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.
Ordered punctures, clock order4, full lift conjugation. No extra meridian square relation; distinct from central Z4extension.
Read the exact result, assumptions and evidence
Result
All62original clock objects admit strict C4*C4 lifts into the inherited lattice extension. Full conjugation gives11356classes=6856+4272+228. Every normal form and automorphism order is stored; independent Burnside verifies all62counts. Clocks0/3and0/5 give212and176classes.
Required assumptions
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.
Ordered G31reflection quadruples and chosen quarterclock
U=P1minusmu4, four ordered positive meridians in the60reflection class with product1. Chosen clocks have exact order4and conjugate meridians cyclically. Equivalence is simultaneous inner conjugation; central relaxations are separately defined relation contracts.
Original cocycle and actual G31module. H is inclusion-minimal; no claim of minimum order among all witnesses.
Read the exact result, assumptions and evidence
Result
The original A=Hom(E8,Z2) extension is nonsplit over all G31, with order-two H2 class. Seven original relation equations give0=1 on an order16 subgroup H, while all22proper subgroups split. The exact order512 lattice quotient encodes the same lift problem.
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.
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.
Exact lattice quotients and non-identifiability of an extra normalization map. No identification of original modular index I with m in each projection, no selection of a law or E7 coupling/scale prediction. The historical7pi window is no independent measurement. The established P1/P2→D5+A3→E8 construction and phi_base=(4/3)c3 remain intact. This comparison concerns only an additionally proposed E7 continuation and is not a missing step in the existing compiler.
Read the exact result, assumptions and evidence
Result
The native E8 quotients for D6+A1+A1, E7+A1, E6+A2 and D5+A3 are C2²,C2,C3,C4. The E7-pair index4 is not cyclic C4 glue. Conditional continuations1/(pi*r) and1/(2pi*m), with selected exceptional rank r and lattice extension index m, agree at E6/E8 but differ at E7:1/(7pi) versus1/(4pi). The anchors do not determine the continuation. Complete conformal windows retain total rank and central charge8.
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.
Fibre holonomy at4 removed branch points remains-I; no ordinary flat extension to the compact elliptic curve. The intrinsic family does not yet identify the original seam/peripheral data. No necessary selection of an individual chart.
Read the exact result, assumptions and evidence
Result
Each of60 Gaussian root lines gives a representative-independent square marking and w²=(v/alpha)^4-1 with elliptic compactification j1728. For the separately supplied all-J representation on CP1 minus mu4, the parity double cover has ±I holonomy and permits chosen parallel real E6/E7 root reductions. E7 descends as an unordered pair. The parity character has8 C4 lifts; cyclic marking symmetry leaves2 uniform orientations. Deck, base rotation and fibre J are different maps. The loop around all4 finite marks is trivial here but not in the existing uniform mu3 cover.
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.
The discriminant proof covers all E6 root subsystems. The finite census covers the specified regular A2 complements. No claim about arbitrary non-root Lie subalgebras or physical scales. This does not forbid transported real E6/E7 reductions on the parity double cover with remaining ±I holonomy.
Read the exact result, assumptions and evidence
Result
No E6 root subsystem is preserved by J: its induced action on K*/K=Z3 would square both to identity and, from J²=-I, to nontrivial inversion. The same argument applies to A2. All1120 regular A2 complements form560 J pairs; their E6 intersections have12 roots in640 and24 roots in480 subsystem cases. An explicitly marked D5<E6<E7<E8 chain with40/72/126/240 roots exists but is not J invariant.
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.
Fixed native Gaussian J and actual root subsystems; no arbitrary Lie-subalgebra embedding classification, new lattice-lift class or physical identification.
Read the exact result, assumptions and evidence
Result
In the given E8 root system J with J²=-I exchanges the120 root-normal E7 slices in60 pairs. Their intersections are D6 with60 roots and their normal planes A1+A1. The operator s_alpha s_Jalpha is literally the original complex Gaussian root reflection of v634; the60 distinct operators generate the existing G31 realization of order46080. Operator equality, not group order alone, supplies the dictionary. The two actual E7 subalgebras contain192 roots; their mixed brackets generate all48 missing roots in one round. Their integral lattice sum is E8 at index1. The60 pairs form a transitive G31 orbit.
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.
Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.
Read the exact result, assumptions and evidence
Result
Complete invariants are canonical ordered tuple orbit, centralizer clock orbit and ordered central phases. Clocks0/3and0/5share monodromy,spectrum,lattice cohomology and obstruction but have centralizers64and128. Without clocks there are1088central lift classes. Phase tuples(2,0,0,0)and(0,2,0,0)over the same tuple are not conjugate.
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.
Original G31reflections and central determinant class
The60original reflections generate G31of order46080. Use exactly E=Z4xG31with determinant cocycle from original holonomy proof§4; distinguish E,effective diagonal quotient and noncentral lattice lifts. Mark z.
Ordered G31reflection quadruples and chosen quarterclock
U=P1minusmu4, four ordered positive meridians in the60reflection class with product1. Chosen clocks have exact order4and conjugate meridians cyclically. Equivalence is simultaneous inner conjugation; central relaxations are separately defined relation contracts.
Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.
Read the exact result, assumptions and evidence
Result
An explicit lattice isometry identifies Gaussian and standard E8roots and preservesJ. Gamma_sign projects to<J>inG31. Neither group embeds in the other; common quotients are precisely1,C2and cannot retain order4clock. Zero invariant lattice mod2space forbids a nonzero equivariant sign-kernel map to trivialZ4and that central pushout identification.
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.
Original G31reflections and central determinant class
The60original reflections generate G31of order46080. Use exactly E=Z4xG31with determinant cocycle from original holonomy proof§4; distinguish E,effective diagonal quotient and noncentral lattice lifts. Mark z.
Ordered G31reflection quadruples and chosen quarterclock
U=P1minusmu4, four ordered positive meridians in the60reflection class with product1. Chosen clocks have exact order4and conjugate meridians cyclically. Equivalence is simultaneous inner conjugation; central relaxations are separately defined relation contracts.
Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.
Read the exact result, assumptions and evidence
Result
All62original order4clock classes obstruct:30have(0,2),32have(2,0). Central clock power alone still gives zero. Defined central covariance with product1requires d=1,3and gives960classes with strict clock or1984with central clock power.992projective orbifold pairs instead have meridian productz2. Same existing extension, explicitly separate contracts.
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.
Original G31reflections and central determinant class
The60original reflections generate G31of order46080. Use exactly E=Z4xG31with determinant cocycle from original holonomy proof§4; distinguish E,effective diagonal quotient and noncentral lattice lifts. Mark z.
Ordered G31reflection quadruples and chosen quarterclock
U=P1minusmu4, four ordered positive meridians in the60reflection class with product1. Chosen clocks have exact order4and conjugate meridians cyclically. Equivalence is simultaneous inner conjugation; central relaxations are separately defined relation contracts.
Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.
Read the exact result, assumptions and evidence
Result
Complete local rerun confirms14760ordered quadruples,17conjugacy classes and5Hurwitz orbits. All17integral Smith forms give one Z22H1, three Z20+Z/2(typeA), thirteen Z20; H0ranks6or4and higher cohomology zero. Central lifts through the same lattice projection cannot change this cohomology.
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.
Original G31reflections and central determinant class
The60original reflections generate G31of order46080. Use exactly E=Z4xG31with determinant cocycle from original holonomy proof§4; distinguish E,effective diagonal quotient and noncentral lattice lifts. Mark z.
Ordered G31reflection quadruples and chosen quarterclock
U=P1minusmu4, four ordered positive meridians in the60reflection class with product1. Chosen clocks have exact order4and conjugate meridians cyclically. Equivalence is simultaneous inner conjugation; central relaxations are separately defined relation contracts.
Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.
Read the exact result, assumptions and evidence
Result
Central orbifold H2 decides lifting; based lifts form an H1torsor and unbased lifts require centralizer quotient. Cone power defects are complete on oriented punctured quotient surfaces. The existing determinant class on C_m*C_mwith opposite parity lifts exactly for8|m, then16lifts; an original order8matrix example is given.
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.
Original G31reflections and central determinant class
The60original reflections generate G31of order46080. Use exactly E=Z4xG31with determinant cocycle from original holonomy proof§4; distinguish E,effective diagonal quotient and noncentral lattice lifts. Mark z.
Ordered G31reflection quadruples and chosen quarterclock
U=P1minusmu4, four ordered positive meridians in the60reflection class with product1. Chosen clocks have exact order4and conjugate meridians cyclically. Equivalence is simultaneous inner conjugation; central relaxations are separately defined relation contracts.
Cohomology of specified local systems on four-punctured sphere; no identification of differently dimensional representations or invariants of every bundle/bordism.
Read the exact result, assumptions and evidence
Result
Existing inclusions P5->Sym4(V4) and exterior²V4->V4⊗V4 are G0-equivariant. J versus JR gives complex H1 dimensions8/8 onV4,12/15 on exterior²V4,15/14 onP5. Integral E8 system with fourJ has H0=0,H1=Z16⊕(Z/2)^4,Hk=0 fork>=2; Smith form1^4,2^4. Local coefficients distinguish marked systems of equal clock order.
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.
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.
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.