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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 | 
000110 | 3. Six rays reconstructed over C, not from a real minimum
000111 | 
000112 | Define the algebraic configuration intrinsically by the actual product
000113 | component
000114 | 
000115 |   Z={ [v] in CP4 : Pi9(v tensor v)=0 }.
000116 | 
000117 | M is injective and its9-sector has nonzero eigenvalue18; this is an actual
000118 | vanishing source-product component. Using its zero locus to RECONSTRUCT
000119 | a configuration is not a rule favoring those rays as physical states.
000120 | 
000121 | Direct row reduction from the ORIGINAL H, without introducing N or W6,
000122 | gives these nine quadrics. Write v=(a,sqrt3 b,sqrt3 c,sqrt3 d,sqrt6 e):
000123 | 
000124 |   a²-d²-4e²=0,
000125 |   b²-d²=c²-d²=0,
000126 |   ab+cd=ac+bd=ad+bc=0,
000127 |   be=ce=de=0.
000128 | 
000129 | This is solved over COMPLEX numbers. If e !=0, then b=c=d=0 and a=+/-2e:
000130 | two projective rays. If e=0, then a²=b²=c²=d². A nonzero solution has a!=0;
000131 | normalize a=1, so b,c,d=+/-1 with bcd=-1: four rays. No solutions lie on
000132 | a=0 except the zero vector. Exact Groebner bases independently solve the
000133 | a=1 chart and exclude all four nonzero charts on a=0.
000134 | 
000135 | Thus Z has exactly SIX complex projective rays. No reality assumption,
000136 | clock, q, target polar angle or original simplex coordinates entered.
000137 | Normalize/orient representatives w_q so sum_q w_q=0. Their Gram is
000138 | 
000139 |   w_q* w_r=(6/5)delta_qr-1/5.
000140 | 
000141 | Only now compare the original N=G_raw^(1/2) W6/sqrt48, with
000142 | G_raw=diag(4,12,12,12,24): every projector satisfies literally
000143 | w_q w_q*=(6/5)n_pi(q)n_pi(q)*, with permutation only and no fitted rotation.
000144 | 
000145 | This strengthens the earlier real equality-set reconstruction. Complex
000146 | minima of the earlier DIFFERENT fourth-moment functional still exist;
000147 | they are not retroactively excluded as physical states. They are simply
000148 | not all zeros of this distinct holomorphic product component.
000149 | 
000150 | The5-sector is span{w_q tensor w_q-I5/5}. This span has dimension5,
000151 | and the linear extension
000152 | 
000153 |   I5_map(w_q)=(sqrt5/2)(w_q tensor w_q-I5/5)
000154 | 
000155 | is a well-defined isometry L -> Ran Pi5: both six-tuples sum to0 and their
000156 | Grams differ by factor4/5. Actual event transpositions give
000157 | 
000158 |   Sym²(T) I5_map=I5_map T,
000159 |   H I5_map=10 I5_map.
000160 | 
000161 | Consequently M I5_map/sqrt10 embeds the SAME actual native five into full
000162 | weight8 source fields. The checker verifies all15 event images, hence
000163 | their entire generated action; their fourfold native lifts were fixed
000164 | in previous original-source checks. This is operator equivariance, not