TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/LOCAL-CURRENT-ERASURE-AND-ORDERING.txt
Original file SHA-256: 4c3397bb7c0bc75c790b0cca29f5a38abfdff9a564ca5179203ef5a801c554de
Inclusive original lines: 97-146
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000097 | 4. Full response to every smooth real vector profile
000098 | 
000099 | Let V(theta) be an arbitrary smooth REAL eight-component Cartan profile.
000100 | Set Vhat_n=int V(theta)e^(-in theta)dtheta/(2pi), so
000101 | Vhat_-n=bar(Vhat_n), and J(V)=sum_n h_Vhat_n(n) is Hermitian.
000102 | Smoothness implies sum_(m>=1)m||Vhat_m||^2<infinity.
000103 | Write Vhat_1=sum_a(p_a u_a+q_a bar(u_a)), p,q complex4.
000104 | All norms on these complex vectors are Hermitian Euclidean norms.
000105 | 
000106 | For ANY unit vector psi in the actual five-code:
000107 | 
000108 |  J(V)psi = h_Vhat1(1)psi +sum_(m>=1)h_Vhat(-m)(-m)psi.
000109 | 
000110 | The first term has conformal grade3, norm squared ||q||^2. The m=1
000111 | creation term has grade5, norm squared ||p||^2+2||q||^2: commuting its
000112 | adjoint through gives ||Vhat1||^2 plus the annihilation norm squared.
000113 | For m>=2 the creation term has grade4+m and norm squared
000114 | m||Vhat_m||^2, since the adjoint annihilates the input code.
000115 | All these different grades are orthogonal. Consequently, exactly,
000116 | 
000117 |  ||J(V)psi||^2=3||q||^2+||p||^2+Tail,
000118 |  Tail=sum_(m>=2)m||Vhat_m||^2.
000119 | 
000120 | Polarization of the same calculation gives the full logical Gram:
000121 |  P_code J(V)^2 P_code=(3||q||^2+||p||^2+Tail) I5.
000122 | Here P_code is only a way of stating matrix elements, not an executed
000123 | intervention or identification with original P_adm.
000124 | 
000125 | For q!=0, the erasure-grade3 share of the squared response norm is
000126 | 
000127 |  eta=||q||^2/(3||q||^2+||p||^2+Tail)<=1/3.
000128 | 
000129 | The bound is sharp for a GLOBAL rotating first harmonic J=a_a+a_a*:
000130 | annihilation norm1, creation norm2, full norm3. If V has support inside
000131 | a proper interval and is notzero, equality is impossible: Tail=0 would
000132 | make V a vector trigonometric polynomial of degree at most1, which cannot
000133 | vanish on the complementary open arc unless identicallyzero. Thus for a
000134 | genuinely local profile with q!=0, eta<1/3. For q=0 there is no erasure
000135 | term. A nonzero smooth properlocal profile always has positive Tail.
000136 | 
000137 | Special narrower case V(theta)=v f(theta), v real constant and f real:
000138 | ||p||=||q||, hence eta<=1/4, again strict for properlocal support. Do not
000139 | use1/4 for arbitrary vector profiles. The principal result is1/3.
000140 | These fractions describe orthogonal source amplitudes; they are NOT
000141 | measurement probabilities, erasure rates, or a proposed instrument.
000142 | 
000143 | In particular a single such Hermitian current response has the same
000144 | total norm for every logical input. Its positive return Gram is scalar
000145 | I5, so that Gram alone cannot select a marked3+2 splitting.
000146 | This does not assert all joint field correlations are scalar.