Translation-free E8 edge protocol
Date: 26 October 2026
Status: exact effective-edge construction; microscopic random-point bulk and E8-specific detector remain open
Parent: translation-free exceptional topology
Evidence: edge ledger
Certificate: disordered edge verifier
Effective carrier
Use eight compact chiral bosons with edge action
where \(K=K_{E_8}\) is the positive Cartan form and \(V(x)\) is any local positive-definite interaction/velocity matrix. Spatial variation of \(V\) removes translation symmetry while leaving the integral commutator form, bulk invertibility and gravitational anomaly unchanged.
The exact topological data are
Thus the bulk has no fractional anyon quotient and the edge is entirely co-propagating. Positive spatial disorder can mix and rescale the modes but cannot introduce a counterpropagating channel inside this effective theory.
What is protected and what is not
Protected by the declared effective carrier:
- the integral even unimodular form;
- the absence of a nontrivial anyon quotient;
- the net chiral central charge and thermal Hall coefficient;
- the sign of every generalized mode velocity for positive \(V(x)\).
Not protected or identified by those data alone:
- equal velocities or a visible affine \(E_8\) current symmetry;
- microscopic translation symmetry;
- access to all 240 norm-two currents through the detector;
- a unique coupled-wire, spin, photonic or mechanical realization;
- the rank-24 Leech or Monster enrichment.
Thermal Hall response detects \(c_-=8\), not \(E_8\) provenance. The E8-specific promotion therefore requires reconstruction of the integral current lattice or a finite set of vertex-operator products that excludes a smaller or differently normalized chiral theory.
Disorder protocol
- Freeze \(K_{E_8}\) and the compactification lattice before generating any records.
- Draw or fabricate a spatial sequence of positive-definite matrices \(V(x_a)\), retaining their calibration provenance.
- Inject a tomographically complete set of edge pulses and measure the full complex eight-port transfer matrix versus frequency.
- Verify one-way energy propagation, the total thermal coefficient and the absence of a hidden counterpropagating pole.
- Reconstruct the integer charge/current lattice from source composition and multiport interference rather than from fitted eigenmode labels.
- Test held-out norm-two current products or fusion/OPE coefficients.
Promotion ladder
| rung | required witness | interpretation |
|---|---|---|
| E0 | eight co-propagating channels | chiral rank-eight response |
| E1 | \(c_-=8\), no fractional quotient | bosonic invertible bulk class |
| E2 | even unimodular integral current lattice | unique rank-eight lattice class |
| E3 | calibrated norm-two current/OPE witness | E8-specific boundary provenance |
| E4 | survival across spatial disorder | translation-free E8 edge |
The verifier closes E1 mathematically and stress-tests the same-chirality part of E4 for a frozen ensemble of positive random \(V(x)\). E2--E3 remain physical reconstruction gates.
Next join
Stack three independently calibrated E8 units and present their coarse \((24,1,24,24)\) record blindly beside the Leech edge. The first detector that separates them must involve root/current or defect data; thermal transport and bulk anyon content are guaranteed to collide.