Translation-free E8 coupled-wire and tomography protocol
Date: 26 October 2026
Status: continuum-local derivation and held-out current test complete; finite-qudit few-body surrogate closed; continuum gap open
Parent: translation-free exceptional topology
Evidence: frozen carrier and detector ledger
Certificate: null-vector and root-shell verifier
Derived edge rather than imposed edge
Place a nonchiral pair (K{\rm wire}=K{E8}\oplus(-K{E_8})) on each of seven one-dimensional wires. Between consecutive wires pin the eight local integer interactions
Every pair of interaction vectors obeys the Haldane null condition. The 48 independent terms gap the counterpropagating interior pairs and leave exactly one (-E8) edge and one (+E8) edge. Thus the chiral boundary is the ungapped remainder of a two-dimensional local array; it is no longer merely declared as an isolated boundary action.
The frozen realization uses unequal transverse spacings and 48 unequal, strictly positive pinning strengths. It therefore has no transverse translation symmetry. This is a translation-free path graph, not yet random connectivity. Nonuniform positive amplitudes retain the same strong-coupling pinning pattern, but this algebraic statement is not substituted for a thermodynamic many-body gap calculation.
The E8-specific detector
Thermal transport sees only (c_-=8). The next detector measures the short-distance product exponents of eight calibrated simple currents. For vertex currents labelled by simple-root coordinates (a,b), the exponent is
The eight-by-eight training record reconstructs the integral Gram form. The test then withholds every nonsimple current. Weyl reflection of the recovered simple roots predicts a shell of exactly 240 distinct norm-two vectors, all with scaling weight one. It also predicts the complete ordered-pair exponent histogram
The carrier passes only if independently excited held-out resonances reproduce that shell and its product exponents without refitting (K). This is stronger than recognizing eight modes or a thermal coefficient: a wrong rank-eight current lattice will generally fail the withheld root closure.
Apparatus translation
The abstract protocol requires:
- seven multi-channel one-dimensional units with separately addressable right- and left-moving bosonic fields;
- 48 neighboring-wire phase-locking controls compiled from the eight columns of (K{E8});
- irregular wire spacing or deliberately nonperiodic bond calibration;
- edge calorimetry for the propagation sign and total (c_-);
- coherent multiport edge injection and frequency-resolved complex response;
- nonlinear or coincidence spectroscopy capable of extracting current-product exponents for the training and held-out sources.
The likely platforms are synthetic resonator networks, superconducting circuits, cold-atom wires or strongly interacting electronic wires. No one of those platforms is selected here because the remaining compilation cost—the many-field cosine to few-body controls—is platform dependent.
Claim boundary and next gate
This tranche closes the continuum-local B1 construction and opens a finite E3 detector. The finite-qudit compiler now closes the bounded-degree hardware and exact many-body-gap gate for a digital clock surrogate. It does not close random connectivity, clock-to-rotor convergence, continuum E8 response or laboratory feasibility. Failure of that next promotion demotes the few-body result to a locking-incidence carrier while retaining the exact detector.
Primary anchors
- Lopes, Quito, Han and Teo, A non-Abelian twist to integer quantum Hall states.
- Lim, Mulligan and Teo, Partial fillings of the bosonic E8 quantum Hall state.
- Leech coupled-wire blueprint.