Only the named unchanged CAR/QWZ representation with the same time. No TFPT no-go; choosing a state removes no added zero vector.
Read the exact result, assumptions and evidence
Result
All four hits add the same half-integer Cartan sectors and second K-zero vector. The already known integer-Cartan/ground-space obstruction therefore excludes equivalence to the unchanged source; positive algebraic extension remains possible.
Required assumptions
Fixed CAR source and adjoined half sector
Ordered eight-channel cocycle, s=(1/2)^8, Q, quarter phase and K_alpha=N+|p|²/2+delta_alpha.p are supplied; alpha=1/4,3/4. Parity equivalence is in the global operator envelope without a separately marked physical X. Neither source nor extension is derived from P1/P2.
Unchanged specified CAR/QWZ source
Compare the known quarter-CAR source or full mass=1, ny=8 QWZ operator with the identical Hilbert representation, Cartan action and physical time. Preserve its previously proved unique ground state and spectral gap. This does not compare all P1/P2 realizations.
Normal ground densities in the supplied K space; no state selection from P1/P2. The ground space was already known.
Read the exact result, assumptions and evidence
Result
All dual-invariant ground mixtures diag(a,1-a) on {0,r0} give the same shift functional. Original Cartan vacuum values select |0><0|; additional Rc invariance requires I2/2. These two conditions conflict.
Required assumptions
Fixed CAR source and adjoined half sector
Ordered eight-channel cocycle, s=(1/2)^8, Q, quarter phase and K_alpha=N+|p|²/2+delta_alpha.p are supplied; alpha=1/4,3/4. Parity equivalence is in the global operator envelope without a separately marked physical X. Neither source nor extension is derived from P1/P2.
Positive base functional and explicit observable algebra
For finite crossed words omega0 is positive; dual Z2 invariance is additionally required for unique algebraic extension. Comparing physical densities must retain the actual Cartan observables and the common normal ground space of supplied K.
Only the stipulated even E8 lift; full odd-Z8 sectors remain additional. No transfer of the different (9,1) source. The ±i path orientation is not selected.
Read the exact result, assumptions and evidence
Result
In the supplied positive E8 module X=E_s commutes with SU8 root currents because alpha.s=Q(alpha)=0; oscillators are fixed. C=Z_e0 R gives CXC^-1=±iX and P sign+1. This is a direct global operator bridge, not X=Y(s).
Required assumptions
Stipulated even E8 completion with actual Klein signs
Assume the E8 extension of the even D8 CAR sector constructed in seam-joint-ward-state, with original ordered cocycle, retained quarter transition and explicit P-lift convention. An ordinary-local inclusion of odd CAR fields is not assumed.
Fixed CAR source and adjoined half sector
Ordered eight-channel cocycle, s=(1/2)^8, Q, quarter phase and K_alpha=N+|p|²/2+delta_alpha.p are supplied; alpha=1/4,3/4. Parity equivalence is in the global operator envelope without a separately marked physical X. Neither source nor extension is derived from P1/P2.
No complete concrete Finsupp kernel instantiation, physical state selection or automatic K-invariance of the bare shift algebra.
Read the exact result, assumptions and evidence
Result
E0(y*y)=a0*a0+phi^-1(a1*a1) proves positivity of omega0 o E0. General algebra/star/positivity and dual uniqueness theorems are Lean-checked; concrete finite-lattice transfer is analytically proved.
Required assumptions
Fixed CAR source and adjoined half sector
Ordered eight-channel cocycle, s=(1/2)^8, Q, quarter phase and K_alpha=N+|p|²/2+delta_alpha.p are supplied; alpha=1/4,3/4. Parity equivalence is in the global operator envelope without a separately marked physical X. Neither source nor extension is derived from P1/P2.
Positive base functional and explicit observable algebra
For finite crossed words omega0 is positive; dual Z2 invariance is additionally required for unique algebraic extension. Comparing physical densities must retain the actual Cartan observables and the common normal ground space of supplied K.
Global crossed operators; no local field equivalence or physical selection.
Read the exact result, assumptions and evidence
Result
262144 candidates on two H200: four universally Lean-checked presentations. X_plus=Pi X_minus preserves source,Q,K,quarter marker and canonical state; two marked P classes remain. Full T1 and Ward are checked. Graded C lifts realize both P signs; scalar X covariance does not follow from the integer endpoint.
Required assumptions
Fixed CAR source and adjoined half sector
Ordered eight-channel cocycle, s=(1/2)^8, Q, quarter phase and K_alpha=N+|p|²/2+delta_alpha.p are supplied; alpha=1/4,3/4. Parity equivalence is in the global operator envelope without a separately marked physical X. Neither source nor extension is derived from P1/P2.
E8 current group conditionally determines the P sign
Only the stipulated even E8 lift; full odd-Z8 sectors remain additional. No transfer of the different (9,1) source. The ±i path orientation is not selected.
Reuse algebra, winding and positive extension. The common source candidate must first supply actual Cartan, ground-space and time compatibility; an identical shift functional is insufficient.
Original historical attempts linked to this question
These are original research records. Their reported verdicts are distinct from the reviewed results above.
quotient locality e8 20261014
Does the proposed eightfold fold/multi-interval completion derive physical quotient locality and an E8 field extension from the same source while preserving the quarter phase, state and time?
Recorded verdict: EXCLUDED_IN_UNCHANGED_SOURCE
Historical inventory; not independently reviewed by this export.
Which full charged crossed-defect extensions of the fixed ordered CAR source preserve its marking, W4 composition and supplied time law, and which obligations remain for a joint source/state/event derivation?
Recorded verdict: PARTIAL
Historical inventory; not independently reviewed by this export.