TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/FULL-SOURCE-HULL-AND-QUARTIC-OPE.txt
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000033 | 2. Exact minimal source closure; no finite five/twenty/ten-dimensional hull
000034 | 
000035 | The source vector space and its energy grading are
000036 |  V_E8=direct_sum_(lambda in E8) M(1)e^lambda,
000037 |  h(h_v1(-m1)...h_vk(-mk)e^lambda)=sum m_i+lambda²/2.
000038 | Its complete finite weight spaces are V_n; its Hilbert completion is H_E8.
000039 | The eight Cartan currents and ALL240 root currents generate this algebra.
000040 | 
000041 | A suitable four-annihilator monomial maps any nonzero homogeneous quartic
000042 | to a nonzero scalar vacuum. In particular the whole C5 generates vacuum.
000043 | For a root alpha and any lattice lambda the leading coefficient
000044 |  E_alpha(z)e^lambda=epsilon(alpha,lambda)z^(alpha.lambda)
000045 |    [e^(lambda+alpha)+higher powers of z]
000046 | generates e^(lambda+alpha) by a current mode. All roots together generate
000047 | the original lattice; no root is selected as a physical interaction.
000048 | Negative Cartan modes then generate every oscillator state over each charge.
000049 | Thus the exact smallest algebraic CURRENT-MODE-invariant hull of C5 is V_E8.
000050 | Both twenties and all their descendants are included automatically.
000051 | 
000052 | For actual local smooth smearings, use a closed Hilbert (or compatible
000053 | smooth-vector) hull, because a smooth field typically has infinitely many
000054 | energy components. Partition a global Fourier profile into finitely many
000055 | locally supported smooth pieces. Hence invariance under all supported
000056 | currents implies invariance under their global modes, without making an
000057 | individual global mode local. The closed hull is H_E8. It is irreducible
000058 | for the full represented field algebra; its bounded commutant is C I.
000059 | This is not the physical H_adm derivation, a particle count, or a claim
000060 | that the earlier Hankel lower bound10 was an exact complete source size.
000061 | 
000062 | 3. Full operator product algorithm and controlled finite grading
000063 | 
000064 | For a source state p(h_-m)e^mu and ANY lattice lambda,
000065 |  Y(e^lambda,z)[p(h_-m)e^mu]
000066 |  =epsilon(lambda,mu) z^(lambda.mu)
000067 |   exp(sum_(m>=1)lambda(-m)z^m/m)
000068 |   p(h_v(-m)-(v.lambda)z^-m)e^(lambda+mu).
000069 | For a descendant field, take normal ordered derivatives of the currents
000070 | times this vertex field. This determines every OPE coefficient and every
000071 | finite ordered mode word in the full source, including all charged and
000072 | higher-mode intermediates. It is not only a dimension or graph argument.
000073 | 
000074 | For a word J_(n1)...J_(nk) acting on weight h, its successive weights are
000075 |  h, h-nk, h-nk-n(k-1),...,h-sum n_j.
000076 | A cutoff above their maximum, retaining ALL charges and oscillator PBW
000077 | states at those weights, computes that word exactly. No state is projected
000078 | back into C5 after any step. Enlarging the cutoff cannot change the result.
000079 | The same rule includes weight shifts h_field-1-n for general VOA modes.
000080 | 
000081 | A finite Fourier truncation is NOT an exactly local field. For each real
000082 | orthonormal current, Sugawara positivity gives on the finite-energy core
000083 |  ||J_n psi|| <=sqrt(62)sqrt(1+|n|)||(L0+1)^(1/2)psi||.
000084 | The negative-mode bound follows from the positive one and its level1 CCR.
000085 | Thus the tail of a smooth smeared current is bounded by that graph norm
000086 | times sqrt(62)sum_(|n|>M)|fhat_n|sqrt(1+|n|), which tends to zero faster
000087 | than any fixed power for smooth profiles. Stronger energy graph norms
000088 | control successive products; current-generated descendant fields have
000089 | polynomial energy bounds. This justifies convergence, not exact locality
000090 | of a truncated matrix. No claim that arbitrary infinite histories are
000091 | already numerically controlled follows from a finite test.
000092 | 
000093 | The complete character, including its horizontal weight variables, is
000094 |  ch V_E8(q,z)=prod_(m>=1)(1-q^m)^(-8)
000095 |               sum_(lambda in E8)q^(lambda²/2)e^lambda(z).
000096 | It determines every finite horizontal E8 decomposition by Weyl characters.
000097 | Direct full-lattice counting gives weights0...6 dimensions
000098 |  1,248,4124,34752,213126,1057504,4530744.
000099 | The neutral weight4 dimension is only726, not213126. Its use for a single
000100 | neutral return is exact; it is not the full coherent multi-charge source.