TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/ORIGINAL-DENSITY-GEOMETRY-AND-TIME-HISTORIES.txt
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000100 | 3. An actually specified original local observable
000101 | 
000102 | Use the original single-circle CAR modes a_r, filled r<0. Its local number
000103 | density has modes
000104 | 
000105 |  J_m=sum_r :a_r^* a_(r+m):,
000106 |  J_m^*=J_-m, [J_m,J_n]=m delta_(m+n,0) I,
000107 |  [K_alpha,J_m]=-m J_m/8, [Q,J_m]=0.
000108 | 
000109 | The definition follows from the actual CAR. It fixes signs and normalization,
000110 | not a new root-current coupling. For a smooth real profile phi, its source
000111 | generator J(phi)=sum_m phi_m J_-m implements multiplication exp(i phi) on
000112 | one-particle fields. Properly supported profiles are genuine local density
000113 | operations in this GIVEN source net. Their physical selection from P1, or
000114 | the choice of a probe profile/detector, is not thereby proved.
000115 | 
000116 | The ORIGINAL fold sends the scalar multiplier to a diagonal matrix in the
000117 | arc basis, and the already fixed Fourier eigenchannels give
000118 | 
000119 |  V_phi,kl(x)=1/8 sum_r phi((x+2pi r)/8) exp(2pi i(l-k)r/8).
000120 | 
000121 | For phi(theta)=exp(i m theta), exactly m=k-l modulo8 survives and its
000122 | coefficient is exp(i m x/8). The positive profile frequency m belongs to
000123 | J_-m. Therefore a source-local scalar phase is generally OFF-DIAGONAL in
000124 | the folded charge channels. It is not the earlier class of independently
000125 | diagonal folded Cartan profiles. Those scoped exclusions cannot be applied
000126 | without this actual source map. The coefficients of these channel changes
000127 | are correlated by a single original profile, not freely chosen root weights.
000128 | 
000129 | In literal source CAR bilinear conventions the first raising mode is
000130 | 
000131 |  J_-1=sum_(k=1..7) E_(k,k-1),0+E_(0,7),-1.
000132 | 
000133 | E_(i,j) means the actual :psi_i^* psi_j: here. Conversion to another lattice
000134 | root basis must transport its cocycle phases. On the original sea only the
000135 | wrap term survives, with charge e0-e7,source energy1,fold energy1/8.
000136 | It is a D8 root current in the even source, not an added odd spinor field.
000137 | 
000138 | 4. Full state and native erasure response, computed directly in CAR
000139 | 
000140 | Use the same positive vectors U and source matrix V=M^T O U. Its entries
000141 | are Gaussian rationals. Native h_a(-n) maps to sum_k V_ka h_k(-n), where
000142 | 
000143 |  h_k(-n)=sum_(r=k mod8) :a_(r+8n)^* a_r:.
000144 | 
000145 | These bilinears reconstruct the previous full response
000146 | R_(b,4)f_i=[t^4]f_i(b+sum_(n>=1)h_+(-n)t^n), b=V^T delta.
000147 | They retain every partition4,31,22,211,1111. The independently computed
000148 | CAR Gram agrees entrywise with the previous complete bosonic Fock Gram.
000149 | Each state has charge0 and ORIGINAL source energy32, not source energy4.
000150 | 
000151 | J1 gives energy31, J-1 energy33. All eight forward and eight backward
000152 | cyclic root charge sectors are retained, including their sea signs.
000153 | Both response Grams have rank5, and the full VECTOR relation is
000154 | 
000155 |  (J1 J-1-J-1 J1)R_(b,4)f_i=R_(b,4)f_i.
000156 | 
000157 | Consequently G_+-G_-=B4. In BOTH original quarters G_- is not proportional
000158 | to B4: this current acts directionally even after accounting for the actual
000159 | input metric. The generalized polynomial det(G_--lambda B4) is squarefree
000160 | of degree5. All five eigenvalues are distinct; no degenerate 3+2 block is
000161 | supplied by this response. A further grouping would need justification.
000162 | 
000163 | The actual erasure a_a=h_(bar V_a)(1) still gives exactly
000164 | 
000165 |  a_a R_(b,4) f_i=R_(b,3) partial_a f_i.
000166 | 
000167 | Its commutator with the new original current is fixed without coefficients:
000168 | 
000169 |  [a_a,J_-1]=sum_r (bar V_(r+1,a)-bar V_(r,a)) a_(r-7)^* a_r.
000170 | 
000171 | All20 full source Ward outputs are nonzero, with every charged component
000172 | retained. This links the original density, its actual quartic response and
000173 | the existing derivative-erasure20. It is not a physical fermion/Yukawa map.
000174 | 
000175 | The complete original oscillator decomposition supplies a further identity:
000176 | 
000177 |  8K_alpha=Q(Q-1)/2+alpha Q+sum_(m>0) J_-m J_m.
000178 | 
000179 | For the actual neutral energy32 inputs this is the finite EXACT statement
000180 | sum_(m=1..32) Gram(J_m R_(b,4))=32 B4. Every one of the32 matrices is saved;
000181 | all m>32 annihilate by positive source energy. This is an energy bound, not
000182 | a chosen mode cutoff or discarded memory. Individual nonscalar responses
000183 | and the fixed total energy coexist.
000184 | 
000185 | The exact CAR calculation uses [-34,34]. Charge0 energy<=33 implies every
000186 | particle index<=32 and every hole index>=-33. All operator paths used here
000187 | stay below that energy. Every omitted edge bilinear annihilates the input;
000188 | signs are (-1) to the number of occupied modes strictly between its endpoints.
000189 | Thus the window implements these complete finite-energy responses exactly.
000190 |