TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/SCALAR-FIRST-ORDER-SOURCE-FIBER-AND-SEAM-WARD.txt
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000031 | 2. Complete normal form with domains and half-density normalization
000032 | 
000033 | In the periodic chart L2(S1,dtheta), every formally selfadjoint smooth
000034 | scalar first-order differential operator with positive principal symbol is
000035 |  B_(a,c)=-i[a(theta)partial_theta+a'(theta)/2]+c(theta),
000036 |  a>0, c real, a,c smooth periodic, Dom B=H1_per.
000037 | For -ia partial+b, formal adjunction forces Im b=-a'/2; Re b=c.
000038 | The a'/2 term is therefore derived from the GIVEN measure, not a coupling.
000039 | A different fixed positive density changes the divergence term; its
000040 | unitary density transport must be retained rather than resetting metrics.
000041 | The quarter twist is ALREADY in c/holonomy in this periodic chart. If a
000042 | separate boundary twist kappa is retained, the spectrum below has
000043 | n+kappa+beta instead; never add the same quarter twice.
000044 | 
000045 | Define
000046 |  T=integral_0^(2pi) dtheta/a, omega=2pi/T,
000047 |  beta=(1/2pi) integral_0^(2pi) c/a dtheta,
000048 |  x(theta)=omega integral_0^theta dt/a(t),
000049 |  eta(theta)=integral_0^theta [omega beta-c(t)]/a(t) dt.
000050 | Then x(theta+2pi)=x(theta)+2pi, eta(theta+2pi)=eta(theta), and
000051 |  (Uf)(theta)=sqrt(omega/a(theta)) exp(i eta(theta)) f(x(theta))
000052 | is unitary from L2(dx) onto L2(dtheta). It maps H1_per onto H1_per.
000053 | Direct differentiation gives the full operator identity, for arbitrary f,
000054 |  B_(a,c) U=U omega(-i partial_x+beta).
000055 | The derivative of sqrt(omega/a) cancels a'/2, while a eta'+c=omega beta.
000056 | Thus B is selfadjoint on the stated domain, its complete spectrum is
000057 |  omega(Z+beta), and these eigenfunctions are a complete transported
000058 | Fourier basis. This is not a finite spectral sample or small perturbation.
000059 | 
000060 | The unmarked positive-orientation source class is therefore determined
000061 | up to this coordinate/gauge transport by omega and beta modZ. Actual
000062 | quarter holonomy beta=alpha modZ, alpha1/4 or3/4, together with the given
000063 | gap1/4 forces omega=1. All such UNMARKED one-particle representatives are
000064 | unitarily equivalent to the original B_alpha. This is a positive source-
000065 | fiber classification, not a family of inequivalent completed P1 models.
000066 |