TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/GRADED-DETERMINANT-SOURCE.txt
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000289 | Appendix: exact carrier compression test
000290 | ========================================
000291 | 
000292 | Status: PROVED analytically under the stated operator assumptions.
000293 | 
000294 | Duplicate check
000295 | ---------------
000296 | The Atlas already contains two nearby but different results:
000297 | 
000298 | * `finding:old-quarter-heat-transfer-has-controlled-rank-one-limit-and-odd-product`
000299 |   proves the controlled heat-transfer limit to a rank-one ground-mode factor.
000300 | * `finding:historical-em-rank-one-and-full-heat-traces-have-different-contracts`
000301 |   separates the historical scalar rank-one response from the full finite heat
000302 |   determinant.
000303 | 
000304 | Neither finding states the carrier-compression equivalence below or its
000305 | faithfulness consequence. The strict chart used for the active covariance
000306 | calculus is recorded separately in
000307 | `premise:faithful-differentiable-covariance-chart`, with 0<C<I and
000308 | h(C)=log(I-C)-log C. The following test links these scopes without identifying
000309 | them.
000310 | 
000311 | Assumptions and compression
000312 | ---------------------------
000313 | Let H be a complex Hilbert space. Let C=C*=C^2 be the old projective circle
000314 | covariance. Let psi_0 be a unit vector in its empty complement,
000315 | 
000316 |   C psi_0 = 0,
000317 |   Pi = |psi_0><psi_0|,
000318 |   C' = C+Pi.
000319 | 
000320 | Let P_F=P_F*=P_F^2 be an orthogonal carrier projection, F=Ran(P_F), and define
000321 | operators on F by
000322 | 
000323 |   C_F  = P_F C P_F |_F,
000324 |   C'_F = P_F C' P_F |_F.
000325 | 
000326 | Put v=P_F psi_0 and w=||v||^2. Then
000327 | 
000328 |   C'_F-C_F = |v><v|.
000329 | 
000330 | Consequently its rank is zero when w=0 and one when w>0. For w>0 it is
000331 | 
000332 |   |v><v| = w |v/sqrt(w)><v/sqrt(w)|.
000333 | 
000334 | Its only nonzero eigenvalue is w. Thus it is a normalized orthogonal projector
000335 | exactly when w=1. Merely knowing “rank one” does not justify replacing the
000336 | compression by a unit rank-one projector.
000337 | 
000338 | Relative determinant
000339 | --------------------
000340 | For every z for which I_F-z C_F is invertible,
000341 | 
000342 |   (I_F-z C'_F)(I_F-z C_F)^(-1)
000343 |     = I_F-z |v><v|(I_F-z C_F)^(-1).
000344 | 
000345 | The difference from the identity has rank one, so its Fredholm determinant is
000346 | well defined even when F is infinite dimensional. The matrix determinant lemma
000347 | gives
000348 | 
000349 |   D_F(z)
000350 |    := det_F[(I_F-z C'_F)(I_F-z C_F)^(-1)]
000351 |     = 1-z <v,(I_F-z C_F)^(-1)v>.
000352 | 
000353 | No commutation of P_F with C is needed for this identity.
000354 | 
000355 | Exact 1-z criterion
000356 | -------------------
000357 | Under the assumptions above, the following are equivalent:
000358 | 
000359 |   (i)  D_F(z)=1-z on a nonempty open subset of the resolvent domain which is
000360 |        analytically connected to z=0;
000361 |   (ii) D_F(z)=1-z throughout that connected resolvent domain;
000362 |   (iii) psi_0 lies in Ran(P_F).
000363 | 
000364 | For a positive contraction C_F the singular z values lie on the positive real
000365 | axis at reciprocals of nonzero spectral values, so the natural resolvent domain
000366 | containing zero is connected. In finite dimension the same statement follows
000367 | immediately from rational-function identity. Equality on a real open interval
000368 | with an accumulation point also suffices by real analyticity and continuation.
000369 | 
000370 | Proof. Set
000371 | 
000372 |   m_F(z)=<v,(I_F-z C_F)^(-1)v>.
000373 | 
000374 | If D_F(z)=1-z on an open set, then m_F(z)=1 there away from z=0. The identity
000375 | theorem extends this to z=0. Hence
000376 | 
000377 |   1=m_F(0)=||v||^2=w.
000378 | 
000379 | Because P_F is an orthogonal projection and ||psi_0||=1, equality
000380 | ||P_F psi_0||=1 holds exactly when P_F psi_0=psi_0, equivalently
000381 | psi_0 in Ran(P_F).
000382 | 
000383 | Conversely, if psi_0 in Ran(P_F), then v=psi_0 and
000384 | 
000385 |   C_F psi_0=P_F C psi_0=0.
000386 | 
000387 | Therefore (I_F-z C_F)^(-1)psi_0=psi_0, so m_F(z)=1 and D_F(z)=1-z wherever the
000388 | relative determinant is defined. This proves the equivalence.
000389 | 
000390 | The projectivity of C ensures that C and C'=C+Pi are pure covariances and makes
000391 | the physical endpoint interpretation immediate. The algebraic equivalence
000392 | itself uses only C psi_0=0, unit normalization of psi_0 and orthogonality of P_F.
000393 | 
000394 | Faithfulness and log-odds consequence
000395 | -------------------------------------
000396 | If the exact factor 1-z is retained, the criterion gives P_F psi_0=psi_0. On the
000397 | same vector,
000398 | 
000399 |   C_F psi_0=0,
000400 |   C'_F psi_0=(C_F+|psi_0><psi_0|)psi_0=psi_0.
000401 | 
000402 | Thus C_F has an eigenvalue 0 and C'_F has an eigenvalue 1. Neither covariance is
000403 | strictly faithful on F. For
000404 | 
000405 |   h(C)=log(I-C)-log C,
000406 | 
000407 | the psi_0 eigenvalue is +infinity for C_F and -infinity for C'_F. In finite
000408 | regularization this is precisely the divergence of the log odds at occupation
000409 | 0 and 1; in the operator formulation the logarithms are singular on that mode.
000410 | 
000411 | The other compression regimes are also exact:
000412 | 
000413 | * w=0: the carrier erases the added mode, C'_F=C_F and D_F(z)=1.
000414 | * 0<w<1: the perturbation is rank one but not a normalized projector. Its
000415 |   determinant begins D_F(z)=1-w z+O(z^2), and in general contains the full
000416 |   compressed resolvent m_F(z). It cannot equal 1-z on an open z range.
000417 | * w=1: the factor is exactly 1-z, but the two endpoint covariances are
000418 |   nonfaithful as just proved.
000419 | 
000420 | Consequent scope separation
000421 | ----------------------------
000422 | The pure old-circle update and the active strict KMS covariance cannot be
000423 | literally the same covariance on the same compressed carrier if the exact
000424 | rank-one determinant factor 1-z is demanded. The former has an exactly empty
000425 | mode before the update and an exactly filled mode after it. The latter requires
000426 | 0<C<I so that the finite log-odds chart is defined.
000427 | 
000428 | This is not a contradiction between the two original results. It identifies the
000429 | missing map between them. A heat or KMS regularization may approach the pure
000430 | rank-one factor, and a different faithful covariance may encode a finite
000431 | response, but then the object is a limit or a changed covariance. It is not the
000432 | same strictly faithful compression of the projective circle state. The present
000433 | test adds no new dynamics, carrier projector or source choice.