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Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/INTRINSIC-EVENT-GEOMETRY-AND-COMPOSITION.txt
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000043 | 2. Actual local holomorphic multiplication: no fitted geometry
000044 | 
000045 | Use the actual source polynomials, in orthonormal mode1 positive currents:
000046 | 
000047 |   P0=sum_a x_a^4,
000048 |   P1=x0²x1²+x2²x3², P2=x0²x2²+x1²x3²,
000049 |   P3=x0²x3²+x1²x2², P4=x0x1x2x3,
000050 |   f_i=P_i/sqrt(N_i), N=(96,8,8,8,1).
000051 | 
000052 | These are the original process._code_basis polynomials, not an arbitrary
000053 | five-space. Since u_a.u_b=0, F_i(z)F_j(w) has no contractions and its
000054 | coincident regular product is exactly :f_i f_j:. Let
000055 | 
000056 |   M:Sym² L -> H_E8[weight8],       M(v sym w)=f_v f_w,
000057 |   L=span(f_i),
000058 | 
000059 | with the multiplication induced from the ordered tensor map f_i tensor f_j
000060 | ->f_i f_j. On normalized Sym² basis the diagonal columns are f_i² and
000061 | off-diagonal columns sqrt2 f_i f_j. All degree8 monomials are retained.
000062 | 
000063 | Exact source Fock arithmetic gives H=M* M with
000064 | 
000065 |   spec H=28^1,10^5,18^9,
000066 |   Pi1=(H-10I)(H-18I)/180,
000067 |   Pi5=(H-28I)(H-18I)/144,
000068 |   Pi9=(H-28I)(H-10I)/(-80).
000069 | 
000070 | They are orthogonal spectral projectors summing to I15. In particular M
000071 | is injective. The1-sector is |I5><I5|/5 in symmetric-tensor coordinates;
000072 | it also determines the symmetric bilinear real structure projectively.
000073 | The construction of H/Pi9 itself does NOT require a real-form choice.
000074 | 
000075 | There is a literal source-product relation to the earlier full local
000076 | quartic-conjugate-quartic Wick ladder. Denote its leading double-contraction
000077 | map by S2. With the intrinsic real structure used solely to compare
000078 | Sym² L with the symmetric part of L tensor bar(L), exact matrices obey
000079 | 
000080 |   H = 10 I15 + (S2* S2)|Sym²,
000081 |   (S2* S2)|Sym² =18 Pi1+8 Pi9.
000082 | 
000083 | This relates two actual products of the same original fields, not just
000084 | matching dimensions or separate spectra. Their spaces are different
000085 | source grades (8 and4); the identity compares Gram tensors on common
000086 | input labels, not equal physical output states. All old Taylor-term and
000087 | mode-versus-local-field qualifications remain in force.
000088 | 
000089 | These normalized two-point Grams can equivalently be read as CENTRAL
000090 | leading OPE coefficients. For A_alpha=:f_i f_j: in the above Sym² basis,
000091 | 
000092 |   A_alpha#(z) A_beta(w)=H_alpha,beta I/(z-w)^16
000093 |                        + lower poles and regular full source fields.
000094 | 
000095 | For F_i the analogous central tensor is delta_ij/(z-w)^8. These coefficients
000096 | are data of the given normalized *-field algebra, before evaluating a
000097 | state. Every normalized functional for which the formal coefficients are
000098 | defined has omega(I)=1; it leaves these central coefficients unchanged.
000099 | This is NOT a uniform analytic short-distance-limit claim for arbitrary
000100 | singular infinite-energy states, nor proof of P1 state admissibility.
000101 | 
000102 | The composite states are holomorphic isotropic degree8 primaries. Thus
000103 | their full common smooth local-smear VACUUM Gram is c_f H, analytically
000104 | from <L_-1^n A_alpha,L_-1^m A_beta>=delta_nm n!(16)_n H_alpha,beta.
000105 | All infinite Fourier tails are retained; no finite Fourier fixture is
000106 | called a supported local field. Hence the same projective1+5+9 spectral
000107 | decomposition survives common nonzero smearing of these local composites.
000108 | Product of separately smeared F's is distinguished in section6.
000109 |