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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000102 | 4. The complete local QUARTIC product, not only single current pulses
000103 | 
000104 | Set F_i(z)=:f_i(u(z)):. Let F_j# be the conjugate local field
000105 | :bar(f_j)(bar(u)):, with the usual unitary radial-coordinate convention
000106 | when interpreted as an adjoint. Then Wick's identity gives EXACTLY
000107 | 
000108 |  F_i(z) F_j#(w)
000109 |   =sum_(r=0)^4 (z-w)^(-2r)
000110 |     sum_(|alpha|=r) :partial_alpha f_i(u(z))
000111 |                       partial_alpha bar(f_j)(bar(u(w))): /alpha!.
000112 | 
000113 | Retain the FULL Taylor expansion in z-w. Denote the leading state of
000114 | each r-contraction component by C^r_ij. Its weights and source Gram are
000115 |  r0: weight8, Gram I25;
000116 |  r1: weight6, Gram4 I25;
000117 |  r2: weight4, Gram spectrum18^1,8^9,0^15;
000118 |  r3: weight2, C^3_ij=delta_ij T_E8;
000119 |  r4: weight0, C^4_ij=delta_ij vacuum.
000120 | 
000121 | In particular C^1_ij=sum_a g_ia(u)bar(g_ja)(bar(u)). The actual erasure20
000122 | gives its Gram4 I25, with no independently chosen carrier coupling. Its
000123 | three-creator/three-annihilator MODE COMPONENT acts on grade3 as
000124 |  sum_a |g_ia><g_ja|=Eij tensor I4 on the actual native20.
000125 | This is not the full local field zero mode, a selected physical execution,
000126 | or a simultaneous chosen clock on every source grade. The new primary20
000127 | is already in the current/quartic OPE from the prior addendum.
000128 | 
000129 | The r2 leading state S_ij=C^2_ij is NOT alone the whole pole-four
000130 | coefficient: the r3 Taylor term at second order also contributes there.
000131 | Its all-mode1 two-u/two-bar-u component is an intrinsic oscillator
000132 | decomposition of the given source, not an executed projection. The full
000133 | identity above keeps the other terms and every regular contribution.
000134 | 
000135 | 5. Nontrivial logical action exists in the full source algebra
000136 | 
000137 | Let A_i=P_i(u(-1)), D_j=bar(P_j)(bar(u)(1))=A_j*.
000138 | On the ENTIRE weight4 space define the existing mode-word operator
000139 |  Eij=A_i D_j/sqrt(N_i N_j).
000140 | Four annihilators first pass through weights3,2,1,0; only then four
000141 | creators return to weight4. No vacuum/code projection is inserted.
000142 | Exactly Eij=|f_i><f_j| plus0 on the orthogonal weight4 complement.
000143 | Higher modes, opposite polarization or a nonzero lattice charge at
000144 | weight4 leave fewer than four matching mode1 positive-polarization quanta
000145 | and are killed. The neutral726 check validates the full neutral block;
000146 | the charged-block assertion follows from this energy/occupation argument.
000147 | 
000148 | Therefore all25 matrix units, and hence the full M5(C) logical algebra,
000149 | are AVAILABLE in the source-generated mode algebra. It is irreducible5,
000150 | with centerC, not an intrinsically selected M3+M2 block algebra.
000151 | The matrixunit identities do NOT hold as identities on all higher grades.
000152 | These are products of global modes of local fields, not themselves a
000153 | newly proved locally supported instrument. They do not contradict the
000154 | older theorem about finite Cartan-Weyl unitary histories, a different class.
000155 | Expanding an already specified R5 in these units would implement its
000156 | chosen coefficients mathematically, not derive or select R5 from P1.
000157 | 
000158 | 6. First nontrivial local pair component and its intrinsic geometry
000159 | 
000160 | Use the normalized Sym²V4 Fock monomials q_A=x^A/sqrt(A!), |A|=2.
000161 | The actual quartic2+2 flattening matrices are
000162 |  (H_i)_(AB)=partial_(A+B) f_i(0)/sqrt(6 A! B!).
000163 | They are symmetric, real in the literal original code coordinates,
000164 | HS-orthonormal and mutually commuting. Their ten joint eigenlines are
000165 | distinct. More intrinsically the C^2 pair operators6H_i H_j* form a
000166 | commuting pair-space algebra; that statement is covariant under changes
000167 | of the pair basis. The computation introduces no diagonalizing physical
000168 | coupling or field identification.
000169 | 
000170 | Let h_t=(h_t0,...,h_t4) be the joint eigenvalue rows, t=1...10.
000171 | They obey sum_t h_t h_t^T=I5, ||h_t||²=1/2 and
000172 |  |h_t.h_s|=1/6 (t!=s). In an ON source basis B_t=q_t(u)bar(q_t)(bar(u)),
000173 |  S_ij=6sum_t h_ti h_tj B_t.
000174 | This is checked as a literal equality of original Fock polynomials.
000175 | Its nonzero Gram is36(h_t.h_s)²=8I10+ones10, hence18^1,8^9.
000176 | The logical rank-one projectors P_t=v_t v_t^T, v_t=sqrt2 h_t, satisfy
000177 |  sum_t P_t=2I5, Tr(P_t P_s)=1/9 for t!=s.
000178 | They give a source-derived TEN-RAY geometry. This pair-space10 and the
000179 | earlier moment-Hankel rank10 concern different objects; no identification
000180 | of them by dimension is made.
000181 | 
000182 | The ten projectors span C5 through their ranges and all overlaps are
000183 | nonzero. An antiunitary conjugation fixing each ray exists (all coordinates
000184 | above are real). Any two such antiunitaries differ by a unitary fixing
000185 | each ray; nonorthogonal eigenrays force that unitary to be scalar.
000186 | Thus the configuration itself determines a projective real structure,
000187 | unique up to a common phase. This is a geometric consequence of the
000188 | source tensor, not a new restriction of physical states to real vectors.