TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/INTRINSIC-EVENT-GEOMETRY-AND-COMPOSITION.txt
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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
000165 | the assertion '5=5'. The8 is a conformal weight, not a spacetime dimension.
000166 | 
000167 | Original erasure a_a=bar u_a(1) remains compatible with this product by
000168 | the actual source derivation law
000169 | 
000170 |   a_a(f_i f_j)=(partial_a f_i)f_j+f_i(partial_a f_j).
000171 | 
000172 | There is no independently tuned coupling. This identity does not close
000173 | the products on the old erasure20 or identify new states as particles;
000174 | all grade7 outputs are retained. General conjugate products also retain
000175 | the known singular erasure20 and regular primary carry20.
000176 |