Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Carrier, families and E8 closure
The published foundation in this area
32 authored claim references lead into the original ledger and paper sections. This source record remains available even where no separate reviewed research packet is attached.
The finite compiler combines a five-slot carrier, a selected half-spinor action, D5×A3 and Z4 glue into an E8 lattice closure. This establishes a mathematical construction within its stated choices, not the physical gauge group or a first-principles origin.
Starting structure in the Atlas orientation
Under the stated actions, D5×A3 with Z4 glue gives the even unimodular rank-8 E8 lattice; the half-spinor reading depends on the chosen group action.
Which connection is sought
Open items include physical boundary realization, a local interacting chiral 3+1D implementation, mirror decoupling, and derivation of the action from the origin.
How it connects
E8 carries internal clock, code, flavor, and arithmetic structures. Rank, representation, family count, and spacetime dimension remain distinct types.
Only fixed fold time and fixed 5+3 marking. Covariant marking transport is positively constructed; other sources or dynamical dictionaries are not excluded.
Read the exact result, assumptions and evidence
Result
After actual quarter conjugation the D5 projections of the time axis have distinct squared norms 129/256 and 137/256. Hence no orthogonal T satisfies T d_beta=d_beta and T O(P5)T^-1=P5. This excludes every Weyl correction in this fixed comparison.
Required assumptions
Given chiral source in both quarter domains
D_alpha=-i∂theta on f(theta+2pi)=exp(2pi i alpha)f(theta), filled negative spectrum and relative particle/hole excitation time; alpha=1/4,3/4. This compares two conjugate domains, not opposite charges in one fixed domain or all P1 sources.
Stipulated E8 quarter-fold dictionary
The existing Weyl lift M, affine E8 module, Q'=2(H3+H6) and K_alpha=L0+delta_alpha.H0 with alpha=1/4,3/4 are held fixed for comparison. This representation is not derived from P1/P2; an unbroken SM time action is not assumed.
First 5+3 marking and Z4 character supplied
In the supplied E8 dictionary, D5 on the first five coordinates, D3 on the last three and k=2sum(p1..p5) mod4 are given for the first quarter domain. The other domain may transport these data covariantly; their original physical selection is not presumed.
Exact map of the supplied source and conditional E8 lift; first marking, source realization and physical event law are not derived from P1/P2. Both quarter domains remain explicit; no switch to another branch to solve a fixed-sector question.
Read the exact result, assumptions and evidence
Result
n→−n−1 forces j→7−j and O=−MRM^T with O(x)=(x4,−x5,−x6,x1,−x2,−x3,x8,x7). O d_alpha=d_(1−alpha); P5_beta=O P5_alpha O^-1 and k_beta=k_alpha∘O^-1 need no new marking choice. Sea state and positive time are transported; the E8 current lift assumes the existing extension.
Required assumptions
Given chiral source in both quarter domains
D_alpha=-i∂theta on f(theta+2pi)=exp(2pi i alpha)f(theta), filled negative spectrum and relative particle/hole excitation time; alpha=1/4,3/4. This compares two conjugate domains, not opposite charges in one fixed domain or all P1 sources.
Stipulated E8 quarter-fold dictionary
The existing Weyl lift M, affine E8 module, Q'=2(H3+H6) and K_alpha=L0+delta_alpha.H0 with alpha=1/4,3/4 are held fixed for comparison. This representation is not derived from P1/P2; an unbroken SM time action is not assumed.
First 5+3 marking and Z4 character supplied
In the supplied E8 dictionary, D5 on the first five coordinates, D3 on the last three and k=2sum(p1..p5) mod4 are given for the first quarter domain. The other domain may transport these data covariantly; their original physical selection is not presumed.
Correction of the finite selection inference; no identification of I with the actual Calderon operator and no P1 countermodel.
Read the exact result, assumptions and evidence
Result
CAR.PAIR.01 counts complex-bilinear singlets. The Hermitian scalar <f,g> already exists on S+; CAR.PAIR.02's 'scalar iff sheet-odd' inference is false in this reading. The previous J_C includes reversion signs and is fully Spin(10)-equivariant.
Required assumptions
Original oriented exterior carrier
Given E=C5=E3⊕E2, a Hermitian metric, volume orientation, S± and Y=(-1/3)^3⊕(1/2)^2. Internal carrier labels are not physical vacua; geometric seam reflection is not identified with carrier conjugation.
Given chiral source. No selection from P1/P2, no equal responses within one fixed twist, and no identification with affine r0.
Read the exact result, assumptions and evidence
Result
n→−n−1, alpha→1−alpha gives C D_alpha C^-1=−D_(1−alpha). Particles become holes and conversely; normal charge reverses and positive excitation energy is preserved. The unique sea state maps to the other sea state without adding a zero vector within one domain.
Required assumptions
Given chiral source in both quarter domains
D_alpha=-i∂theta on f(theta+2pi)=exp(2pi i alpha)f(theta), filled negative spectrum and relative particle/hole excitation time; alpha=1/4,3/4. This compares two conjugate domains, not opposite charges in one fixed domain or all P1 sources.
Representation with metric/orientation; J_C is not yet the actual boundary/OS conjugation. Two carrier labels are not two vacua.
Read the exact result, assumptions and evidence
Result
J_C=∏(epsilon_i+iota_i)K is antiunitary, exchanges S± and reverses Y. Exactly Lambda0 and Lambda5 have Y=0. J_C dGamma(h) J_C^-1=tr(h)I−dGamma(h). It is fully Spin(10)-equivariant. Reuse CAR.PAIR.01's bilinear pairing; CAR.PAIR.02's Hermitian selection inference has been corrected and is not evidence.
Required assumptions
Original oriented exterior carrier
Given E=C5=E3⊕E2, a Hermitian metric, volume orientation, S± and Y=(-1/3)^3⊕(1/2)^2. Internal carrier labels are not physical vacua; geometric seam reflection is not identified with carrier conjugation.
Only the supplied affine dictionary. Backgrounds or time-dependent covariant symmetries have different premises; no general gauge-physics no-go.
Read the exact result, assumptions and evidence
Result
Mdelta=(-1/2,0,alpha/4,-1/2,0,(alpha-1)/4,-1/4,-1/4). The six (3,2) root energies are (0,1/2,1/2,1,1/2+alpha/4,1+alpha/4). K therefore does not commute with the fixed full SM action. The quarter phase is retained.
Required assumptions
Original oriented exterior carrier
Given E=C5=E3⊕E2, a Hermitian metric, volume orientation, S± and Y=(-1/3)^3⊕(1/2)^2. Internal carrier labels are not physical vacua; geometric seam reflection is not identified with carrier conjugation.
Stipulated E8 quarter-fold dictionary
The existing Weyl lift M, affine E8 module, Q'=2(H3+H6) and K_alpha=L0+delta_alpha.H0 with alpha=1/4,3/4 are held fixed for comparison. This representation is not derived from P1/P2; an unbroken SM time action is not assumed.
Original carrier plus supplied E8 charge dictionary; no physical field identification.
Read the exact result, assumptions and evidence
Result
epsilon1 epsilon4 produces u1∧u4 in the carrier: Y=1/6 and source charge change0. Root operator, spinor carrier vector and affine root-current state have different types. The exterior unit is the internal nu^c label, not the affine vacuum.
Required assumptions
Original oriented exterior carrier
Given E=C5=E3⊕E2, a Hermitian metric, volume orientation, S± and Y=(-1/3)^3⊕(1/2)^2. Internal carrier labels are not physical vacua; geometric seam reflection is not identified with carrier conjugation.
Stipulated E8 quarter-fold dictionary
The existing Weyl lift M, affine E8 module, Q'=2(H3+H6) and K_alpha=L0+delta_alpha.H0 with alpha=1/4,3/4 are held fixed for comparison. This representation is not derived from P1/P2; an unbroken SM time action is not assumed.
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.