TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/primitive-seam-operator-reconstruction-20261025/TOEPLITZ-SEAM-RECONSTRUCTION.txt
Original file SHA-256: 0ca3b6733b10723eb97caba84f2426396dab98120852cdbf4d2364df8e8a65c4
Inclusive original lines: 201-287
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 | 
000202 | Under (8), Pi_Sigma=P_C-u_Sigma P_C u_Sigma* commutes with the full time
000203 | group.  Its rank-one range is therefore a one dimensional reducing spectral
000204 | subspace of B_Sigma.  For a unit psi in that range there is a real epsilon
000205 | such that
000206 | 
000207 |     B_Sigma psi = epsilon psi.                               (9)
000208 | 
000209 | Equations (5), (8), and (9) give directly in the original Hilbert space
000210 | 
000211 |     B_Sigma e_n = (epsilon+n)e_n, n in Z.                   (10)
000212 | 
000213 | On H_circle this is precisely
000214 | 
000215 |     K_epsilon = -i partial_theta + epsilon,
000216 | 
000217 | under the Fourier map e_n -> exp(in theta).  The nonnegative part inside
000218 | P_C H is its Hardy restriction.  The bilateral circle is already embedded in
000219 | the original H; no new product geometry, channel, or dilation is introduced.
000220 | The original collar formula alone proves neither (8) nor (9).
000221 | 
000222 | The offset epsilon is independent of the index.  For every real epsilon the
000223 | same Hardy projector C, the same shift S, the same defect projector, and the
000224 | same Fredholm index coexist with B_epsilon=-i partial_theta+epsilon.  If the
000225 | original source additionally proved on H_circle that
000226 | 
000227 |     exp(2 pi i B_Sigma)=i identity,  with 0<epsilon<1,         (11)
000228 | 
000229 | then (9) would give exp(2 pi i epsilon)=i and hence epsilon=1/4.  Neither the
000230 | Ward covariance (8) nor the quarter monodromy (11) follows from wind(u)=1 or
000231 | SF=1.  They require original operator and boundary-domain statements.
000232 | 
000233 | Ordinary versus graded determinant
000234 | ==================================
000235 | 
000236 | There is an exact index-only scalar construction.  Regard a Fredholm S as the
000237 | two term complex H --S--> H, with H^0=ker(S) and H^1=coker(S).  With the
000238 | even-over-odd convention,
000239 | 
000240 |     sdet(1-q | H^bullet)
000241 |       = det(1-q | H^0) / det(1-q | H^1)
000242 |       = (1-q)^ind(S).
000243 | 
000244 | For ind(S)=-1 this is (1-q)^(-1).  Therefore
000245 | 
000246 |     (5/4) log sdet(1-q | H^bullet) = -(5/4) log(1-q),
000247 | 
000248 | which reproduces the scalar seam factor without choosing a vector direction.
000249 | This is a mathematically valid relative graded determinant repair.
000250 | 
000251 | It is not the trace prescription used in the original electromagnetic
000252 | closure.  paper-source:c439790c9d6f51b9bcaa037f, lines 94-122 and 219-224,
000253 | uses the ordinary admissible trace
000254 | 
000255 |     -(5/4) tr_adm log(1-q |psi_Sigma><psi_Sigma|)
000256 | 
000257 | and explicitly invokes the ordinary rank-one determinant identity
000258 | det(1-q |psi><psi|)=1-q.  The later infrared formula in
000259 | documents/archive--paper-latex--tfpt-42/sections/
000260 | 041-section-relative-aps-and-superconnection-setup.tex, lines 215-230, is also
000261 | an ordinary Fredholm determinant of a positive trace-class transfer operator.
000262 | The STr log in the general 1PI expansion at
000263 | documents/archive--paper-latex--tfpt-42/sections/
000264 | 035-subsection-renormalized-physical-and-scheme-readout-laye.tex, lines
000265 | 116-128, is a field-statistics supertrace.  It does not define a grading on
000266 | ker(S) and coker(S), and it is not connected to the seam Fredholm index.
000267 | 
000268 | Final status
000269 | ============
000270 | 
000271 | PROVED CONDITIONALLY: If the original pair (P_C,u_Sigma) satisfies leakage
000272 | zero and Toeplitz index -1, Pi_Sigma=P_C-SS* is a canonical rank-one
000273 | projector.  Its bilateral u_Sigma orbit is a canonical reducing circle sector
000274 | inside the original Hilbert space.  No free psi_Sigma direction and no
000275 | external unitary dilation are needed.  Purity is required only to identify
000276 | the whole Calderon range with that Hardy sector.
000277 | 
000278 | NOT DERIVED FROM THE PRESENT ORIGINALS: The source proves the K^1 class and
000279 | spectral flow, but not leakage zero or the strong B_Sigma covariance.  It also
000280 | does not prove purity for the whole Calderon range or the quarter monodromy
000281 | needed to fix epsilon=1/4.  The d1 and h_Sigma^red minimizations do not fill
000282 | these typed gaps.
000283 | 
000284 | AVAILABLE BUT NEW PRESCRIPTION: A graded determinant on Fredholm cohomology
000285 | does recover (1-q)^(-1) from ind(S)=-1 alone.  The current scalar electromagnetic
000286 | formula does not use that prescription, so it cannot be claimed as an
000287 | originally derived replacement without changing the action definition.