TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/compiler-vacuum-current-cubic-20260918/PROOF.txt
Original file SHA-256: d65189b806facbb1eb718eec50221bb85b3a8c3417f98fb6627e01c72c98ef62
Inclusive original lines: 20-105
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000020 | The checker extracts the unmodified Chevalley and Affine classes of v498 and
000021 | the original root generator of v1, in the same pinned E8 chamber as contracts
000022 | .03-.05. All original asserts are translated to unconditional checks so that
000023 | python -OO does not disable them. No action, oscillator or Hamiltonian is added.
000024 | 
000025 | In the source convention [e_a,e_-a]=-H_a and kappa(e_a,e_-a)=-1. Thus the
000026 | compact adjoint is e_a^dagger=-e_-a, H_i^dagger=H_i. The full bracket
000027 | anti-involution is checked, not inferred from root dimensions. The affine rule is
000028 | 
000029 |  [J^a_m,J^b_n] = J^[a,b]_(m+n) + m k kappa(a,b) delta_(m+n,0).
000030 | 
000031 | At k=1 the original vacuum obeys J^a_m Omega=0 for m>=0. Consequently
000032 | 
000033 |  G_ab = <Omega|(J^a_-1)^dagger J^b_-1|Omega> = kappa(a^dagger,b).
000034 | 
000035 | The full 248-by-248 Gram consists of I_240 on root vectors, the positive E8
000036 | Cartan matrix on simple coroots, and zero cross blocks. The actual affine PBW
000037 | engine evaluates every matrix entry. Positive Cartan principal minors are exact.
000038 | This is a grade-one result; construction of the full unitary affine quotient
000039 | is part of the original WZW/net premise, not newly proved by this finite check.
000040 | 
000041 | 2. FAMILY RESPONSE IS NOT A FAMILY VACUUM STATE
000042 | The same marked 64 root currents are indexed by carrier dimension 16 and four
000043 | last-three-coordinate weights. In doubled-root convention those weights are
000044 | (-1,-1,-1), (-1,1,1), (1,-1,1), (1,1,-1). The last three are exactly the old
000045 | 48 carrier/family roots; the first is the determinant-complement sector.
000046 | 
000047 |  G_(A,i)(B,j) = delta_AB delta_ij.
000048 |  Sum_A G_(A,i)(A,j) = 16 delta_ij.
000049 | 
000050 | Normalizing this response Gram gives I_4/4. Conditioning the Gram on the
000051 | three marked family directions gives I_3/3. In the old D4 simplex notation this
000052 | one normalized Gram has (a,b,c)=(1/4,1/4,1/2).
000053 | 
000054 | Omega has grade zero. Every J_-1 Omega here has grade one, orthogonal to Omega;
000055 | the vacuum's probability in that grade-one subspace is zero. Thus I_4/4 is NOT
000056 | obtained by tracing a tensor factor of |Omega><Omega|. A prepared current
000057 | superposition, non-vacuum state, detector coupling or thermal weighting is
000058 | additional data; its populations are not fixed by this response calculation.
000059 | Equal current norms do not predict equal particle abundances or masses.
000060 | 
000061 | This also avoids mixing rotational Gibbs states, translation KMS states and
000062 | the conformal vacuum. The source holomorphic KMS manuscript distinguishes
000063 | these and begins with an existing net/vacuum representation. Its uniqueness
000064 | statements do not manufacture the raw source state.
000065 | 
000066 | 3. THE ACTUAL CUBIC CORRELATION
000067 | The same source vacuum and commutator give, exactly,
000068 | 
000069 |  C(a,b,c)=<Omega|J^a_1 J^b_0 J^c_-1|Omega>=kappa(a,[b,c]).
000070 | 
000071 | Restrict to the parent's (27,3) sector of E8 under E6+A2. Use the original A2
000072 | root actions to transport one 27 basis into the other two, including every
000073 | cocycle sign. No independent re-fitting of bases is done. Then
000074 | 
000075 |  C_(A,i)(B,j)(C,k) = d_ABC epsilon_ijk.
000076 | 
000077 | The factorization is checked on all 27^3 coefficient triples and all six
000078 | distinct-family permutations, including all zeros. A2 Cartan neutrality
000079 | excludes every repeated-family triple. There are 1,620 nonzero ordered
000080 | amplitudes in the full 81-current sector. The original affine PBW engine
000081 | independently evaluates these amplitudes. d is completely symmetric and has
000082 | 270 nonzero ordered entries. P(x)=d(x,x,x)/6 has 45 squarefree monomials with
000083 | coefficients +/-1 in this source-adapted basis.
000084 | 
000085 | Under 27=16_1+10_-2+1_4 these 45 monomials split into:
000086 |  40 of type 16_1 * 16_1 * 10_-2;
000087 |   5 of type 10_-2 * 10_-2 * 1_4.
000088 | The original seam deck and cocycle-corrected reflection from .05 preserve
000089 | every nonzero amplitude. Their actions on the entire 27 blocks are used;
000090 | applying only a family phase and omitting the E6 component would be wrong.
000091 | 
000092 | 4. UNIQUENESS, NOT JUST ONE FITTED CUBIC
000093 | Enumerate ALL degree-three monomials in the 27 coordinates (with repetitions
000094 | allowed initially). The rank-six E6 Cartan neutrality condition leaves 45.
000095 | Impose invariance under all 72 original E6 root operators. The resulting
000096 | integer Ward matrix has shape 1080 x 45, exact rank 44 and nullity one.
000097 | Its kernel contains the cubic already extracted from the source currents.
000098 | Hence its relative coefficients are forced; only common normalization remains.
000099 | The source level and fixed current normalization supply that normalization.
000100 | A negative control changing one relative sign fails the same Ward equations.
000101 | 
000102 | This is an explicit source-level identification of a known E6 invariant, not
000103 | the discovery of a new exceptional algebra or a new physical force. Independent
000104 | literature: X. Xu, Polynomial Representation of E6 and Its Combinatorial and PDE
000105 | Implications, arXiv:0811.1399; T. Deppisch, E6Tensors, arXiv:1605.05920.