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.
Full cocycle and global trivialization on G0 only; no splitting decision for full G31 or W(E8).
Read the exact result, assumptions and evidence
Result
For actual G0=<J,R>=C4×C2 the full8×8factor table of the original ordered lattice lift is computed. Its Hom(L,F2) class vanishes: b=(0,86,6,80,136,223,142,217), ordered1,R,J,JR,J²,J²R,J³,J³R, satisfies b(g)+g.b(h)+c(g,h)=b(gh) on64pairs. All512cocycle triples checked.
Required assumptions
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.
Actual clock/complex-reflection subgroup and native quartic space
R=diag(-1,1,1,1) on Gaussian C4 is the complex reflection of root line(1+i)e0; G0=<J,R>=C4×C2. P5 is the existing five-dimensional symmetric quartic subspace in Sym4(V4), W6=exterior²V4. R is not a real codim1 E8 Weyl reflection; P5 is not the full fourth tensor power.
Central quotient extension with trivial coefficient action, distinct from split noncentral extension over<J> and the G0 factor.16classes describe the Z-induced family, not all H2(F2^5,U1).
Read the exact result, assumptions and evidence
Result
Z(Gamma)=F2^5 and Gamma/Z=F2^5, with full central cocycle kappa((u,e),(a,f))=(ea,ef). Restriction vanishes exactly on subgroups in e=0; otherwise a nontrivial square obstructs an order2 lift.32central characters yield16marked U1 classes,15nonzero: pure clock carry is a U1 coboundary, other classes are distinguished by commutators.
Required assumptions
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.
Complete only for declared ordered quarter-clock/sign-lift stratum under Gamma conjugation; not full G31/W(E8) reflection moduli or additional equivariant structures.
Read the exact result, assumptions and evidence
Result
Gamma=F2^8 semidirect C4 has order1024. Peripheral classes C_u over J are indexed by F2^4. Four ordered classes admit product1 iff sum ui=0. Normal forms(x,y) modulo translation(u1+u0,u2+u0) give256classes for diagonal profiles,128otherwise. Total526336;4096orbits size16 with stabilizer C4×C2^4 and522240size32 with stabilizer C2^5. Image-order census4:16,8:720,16:14880,32:80640,64:430080; no surjection.
Required assumptions
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.
Declared definitions and derived standard extension/gluing algebra, not a classification of all object groups.
Read the exact result, assumptions and evidence
Result
Marked lattice transport functors with explicit reflection/clock markings define a category, products and boundary gluing under inverse-conjugate holonomies. Ordered generator phase eta gives the full factor c(g,h)(y)=eta_h(x)eta_g(hx)/eta_gh(x), x=(gh)^-1y. It is Hom(L,F2)-valued with coefficient action, not generally scalar or central.
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.
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.