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.
One selected actual admissible chat and its whole power family. All four admissible sign lifts have the same cycle character. No frozen full marked-tuple equivalence or microscopic origin is proved.
Read the exact result, assumptions and evidence
Result
After toralizing actual chat and aligning its ninth power to kappa, all candidates c_t=kappa sigma_t are covered by3^8=6561 cubic torus points. Exactly480 have the full original weight-one character and form one orbit under reflections in52 U roots. Thus chat,Jhat=chat^9,sigmahat=chat^4 are simultaneously conjugate to c_t,kappa,sigma_t. Combining kappa with the old frozen sigma already fails on an explicit root.
Required assumptions
Full unitary E8 field conjugacy with transported marks
Start with the existing unitary E8 lattice VOA, original cocycle and actual chat,Jhat,sigmahat. Full inner unitary E8 automorphisms are allowed for this conjugacy comparison. Cartan, Gaussian V4, Y, P5 and other marks are transported rather than frozen. This category is explicitly larger than original sign lifts and implies neither a G31 splitting nor a preferred microscopic conjugator.
Complete torus and Weyl classification plus compact E8 conjugacy, not mere matching spectra. C is not preferred and retains other marks only through their joint transport.
Read the exact result, assumptions and evidence
Result
Five-carrier parity kappa=exp(2pi i v(0)), v=(1/2^5,0^3), acts on the four U glue sectors as1,i,-1,-i and fixes precisely U. Among all65536 torus points modulo4 exactly15120 match actual Jhat; they form one actual W(E8) orbit. Thus C kappa C^-1=Jhat for a full VOA automorphism C. Each two inverse primitive characters share a fixed root lattice D5+A3; the7560 kernels are equivariantly the actual7560 compiler/tetrahedron flags.
Required assumptions
Full unitary E8 field conjugacy with transported marks
Start with the existing unitary E8 lattice VOA, original cocycle and actual chat,Jhat,sigmahat. Full inner unitary E8 automorphisms are allowed for this conjugacy comparison. Cartan, Gaussian V4, Y, P5 and other marks are transported rather than frozen. This category is explicitly larger than original sign lifts and implies neither a G31 splitting nor a preferred microscopic conjugator.
Precisely the given scalar character restricted to fixed D8 subnet. Other source-derived field dictionaries are not excluded.
Read the exact result, assumptions and evidence
Result
The canonical scalar torus character g_q on V_E8 has order2 and fixed VOA V_E7 tensor V_A1. Over fixed V_D8 exactly one further extension gamma g_q exists, with fixed VOA V_D8prime and order2. Their weight-one spectra136/0/112/0 and120/0/128/0 exclude identification with actual native Jhat with60/64/60/64, even under full field conjugacy.
Required assumptions
Given scalar quarter twist and explicit charge chart
Compare the given eight complex current channels, q=(1/4)^8, standard D5+D3 lattice and F=I5+T with T(x,y,z)=(x+y+z,x-y-z,-x+y-z,-x-y+z)/2. Eight channels and their normalization are not derived here from P1. Distinguish a local charged field, scalar monodromy character, flux register and native clock.
Full unitary E8 field conjugacy with transported marks
Start with the existing unitary E8 lattice VOA, original cocycle and actual chat,Jhat,sigmahat. Full inner unitary E8 automorphisms are allowed for this conjugacy comparison. Cartan, Gaussian V4, Y, P5 and other marks are transported rather than frozen. This category is explicitly larger than original sign lifts and implies neither a G31 splitting nor a preferred microscopic conjugator.
Given eight-channel chart and marked lattice generator. Squaring a finite disorder operator is not identified with the local vertex field.
Read the exact result, assumptions and evidence
Result
F(2q)=lambda is an actual lattice identity, not only equal norm. s=2q has order4 modulo D5+D3 and order2 modulo D8. The tower U to V_D8 to V_E8 has indices2,2 and mu-indices16,4,1. The older SO16 base with index4 mixes two different inclusions.
Required assumptions
Given scalar quarter twist and explicit charge chart
Compare the given eight complex current channels, q=(1/4)^8, standard D5+D3 lattice and F=I5+T with T(x,y,z)=(x+y+z,x-y-z,-x+y-z,-x-y+z)/2. Eight channels and their normalization are not derived here from P1. Distinguish a local charged field, scalar monodromy character, flux register and native clock.
Original ordered lattice cocycle, fixed Jhat marking and explicitly induced Burau deck local system; no independent Gauss-Manin identification or splitting of the full G31 character extension.
Read the exact result, assumptions and evidence
Result
In the original lattice chart SNF(I-J) has invariants1,1,1,1,2,2,2,2:256 J sign lifts yield16 sign-gauge classes and one class if only gauge equivalence is enlarged to quarter phases. Enlarging candidates too gives16 classes. The original additional equation chat^9=fixed S_J has exactly four solutions and one orbit under16 S_J-preserving character gauges. On all five fundamental peripheral loops the original induced Burau comparison has rho_rel=A^(7nu), nu=(1,1,1,1,-4). Faithful order12 proves globally ker(rho_rel)=ker(nu mod12), hence minimal C12. The general C12xC12 degree144 counterexample does not satisfy this factorization.
Required assumptions
Joint category of original compiler flags and cover actions
Retain original E8 lattice VOA products, J/sigma/c/Y, D5+A3 root factors and quartic/erasure diagram. Transitions transport all marks through actual lifts in the original sign extension; they are not fixed-J stabilizers. The Burau peripheral comparison uses the associated local system induced from the existing marked C3 cover and its retained deck module. No independent geometric Gauss-Manin identification, added line correction or selected E6 vertex.
Full original G31action and inherited lattice sign extension
The60actual original complex reflections act on the same original even E8lattice. Study exactly their full preimage in the automorphism group of the lattice group defined by unchanged eps_e8. Kernel A=Hom(E8,Z2), with g.a=a composed g^-1. No scalar replacement, enlarged coefficients or extra square relation for lifted puncture meridians.
Linear data reconstruction on the same finite E6 set with counting inner product. Preserves supplied data; does not generate a distinguished corner, source or physical state law.
Read the exact result, assumptions and evidence
Result
For E6/E7 incidence N and E6/tetrahedron incidence M, every entry satisfies NNt=A_D5+3I and MMt=A_D5+27I. Hence MMt=NNt+24I and for every signal f, norm(Mt f)^2=norm(Nt f)^2+24norm(f)^2. N has rank120 and loses1000 directions300+700; M has rank1120. All tetrahedron sums recover f exactly as (NNt+24I)^(-1)M(Mt f), also through an explicit rational degree3 polynomial. The information theorem does not require irreducible decomposition.
Required assumptions
Regular E8 cuts and full compiler flags
Fix the standard E8 roots with their actual Weyl action. E6 objects are complements of regular A2 root systems and E7 objects complements of root lines. A tetrahedron comprises the four A2 faces of a regular A3 normal system with common D5 core; its D8 incidence requires both factors. Relations are literal root inclusions and intersection sizes. Signals are rational, real or complex functions with counting inner product. Equivalence transports all flags by the Weyl action, without a selected corner or physical source.
Actual full D5+A3 root flags, not uniqueness from the A3 root subsystem alone.
Read the exact result, assumptions and evidence
Result
All135 D8 frames,7560 A3 tetrahedra and their actual incidences were generated. Each full pair of A3 normal system and common D5 core lies in exactly one D8; each D8 contains56 such flags. Bare A3 containment instead gives616 A3 systems per D8, consisting of56 coordinate D3 and560 four-axis A3 chains, and11 D8 per bare A3. Unique tetrahedron-to-frame incidence is therefore a full compiler-flag statement.
Required assumptions
Regular E8 cuts and full compiler flags
Fix the standard E8 roots with their actual Weyl action. E6 objects are complements of regular A2 root systems and E7 objects complements of root lines. A tetrahedron comprises the four A2 faces of a regular A3 normal system with common D5 core; its D8 incidence requires both factors. Relations are literal root inclusions and intersection sizes. Signals are rational, real or complex functions with counting inner product. Equivalence transports all flags by the Weyl action, without a selected corner or physical source.
Complete for regular A2-normal E6 root cuts in standard E8 under its actual Weyl action; not arbitrary Lie-subalgebra embeddings.
Read the exact result, assumptions and evidence
Result
The actual Weyl action is transitive on1120 E6 cuts. An explicit stabilizing E6/A2 reflection subgroup is transitive on each intersection class: D5 with81,D4 with270,A4 with648,A2+A2 with120 neighbours. Including equality these are the full five orbitals. All125 structure constants yield the commutative full commutant. Normalized primitive idempotents have ranks1,35,84,300,700, proving the complete irreducible Weyl decomposition.
Required assumptions
Regular E8 cuts and full compiler flags
Fix the standard E8 roots with their actual Weyl action. E6 objects are complements of regular A2 root systems and E7 objects complements of root lines. A tetrahedron comprises the four A2 faces of a regular A3 normal system with common D5 core; its D8 incidence requires both factors. Relations are literal root inclusions and intersection sizes. Signals are rational, real or complex functions with counting inner product. Equivalence transports all flags by the Weyl action, without a selected corner or physical source.
Ordered four J sign lift stratum and pure mapping classes. No puncture permutations, silently frozen reflection marking quotient, or physical origin theorem.
Read the exact result, assumptions and evidence
Result
For N=I+A, N²=0, b_i² adds N(ci+c_(i+1)) to both adjacent characters. The pure mapping class action on original Gamma classes factors faithfully through C2×C2. Burnside and independent enumeration of all2^20 character gauge classes give157696 orbits with original Gamma class size census1:4096,2:46080,4:107520, weighted total526336. Every normal form is saved.
Required assumptions
Active purely mathematical marked-seam program
Within the stated mathematical scope, marked pairs, lattice transport systems, reflections, clocks and cohomology are independent mathematical objects. Earlier origin/completion tasks are historical, not active. Distinguish new definitions, derived theorems and optional markings.
Existing even E8 lattice, Gaussian J and sign lifts
Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts.
Four ordered punctures with fixed quarter-clock projection
X is CP1 minus ordered(1,i,-1,-i), positive meridians product1, fixed rho(gamma_i)=J. Classify all sign lifts into full preimage Gamma of<J>, modulo simultaneous Gamma conjugation with optional prescribed peripheral conjugacy classes. No mapping-class, puncture permutation or additional spatial reflection-equivariance quotient.
Pure Hurwitz quotient of the ordered quarter clock stratum
In addition to original Gamma conjugacy identify ordered four J tuples by pure mapping classes generated by adjacent squared Hurwitz moves b0²,b1², with b_i(x,y)=(xyx^-1,x). No puncture permutations or silently frozen extra reflection marking.
Complete only in the explicitly fixed native projection and morphism classes. Not all normalizer reflections or primitive physical source/state selection.
Read the exact result, assumptions and evidence
Result
Exact original field relations give binary ranks20,20,24,30 on40,40,40,48 variables for reflections s=0,1, rotation and joint s=0 marking. Character gauge rank4 gives65536,65536,4096,16384 marked classes. Complete forgetful images in original Gamma moduli have2176,2560,256,64 classes. All168 columns independently checked in original cocycle field action and ranks with SymPy GF2; all forgetful image orbits separately enumerated.
Required assumptions
Active purely mathematical marked-seam program
Within the stated mathematical scope, marked pairs, lattice transport systems, reflections, clocks and cohomology are independent mathematical objects. Earlier origin/completion tasks are historical, not active. Distinguish new definitions, derived theorems and optional markings.
Existing even E8 lattice, Gaussian J and sign lifts
Given existing even unimodular E8 with actual J(x0,x1,...)=(-x1,x0,...) and J²=-I, and the original ordered sign lattice cocycle. Sign automorphisms have kernel Hom(L,F2). Retain original integral ordered basis and generator phases; no enlargement to U1 characters except explicitly defined scalar pushouts.
Four ordered punctures with fixed quarter-clock projection
X is CP1 minus ordered(1,i,-1,-i), positive meridians product1, fixed rho(gamma_i)=J. Classify all sign lifts into full preimage Gamma of<J>, modulo simultaneous Gamma conjugation with optional prescribed peripheral conjugacy classes. No mapping-class, puncture permutation or additional spatial reflection-equivariance quotient.
Fixed native spatial reflection and rotation markings
In the original E8 sign lift stratum M_i=D_ci j, additionally mark T=D_t j^s t0 for s=0,1 and/or R=D_r j. T0 is the existing alternating sign isometry. Impose the stated peripheral equivariances and dihedral relation at s=0. Marked morphisms are simultaneous character conjugations with fixed projected operators. Forgetful images use full Gamma conjugacy. No arbitrary normalizer isometries or physical selection.
Complete classification within connected local unitary SAME-CONFORMAL extensions. Corrects the overstrong remark in THEOREMS.txt Section5: merely containing the four sectors already suffices under these remaining hypotheses. A tensor spectator changes the conformal vector. The two E8 classes distinguish fixed U embeddings and become isomorphic after allowing A3 diagram reversal. No derivation of this local realization from primitive P1 or physical completion.
Read the exact result, assumptions and evidence
Result
Regularity of U and common L0 make W automatically a finite U module. Its connected separable algebra object B has simple regular B module. Every invertible M_a tensor B is simple, so free-module adjunction forces n_a=dim Hom_B(M_a tensor B,B) in{0,1}. Support S is the stabilizer subgroup of B and is isotropic by trivial twist. M_h in W yields H subset S subset Hperp=H, hence exactly four sectors once and the unique marked E8 extension with vacuum and conformal data. Before specifying the generator there are exactly four local conformal extension classes over fixed U: U, V_D8 and the two oriented V_E8 embeddings.
Required assumptions
Original D5+A3 compiler split and established seed
Retain the established P1/P2 compiler with cyclic C4 glue to E8, D5 on the first5 and A3=D3 on the last3 standard coordinates, c3=1/(8pi) and phi_base=(4/3)c3=1/(6pi). No replacement by an optional E7 readout. Cut equivalence is the actual W(D5)xW(A3) action in the fixed chart, without conjugating J.
Local conformal compiler extension containing one marked generator
U=V_D5 tensor V_A3 has its fixed unitary conformal structure and marked discriminant chart. W is a connected local unitary VOA extension sharing its conformal vector and finite dimensional weight spaces; isomorphisms fix U. The marked module M_h for h=(1,1) occurs. Exact sector support, multiplicities, finite extension index, holomorphicity and an additional minimality rule are not assumed. Membership of primitive P1 in this realization is not claimed.
Exactly the existing marked associated E8 object and its standard conformal vacuum realization. No general physical time no-go, base/deck/fibre identification or additional state marking.
Read the exact result, assumptions and evidence
Result
Every original lattice-field lift fixes the canonical vacuum and Sugawara vector and commutes with every L_n. Thus the existing d field lifts126/127 remain automorphisms after including these canonical data and still fix no E6 corner. Internal Jhat has eigenvalues+i and-i four times each on the eight weight1 Cartan currents; exp(i theta L0) is scalar there. Hence Jhat is not a conformal time rotation for any theta. The joint cover construction and nonsplit transition carries remain functorial.
Required assumptions
Exact local reconstruction class of the marked C4 glue
Mathematical object class Ext_H(U): U=V_D5 tensor V_A3 with fixed embedding and sector chart, H=<(s,f)>=C4. Objects are connected local unitary conformal VOA extensions with the same conformal vector and underlying labelled U module exactly the sum of M_h for h in H, each once. Isomorphisms fix U and sector labels. This is an explicit mathematical reconstruction class; membership of primitive P1 is neither assumed nor derived.
Joint category of original compiler flags and cover actions
Retain original E8 lattice VOA products, J/sigma/c/Y, D5+A3 root factors and quartic/erasure diagram. Transitions transport all marks through actual lifts in the original sign extension; they are not fixed-J stabilizers. The Burau peripheral comparison uses the associated local system induced from the existing marked C3 cover and its retained deck module. No independent geometric Gauss-Manin identification, added line correction or selected E6 vertex.
Full original G31action and inherited lattice sign extension
The60actual original complex reflections act on the same original even E8lattice. Study exactly their full preimage in the automorphism group of the lattice group defined by unchanged eps_e8. Kernel A=Hom(E8,Z2), with g.a=a composed g^-1. No scalar replacement, enlarged coefficients or extra square relation for lifted puncture meridians.