TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/E8-BOUNDARY-DICTIONARY.txt
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000026 | 2. Concrete current dictionary of the stipulated local E8_1 realization
000027 | 
000028 | This section uses the standard positive E8 lattice vacuum representation.
000029 | It is NOT an identification of that representation with the raw P1 kernel.
000030 | Use orthonormal R8 coordinates, split5+3, with roots
000031 |   +/-e_i +/-e_j (112 roots),
000032 |   (epsilon_1,...,epsilon_8)/2 with an even number of minuses (128 roots).
000033 | Let H^a=i partial X^a and V_p=e_p :exp(i p.X):. The local currents are the
000034 | eight H^a and240 root vertices. Their total conformal weight is1.
000035 | Under Spin10 x SU4 they decompose as
000036 |   (45,1)+(1,15): 60 currents, internal quarter grade0;
000037 |   (10,6): 60 currents, weights(1/2,1/2), quarter grade2;
000038 |   (16,4) and its conjugate: 64 each, weights(5/8,3/8), grades1/3.
000039 | The labels of conjugate spinors depend on the stated orientation convention.
000040 | The actual code exports every charge, grade, conformal subweight and adjoint
000041 | sign. None of these248 weight-one bosonic currents is silently declared an
000042 | elementary four-dimensional fermion or an independent native CCR mediator.
000043 | The operator/statistics distinction was already established in
000044 | seam-joint-ward-state-20261014:858-895 and is reused, not claimed as new.
000045 | 
000046 | 3. Exact cocycle, all distinct-point current responses and source series
000047 | 
000048 | The script fixes a unimodular simple-root basis B, Gram G=B^t B, and integral
000049 | coordinates p=Bm. Its real ordered cocycle is
000050 |   epsilon(p,q)=(-1)^[sum_(i>j) m_i n_j G_ij].
000051 | Bimultiplicativity proves associativity. Since G has even diagonal,
000052 | epsilon(p,q)/epsilon(q,p)=(-1)^(p.q). This cancels the coordinate exchange
000053 | sign of integral powers, giving bosonic root-current locality. With these
000054 | basis phases V_p^dagger=epsilon(-p,p) V_-p; conjugate two-point norms are
000055 | positive in the stipulated lattice Fock representation.
000056 | 
000057 | For radially ordered insertions at distinct points the exact vacuum formula is
000058 |   <product_i V_pi(z_i)> = delta_(sum_i pi,0)
000059 |       product_(i<j) epsilon(pi,pj) (z_i-z_j)^(pi.pj).
000060 | All mixed Cartan insertions follow from the Heisenberg Ward recursion. For
000061 | H_di(u_i), set mu_i=sum_j di.pj/(u_i-z_j) and
000062 | C_ij=di.dj/(u_i-u_j)^2. Remove the first Cartan insertion, multiply by mu_i,
000063 | and sum its contractions C_ij times the recursion without both i,j. The
000064 | root response multiplies the result. This gives all248 current responses,
000065 | not just root-vertex insertions or a two-point Gram matrix.
000066 | 
000067 | Thus, writing O_J=sum_p J_p V_p+sum_a J_a H^a, all separated-point Taylor
000068 | coefficients of the formal normalized source series
000069 |   Z[J]/Z[0]=sum_n (1/n!) integral <product_i O_J(z_i)>
000070 | are fixed in this representation. An integrated distributional functional
000071 | still needs contact/regularization data; no convergence theorem for arbitrary
000072 | J is asserted. Z[0], determinant-line phase, geometry and state/time pullback
000073 | are not reconstructed by the normalized series.
000074 | 
000075 | 4. Retained quarter grading and an explicitly non-Gaussian response
000076 | 
000077 | With v=(0,0,0,0,0,1/2,1/2,1/2), g V_p g^-1=exp(2pi i v.p)V_p.
000078 | The grade4 v.p modulo4 gives root counts52/64/60/64 and, including Cartan,
000079 | current counts60/64/60/64. Neutral multipoint products have total grade0.
000080 | This is the internal algebra grading. It is not identified with the actual
000081 | charged-source time or its vacuum phase. The previously checked particle/hole
000082 | quarter-time equation and sea/central factors are preserved separately.
000083 | 
000084 | For a=(1/2)^8, b=(-1,-1,0,0,0,0,0,0), c=-(a+b), the grades are(3,0,1).
000085 | At positions(3,2,1), the ordered three-root response is-1/2. For the four
000086 | charges(a,-a,b,-b) at(4,3,2,1), the raw-cocycle response is-3/4 and its
000087 | connected part is+1/4. With adjoint-normalized conjugates, the four-point
000088 | and connected values change by the common adjoint factor; nonvanishing is
000089 | basis-phase independent. The adjoint-normalized H_a(4)V_a(3)V_-a(1)
000090 | response is+1/3. All six joint exchanges of the neutral triple agree.
000091 | Free boson/fermion realization does not make nonabelian current variables
000092 | Gaussian. These source Taylor coefficients do not prove a physical Yukawa
000093 | coupling, native pair-to-boson transfer or four-dimensional interaction law.
000094 | 
000095 | The script checks 57,600 root-pair locality/grade relations and240 positive
000096 | adjoint two-point coefficients. Independent review also checked cocycle
000097 | associativity and the Cartan rules, caught and corrected a serialization
000098 | variable-shadowing error, and replayed the repaired output. The published
000099 | probe is now read back and recomputed by a regression test. All45 tests in
000100 | this research directory passed. The known E8 character values[1,248,4124]
000101 | are explicitly attributed to v156, not newly computed by the response script.
000102 | 
000103 | 5. Relation to the original field-dependent P1 kernel
000104 | 
000105 | The original source01:302-316 defines the spectral comparison using its given
000106 | D_rel, D_ref, profile f and eta phase. Its field insertion algebra is an
000107 | additional structure at319-343. At source04:2160-2184,2272-2302 the physical
000108 | S_adm[Phi] and its derivatives are written symbolically. The discrete reverse
000109 | bootstrap above constrains numerical and lattice closure data; its cited
000110 | calculations contain no reconstruction of these field-dependent operators.
000111 | To equate the response series in section3 with that kernel requires an actual
000112 | local source/insertion pullback preserving charges, statistics, domains,
000113 | state and time, plus the original Phi dependence and absolute/contact data.
000114 | Writing a substitution J=F(Phi), choosing a kinetic operator or choosing a
000115 | spectral profile would impose those data rather than derive them here.
000116 | 
000117 | The present positive outcome is an executable complete separated-point E8_1
000118 | current dictionary under its explicit realization premise. It supplies exact
000119 | nonlinear response targets for that physical identification. It does not
000120 | close the raw P1/P2 field dictionary or complete TFPT. No physical gates,
000121 | ledger statuses, paper claims or Lean theorems are promoted by this result.