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.