# Translation-free exceptional topology research docket

**Status:** active  
**Parent:** [programme arm](README.md)

## 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.