Richard Hoekstra
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Confinement from the conductor: the defect spectrum of a singular Hecke order

Paper IX in the singular Hecke node series
Richard Hoekstra · 2026 · PDF

The singular order

The characteristic polynomial of T₂ on S₂(Γ₀(163)) factors over Q as x · g₅(x) · g₇(x) with g₅, g₇ irreducible of degrees 5 and 7. The root x = 0 corresponds to the rational newform f (since a₂(E) = 0). Over Q₃, g₅ acquires a linear root α₅ with α₅ ≡ 12 (mod 27), and g₇ acquires a linear root α₇ with α₇ ≡ 3 (mod 27).

The congruence

The congruence underlying the singularity is ap(fV₁) ≡ ap(fV₅) (mod 3) for all primes p, where fV₁ = f is the rational newform attached to E : y² + y = x³ − 2x + 1, and fV₅ is a degree-5 eigenform defined over Q[x]/(x⁵ + 5x⁴ + 3x³ − 15x² − 16x + 3) with discriminant 65657.

The pair factor of the singular Hecke order is:

R = {(σ₁, σ₂) in Z₃² : σ₁ ≡ σ₂ (mod 3)} ≅ Z₃[η]/(η² − 3η)

with conductor c = 3R and normalisation R̃ = Z₃ x Z₃. The conductor exact sequence is 0 → R → Z₃ x Z₃ → F₃ → 0.

The three sectors

The residue classes R/3R decompose into three sectors determined by the conductor boundary:

SectorDefinitionRepresentativeSpectral property
B₀ (vacuum)c/3c — sections supported at the node(3, 0)Energy converges: E₀ = 1215 = 5 · 3⁵, stable for k ≥ 4
B₁ (matter)R \ c, branch 1(1 + 3a, 1 + 3b)Energy diverges: Emin(k) = Θ(32k)
B₂ (matter)R \ c, branch 2(−1 + 3a, −1 + 3b)Spectrally identical to B₁

The vacuum energy is computed from the Hecke invariant: Σ ap(E)² = 0 + 4 + 36 + 36 + 49 + 4 = 129 = 3 · 43, giving E₀ = 9(2d + 129) = 9 · 135 = 1215. The vacuum tower is exactly geometric: E(3n, 0) = 9n · 135 for 1 ≤ n ≤ k − 3.

The Z/2Z symmetry and Dsg(R)

Z/2Z spectral symmetry The map (σ₁, σ₂) ↦ (−σ₁, −σ₂) sends B₁ ↔ B₂ and preserves the defect energy E. The spectra of B₁ and B₂ are identical at every precision k.
Σ² = Id The suspension functor Σ on the singularity category Dsg(R) satisfies Σ² = Id. This 2-periodicity is realised by the negation map above. The Grothendieck group K₀(Dsg(R)) ≅ Z/2Z, with the σ₁-soliton (3, 0) carrying charge +1 and the σ₂-soliton (0, 6) carrying charge −1.

Proof sketch. The singularity category Dsg(R) = Db(R-mod)/Perf(R) captures the non-free modules. Since R is a curve singularity (two copies of Spec(Z₃) meeting at a point), Dsg(R) has period 2. The negation symmetry (σ₁, σ₂) ↦ (−σ₁, −σ₂) interchanges the two matter sectors while fixing the vacuum. Combined with the quadratic nature of E, it preserves the energy functional. The Grothendieck group detects only the parity of the branch, hence K₀ ≅ Z/2Z. Cross-branch annihilation is algebraic: (3, 0) · (0, 6) = (0, 0).

Confinement = Hensel obstruction

Confinement The vacuum sector lifts to Z₃: the element (3, 0) is a fixed integer. The matter sector does not: the optimal element at precision k changes at every k, and Emin(k) = Θ(32k) → ∞. The conductor c = 3R is the sharp boundary — Hensel lifting converges on c and fails on R \ c.

Numerical verification at successive precisions:

k3kEvacEmatterGap
327324657333
481121530871872
524312152344522230
67291215185877184662

The vacuum energy stabilises at k = 4. The matter energy grows as Em ≈ 0.35 · 32k. The Z/2Z symmetry is exact at every k.

This paper gives the defect spectrum of the singularity. The algebraic structure — three sectors, 2-periodic suspension, K₀ = Z/2Z — is the skeleton on which the Hamming code (Paper IV) and the automaton (Paper V) are built.
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