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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Inclusive original lines: 158-227
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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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.
000189 | 
000190 | 7. Reconstruct the ORIGINAL unmarked W6 simplex, with no W6 input
000191 | 
000192 | On the intrinsic projective REAL locus, for unit w define only as an
000193 | algebraic diagnostic M(w)=sum_t(v_t.w)^4. Tightness and Cauchy imply
000194 |  M(w)>= (sum_t(v_t.w)²)²/10=2/5,
000195 | with equality precisely when every (v_t.w)²=1/5.
000196 | No physical potential, energy-minimization or state-selection rule is
000197 | postulated here. We determine a distinguished geometry of the tensor.
000198 | 
000199 | Write H for the10x5 row matrix h_t. Since H^T H=I5, equality is equivalent
000200 | to H w=s/sqrt10, s in{+1,-1}^10, HH^T s=s.
000201 | The matrix6HH^T has diagonal3 and off-diagonal entries+/-1. Exhaustion of
000202 | all1024 sign vectors gives exactly12 solutions; quotient by overall sign
000203 | gives six rays w=H^T s/sqrt10. One orientation makes their Gram
000204 |  (6/5)I6-(1/5)ones6 and their sum0: a regular6-vertex simplex in real5.
000205 | The overall antipode and permutation are not selected.
000206 | 
000207 | ONLY AFTER this reconstruction, introduce the original existing matrix
000208 |  N=diag(sqrt(4,12,12,12,24)) R/sqrt48,
000209 |  R rows=(2,2,-1,-1,-1,-1),(0,0,-1,-1,1,1),
000210 |         (0,0,-1,1,-1,1),(0,0,1,-1,-1,1),(1,-1,0,0,0,0).
000211 | For all six reconstructed rays, EXACTLY
000212 |  w_j w_j^T=(6/5)N_(pi(j))N_(pi(j))^T,
000213 | with only a permutation pi, no fitted rotation or matching metric.
000214 | N is the original W6 simplex chart used in the previous native contracts.
000215 | It is absent from the derivation of H, its rays, the real structure and
000216 | the12 equality points. The literal quartic constructor also has no N input.
000217 | 
000218 | This meets the structural success target: a previously separately
000219 | supplied unmarked native W6 geometry is recovered from the full local
000220 | quartic OPE. It does NOT derive the positive polar transport or marked P2.
000221 | 
000222 | Essential complex control: on the full complex code the function
000223 |  M_C(w)=sum_t |v_t* w|^4 has additional minima. For two different real
000224 | simplex vectors w1,w2, w_theta=cos(theta)w1+i sin(theta)w2 has norm1 and
000225 | all squared overlaps1/5. The checker retains a nonreal example. Thus
000226 | the six-ray theorem concerns the source's intrinsic projective real
000227 | geometry; it is not a claim of six unique complex states or physical vacua.