30 have non-zero eigenvalues: the trivial character, the quadratic character (·/5) and the two even quartic characters of conductor 15. • Rigidity: every odd character, and every character whose conductor does not divide 30, has eigenvalue 0. The tower is a rigid carrier of the 16-cell and adds no new spectral information on higher levels. • Symmetry: Aut(G_T) = S_m ≀ S₄, and G_T is strongly regular with parameters (4m, 3m, 2m, 3m). • Icosahedron comparison: every eigenvalue of the tower is an integer, so neither the golden ratio nor √5 occurs in its spectrum. The icosahedron has spectrum {5, √5 (three times), −1 (five times), −√5 (three times)}. Three towers, clearly separated The paper separates three constructions that share vocabulary but are different mathematical objects: • the Witt tower G_T on the primorial unit groups (this paper); • the cross-polytope reduction tower β_n → β₄ of The Geometry of Unity, a cocktail-party graph with a different spectrum (at 48 vertices the degrees are 36 and 46); • the power tower on the moduli 30^j, whose inverse limit is ℤ₂^× × ℤ₃^× × ℤ₅^×. Results are not transferred between these towers without proof. Figures 1. The 16-cell β₄ on the units modulo 30, with its four antipodal axes. 2. The first three levels of the primorial chain (8 → 48 → 480 units), coloured by axis. 3. The spectra of the Witt graph G₂ and of the cross-polytope graph β₂₄ on the same 48 vertices. Changes from Version 2 • Product index in the abstract corrected. The formula in Version 2 was shifted by one and gave Π₂ = 4; the correct value is Π₂ = 6. The tables and proofs were not affected. • Standard indexing p₁ = 2 throughout, with a dictionary T ↔ k = T + 2 to The Shape of Numbers. • New: the axis rule, the complete four-partite Cayley form, characters as eigenvectors, the rigidity corollary, the exact automorphism group and strong regularity. • New: an explicit separation of the Witt tower, the cross-polytope reduction tower and the power tower. • Icosahedron spectrum corrected. • Withdrawn: • the “structural difference Δ”; • the claim that the lift is the unique bilateral, antipode-preserving, degree-regular blow-up; • the applications to expanders, statistical physics, coding and prime enumeration. • Three new figures. All numerical statements were re-run with the attached script. Verification The attached script witt_tower_check.py (Python 3, NumPy, SciPy, SymPy) checks: • for T = 1–4: the construction from the stagewise rule, its equality with the axis rule and the Cayley rule, the one-step tensor identity, and the full spectrum; • for T = 5 (92 160 vertices): a matrix-free Lanczos computation returning 69 120 and three times −23 040; • the unit windows, the icosahedron spectrum and the spectrum of β₂₄. It runs in under a minute on a standard laptop.