Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Active mathematical program: marked seam monodromies and lattice lifts
Which marked seam/lattice transport geometries, lift classes and local-coefficient invariants arise from existing clock/reflection/quartic data? First complete case: sign lifts of fourJ representation on CP1 minusμ4.
Exactly the fixed native root chart and the given eight-channel time operator. The tetrahedron covers four of1120 E6 and six of120 E7. The Q-neutral E7 has a half-root normal and is not one of those six; Q is not automatically Y. Only ordinary horizontal n=0 modes are classified, not all affine conserved modes. Existing shifted p* sl2 modes and the solved color-center-neutral vacuum selection remain valid. The projector counterexample does not exclude genuine subalgebra inclusions, state restrictions or separately justified physical reductions.
Read the exact result, assumptions and evidence
Result
In the fixed original D5+A3 cut, each of the six pairs among the four E6 systems generates precisely its unique E7:104 union roots close to126, with a common40-root D5. Orthogonal root/Cartan projectors onto E6 and E7 are not Lie homomorphisms, as removed opposite-root pairs with nonzero projected coroot show. For the existing H8=8L0-rho_A7.a0+(2a-1)Q/2, Q-neutral E7 membership is not full horizontal energy symmetry: [H8,J_(e0-e1,0)]=-J_(e0-e1,0). At a=1/4 and3/4 exactly8 ordinary n=0 root modes commute, of type4A1; with8 Cartan modes this horizontal centralizer has dimension16. Complete conformal E7+A1 and E6+A2 windows retain their complements and glued modules. A cut sequence is not a derived physical breaking or time history.
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.
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.
Four D8 source modules with fixed affine time
Mathematical parent of the four D8*/D8 modules 0,v,s,c with eight identical Heisenberg oscillators, source charge chart p and N≥0. H=8(N+|p|²/2+delta.p),delta_j=(j+a)/8−1/2,0<a<1,Q=sum p_j. Original quarter cases are a1/4,3/4. Conformal L0, individual Cartan charges and totalQ remain distinct. This mathematical completion does not assume a physical P1 realization of the Ramond space or a local field map.
Actual embedded current algebra and the given local unitary affine source. U1 subscripts denote internal charges. No derivation of a four-dimensional fermionic Yukawa action, particle population, mass or primitive physical P1 source identification. Recovery of completed original results, not a new mathematical first proof.
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Result
The original18 September proofs identify E6 and A2 as mutual centralizers in the actual Chevalley E8 algebra. The48=(16,3) carrier is not E6 closed; its full closure is81=(27,3) with27=16_1+10_-2+1_4. In the given affine vacuum the actual three-current correlator kappa(a,[b,c]) equals d_ABC epsilon_ijk. Its45 cubic candidates have Ward rank44 and hence a unique relative coefficient line. The positive grade-one Gram gives64/48=4/3 its exact equal-norm current-response meaning. Normalized I4/4 is not a vacuum partial trace. These completed original results are recovered into reviewed findings; missing reviewed coverage was not an open derivation.
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.
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.
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.
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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.
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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.
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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.
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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.
Joint associated object, exact transition composition, minimal Weyl closure, one cyclic lift class and relative12 transport are determined; full field phases do not remove the selection obstruction.
Consequence: For stronger or different goals specify the changed object class or an actually supplied new marking. No freely correcting line twist or silent freezing of every field vector.
All products and conformal vacuum data follow within the exact extension class; the internal clock remains distinct from conformal rotation.
Consequence: A stronger origin theorem requires an actual local identification of the primitive source with this class, not another freely constructed field model.
Reuse complete finite cut algebra and tetrahedron reconstruction
The full five orbitals are classified. MMt=NNt+24I directly proves lossless reconstruction from tetrahedron sums and locates the rank loss of the E7 view.
Consequence: Other cut classes or a distinguished input require explicit new premises; do not replace the proved incidence algebra by numerical coincidences.
Classify additional reflection/clock equivariance and mapping-class quotients
Define extra markings and morphisms and prove remaining classes relative to the present moduli.
Next investigation: Fixed native T0/JT0 reflection lifts, R=J rotation, joint native case and pure mapping class quotient are now decided in MATH.SEAM.EQUIVARIANCE.SEARCH.01. A larger quotient over all allowed normalizer isometries or jointly transported reflection markings still needs its explicit category and complete classification. Do not promote scoped results to that stronger statement.
Builds on: Completely classify quarter-clock four-puncture lifts
Resolved in scope
Equivariant lifting and divisibility by eight
Full proof and finite evaluation under explicit equivalence.
Completely decide joint compiler-field-cover construction
Actual common carrier with composing maps, minimal closure and complete lift/periphery/selection compatibility decision in the stated category.
Builds on: Determine E7-pair action on all original lattice fields; Completely connect E6/E7 cuts to the original compiler; Compare common covers and actual clock carriers; Prove shared representation consequence for48,4/3 and5/4
Determine complete cut algebra and lossless tetrahedron transform
Determine all full orbitals, structure constants, incidences and information losses on actual root sets and reconcile the original clock comparison with the independent report.
Builds on: Completely decide joint compiler-field-cover construction
Original historical attempts linked to this question
These are original research records. Their reported verdicts are distinct from the reviewed results above.
e series compiler closure 20261023
Complete the four compiler follow-ups: full lattice-field lifts of E7 pairs, all E6/E7 compatibility with the native D5+A3 split, exact common covers and actual clock intertwiners, and a common representation dictionary for the existing48,4/3 and5/4.
Recorded verdict: PASS
Historical inventory; not independently reviewed by this export.
Close the supplied independent E6/E7/E8 finite incidence classification against the original marked field lifts and global cover monodromies, and identify whether the tetrahedral incidence retains all E6 information.
Recorded verdict: PASS
Historical inventory; not independently reviewed by this export.
Determine the complete joint mathematical compatibility object of the existing compiler flags, E7 pair operations, full lattice fields and marked covers, preserving their actual transitions and all previously proved obstructions.
Recorded verdict: PASS
Historical inventory; not independently reviewed by this export.
Reconstruct the canonical local E8 field completion from the already established marked D5+A3/C4 compiler glue; determine which original source and dynamics implications are actually proved.
Recorded verdict: PARTIAL
Historical inventory; not independently reviewed by this export.
Determine strict, peripheral and equivariant central lifts; evaluate all original G31 quarterclocks, integral monodromies, and compare actual sign-lift and reflection groups.
Recorded verdict: PASS
Historical inventory; not independently reviewed by this export.
Decide splitting of the actual character-valued lattice lift extension over all original G31; completely classify lifts of the original17monodromies and62quarterclock objects.
Recorded verdict: PASS
Historical inventory; not independently reviewed by this export.
Classify all sign lifts of the native quarter-clock representation on the ordered four-punctured sphere, and determine their actual extension classes and native quartic/exterior cohomology.
Recorded verdict: PASS
Historical inventory; not independently reviewed by this export.
Classify native marked quarter-clock reflection and rotation lifts, their forgetful images and the pure mapping class quotient; compare local model construction with exact active hypothesis search.
Recorded verdict: PARTIAL
Historical inventory; not independently reviewed by this export.