TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/PROOF.txt
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Inclusive original lines: 130-176
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000130 | 6. The positive collar control and correct thermal representation
000131 | 
000132 | For the repeated-mode calibration use TWO DECLARED IDENTICAL COPIES of the
000133 | lowest alpha=1/4 source mode, B=(1/4)I2, and DECLARED length l=4 log4.
000134 | This is not the particle/hole pair of the full chiral source. Then R_l=(1/4)I2 and
000135 |   G=(1/17)[[16I,4I],[4I,I]].
000136 | The boundary CAR has four modes. Its pure state is a vector in dimension16;
000137 | the source amplitude Q acts on TWO modes, dimension4. These are different
000138 | spaces. The reconstructed Q has diagonal (1,1/4,1/4,1/16), and the restricted
000139 | thermal state is Q^2/Tr Q^2. Its one-particle occupation is 1/17, so
000140 |   log((1-n)/n)=log16=2 l B.
000141 | This is state and heat amplitude from the same full graph in this canonical
000142 | fermionic realization; no many-body H, density matrix or target Q is supplied.
000143 | 
000144 | A separate one-mode thermal GNS implementation uses left CAR on HS operators,
000145 | Omega_rho=vec(sqrt rho) and L=H tensor I-I tensor H^T. It has both signs.
000146 | For the same ratio r=1/4, exp(-l L) has eigenvalues {1,1,1/4,4} and fixes
000147 | Omega_rho. It is positive but not a contraction. The repaired search accepts
000148 | this source control with the source-derived finite ceiling4; ceiling1 is a
000149 | different declared class. We do NOT replace this actual thermal generator by a
000150 | chosen positive pure-state parent for the boundary purification.
000151 | 
000152 | The actual lowest particle/hole pair of the single chiral A_alpha component has
000153 | energies
000154 | (alpha,1-alpha)=(1/4,3/4). At the same given length its response is
000155 | R=diag(1/4,1/64). The same boundary-projection and oriented-pairing construction
000156 | therefore gives Q=diag(1,1/4,1/64,1/256) and the thermal density
000157 | diag(65536,4096,16,1)/69649. Here the input-dual CAR covariance is
000158 | diag(16/17,4096/4097); the output covariance is diag(1/17,1/4097).
000159 | This is not an insertion of |D| as a new time operator: the signed source pair
000160 | has D=diag(1/4,-3/4), R_raw=diag(1/4,64), Q_raw=diag(1,1/4,64,16).
000161 | The second-mode CAR particle-hole conjugation, followed by division by the
000162 | filled-sea eigenvalue 64, gives precisely the positive relative Q above.
000163 | Explicitly C=c_-+c_-^* is unitary in this finite sector. With N=N_++N_-,
000164 | C N C^*-I=N_+-N_- and
000165 | C[(1/4)N_+-(3/4)N_-]C^*+(3/4)I=(1/4)N_++(3/4)N_-.
000166 | The same conjugation applied to the original quarter Gamma(iI), divided by the
000167 | filled-reference phase i, gives the charged relative W below. Thus charge,
000168 | energy and quarter action use the same source map and the same filled reference.
000169 | These subtracted reference scalars do not recover an absolute partition factor.
000170 | The charged quarter is W=diag(1,i,-i,1), with opposite creator phases i and -i,
000171 | and equals exp(2pi i H) for H=diag(0,1/4,3/4,1). It is not the i^N rotation
000172 | of two equally charged creation operators. The actual source-pair control and
000173 | the duplicated-mode calibration are stored separately.
000174 | The actual-pair SCS run returns optimal_inaccurate at floor0 and infeasible at
000175 | floor1/4. The exact reconstructed matrix independently proves feasibility;
000176 | the solver warning is retained and is not a clean numerical optimality proof.