Established within the stated premises
finding:finite-carrier-glue-elements-orbits-and-halfway
In Fin4 x Fin4 with qZ=(5x^2+3y^2) mod8, named Lean declarations classify all six isotropic elements and four odd-coordinate isotropic generators. Explicit orbit identities give H1 and H2; klein_not_isotropic gives the nonisotropic Klein witness, spinor_swap_exchanges swaps the two orbits, and unique_halfway_stage classifies the only nonzero isotropic order-two element (2,2). Index and sector arithmetic are additional separate declarations.
Scope: The concrete element/orbit and arithmetic statements are available. No fresh kernel run, nor a newly checked single Lean theorem about all abstract subgroups or lattice isomorphisms. The general subgroup conclusion also uses finite group theory; concrete E8 basis embedding and physical carrier selection are separate.
Required premises and original source excerpts
Established within the stated premises
finding:marked-seam-category-and-character-valued-full-lift-factor
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.
Scope: Declared definitions and derived standard extension/gluing algebra, not a classification of all object groups.
Required premises and original source excerpts
- 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.
premise:marked-seam-pure-mathematical-program - 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.
premise:native-even-e8-gaussian-clock-and-sign-extension
Established within the stated premises
finding:quarter-clock-sign-lift-complete-four-puncture-classification
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.
Scope: Complete only for declared ordered quarter-clock/sign-lift stratum under Gamma conjugation; not full G31/W(E8) reflection moduli or additional equivariant structures.
Required premises and original source excerpts
- 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.
premise:native-even-e8-gaussian-clock-and-sign-extension - 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.
premise:ordered-four-punctures-fixed-quarter-projection
Established within the stated premises
finding:quarter-lift-central-quotient-class-and-all-subgroup-restrictions
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.
Scope: 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).
Required premises and original source excerpts
- 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.
premise:native-even-e8-gaussian-clock-and-sign-extension
Established within the stated premises
finding:native-clock-reflection-full-factor-cocycle-trivialization
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.
Scope: Full cocycle and global trivialization on G0 only; no splitting decision for full G31 or W(E8).
Required premises and original source excerpts
- 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.
premise:native-even-e8-gaussian-clock-and-sign-extension - 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.
premise:native-clock-complex-reflection-and-quartic-subspace
Established within the stated premises
finding:native-quartic-exterior-local-cohomology-distinguishes-clock-monodromies
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.
Scope: Cohomology of specified local systems on four-punctured sphere; no identification of differently dimensional representations or invariants of every bundle/bordism.
Required premises and original source excerpts
- 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.
premise:native-even-e8-gaussian-clock-and-sign-extension - 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.
premise:ordered-four-punctures-fixed-quarter-projection - 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.
premise:native-clock-complex-reflection-and-quartic-subspace
Established within the stated premises
finding:orbifold-lift-criterion-and-determinant-eight-divisibility
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.
Scope: Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.
Required premises and original source excerpts
- 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.
premise:marked-seam-pure-mathematical-program - 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.
premise:original-g31-central-determinant-extension - 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.
premise:ordered-original-g31-reflection-and-clock-markings
Established within the stated premises
finding:original-g31-reflection-census-and-all-integral-cohomology
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.
Scope: Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.
Required premises and original source excerpts
- 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.
premise:marked-seam-pure-mathematical-program - 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.
premise:original-g31-central-determinant-extension - 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.
premise:ordered-original-g31-reflection-and-clock-markings
Established within the stated premises
finding:all62-original-clock-obstructions-and-central-variants
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.
Scope: Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.
Required premises and original source excerpts
- 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.
premise:marked-seam-pure-mathematical-program - 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.
premise:original-g31-central-determinant-extension - 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.
premise:ordered-original-g31-reflection-and-clock-markings
Established within the stated premises
finding:actual-sign-lift-g31-group-comparison
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.
Scope: Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.
Required premises and original source excerpts
- 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.
premise:marked-seam-pure-mathematical-program - 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.
premise:original-g31-central-determinant-extension - 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.
premise:ordered-original-g31-reflection-and-clock-markings
Established within the stated premises
finding:complete-g31-lift-invariants-and-explicit-collisions
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.
Scope: Original60reflection class, ordered punctures, marked central generator and simultaneous inner conjugation. General theorems use their explicitly stated hypotheses.
Required premises and original source excerpts
- 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.
premise:marked-seam-pure-mathematical-program - 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.
premise:original-g31-central-determinant-extension - 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.
premise:ordered-original-g31-reflection-and-clock-markings
Established within the stated premises
finding:e7-quarter-pairs-literal-native-g31-reflections
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.
Scope: Fixed native Gaussian J and actual root subsystems; no arbitrary Lie-subalgebra embedding classification, new lattice-lift class or physical identification.
Required premises and original source excerpts
- 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.
premise:marked-seam-pure-mathematical-program - 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.
premise:native-even-e8-gaussian-clock-and-sign-extension
Established within the stated premises
finding:e6-root-subsystem-quarter-clock-discriminant-obstruction
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.
Scope: 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.
Required premises and original source excerpts
- 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.
premise:marked-seam-pure-mathematical-program - 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.
premise:native-even-e8-gaussian-clock-and-sign-extension
Established within the stated premises
finding:e-series-square-cover-family-and-quarter-descent
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.
Scope: 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.
Required premises and original source excerpts
- 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.
premise:marked-seam-pure-mathematical-program - 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.
premise:native-even-e8-gaussian-clock-and-sign-extension
Established within the stated premises
finding:e-series-integral-glue-and-normalization-nonidentifiability
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.
Scope: 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.
Required premises and original source excerpts
- 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.
premise:marked-seam-pure-mathematical-program - 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.
premise:native-even-e8-gaussian-clock-and-sign-extension
Established within the stated premises
finding:global-original-g31-character-lift-nonsplitting
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.
Scope: Original cocycle and actual G31module. H is inclusion-minimal; no claim of minimum order among all witnesses.
Required premises and original source excerpts
- 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.
premise:native-even-e8-gaussian-clock-and-sign-extension - 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.
premise:original-full-g31-action-and-inherited-lattice-sign-extension
Established within the stated premises
finding:all62-original-character-clock-lifts-complete
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.
Scope: Ordered punctures, clock order4, full lift conjugation. No extra meridian square relation; distinct from central Z4extension.
Required premises and original source excerpts
- 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.
premise:original-full-g31-action-and-inherited-lattice-sign-extension - 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.
premise:ordered-original-g31-reflection-and-clock-markings
Established within the stated premises
finding:all17-original-character-monodromy-lifts-complete
Full conjugation over the17original reflection quadruple classes gives835456character-valued monodromy lift classes, with a complete affine normal-form criterion. V4embeddings with and without integral Z2torsion give26128and42240classes per ordered orbit.
Scope: F3representations, ordered punctures, full lift conjugation. Census and normal-form criterion; no stored835456-representative list.
Required premises and original source excerpts
- 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.
premise:original-full-g31-action-and-inherited-lattice-sign-extension - 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.
premise:ordered-original-g31-reflection-and-clock-markings
Established within the stated premises
finding:e7-pair-full-lattice-field-character-torsor
For all60 native Gaussian root lines, r=s_alpha s_Jalpha is the original complex reflection. Its actual Tits lift T=n_alpha n_Jalpha has T²=D_k, k(lambda)=<alpha+Jalpha,lambda>mod2, negative on128 roots. T=S_r D_chi holds on every lattice field with explicit input character. Exactly eight corrections per line are involutive and commute with the fixed J lift, a torsor under three-dimensional A^{<r,J>}. They agree on the actual neutral4/5/20 diagram and differ on charged sectors. The full G31 extension remains nonsplit.
Scope: Unchanged eps_e8, ordered integral basis and original lattice VOA. All-lattice equality follows from the character lemma and eight generators; all240 roots and13680 nonzero brackets per line checked separately. Individual lifts do not provide a global group section.
Required premises and original source excerpts
- 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.
premise:original-full-g31-action-and-inherited-lattice-sign-extension - 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.
premise:native-clock-complex-reflection-and-quartic-subspace
Established within the stated premises
finding:e-series-complete-native-compiler-cut-dictionary
The120 E7 cuts form four W(D5)xW(A3) orbits of sizes20,6,30,64, with exactly six containing the original D5. The1120 regular E6 cuts form seven orbits of sizes40,120,320,60,480,4,96. Their560 J pairs intersect320 times in A2+A2 and240 times in D4. Exactly four E6 cuts contain D5, a canonical unordered S4 tetrahedral family with actual weightwise E8/D5/A3/E6/A2 branchings. J sends all four outside this family with common first-four-coordinate D4. The3+2 marking fixes the family but selects no cut.
Scope: Complete finite classification of regular root cuts under the fixed compiler Weyl group, with stabilizers and actual actions. No classification of arbitrary Lie algebra embeddings; retain the original clock.
Required premises and original source excerpts
- 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.
premise:fixed-native-d5-a3-compiler-split-and-seed - 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.
premise:native-even-e8-gaussian-clock-and-sign-extension
Open questions preserve their achieved subresults
Each research problem has its own scope. A resolved task does not close its whole physical parent. Historical tasks remain explicitly historical in the source record.
Future quotient and positive GNS reconstruction
What is determined by complete future behavior, and what additional positive word-kernel source is needed to reconstruct a cyclic operator realization?
- Resolved in scope: Future quotient from complete rRead result
- Resolved in scope: Conditional GNS reconstruction from supplied KRead result
- open: Select complete K from the original source
Historical: common source, state and event selection
Outside the active mathematical research program since the user's23Oct2026 decision. Historical question: Do the original principles derive a common physical source together with its state, dynamical events, and physically observed fields?
Seam connection and family action
Does the joint discrete theorem determine a concrete connection and a family coupling compatible with the determinant seam action?
- Resolved in scope: Separate classes from actual transportRead result
- open: Derive the common connection from the original source
Physical time and complete transfer
Do the original reflection/projector structures determine the full physical time generator and required field transfer?
- Resolved in scope: Reuse the sign-projector counterexampleRead result
- Resolved in scope: Reuse conditional full-transfer reconstructionRead result
- open: Identify the physical complete TFPT transfer
- Resolved in scope: Reuse complete collar reconstructionRead result
- Resolved in scope: Classify homogeneous GKLS embedding of the shared register familyRead result
- Resolved in scope: Decide every homogeneous GKSL logarithm of the fixed M4 channelRead result
Flavor ratio and absolute amplitude scale
Which flavor quantity is fixed on the derived discrete stratum, and which requires the continuous holonomy anchor?
- Resolved in scope: Reuse the derived flavor ratioRead result
- open: Determine the absolute holonomy amplitude
Alpha: derived terms, unique root and physical EM identification
Where do the factors and closed alpha equation come from, which provenance steps are proved, and which precise physical EM identification remains open?
- Resolved in scope: Separate root calculation from source selectionRead result
- open: Derive the actual EM functional variation and Thomson identification
- Resolved in scope: Establish the unique global positive rootRead result
Geometric flow and gravity normalization
Do common matter-only observations derive geometric horizon flow and the area response from the original gravity source?
- Resolved in scope: Record scope of fixed original functionalRead result
- open: Derive geometric modular response and area coefficient
Clocks and complete neutral source response
Which complete quadratic response can the actual clock images reconstruct?
- Resolved in scope: Reconstruct all neutral quadratic matrix directionsRead result
- open: Derive physically admissible probe access
Clock action on charge, time and quarter transition
How does the actual Gaussian J act on the complete marked source?
- Resolved in scope: Construct original J lift with complete time/charge actionRead result
- open: Derive physical meaning of the frame orbit
Which responses distinguish charged time action?
Which data separate sources with identical complete neutral vacuum response?
- Resolved in scope: Determine twist blindness completelyRead result
- Resolved in scope: Reconstruct twist from charged responsesRead result
- open: Derive actual charged source probes
Which seam action distinguishes native opposite charges?
Can existing native charge fields produce the quarter-twist time response of the seam?
- Resolved in scope: Check native fields and response discriminatorRead result
- open: Derive actual marked seam/state action
Constructive seam search: operators, state and source compatibility
What does the complete phase search decide, and which additional state information prevents a false source solution?
- Resolved in scope: Finite search closed, actual winding and parity equivalenceRead result
- Resolved in scope: Positive finite-word state extensionRead result
- Resolved in scope: E8 current group conditionally determines the P signRead result
- Resolved in scope: The same shift functional hides distinct ground statesRead result
- Resolved in scope: No phase hit realizes the unchanged sourceRead result
Which partner actually belongs to the original seam?
Which charge, state type and time action belong to the root partner and original sheet pairing?
- Resolved in scope: Concrete pair operator; source neutrality is not hypercharge neutralityRead result
- Resolved in scope: Fixed quarter time splits the color/weak multipletRead result
- Resolved in scope: Explicit conjugation for the existing sheet pairingRead result
- Resolved in scope: Actual partner map between quarter domainsRead result
- Resolved in scope: Bilinear sheet pairing is not Hermitian kernel selectionRead result
- Resolved in scope: Actual quarter conjugation transports the supplied markingRead result
- Resolved in scope: No time-stabilizing orthogonal repair of the fixed markingRead result
- Resolved in scope: Spinor16 is a state sector, not a 16-real one-particle fieldRead result
Automated construction of operator maps and sources
Which tools generate operator maps, learn counterexamples, and can search for the full positive source?
- Resolved in scope: Calibrate new search without supplying the target mapRead result
- open: Synthesize a joint field, marking and transfer dictionary
- open: Search for the positive common source kernel itself
- Resolved in scope: Automatically construct, certify and distinguish finite kernelsRead result
- Resolved in scope: Repair search conditions against source types and collar controlsRead result
- Resolved in scope: Reconstruct multi-field sewing of the canonically identified boundary sourceRead result
- Resolved in scope: Retain absolute reference amplitude in searchRead result
- Resolved in scope: Integrate supplied free source including boundary response and scalarRead result
- Resolved in scope: Convert the full finite CAR kernel into a link functionalRead result
- Resolved in scope: Close original recursive word reconstruction with memoryRead resultRead result
- Resolved in scope: Retain whole original caller through scale and readoutRead resultRead result
- Resolved in scope: Execute actual shared Higgs source with full matter words and local RPRead resultRead result
- Resolved in scope: Reduce whole constant original Higgs spectral fiberRead result
- open: Resolve original geometric Higgs residue sign and field map
- Resolved in scope: Evaluate local original Higgs heat identification and whole finite consumerRead resultRead result
- Resolved in scope: Execute whole variable bound Higgs reference and original scale mapRead resultRead result
- Resolved in scope: Execute whole inner Higgs scale/source and normal kernel transportRead resultRead result
- Resolved in scope: Execute given whole superconnection and normal spectral readerRead resultRead result
- Resolved in scope: Determine own-admissibility and whole total-chiral spectral readersRead resultRead result
- Resolved in scope: Derive whole scale-energy map,normal profile and relative flowRead resultRead result
- Resolved in scope: Execute original Green energy producer and mixed scale readoutRead resultRead result
- Resolved in scope: Match current action argument and common Higgs/gravity weightsRead resultRead result
- Resolved in scope: Execute whole original curvature consumers and Higgs radial sourceRead resultRead resultRead result
- Resolved in scope: Execute original whole four-ghost source and geometric type testRead resultRead result
- Resolved in scope: Execute original moving normal kernel derivative and whole source responseRead result
- Resolved in scope: Decide fixed original O1 source and normal operator placementRead result
- Resolved in scope: Compare whole Higgs mixture and covariance reader on invariant four-field witnessRead result
- Resolved in scope: Test exact bounded cutoff on whole ordinary Higgs volumeRead result
- Resolved in scope: Test whole original circle joint mode and IR Higgs normalizationRead result
- Resolved in scope: Test original O1 mode covariance against whole Higgs-law classRead result
- open: Reconstruct original localization from the whole process
- Resolved in scope: Test bare native-code word compression against actual localityRead result
- Resolved in scope: Decide original geometric local-time class and execute actual interval readerRead resultRead result
- Resolved in scope: Execute actual regional two-face partner and supported charged wordRead resultRead result
- Resolved in scope: Execute original color marginal, modular field law and complete class readerRead resultRead resultRead result
- Resolved in scope: Reconstruct common Higgs law from whole original quarter source towerRead resultRead resultRead result
- Resolved in scope: Execute original record action on Doily and ordered whole source responseRead resultRead resultRead resultRead result
- Resolved in scope: Locally implement original record process and test unchanged exterior sourceRead resultRead resultRead result
- Resolved in scope: Execute original elliptic insertion,whole source and causal responseRead resultRead resultRead result
- Resolved in scope: Evaluate actual joint Higgs and color group sourceRead resultRead result
- Resolved in scope: Execute whole original graph color reader with typed original center selectorRead resultRead result
- Resolved in scope: Execute original color fixed source and first inherited grading testRead resultRead result
- Resolved in scope: Execute actual original color-spinor CAR four-word and complete spin-sector sewingRead resultRead result
- Resolved in scope: Decide actual color admissibility of density operations and coordinate regionsRead resultRead result
- Resolved in scope: Execute actual whole seam-operation class and two-face responseRead resultRead result
- Resolved in scope: Execute entire original C6 word expansion with source amplitudeRead resultRead result
- Resolved in scope: Execute original collective Higgs force source and entire mean dualRead resultRead resultRead result
- Resolved in scope: Execute original all-mode force with actual time and whole sourceRead resultRead result
- Resolved in scope: Execute original local space/time force source and causal liftRead resultRead result
Origin reverse selection, field dictionary and original P1 kernel
What does reverse selection of P1/P2 give for concrete local fields and multi-field responses, and which identification with the original P1 kernel has actually been proved?
Observable reality: minimal conditions and counterworlds
Which operational conditions are independently required, which inequivalent models satisfy them, and which observable distinctions remain for selection?
Formal carrier core: unbounded Pascal uniqueness and finite glue
Which carrier and glue statements are already present in the papers and Lean sources, and which physical selection does not follow?
- Resolved in scope: Classify the unbounded Pascal proofRead result
- Resolved in scope: Classify the formal finite glue scopeRead result
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.
- Resolved in scope: Define marked seam category and factor formulaRead result
- Resolved in scope: Completely classify quarter-clock four-puncture liftsRead result
- Resolved in scope: Determine central class and all subgroup restrictionsRead result
- Resolved in scope: Determine full G0 cocycle and classRead result
- Resolved in scope: Determine native4/5/6local cohomologyRead result
- Resolved in scope: Classify unrestricted G31 reflection monodromiesRead resultRead result
- Resolved in scope: Determine lift class beyond G0Read result
- open: Classify additional reflection/clock equivariance and mapping-class quotients
- Resolved in scope: Equivariant lifting and divisibility by eightRead result
- Resolved in scope: All62clock obstructions and central transport variantsRead result
- Resolved in scope: Actual group comparison and central pushout exclusionRead result
- Resolved in scope: Complete invariants and explicit collisionsRead result
- Resolved in scope: Compare E6 and E7 slices with the actual quarter clockRead resultRead resultRead resultRead result
- Resolved in scope: Classify original character-valued G31monodromy liftsRead resultRead result
- Resolved in scope: Determine E7-pair action on all original lattice fieldsRead result
- Resolved in scope: Completely connect E6/E7 cuts to the original compilerRead result
- Resolved in scope: Compare common covers and actual clock carriersRead result
- Resolved in scope: Prove shared representation consequence for48,4/3 and5/4Read result
- Resolved in scope: Completely decide joint compiler-field-cover constructionRead resultRead resultRead resultRead result
- Resolved in scope: Reconstruct canonical local compiler field source from C4Read resultRead result
- Resolved in scope: Eliminate redundant sector and multiplicity assumptionsRead result
- Resolved in scope: Classify native equivariant markingsRead result
- Resolved in scope: Classify pure quarter clock mapping class quotientRead result
- Resolved in scope: Determine complete cut algebra and lossless tetrahedron transformRead resultRead resultRead resultRead result
- Resolved in scope: Completely compare scalar glue and actual native compiler classRead resultRead resultRead resultRead result
Common clock sectors, quartic geometry and operational reconstruction
Which previously separate data of the original TFPT core follow from a common clock and representation geometry, and which complete finite measurement classification follows without asserting physical source selection?
Automatic joint origin hypotheses and source responses
Which simple joint rules connect the existing P1 source chain,double cover,information and compiler,and which whole candidate classes can be decided automatically using actual source consequences?
- Resolved in scope: Decide even-matter word class and check actual AI operator expressionsRead resultRead result
Existing TFPT derivations and correction of repeated gap diagnoses
Which common source/reconstruction chain already exists in the old original papers, and which purported global gaps arose from reducing P1 or choosing an optional CAR route?
Complete physical Hamiltonian source with Gauss sector and thermal word limit
Which whole source is constructed on the original local Hamiltonian route, how do physical constraints and thermal words follow, and how is its actual IR response distinguished from the original covariant geometric source branch?
Original TFPT source versus DET16 mirror vertex
Does the same original TFPT source produce the concrete DET16 generator and whole responses,or is a computable parent supplied as extra input?
Whole original source and canonical vacuum attachment
Which vacuum joins are already in the whole original source,and how do H ground source and R104 field/Gauss source join completely without second state choice or parent change?
Craig chiral measure and the original whole TFPT source
Which concrete operator and Berezin measure map preserves the original TFPT source when connecting Craig overlap measure?
- open: Compare original4D fermion Hessian and measure line with Craig
Cyclic quantum universe and original Genesis connection
Does original TFPT dynamics yield the exact cyclic quantum return in the September/October press lead and a geometric Genesis transition?
- open: Test periodicity and Genesis response on the same whole source
Physical constants as joint constraints on the TFPT source
Which independent constants constrain the original common source, and can one relative heat product supply gravity and vacuum energy together?
- open: Evaluate original volume, curvature and gauge responses together