Translation-free exceptional topology research docket
Status: active Parent: programme arm
Tranche A — establish the translation-free trunk
Progress: A1 has a fixed-connectivity structurally disordered realization; A2 is closed for Bott/localizer agreement on the frozen finite family; A3 is open beyond the certified four-point jitter interval and toward random connectivity, ensemble and size scaling.
| ID | task | output | gate |
|---|---|---|---|
| A1 | choose a two-dimensional local Hamiltonian with crystalline and random-point realizations | frozen carrier and parameter file | same local degrees and couplings in both presentations |
| A2 | implement Bott, local Chern and spectral-localizer diagnostics | cross-checked invariant notebook/script | agreement away from finite-size transition windows |
| A3 | sweep structural and coupling disorder | phase tube with uncertainty | invariant plateau plus mobility/localizer gap |
| A4 | open a boundary and inject a local drive | bulk--boundary record | oriented edge transport matches bulk sign |
| A5 | adversarially vary detector resolution | quotient ledger | topology distinguished from detector survival |
Tranche B — insert the exceptional chiral unit
Progress: B1 is closed at continuum-local coupled-wire level and at finite-qudit surrogate level: an integral dual-basis compiler reduces every column cosine to two-body locks, whose commuting completion has graph degree one. B2 now has an exact width-independent gap above the edge-degenerate ground space for that clock carrier across 28 frozen width/disorder/local-dimension instances. Clock-to-rotor convergence, continuum E8 response and a material-native Hamiltonian remain open. Local fixed-coupling clock-to-rotor convergence is now certified through (q=129); the coupled eight-channel metric is no longer treated as uniquely implied by topology: the (K) and (K^{-1}) constitutive branches have exact, distinct Coxeter resonance fingerprints, and full eight-port calibration is the next gate. The E8 theta series now also supplies a shared complete-shell sparse basis through 3,721,681 states for the conditional nonlinear eigensolve; eigenvalue reconnaissance has reached 117,361 states, but the first-gap drift still fails the one-percent promotion gate. A restarted Davidson solve has now reached 340,321 states at shell eight; 3.11% and 7.80% drift still reject promotion. Shells 10--12 and leakage/transition estimators remain open. The (K)-kernel and inverse-kernel paths now bifurcate: all (K)-kernel cosine steps are present from shell one, while the inverse step set is not complete until shell 15 (3,721,681 states). Lower inverse rungs remain reconnaissance. The eligible (K)-kernel has reached shell 12 (1,604,401 states): two successive sub-one-percent increments promote the provisional unit-coupling compact-cell gap (0.1019267219), with boundary probability below (3\times10^{-5}). Parity requires an odd sine/phase drive to see the first gap. Five scalar coupling ratios and two non-collinear unequal-channel directions now pass the compact-cell gate; one positive kinetic-metric direction also passes. An open perturbation tube and intercell-width promotion remain open; one mixed amplitude-plus-metric corner passes and closes the planned finite stencil. A four-state-per-cell projected dimer now has a cutoff-stable gap and resolves collective from relative-cell modes through J=0.1. The projected gap now also passes through twelve roots; its splitting is quantitative through J=0.05. At that coupling, widths 3--5 and roots 6--10 close a 3x3 projected gap convergence grid. Periodic widths 3--5 form a flat odd collective gap plateau, and open/periodic disagreement falls below one percent by width four. A thermodynamic or full-charge-chain gap remains open. The cutoff-eight/eight-root signed local operator dictionary is now serialized and replay-certified, so larger-width scans no longer repeat the charge-shell solve or silently change the local carrier. The E8-specific detector predicts a held-out 240-current root shell. B4 is closed on the declared E8-cubed/Leech/Moonshine family: the bulk detector collides and \(\dim V_1=(744,24,0)\) separates the three candidates.
| ID | task | output | gate |
|---|---|---|---|
| B1 | compile an \(E_8\) coupled-wire/network Hamiltonian | source-native action | continuum schedule and two-local clock surrogate closed; material carrier open |
| B2 | disorder positions and couplings without closing the gap | disorder ensemble | finite-qudit uniform gap and local rotor limit closed; constitutive branch calibration and joined nonlinear gap open |
| B3 | construct three-copy rank-24 carrier | \(E_8^3\) benchmark | rank 24, determinant one, positive signature |
| B4 | compare to the corpus Leech \(K\)-matrix | blind collision benchmark | coarse bulk detector collides; root/current detector separates |
Tranche C — boundary enrichment and Monster gate
| ID | task | output | gate |
|---|---|---|---|
| C1 | implement coherent inversion and projection | untwisted orbifold rail | calibrated gauge/projection record |
| C2 | add the twisted sector | joined Hilbert carrier | sector dimensions and anomaly normalization pass |
| C3 | reconstruct finite witness stack | trace/defect/OPE bundle | rejects generic \(c=24\), \(E_8^3\) and Leech-only rivals |
| C4 | disorder the microscopic realization | RG/stability study | same infrared witness or explicit enrichment-loss boundary |
| C5 | close the return test | held-out prediction | same carrier reconstructs and predicts an unseen defect history |
Tranche D — quasicrystal material backplane
Progress: D0 is closed as a typed protocol. The source proposal's acoustic growth, phason and multi-sensor ideas have been retained, while automatic E8, new-gravity and room-temperature-superconductivity promotions are quarantined. Fabrication and measurement remain open.
| ID | task | output | gate |
|---|---|---|---|
| D1 | fabricate matched periodic, quasicrystalline and amorphous backplanes | blinded 3-class material set | diffraction and composition identify every class independently |
| D2 | resolve phason and ordinary mechanical channels | mode ledger | phason coordinate survives mounting and thermal nulls |
| D3 | install identity and E8 relation programmes | balanced 3x2 array | full eight-port transfer matrix matches assigned programme |
| D4 | measure gap, edge and root records | blind interaction contrast | quasicrystal-by-E8 contrast survives all 232 held-out currents |
| D5 | open material response channels | transport/thermal/nonlinear bundle | effect does not reduce to mass, temperature, drive or scalar transport |
| D6 | open precision residual channel | upper bound or replicated anomaly | conventional stress-energy and apparatus backgrounds rejected first |
Earliest useful result
The earliest publishable result need not reach Monster. A complete result is:
A translation-free local Hamiltonian retains a certified real-space topological index and bulk--boundary response across a disorder tube, while a declared finer edge observable is lost before the bulk topology changes.
That would experimentally and mathematically separate stable topology from reconstructive boundary memory—the exact distinction needed by the larger programme.
Stop rules
- Stop calling a Hall response topological if no real-space invariant or protected boundary response can be defined.
- Stop the \(E_8\) promotion if a smaller unimodular edge theory spans every measured record.
- Stop the Monster promotion if the data factor through central charge, untwined character or a smaller defect quotient.
- Retain a negative result as a quantified enrichment-loss boundary rather than weakening detector or provenance requirements.