TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/PROOF.txt
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Inclusive original lines: 482-517
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000482 | 11.2 Repair of an invalid original order-one perturbation inference
000483 | 
000484 | The implication at01:397-405 from an H1 graph estimate to an arbitrarily small
000485 | relative bound is invalid. A bounded-coefficient first-order perturbation need
000486 | not be infinitesimally bounded relative to a first-order anchor.
000487 | Already P=aA on any unbounded selfadjoint anchor has best relative bound|a|.
000488 | The estimate |a lambda_n|<=eps|lambda_n|+C_eps fails at large eigenvalues for
000489 | eps<|a|. For P=-A, the sum restricted to Dom(A) is the zero operator on that
000490 | proper dense domain. It is not closed or selfadjoint, though its closure is the
000491 | bounded zero operator on the whole Hilbert space; compact resolvent is lost.
000492 | 
000493 | A diagnostic retains the quarter domain AND the even/odd reflection types.
000494 | On the compact product torus, psi(theta+2pi,r)=i psi(theta,r), periodic r, let
000495 |   A=sigma_x(-i partial_theta)+sigma_y(-i partial_r),
000496 |   P=a I(-i partial_r).
000497 | In periodic theta coordinates k=n+1/4. Reflection r->-r with fiber lift sigma_x
000498 | makes A even and P odd. At fixed k=1/4 and r-frequency m->infinity,
000499 |   ||P psi_m||/||A psi_m||=|a|m/sqrt(m^2+1/16)->|a|.
000500 | At a1 one eigenvalue of A+P is m-sqrt(m^2+1/16)->0. Choose its unit eigenvectors
000501 | v_m. The partial sums of sum_m v_m/m are graph-Cauchy for A+P, with L2 limit
000502 | outside H1. Hence the realization on Dom(A)=H1 is not closed.
000503 | 
000504 | Correct constructive sufficient conditions include:
000505 |   (i) a symmetric bounded order-zero perturbation on the anchor domain;
000506 |   (ii) an actually established relative bound<1 on that same domain; or
000507 |   (iii) an independent elliptic-total-operator theorem with a compatible
000508 |         selfadjoint elliptic boundary condition, not automatically the old one.
000509 | For the displayed torus family |a|<1 gives the relative-bound theorem. Requiring
000510 | the SAME original Dirac/Clifford principal metric in this family is stronger:
000511 |   (sigma_y+a I)^2-I = a^2 I+2a sigma_y,
000512 | so it forces a0. General order-one perturbations cannot be treated as canonical
000513 | inner fluctuations: on a smooth compact spectral triple a[D,b] is bounded,
000514 | whereas a change of the principal symbol must be checked separately.
000515 | The diagnostic is NOT an exclusion of the actual P1/P2 source. At a1 it is not
000516 | an admissible elliptic Dirac source; that is exactly what the false general
000517 | perturbation inference failed to test. No historical source is overwritten.