TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/NS-CONTINUUM-WINDING-LIMIT.txt
Original file SHA-256: 4a61f9604bb51f0d13419606d7d8c64fa2cfc840b417a7297403db7d6706f916
Inclusive original lines: 201-304
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
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000201 |   J_N u_N J_N^* - P_{H_N} u P_{H_N}
000202 |     = |psi_{-M}><psi_{M-1}|.                             (12)
000203 | 
000204 | This rank-one wrap has operator norm one for every N, but annihilates every
000205 | fixed finite Fourier core once N is large. Hence the map is locally exact and
000206 | converges strongly, while it cannot converge in norm as a full finite cyclic
000207 | representation.
000208 | 
000209 | The same distinction controls the determinant. Under the continuum embedding,
000210 | 
000211 |   |psi_{-M}><psi_{-M}| -> 0 strongly,
000212 | 
000213 | but every such projector has trace norm one. Therefore (4) is exact. The trace
000214 | cannot jump from zero to one under trace-norm convergence, so no theorem about
000215 | Fredholm determinant continuity can turn (3) into 1-z. The continuum factor is
000216 | obtained by the old directed relative pair
000217 | 
000218 |   (I-z u C u^*)(I-zC)^{-1}=I-zPi_0,
000219 |   det_F=1-z,                                             (13)
000220 | 
000221 | not by taking the numerical limit of the finite ordinary determinant.
000222 | 
000223 | Three cancellations that must remain distinct
000224 | ---------------------------------------------
000225 | 
000226 | 1. Finite NS cyclic cancellation
000227 | 
000228 | The two factors in (3) live on one finite chiral circle. The second factor is
000229 | the Fourier alias at the cutoff. It escapes strongly as N grows, but it is
000230 | present with unit trace weight for every finite N.
000231 | 
000232 | 2. Finite carrier compression of an infinite source
000233 | 
000234 | KMS-AND-CARRIER-TRANSPORT.txt:194-220 begins with the infinite source. A naive
000235 | finite compression loses part of its trace-class response. The exact Schur
000236 | complement plus the unchanged complement restores the original factor 1-z.
000237 | This is not the cyclic determinant (3), whose winding has index zero at every
000238 | finite N.
000239 | 
000240 | 3. Reflected face or full Dirac cancellation
000241 | 
000242 | RELATIVE-PAIRING-REVIEW.txt:66-115,466-475 has two continuum faces with inverse
000243 | directed factors (1-z) and (1-z)^{-1}. That cancellation survives the continuum
000244 | limit and the Quillen norm. Its partner is a reflected physical sector, not the
000245 | UV alias in (2). Removing the finite alias therefore does not remove the
000246 | full-Dirac partner.
000247 | 
000248 | Small decisive certificate
000249 | ---------------------------
000250 | 
000251 | The following is the executed Fourier form of the actual sampled v519 data. It
000252 | uses no fitted matrix. The geometric-sum identity (6) identifies P with the
000253 | v519 Q_N before this finite calculation.
000254 | 
000255 |   import numpy as np
000256 |   for N in (8,16,32):
000257 |       M=N//2
000258 |       modes=np.arange(-M,M)
000259 |       P=np.diag((modes>=0).astype(complex))
000260 |       U=np.zeros((N,N),complex)
000261 |       for j,n in enumerate(modes):
000262 |           target=n+1 if n<M-1 else -M
000263 |           U[np.where(modes==target)[0][0],j]=1
000264 |       D=U@P@U.conj().T-P
000265 |       z=1/3
000266 |       ratio=(np.linalg.det(np.eye(N)-z*U@P@U.conj().T)
000267 |              /np.linalg.det(np.eye(N)-z*P))
000268 |       print(N, [(int(modes[j]),int(round(D[j,j].real)))
000269 |                 for j in range(N) if abs(D[j,j])>1e-12],
000270 |             np.linalg.matrix_rank(D), np.trace(D), ratio)
000271 | 
000272 | Executed output:
000273 | 
000274 |   8  [(-4,+1),(0,-1)]   rank2 trace0  ratio1
000275 |   16 [(-8,+1),(0,-1)]   rank2 trace0  ratio1
000276 |   32 [(-16,+1),(0,-1)]  rank2 trace0  ratio1
000277 | 
000278 | The actual v519 kernel was also evaluated directly before Fourier replacement:
000279 | for N=8 and16, Q_N^2-Q_N had numerical norms 1.19e-16 and6.87e-16, and its
000280 | Fourier occupation list was exactly 0 on -M,...,-1 and1 on0,...,M-1 to twelve
000281 | decimal places. v519 proves the N=8 projector identity exactly; equation (6)
000282 | proves the general even-N statement analytically.
000283 | 
000284 | Consequence for the requested forward map
000285 | -----------------------------------------
000286 | 
000287 | The old and new sources now have an explicit common operator dictionary:
000288 | 
000289 |   old twisted circle at a=1/2
000290 |        --sample at theta_j=2 pi j/N-->
000291 |   v519 finite NS one-particle space,
000292 | 
000293 |   old spectral polarization               -> Q_N or I-Q_N by convention,
000294 |   old u=M_z                                -> diagonal site phase u_N,
000295 |   old Fourier shift n->n+1                 -> cyclic shift plus one UV alias,
000296 |   old rank-one spectral-flow defect        -> IR defect plus inverse UV defect.
000297 | 
000298 | This map does not require P1 to select another free state. It uses the v519
000299 | state already supplied and the fullv21 operation already supplied. What it does
000300 | not provide is a finite OS prescription whose ordinary determinant equals the
000301 | infinite directed determinant. The smallest remaining source question is
000302 | therefore sharply typed: does an original continuum OS or admissible functional
000303 | explicitly invoke the fullv21 polarized relative Fredholm pair (13)? The finite
000304 | v519/v524/v526 construction itself answers no.