Established within the stated premises
finding:common-cover-clock-and-maximal-integral-phi12-intertwiner
C6 and C12 are the minimal marked common covers of C2/C3 and C4/C3, of genera7 and15. J<->Q, sigma<->deck and c=J sigma<->R=Q deck preserve c⁹=J and c⁴=sigma. The unique lift with this full marking is Qn Delta_n^4. Actual real8 has character Phi4² Phi12; the retained rational Burau rank6 model has Phi6 Phi12. Their canonical Phi12 kernels admit an explicit integral Z[C12] intertwiner also transporting J and sigma. Different complementary spectra obstruct full equality; retained alternating forms of types(1,3) and(1,2) obstruct any form-preserving unimodular intertwiner. Peripheral J is deck on C4 while compiler J=c⁹ rotates the base; no phase choice merges these roles.
Scope: Fixed deck orientation, actual matrices and saturated integral Phi12 sublattices. The displayed intertwiner is an explicit certificate, not a uniquely selected identification. Trace form and basis belong to the retained Burau lattice model, without an independent geometric cycle identification. Quartic5, exterior6, erasure20 and permutation characters remain distinct and are completely compared.
Required premises and original source excerpts
- Existing marked cyclic covers and Burau lattice model
On B=P1 minus(mu4 union infinity), Cn:un^n=x^4-1 for n=2,3,4 are the existing marked covers with positive finite exponent1. Fix the mu3 deck orientation omega. Use the original J and sigma with c=J sigma, actual reduced Burau matrices at omega and their retained Z[omega]^3 basis and invariant Hermitian form. Do not identify this integral model and trace form with independently constructed geometric cycles or their intersection form.
premise:existing-marked-cyclic-cover-and-burau-model - 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-four-cut-projectors-unify48-fourthirds-fivequarters
In the original half-root block B=(16,4), the four D5-containing E6 cuts yield literal rank16 projectors P_t, with P_t P_u=delta_tu P_t and sum P_t=I64. Q_t=I-P_t has rank48 and carries exactly(16,3), with checked D5 and A2 root action. Every cut gives Tr(I64)/Tr(Q_t)=64/48=4/3=phi_base/c3. Every weight of the same block has squared D5 projection5/4 and A3 projection3/4. The same48 lies in(27,3)=(16,3)_1+(10,3)_-2+(1,3)_4, hence81=48+30+3. The E7 missing48 instead carries(12,2,2), whose actual Y character rejects identification with three D5 spinors.
Scope: Canonical unordered S4 projector family on the original root-current block. Exact representation interpretation of the existing seed ratio, without an additional readout law or selection of a tetrahedral vertex.
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
Established within the stated premises
finding:complete-covariant-compiler-voa-source-and-weyl-closure
H=W(D5)xW(D3) with actual J generates exactly W(D8):224 E6 cuts in56 tetrahedral frames. Each frame resolves the entire128 half-root block into four rank32 projectors; complements have rank96 and128/96=4/3, retaining squared norms5/4 and3/4. Original E7-pair operations enlarge closure to W(E8):1120 E6 cuts,7560 A3 frames,27 through each cut. The full original sign extension over W(E8) acts on the existing VOA source. F_ghat[p,v]=[p,ghat v] transports all marks and fibre holonomy and composes exactly, retaining the nonsplit character carry.
Scope: Minimal generated transition closure, not a fixed-J/fixed-D5+A3 symmetry claim.128 is fixed only under D8 and transported under the full action. Organizes the given lattice VOA and covers; no physical-origin derivation or freely chosen helper model.
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 - 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 - Joint category of original compiler flags and cover actions
Retain original E8 lattice VOA products, J/sigma/c/Y, D5+A3 root factors and quartic/erasure diagram. Transitions transport all marks through actual lifts in the original sign extension; they are not fixed-J stabilizers. The Burau peripheral comparison uses the associated local system induced from the existing marked C3 cover and its retained deck module. No independent geometric Gauss-Manin identification, added line correction or selected E6 vertex.
premise:joint-native-compiler-cover-covariance-category
Established within the stated premises
finding:complete-fixed-jhat-cyclic-field-lift-and-associated-clock
With original fixed Jhat=S_J, the ninth-power equation has F2 rank6. Exactly four compiler lifts S_c D_a with a=31,62,151,182 have ninth power Jhat; their sigmahat masks are63,30,183,150. The16 Jhat-preserving character gauges give one orbit with stabilizer4. No solution retains both separately generator-normalized J and sigma sections. On Vper=(C12 x V_L)/((Delta12 p,v)~(p,Jhat v)), Chat[p,v]=[R12 p,chat v] satisfies Chat4=sigmahat, Chat12=I and Jcomp=Chat9=Q12 Jper. The two J roles share the same fibre lift while retaining different base components.
Scope: All256 character candidates over the fixed cyclic compiler group; all-field equality from eight generators and unchanged cocycle, with all240 roots checked directly. No full G31/Weyl splitting or identification of different cover automorphisms.
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 - Existing marked cyclic covers and Burau lattice model
On B=P1 minus(mu4 union infinity), Cn:un^n=x^4-1 for n=2,3,4 are the existing marked covers with positive finite exponent1. Fix the mu3 deck orientation omega. Use the original J and sigma with c=J sigma, actual reduced Burau matrices at omega and their retained Z[omega]^3 basis and invariant Hermitian form. Do not identify this integral model and trace form with independently constructed geometric cycles or their intersection form.
premise:existing-marked-cyclic-cover-and-burau-model - Joint category of original compiler flags and cover actions
Retain original E8 lattice VOA products, J/sigma/c/Y, D5+A3 root factors and quartic/erasure diagram. Transitions transport all marks through actual lifts in the original sign extension; they are not fixed-J stabilizers. The Burau peripheral comparison uses the associated local system induced from the existing marked C3 cover and its retained deck module. No independent geometric Gauss-Manin identification, added line correction or selected E6 vertex.
premise:joint-native-compiler-cover-covariance-category
Established within the stated premises
finding:relative-native-burau-clock-hom-faithful12-transport
On the actual common Phi12 lattices, Hom_clock=Q[A12] has dimension4 and integral Hom=Z[zeta12], with integral isomorphisms a unit torsor. Native periphery is A12^9; the Burau periphery induced from the existing mu3 cover deck module is A12^4. Relative Hom transport is therefore H->A12^7 H, with exactly0 parallel clock intertwiners. Every nonzero relative orbit has order12. Exponent vector(7,7,7,7,8) has exactly the original winding kernel modulo12, so the existing C12 is the minimal trivialization.
Scope: Comparison of native and explicitly cover-induced Burau local systems. No independently geometric Gauss-Manin H1 family, forced carrier equality or extra line twist erasing carry. The earlier clock dictionary remains valid but is not a parallel peripheral dictionary.
Required premises and original source excerpts
- Existing marked cyclic covers and Burau lattice model
On B=P1 minus(mu4 union infinity), Cn:un^n=x^4-1 for n=2,3,4 are the existing marked covers with positive finite exponent1. Fix the mu3 deck orientation omega. Use the original J and sigma with c=J sigma, actual reduced Burau matrices at omega and their retained Z[omega]^3 basis and invariant Hermitian form. Do not identify this integral model and trace form with independently constructed geometric cycles or their intersection form.
premise:existing-marked-cyclic-cover-and-burau-model - Joint category of original compiler flags and cover actions
Retain original E8 lattice VOA products, J/sigma/c/Y, D5+A3 root factors and quartic/erasure diagram. Transitions transport all marks through actual lifts in the original sign extension; they are not fixed-J stabilizers. The Burau peripheral comparison uses the associated local system induced from the existing marked C3 cover and its retained deck module. No independent geometric Gauss-Manin identification, added line correction or selected E6 vertex.
premise:joint-native-compiler-cover-covariance-category
Established within the stated premises
finding:e6-vertex-nonselection-survives-full-field-clock-phases
C_H(J) has order64 and cyclic4 image on the four E6 vertices. Its Y stabilizer is{I,d}, d=diag(1,1,1,1,1,1,-1,-1), acting by two transpositions without a fixed vertex. d preserves sigma,c and the actual quartic/erasure diagram. With fixed full Jhat and chat=S_c D31, exactly four d field lifts126,127,128,129 commute with both;126 and127 are involutions. Thus genuine automorphisms of the same full associated object fix no E6 vertex. Character-gauge conjugation transfers this to all four cyclic lifts. The supplied field phases therefore select no vertex.
Scope: Complete stabilizer test inside H preserving root factors setwise, with fixed J/compiler field lifts, Y and covariant quartic/erasure maps. No pointwise freezing of every individual quartic vector or separately chosen charged state. The full transported family is the exact object.
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 - Joint category of original compiler flags and cover actions
Retain original E8 lattice VOA products, J/sigma/c/Y, D5+A3 root factors and quartic/erasure diagram. Transitions transport all marks through actual lifts in the original sign extension; they are not fixed-J stabilizers. The Burau peripheral comparison uses the associated local system induced from the existing marked C3 cover and its retained deck module. No independent geometric Gauss-Manin identification, added line correction or selected E6 vertex.
premise:joint-native-compiler-cover-covariance-category - 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:marked-c4-glue-unique-local-e8-vacuum-source
In exactly Ext_H(U) there is one VOA and normalized local Q system class over U, represented by V_E8. q_D5(s)=5/8 and q_A3(f)=3/8 make H Lagrangian; the full abelian associator/braiding class trivializes only on H. H2(C4,C*)=H2(C4,U1)=0 gives uniqueness, with residual automorphisms Hom(C4,U1)=C4. The actual weight1 branching is60+64+60+64=248. Lattice vacuum, Sugawara vector, L0=N+Norm(alpha)/2, all vacuum correlators, unitary structure over U and associated local conformal net are determined in this class.
Scope: Complete conditional mathematical reconstruction for the specified U module and local unitary conformal extension. No derivation of primitive P1 membership or physical time/state/spacetime identification. H3(C4,U1) is not asserted zero; the original nonsplit G31 character extension is not replaced by scalar coefficients.
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 - Exact local reconstruction class of the marked C4 glue
Mathematical object class Ext_H(U): U=V_D5 tensor V_A3 with fixed embedding and sector chart, H=<(s,f)>=C4. Objects are connected local unitary conformal VOA extensions with the same conformal vector and underlying labelled U module exactly the sum of M_h for h in H, each once. Isomorphisms fix U and sector labels. This is an explicit mathematical reconstruction class; membership of primitive P1 is neither assumed nor derived.
premise:marked-c4-unitary-local-extension-reconstruction-class
Established within the stated premises
finding:canonical-vacuum-clock-preserves-e6-nonselection
Every original lattice-field lift fixes the canonical vacuum and Sugawara vector and commutes with every L_n. Thus the existing d field lifts126/127 remain automorphisms after including these canonical data and still fix no E6 corner. Internal Jhat has eigenvalues+i and-i four times each on the eight weight1 Cartan currents; exp(i theta L0) is scalar there. Hence Jhat is not a conformal time rotation for any theta. The joint cover construction and nonsplit transition carries remain functorial.
Scope: Exactly the existing marked associated E8 object and its standard conformal vacuum realization. No general physical time no-go, base/deck/fibre identification or additional state marking.
Required premises and original source excerpts
- Exact local reconstruction class of the marked C4 glue
Mathematical object class Ext_H(U): U=V_D5 tensor V_A3 with fixed embedding and sector chart, H=<(s,f)>=C4. Objects are connected local unitary conformal VOA extensions with the same conformal vector and underlying labelled U module exactly the sum of M_h for h in H, each once. Isomorphisms fix U and sector labels. This is an explicit mathematical reconstruction class; membership of primitive P1 is neither assumed nor derived.
premise:marked-c4-unitary-local-extension-reconstruction-class - Joint category of original compiler flags and cover actions
Retain original E8 lattice VOA products, J/sigma/c/Y, D5+A3 root factors and quartic/erasure diagram. Transitions transport all marks through actual lifts in the original sign extension; they are not fixed-J stabilizers. The Burau peripheral comparison uses the associated local system induced from the existing marked C3 cover and its retained deck module. No independent geometric Gauss-Manin identification, added line correction or selected E6 vertex.
premise:joint-native-compiler-cover-covariance-category - 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:marked-generator-forces-e8-without-support-multiplicity-or-minimality-input
Regularity of U and common L0 make W automatically a finite U module. Its connected separable algebra object B has simple regular B module. Every invertible M_a tensor B is simple, so free-module adjunction forces n_a=dim Hom_B(M_a tensor B,B) in{0,1}. Support S is the stabilizer subgroup of B and is isotropic by trivial twist. M_h in W yields H subset S subset Hperp=H, hence exactly four sectors once and the unique marked E8 extension with vacuum and conformal data. Before specifying the generator there are exactly four local conformal extension classes over fixed U: U, V_D8 and the two oriented V_E8 embeddings.
Scope: Complete classification within connected local unitary SAME-CONFORMAL extensions. Corrects the overstrong remark in THEOREMS.txt Section5: merely containing the four sectors already suffices under these remaining hypotheses. A tensor spectator changes the conformal vector. The two E8 classes distinguish fixed U embeddings and become isomorphic after allowing A3 diagram reversal. No derivation of this local realization from primitive P1 or physical completion.
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 - Local conformal compiler extension containing one marked generator
U=V_D5 tensor V_A3 has its fixed unitary conformal structure and marked discriminant chart. W is a connected local unitary VOA extension sharing its conformal vector and finite dimensional weight spaces; isomorphisms fix U. The marked module M_h for h=(1,1) occurs. Exact sector support, multiplicities, finite extension index, holomorphicity and an additional minimality rule are not assumed. Membership of primitive P1 in this realization is not claimed.
premise:connected-local-same-conformal-compiler-extension-with-generator
Established within the stated premises
finding:native-reflection-rotation-lift-classes-and-forgetful-images
Exact original field relations give binary ranks20,20,24,30 on40,40,40,48 variables for reflections s=0,1, rotation and joint s=0 marking. Character gauge rank4 gives65536,65536,4096,16384 marked classes. Complete forgetful images in original Gamma moduli have2176,2560,256,64 classes. All168 columns independently checked in original cocycle field action and ranks with SymPy GF2; all forgetful image orbits separately enumerated.
Scope: Complete only in the explicitly fixed native projection and morphism classes. Not all normalizer reflections or primitive physical source/state selection.
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 - 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 - Fixed native spatial reflection and rotation markings
In the original E8 sign lift stratum M_i=D_ci j, additionally mark T=D_t j^s t0 for s=0,1 and/or R=D_r j. T0 is the existing alternating sign isometry. Impose the stated peripheral equivariances and dihedral relation at s=0. Marked morphisms are simultaneous character conjugations with fixed projected operators. Forgetful images use full Gamma conjugacy. No arbitrary normalizer isometries or physical selection.
premise:fixed-native-reflection-rotation-character-markings
Established within the stated premises
finding:quarter-clock-pure-mapping-class-complete-quotient
For N=I+A, N²=0, b_i² adds N(ci+c_(i+1)) to both adjacent characters. The pure mapping class action on original Gamma classes factors faithfully through C2×C2. Burnside and independent enumeration of all2^20 character gauge classes give157696 orbits with original Gamma class size census1:4096,2:46080,4:107520, weighted total526336. Every normal form is saved.
Scope: Ordered four J sign lift stratum and pure mapping classes. No puncture permutations, silently frozen reflection marking quotient, or physical origin theorem.
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 - 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 - Pure Hurwitz quotient of the ordered quarter clock stratum
In addition to original Gamma conjugacy identify ordered four J tuples by pure mapping classes generated by adjacent squared Hurwitz moves b0²,b1², with b_i(x,y)=(xyx^-1,x). No puncture permutations or silently frozen extra reflection marking.
premise:ordered-quarter-clock-pure-hurwitz-equivalence
Established within the stated premises
finding:e8-cut-five-orbital-incidence-algebra-complete
The actual Weyl action is transitive on1120 E6 cuts. An explicit stabilizing E6/A2 reflection subgroup is transitive on each intersection class: D5 with81,D4 with270,A4 with648,A2+A2 with120 neighbours. Including equality these are the full five orbitals. All125 structure constants yield the commutative full commutant. Normalized primitive idempotents have ranks1,35,84,300,700, proving the complete irreducible Weyl decomposition.
Scope: Complete for regular A2-normal E6 root cuts in standard E8 under its actual Weyl action; not arbitrary Lie-subalgebra embeddings.
Required premises and original source excerpts
- Regular E8 cuts and full compiler flags
Fix the standard E8 roots with their actual Weyl action. E6 objects are complements of regular A2 root systems and E7 objects complements of root lines. A tetrahedron comprises the four A2 faces of a regular A3 normal system with common D5 core; its D8 incidence requires both factors. Relations are literal root inclusions and intersection sizes. Signals are rational, real or complex functions with counting inner product. Equivalence transports all flags by the Weyl action, without a selected corner or physical source.
premise:standard-e8-root-cut-full-flag-incidence-category
Established within the stated premises
finding:compiler-d8-frame-requires-full-d5-a3-flag
All135 D8 frames,7560 A3 tetrahedra and their actual incidences were generated. Each full pair of A3 normal system and common D5 core lies in exactly one D8; each D8 contains56 such flags. Bare A3 containment instead gives616 A3 systems per D8, consisting of56 coordinate D3 and560 four-axis A3 chains, and11 D8 per bare A3. Unique tetrahedron-to-frame incidence is therefore a full compiler-flag statement.
Scope: Actual full D5+A3 root flags, not uniqueness from the A3 root subsystem alone.
Required premises and original source excerpts
- Regular E8 cuts and full compiler flags
Fix the standard E8 roots with their actual Weyl action. E6 objects are complements of regular A2 root systems and E7 objects complements of root lines. A tetrahedron comprises the four A2 faces of a regular A3 normal system with common D5 core; its D8 incidence requires both factors. Relations are literal root inclusions and intersection sizes. Signals are rational, real or complex functions with counting inner product. Equivalence transports all flags by the Weyl action, without a selected corner or physical source.
premise:standard-e8-root-cut-full-flag-incidence-category
Established within the stated premises
finding:tetrahedron-incidence-lossless-e6-reconstruction
For E6/E7 incidence N and E6/tetrahedron incidence M, every entry satisfies NNt=A_D5+3I and MMt=A_D5+27I. Hence MMt=NNt+24I and for every signal f, norm(Mt f)^2=norm(Nt f)^2+24norm(f)^2. N has rank120 and loses1000 directions300+700; M has rank1120. All tetrahedron sums recover f exactly as (NNt+24I)^(-1)M(Mt f), also through an explicit rational degree3 polynomial. The information theorem does not require irreducible decomposition.
Scope: Linear data reconstruction on the same finite E6 set with counting inner product. Preserves supplied data; does not generate a distinguished corner, source or physical state law.
Required premises and original source excerpts
- Regular E8 cuts and full compiler flags
Fix the standard E8 roots with their actual Weyl action. E6 objects are complements of regular A2 root systems and E7 objects complements of root lines. A tetrahedron comprises the four A2 faces of a regular A3 normal system with common D5 core; its D8 incidence requires both factors. Relations are literal root inclusions and intersection sizes. Signals are rational, real or complex functions with counting inner product. Equivalence transports all flags by the Weyl action, without a selected corner or physical source.
premise:standard-e8-root-cut-full-flag-incidence-category
Established within the stated premises
finding:restricted-clock-lift-and-global-cover-audit-reconciled
In the original lattice chart SNF(I-J) has invariants1,1,1,1,2,2,2,2:256 J sign lifts yield16 sign-gauge classes and one class if only gauge equivalence is enlarged to quarter phases. Enlarging candidates too gives16 classes. The original additional equation chat^9=fixed S_J has exactly four solutions and one orbit under16 S_J-preserving character gauges. On all five fundamental peripheral loops the original induced Burau comparison has rho_rel=A^(7nu), nu=(1,1,1,1,-4). Faithful order12 proves globally ker(rho_rel)=ker(nu mod12), hence minimal C12. The general C12xC12 degree144 counterexample does not satisfy this factorization.
Scope: Original ordered lattice cocycle, fixed Jhat marking and explicitly induced Burau deck local system; no independent Gauss-Manin identification or splitting of the full G31 character extension.
Required premises and original source excerpts
- Joint category of original compiler flags and cover actions
Retain original E8 lattice VOA products, J/sigma/c/Y, D5+A3 root factors and quartic/erasure diagram. Transitions transport all marks through actual lifts in the original sign extension; they are not fixed-J stabilizers. The Burau peripheral comparison uses the associated local system induced from the existing marked C3 cover and its retained deck module. No independent geometric Gauss-Manin identification, added line correction or selected E6 vertex.
premise:joint-native-compiler-cover-covariance-category - 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:double-scalar-quarter-is-marked-glue-with-two-step-index
F(2q)=lambda is an actual lattice identity, not only equal norm. s=2q has order4 modulo D5+D3 and order2 modulo D8. The tower U to V_D8 to V_E8 has indices2,2 and mu-indices16,4,1. The older SO16 base with index4 mixes two different inclusions.
Scope: Given eight-channel chart and marked lattice generator. Squaring a finite disorder operator is not identified with the local vertex field.
Required premises and original source excerpts
- Given scalar quarter twist and explicit charge chart
Compare the given eight complex current channels, q=(1/4)^8, standard D5+D3 lattice and F=I5+T with T(x,y,z)=(x+y+z,x-y-z,-x+y-z,-x-y+z)/2. Eight channels and their normalization are not derived here from P1. Distinguish a local charged field, scalar monodromy character, flux register and native clock.
premise:scalar-eight-channel-twist-and-explicit-compiler-charge-chart
Excluded in the stated class
finding:scalar-quarter-two-involutive-e8-extensions
The canonical scalar torus character g_q on V_E8 has order2 and fixed VOA V_E7 tensor V_A1. Over fixed V_D8 exactly one further extension gamma g_q exists, with fixed VOA V_D8prime and order2. Their weight-one spectra136/0/112/0 and120/0/128/0 exclude identification with actual native Jhat with60/64/60/64, even under full field conjugacy.
Scope: Precisely the given scalar character restricted to fixed D8 subnet. Other source-derived field dictionaries are not excluded.
Required premises and original source excerpts
- Given scalar quarter twist and explicit charge chart
Compare the given eight complex current channels, q=(1/4)^8, standard D5+D3 lattice and F=I5+T with T(x,y,z)=(x+y+z,x-y-z,-x+y-z,-x-y+z)/2. Eight channels and their normalization are not derived here from P1. Distinguish a local charged field, scalar monodromy character, flux register and native clock.
premise:scalar-eight-channel-twist-and-explicit-compiler-charge-chart - Full unitary E8 field conjugacy with transported marks
Start with the existing unitary E8 lattice VOA, original cocycle and actual chat,Jhat,sigmahat. Full inner unitary E8 automorphisms are allowed for this conjugacy comparison. Cartan, Gaussian V4, Y, P5 and other marks are transported rather than frozen. This category is explicitly larger than original sign lifts and implies neither a G31 splitting nor a preferred microscopic conjugator.
premise:full-unitary-e8-field-conjugacy-with-transported-marks
Established within the stated premises
finding:native-quarter-glue-inner-conjugacy-and-oriented-tetrahedron-flags
Five-carrier parity kappa=exp(2pi i v(0)), v=(1/2^5,0^3), acts on the four U glue sectors as1,i,-1,-i and fixes precisely U. Among all65536 torus points modulo4 exactly15120 match actual Jhat; they form one actual W(E8) orbit. Thus C kappa C^-1=Jhat for a full VOA automorphism C. Each two inverse primitive characters share a fixed root lattice D5+A3; the7560 kernels are equivariantly the actual7560 compiler/tetrahedron flags.
Scope: Complete torus and Weyl classification plus compact E8 conjugacy, not mere matching spectra. C is not preferred and retains other marks only through their joint transport.
Required premises and original source excerpts
- Full unitary E8 field conjugacy with transported marks
Start with the existing unitary E8 lattice VOA, original cocycle and actual chat,Jhat,sigmahat. Full inner unitary E8 automorphisms are allowed for this conjugacy comparison. Cartan, Gaussian V4, Y, P5 and other marks are transported rather than frozen. This category is explicitly larger than original sign lifts and implies neither a G31 splitting nor a preferred microscopic conjugator.
premise:full-unitary-e8-field-conjugacy-with-transported-marks
Established within the stated premises
finding:native-cyclic-compiler-inner-conjugacy
After toralizing actual chat and aligning its ninth power to kappa, all candidates c_t=kappa sigma_t are covered by3^8=6561 cubic torus points. Exactly480 have the full original weight-one character and form one orbit under reflections in52 U roots. Thus chat,Jhat=chat^9,sigmahat=chat^4 are simultaneously conjugate to c_t,kappa,sigma_t. Combining kappa with the old frozen sigma already fails on an explicit root.
Scope: One selected actual admissible chat and its whole power family. All four admissible sign lifts have the same cycle character. No frozen full marked-tuple equivalence or microscopic origin is proved.
Required premises and original source excerpts
- Full unitary E8 field conjugacy with transported marks
Start with the existing unitary E8 lattice VOA, original cocycle and actual chat,Jhat,sigmahat. Full inner unitary E8 automorphisms are allowed for this conjugacy comparison. Cartan, Gaussian V4, Y, P5 and other marks are transported rather than frozen. This category is explicitly larger than original sign lifts and implies neither a G31 splitting nor a preferred microscopic conjugator.
premise:full-unitary-e8-field-conjugacy-with-transported-marks
Established within the stated premises
finding:recovered-e6-closed-81-current-module-and-unique-source-cubic
The original18 September proofs identify E6 and A2 as mutual centralizers in the actual Chevalley E8 algebra. The48=(16,3) carrier is not E6 closed; its full closure is81=(27,3) with27=16_1+10_-2+1_4. In the given affine vacuum the actual three-current correlator kappa(a,[b,c]) equals d_ABC epsilon_ijk. Its45 cubic candidates have Ward rank44 and hence a unique relative coefficient line. The positive grade-one Gram gives64/48=4/3 its exact equal-norm current-response meaning. Normalized I4/4 is not a vacuum partial trace. These completed original results are recovered into reviewed findings; missing reviewed coverage was not an open derivation.
Scope: Actual embedded current algebra and the given local unitary affine source. U1 subscripts denote internal charges. No derivation of a four-dimensional fermionic Yukawa action, particle population, mass or primitive physical P1 source identification. Recovery of completed original results, not a new mathematical first proof.
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 - Exact local reconstruction class of the marked C4 glue
Mathematical object class Ext_H(U): U=V_D5 tensor V_A3 with fixed embedding and sector chart, H=<(s,f)>=C4. Objects are connected local unitary conformal VOA extensions with the same conformal vector and underlying labelled U module exactly the sum of M_h for h in H, each once. Isomorphisms fix U and sector labels. This is an explicit mathematical reconstruction class; membership of primitive P1 is neither assumed nor derived.
premise:marked-c4-unitary-local-extension-reconstruction-class
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