TFPT scientific source excerpt Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/SEAM-STATE-EVENT-POLAR-SELECTION.txt Original file SHA-256: c55c96c733a3ad7c314fe251bc8ff8da8940129fd6e99dfc08e449bf4475c1cf Inclusive original lines: 4-160 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000004 | 1. Question, accepted inputs and exact conclusion 000005 | 000006 | Do the ORIGINAL seam, state and event data select the actual polar transport 000007 | and original 3+2 mark in the complete source? Do not insert a fitting 000008 | coupling, measurement, normalization rule or a closed five-state dynamics. 000009 | 000010 | Retain the accepted E8_1 lattice/current realization, its unitary vacuum, 000011 | the actual J-positive Cartan embedding, original normalized code fields 000012 | F_i=:f_i(u):, and the FULL local field hull H_E8. Reuse the previous 000013 | intrinsic ten-ray reconstruction of the six unmarked simplex axes. Original 000014 | N coordinates are used only to retain the unchanged compiler sigma, 000015 | native event labels and q=5 for comparison. The choice/preparation of q 000016 | is granted as the existing marked INPUT, not derived as a unique state. 000017 | 000018 | NEW POSITIVE RESULT: actual native event differences are precisely the 000019 | original difference FIELDS. Their full vacuum response Gram supplies the 000020 | positive factor in the old polar prescription. Without inserting P2 or 000021 | its angle, its exact matrix is recovered as the positive RELATIVE response 000022 | kernel R^(-1/2) K_W R^(-1/2). The polar formula thus has an intrinsic 000023 | full-field response interpretation, beyond a chosen distance minimizer. 000024 | 000025 | This does NOT show that the original action executes this normalization, 000026 | chooses the associated readout, or prepares the marked state. The direct 000027 | raw responses are nonorthogonal. GNS/reflection positivity supplies their 000028 | metric, not an event-processing or field-coupling law. On a second actual 000029 | source state even the vacuum-normalized fields do not have unit response 000030 | Gram. Native J and the coupled determinant event lift must also be kept 000031 | distinct: the latter has a different coherent response and does not extend 000032 | with its literal Cartan action to the original E8 lattice field algebra. 000033 | 000034 | 2. Actual native event differences are local source fields 000035 | 000036 | Let n_j be original columns of N, with n_j^*n_k=delta_jk-1/6 and sum n_j=0. 000037 | The previous full quartic OPE reconstructs these axes in these coordinates; 000038 | the regular simplex orientation fixes relative signs up to one common sign. 000039 | For a coefficient vector v, put F_v=sum_i v_i F_i. 000040 | 000041 | Each actual native reflection r induces one transposition (jk) on these 000042 | six coherent fields. Every transposition has four actual native lifts. 000043 | Choose the existing marked q=5. An incident (j5) gives 000044 | 000045 | (Ad_r-id) F_n5 = F_nj-F_n5 =: Delta_j, j=0,...,4. 000046 | 000047 | The other ten transposition images give zero. These are exact identities 000048 | of entire local fields, not only their compressed matrices. The native 000049 | root/lattice automorphism has a lift to the field algebra; cocycle lift 000050 | choices do not change its neutral Heisenberg action. Difference of local 000051 | fields with the same support is local. A difference of automorphisms is 000052 | NOT thereby a physical coherent intervention, unitary or detector event. 000053 | 000054 | With D=(n_j-n_5) and G=D^*D, the actual vacuum field contraction gives 000055 | 000056 | G=I5+J5, spec(G)=(6,1,1,1,1), 000057 | =(delta_jk+1)/(z-w)^8. 000058 | 000059 | Here J5 is the all-ones MATRIX, not the native complex-structure clock J. 000060 | Each d_j has squared norm2; separate unit normalization leaves off-diagonal 000061 | overlap1/2. The five responses are not orthogonal records. 000062 | 000063 | This persists for properly supported smooth common profiles. Creativity 000064 | and translation give Y(f_i,z)Omega=exp(z L_-1) f_i and 000065 | 000066 | =delta_nm delta_ik n!(8)_n. 000067 | 000068 | Thus the full smeared-field vacuum Gram is c_f G for a common positive 000069 | profile norm c_f, whenever the response is nonzero. The all-n identity is 000070 | analytic; finite checks n=0,1,2,3 have squared descendant norms1,8,144,4320. 000071 | No finite Fourier polynomial is described as proper local. The infinite 000072 | smooth-profile tail is retained. No new kernel coupling is introduced. 000073 | 000074 | 3. Complete uncontracted products and original clock orbit data 000075 | 000076 | The unchanged sigma=(234) fixes 0,1,5. After retaining q=5, its five event 000077 | labels therefore have the original three moving and two remaining fixed 000078 | slots. Define Q_C=diag(0,0,1,1,1), Q_W=I-Q_C as LABEL cuts. These counts 000079 | come from the specified original sigma, not from Standard Model charges; 000080 | neither sigma nor q is derived anew from raw P1 here. 000081 | 000082 | Keep the full local products sum_(j in C/W) Delta_j(z)Delta_j#(w), with 000083 | every Wick contraction r=0,...,4 and ALL Taylor coefficients. Their 000084 | coefficient tensors in the five-field multiplet are 000085 | 000086 | K_C=D Q_C D^*, K_W=D Q_W D^*, 000087 | R=K_C+K_W=D D^*=I5+6 n_5 n_5^*=I5+5 e0 e0^*. 000088 | 000089 | These coefficient matrices are not full local zero modes or a proposed 000090 | physical effective Hamiltonian. The uncontracted r0 map into actual 000091 | weight8 source fields is injective with Gram I25, as previously proved. 000092 | Thus their difference is visible in complete source tensors, not caused 000093 | by repeatedly projecting to C5. This map is NOT a homomorphism of matrix 000094 | multiplication. The matrix commutator below is not asserted to equal a 000095 | local field commutator. 000096 | 000097 | Exact spectra and coefficient commutator are 000098 | 000099 | spec K_C=(4,1,1,0,0), spec K_W=(3,1,0,0,0), 000100 | ||[K_C,K_W]||_HS^2=72. 000101 | 000102 | Their ranks alone do not give an orthogonal 3+2 split. The Gram matrices 000103 | of the actual leading source tensors (C,W) in each Wick class are 000104 | 000105 | r0: [[18,6],[6,10]], r1: [[72,24],[24,40]], 000106 | r2: [[144,72],[72,72]], r3: [[144,96],[96,64]], 000107 | r4: [[36,24],[24,16]]. 000108 | 000109 | The r3 fields are exactly 6 T_E8 and4 T_E8; r4 scalars6 and4. Leading r2 000110 | is not the whole pole-four coefficient: Taylor descendants from r3 remain. 000111 | The general full Wick identity is the one in the previous addendum. 000112 | 000113 | On E+=span(e0,e1), with the original orthonormal basis 000114 | e0=sqrt(6/5)n5, e1=sqrt(5/6)(n2+n3+n4+3n5/5), one gets 000115 | 000116 | K_C|E+ = [[18,-6],[-6,2]]/5, 000117 | K_W|E+ = [[12, 6],[ 6,3]]/5. 000118 | 000119 | On the color-difference two-space K_C=I,K_W=0; on the weak difference 000120 | line a=(n0-n1)/sqrt2, K_C=0,K_W=I. Their unital generated coefficient 000121 | algebra is exactly C on the two-space, C on the line, and M2 on E+; 000122 | dimension6. A nonzero commutator and the two real symmetric rank-one 000123 | generators give all M2; the scalar blocks are separated by polynomials. 000124 | This is a 2+1+2 symmetry-multiplicity decomposition. A chosen line in E+ 000125 | can join either block, but that grouping is not a central 3+2 superselection 000126 | of this algebra. This is not a new classification of the entire field net. 000127 | 000128 | 4. Exact positive relative response reproduces the actual polar mark 000129 | 000130 | Use only the just-derived event Gram to form its mathematical positive 000131 | normalization. Since D is invertible, 000132 | 000133 | U=D G^(-1/2)=R^(-1/2)D, 000134 | G^(-1/2)=I+(1/sqrt6-1)J5/5, 000135 | R^(-1/2)=I+(1/sqrt6-1)e0 e0^*, 000136 | U^*U=I. 000137 | 000138 | Consequently the relative tensor is an exact orthogonal projector: 000139 | 000140 | P_rel=R^(-1/2) K_W R^(-1/2)=U Q_W U^*, 000141 | P_rel^2=P_rel=P_rel^*, rank P_rel=2. 000142 | 000143 | Only now compare with the original target. With 000144 | c_pol=-sqrt(3/5)e0+sqrt(2/5)e1 and 000145 | F_sigma=a a^*+e0 e0^*+e1 e1^*, one obtains LITERALLY 000146 | 000147 | P_rel=F_sigma-c_pol c_pol^*=P2_original. 000148 | 000149 | No charge eigenvalues, SM representation, fitted angle or fitted rotation 000150 | enter this derivation. Its complement has rank3. For other original fixed 000151 | q the construction is transported by the native fixed-point permutations, 000152 | as in the previously established clock-centralizer orbit. 000153 | 000154 | The unique POSITIVE correction B satisfying B R B=I is R^(-1/2): 000155 | put X=R^(1/2) B R^(1/2)>0, whence X^2=R and X=R^(1/2). 000156 | Without positivity of the correction, completeness C R C^*=I instead 000157 | permits every C=V R^(-1/2), V unitary. Positivity of the source Gram is 000158 | not positivity of this correction, nor a reason to execute any correction. 000159 | The standard polar facts agree with Higham1986 Theorem1.1/Lemma2.1: 000160 | https://eprints.maths.manchester.ac.uk/694/1/high86p.pdf .