TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/mixed-current-origin-20261014/CONTINUOUS-TIME-NONUNITARY-SECTOR-TEST.txt
Original file SHA-256: eb31933e4bbf3c1da2b02661381ddf128d37a1d7ebc507bb66982c2c78944897
Inclusive original lines: 198-271
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.

000198 | 6. Heat regularization: a concrete nonunitary attempt and its trap
000199 | 
000200 | First consider the ACTUAL full QWZ source, not a bosonized replacement.
000201 | Take the original neutral pair P_E, E the eight preimages of a base
000202 | interval. Imaginary-time filtering gives v_(N,s)=exp(-sH_N)P_E Omega_N.
000203 | This is a pure Slater vector, so its complete finite word state is fixed
000204 | by its covariance and CAR/Wick. Write U_-,U_+ for all occupied/unoccupied
000205 | one-body eigenvectors, epsilon_-,epsilon_+ for their positive scaled
000206 | hole/particle energies, and sigma the actual parity sign matrix. In a
000207 | nonsingular Slater chart set
000208 |   A=U_-^* sigma U_-, B=U_+^* sigma U_-, Zp=B A^-1,
000209 |   Zp_s=exp(-s epsilon_+) Zp exp(-s epsilon_-),
000210 |   V_s=U_-+U_+ Zp_s,
000211 |   C_s=V_s (I+Zp_s^* Zp_s)^-1 V_s^*.
000212 | A singular A requires a different chart; it is no sector obstruction.
000213 | The exact normalized ORIGINAL time kernel is
000214 |   F_(N,s)(t)=det(I+Zp_(s+t/2)^* Zp_(s+t/2))
000215 |               /det(I+Zp_s^* Zp_s).                  (5b)
000216 | The prefactor det(A) cancels; all massive states and both edges remain.
000217 | 
000218 | The checker evaluates Nx32,Ny8, quarters1 AND3, s1/4,t1/10, base interval
000219 | (pi/2,pi), retaining all512 one-body modes and256 occupied modes. Both
000220 | quarters give <H_N>=36.8319309197 and F_(N,s)(t)=0.0735083926. The original
000221 | eight JOINT Bloch-residue occupations are unchanged to numerical accuracy;
000222 | they are NOT identified with eight separately left-chiral D8 Cartans.
000223 | Both physical-edge covariance responses survive. These are finite controls
000224 | of an actual nonunitary original operator, not a continuum-sector proof.
000225 | Imaginary time is used as a diagnostic; its physical preparation is not
000226 | claimed to have been selected by P1.
000227 | 
000228 | For a separate analytic SINGLE-CHIRAL comparison the time kernel can also
000229 | be summed in closed form. This is conditional on the stated Weyl map.
000230 | In the GIVEN single-chiral Weyl comparison, a periodic uniform base profile
000231 | phi(theta)=f(8theta) displaces the source oscillator J_(8n) with coherent
000232 | amplitude a_n=i sqrt(8n) f_n; its GIVEN K=H_cover/8 frequency is n.
000233 | The actual full-strip heat filter above uses H_N, converging to cover time.
000234 | This chiral comparison uses the previously fixed constant time conversion;
000235 | it preserves continuity/compactness and does NOT divide primitive flow.
000236 | For a step profile f=pi 1_(x,y), n>0,
000237 |   f_n=(exp(-inx)-exp(-iny))/(2 i n).
000238 | At fixed positive heat time s, the normalized vector proportional to
000239 | exp(-sK) W(phi_eps)Omega has in the sharp profile limit amplitudes
000240 |   a_n(s)=exp(-sn) sqrt(2/n)
000241 |          [exp(-inx)-exp(-iny)].                       (6)
000242 | The vacuum overlap can tend to zero before normalization. Taking ratios
000243 | of finite-cutoff coherent coefficients proves(6); no unregulated infinite
000244 | determinant is used. The resulting coherent vector has finite norm and
000245 | finite energy for every s>0, but is a NEUTRAL PAIR state in the old Fock
000246 | module, not a charged-sector vector.
000247 | 
000248 | Put d=y-x, r=exp(-2s). Its exact original K expectation is
000249 |   E_s=4 sum_(n>=1) r^n[1-cos(nd)]
000250 |      =4[r/(1-r)-Re(r exp(id)/(1-r exp(id)))].           (7)
000251 | For d=pi, E_s=8r/(1-r^2), which diverges like2/s as s down0.
000252 | The normalized time-response kernel is
000253 |   F_s(t)=exp(-sum_n |a_n(s)|^2[1-exp(-tn)]),           (8)
000254 | continuous at zero. In fact the normalized heat-filtered vector is the
000255 | old Weyl coherent vector with Poisson-smoothed profile f_(s,n)=e^(-sn)f_n.
000256 | Both endpoints and the old Cartan charges remain present.
000257 | For d=pi, the exact sum in(8) gives the particularly decisive formula
000258 |   F_s(t)=[(1+r exp(-t))(1-r)/((1-r exp(-t))(1+r))]^4.  (8a)
000259 | For each s>0 this is continuous and equals1 at t0. But at every t>0,
000260 | F_s(t)->0 as s down0. Removing the heat regularization restores the
000261 | sharpness at the cost of a unit jump between the limiting norm at0 and
000262 | the positive-time kernel. This is a concrete example of the failure of D,
000263 | not a normalizable positive-time charged state in the same time limit.
000264 | 
000265 | This is a chiral smooth-Weyl comparison. It is NOT a theorem identifying
000266 | the sharp discrete lattice parity P_E with that sharp Weyl limit. Exact
000267 | finite parity is 2pi-periodic and selfadjoint; the single unwrapped
000268 | bosonic profile need not retain its branch data in a singular limit.
000269 | The actual microscopic parity asymptotics require their own determinant/
000270 | word-limit control. The main theorem in sections2-4 does not need this
000271 | identification and applies to the ACTUAL lattice insertions directly.