TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/GRADED-DETERMINANT-SOURCE.txt
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Inclusive original lines: 696-749
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000696 | 
000697 |   0<=Tr(FD_beta F)<1.
000698 | 
000699 | No commutation between F and D_beta is used. If F is nonzero, the lower bound is
000700 | also strict; allowing the zero projection explains the report's weak lower
000701 | bound.
000702 | 
000703 | For the finite-dimensional compressed relative determinant,
000704 | 
000705 |   D_F(z)=det[(I_F-z F C'_beta F)(I_F-z F C_beta F)^(-1)],
000706 | 
000707 | Jacobi's formula at z=0 gives
000708 | 
000709 |   D_F'(0)=-Tr(F(C'_beta-C_beta)F)=-Tr(FD_beta F)>-1.
000710 | 
000711 | Since (1-z)' at zero is -1, D_F cannot equal 1-z on any open z domain. Moreover
000712 | C_beta and I-C_beta have trivial kernel. On the unit sphere of a finite F this
000713 | implies strictly positive minima for both compressed quadratic forms, so
000714 | 
000715 |   0<F C_beta F<I_F,
000716 |   0<F C'_beta F<I_F.
000717 | 
000718 | Their relative determinant is therefore finite and nonzero at z=1, whereas
000719 | 1-z vanishes. Both arguments in Section 6 are valid. The scope exclusion for an
000720 | infinite-rank spacetime projection with only finite internal rank is necessary
000721 | and correctly stated.
000722 | 
000723 | Section 7: exact Schur transport
000724 | --------------------------------
000725 | For |z|<1, A,A' and their QQ compressions are invertible by the Neumann bound.
000726 | Invertibility of the Schur complements follows from the standard block
000727 | factorization. Because F is finite dimensional, every off-diagonal unipotent
000728 | factor differs from I by finite rank, is nilpotent off the diagonal and has
000729 | Fredholm determinant one. Since A'-A is trace class, its QQ compression is trace
000730 | class, so
000731 | 
000732 |   det_rel(A'_QQ,A_QQ)
000733 | 
000734 | exists. Ordinary Fredholm multiplicativity gives exactly
000735 | 
000736 |   det_rel(A',A)
000737 |    =det(S'_F S_F^(-1)) det_rel(A'_QQ,A_QQ).
000738 | 
000739 | There is no zeta-determinant multiplicative anomaly in this calculation. The
000740 | covariance formula
000741 | 
000742 |   S_F=I_F-zC_F-z^2 C_FQ(I_Q-zC_QQ)^(-1)C_QF
000743 | 
000744 | has the correct sign and power of z. Thus carrier Schur factor plus complement
000745 | retains 1-z for every finite orthogonal F, while naive FCF generally does not.
000746 | 
000747 | Section 5: full NS cover
000748 | ------------------------
000749 | On the antiperiodic cover, modes k+1/2 split by parity as