TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/CONTINUOUS-TIME-NONUNITARY-SECTOR-TEST.txt
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Inclusive original lines: 152-196
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000152 | 5. Full local register removes register-changing loopholes in this class
000153 | 
000154 | At the initial definite quarter r, Z Omega_N=i^r Omega_N and [Z,H_N]=0.
000155 | Suppose a proposed limiting bosonic endpoint field T(f) is local away from
000156 | the retained local EVEN register Z. The full word contract then requires
000157 |   ||(Z_N T_N-T_N Z_N)Omega_N||^2 -> 0.
000158 | Resolve v_N into register grades q=0,1,2,3. Orthogonality gives exactly
000159 |   ||(Z_N-i^r)v_N||^2
000160 |     =sum_q |i^(r+q)-i^r|^2 ||v_(q,N)||^2.             (5)
000161 | All nonzero coefficients are >=2. Thus the total norm of register-changing
000162 | components tends to zero. The surviving vector lies asymptotically in the
000163 | same quarter block, where sections2-4 apply. No definite microscopic
000164 | register grade for T_N was needed.
000165 | 
000166 | The common ORIGINAL microscopic register reference can also be controlled,
000167 | without choosing four independent subtractions. Put
000168 |   e(p)=-Tr|h(p)|/2, E0_N(alpha)=sum_j e(2pi(j-alpha)/N).
000169 | Tr h(p)=0, so this is the actual filled-sea ground energy. The same Bloch
000170 | analysis used in the original exhaustion proof shows that zero occurs
000171 | only at p0, in the two branches with slopes+1,-1. Off p0, e is real
000172 | analytic by spectral functional calculus. Near0 its only nonsmooth part
000173 | is -|p| plus piecewise analytic terms. Thus its distributional second
000174 | derivative has finite total variation V, independent of N and alpha.
000175 | Its Fourier coefficients obey |e_m|<=V/(2pi m^2), m!=0.
000176 | Absolute Fourier convergence gives the exact sampling identity
000177 |   E0_N(alpha)=N sum_(l in Z) e_(lN) exp(-2pi i l alpha).
000178 | For ANY two of the four backgrounds, therefore,
000179 |   |E0_N(alpha)-E0_N(beta)|<=V pi/(3N),
000180 |   |N/(2pi)[E0_N(alpha)-E0_N(beta)]|<=V/6.             (5a)
000181 | The estimate uses sum_(l!=0)l^-2=pi^2/3. Finite-R zero-mode filling affects
000182 | neither Tr|h| nor this ground energy; no Wick replacement is involved.
000183 | 
000184 | Consequently the full register Hamiltonian, with ONE common reference
000185 | equal to the initial quarter vacuum energy, is bounded below by -V/6
000186 | uniformly in N. Its semigroup norms are <=exp(tV/6) at fixed t. Dropping
000187 | the register-changing components in(5) therefore changes the full time
000188 | kernel by an amount tending to zero. The grade0 main part has exactly
000189 | the original nonnegative quarter H_N. Thus A-D reduce to that block even
000190 | for arbitrary grade mixtures. No separate sector energy convention was
000191 | introduced. The actual neutral pair already has grade0 identically.
000192 | 
000193 | A changed Gauss/local-net assignment can make Z a GLOBAL Wilson loop and
000194 | therefore exits this remote-local-Z premise, exactly as the earlier
000195 | conditional completion demonstrated. The theorem does not prohibit that
000196 | move or derive its physical origin.