E8 bounded-degree clock hardware and gap certificate

Date: 26 October 2026

Status: exact finite-qudit hardware gap; continuum E8 promotion open

Parent: E8 coupled-wire tomography

Evidence: hardware and gap ledger

Certificate: exact compiler and gap verifier

The many-field interaction was a basis cost

For the continuum schedule, a column (K e_I) appears on each of two chiralities. In the simple-root basis the eight interactions therefore touch

fields. Since (K{E8}) is unimodular, (K^{-1}) is integral. The compact change of variables

is consequently an allowed integer relabelling, not a real-valued approximation. It sends

Every interwire interaction becomes the two-field lock

The fixed source-to-native mixer follows the E8 Dynkin adjacency and has maximum port degree four. In the native dual basis each bulk register has one partner across an interwire bond.

Exact finite-qudit Hamiltonian

Regularize every compact field by a (q)-level clock register with generalized Paulis (ZX=\omega XZ). For each interior pair define

The two operators commute because their two Weyl phases cancel. With (d_q=1-\cos(2\pi/q)), use

This is a sum of two-body terms. Each pair has a unique generalized-Bell ground state and first excitation energy (min(J{yI},h{yI})). Different bulk pairs are disjoint, while the eight left and eight right edge registers remain free. Therefore the full ground space has dimension (q^{16}), and the gap above that edge-degenerate space is exactly

This is a many-body statement: it is the exact spectrum above the complete ground subspace, not a harmonic normal-mode proxy.

Width, local dimension and disorder sweep

The frozen sweep covers (q=5,7,11,17) and widths (N_y=2,3,5,8,13,21,34), for 28 Hamiltonians and as many as 264 bulk pairs. Every (J) and (h) is independently drawn from ([0.75,1.25]). The exact uniform bound is therefore

at every width and every tested clock dimension; the smallest sampled gap is (0.750080793514). Increasing width adds more gapped Bell pairs but does not soften any existing pair or couple the two free edges.

What has and has not become hardware

Closed here:

• the eight many-field cosines reduce exactly to two-field locks;
• every static term is two-local;
• native bulk graph degree is one and source-mixer degree is at most four;
• the disordered finite-qudit many-body gap has an analytic, width-independent lower bound;
• the two opposite edge register sets survive at every width.

Not closed here:

• convergence of this clock regularization to compact rotors or chiral bosons;
• recovery of (c_-=8), the E8 current OPE or thermal Hall transport from this stabilizer Hamiltonian;
• a material-native realization of the generalized-Pauli couplers;
• stability under interactions that connect different Bell pairs;
• random rather than path connectivity.

The result therefore closes the bounded-degree hardware and finite many-body gap gate at the digital surrogate level. The next promotion must add the continuum kinetic metric and show clock-to-rotor convergence without losing the gap or the held-out 240-root response.

External anchors

• Teo and Kane, From Luttinger liquid to non-Abelian quantum Hall states.
• Lopes, Quito, Han and Teo, A non-Abelian twist to integer quantum Hall states.
• Yang et al., Quantum Double in a Superconducting Wire Array.