TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/DOUBLE-WALL-REVIEW.txt
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Inclusive original lines: 163-227
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000163 | For a Phi-eigenvector of eigenvalue m!=0 and an S-eigenvector of eigenvalue
000164 | s=+/-1, a normal solution is square integrable exactly when s m<0. Thus
000165 | 
000166 |   s=-sign(m),
000167 |   D|_(ker Q)=p S,
000168 |   dim ker(Q)|_(S=+1)-dim ker(Q)|_(S=-1)
000169 |        =n_-(Phi)-n_+(Phi)=-signature(Phi).              (5)
000170 | 
000171 | The opposite kink reverses the sign. A zero eigenvalue of Phi gives a
000172 | constant normal solution, not a localized L2 mode on this infinite line.
000173 | 
000174 | For finite Hermitian Phi satisfying {gamma_F,Phi}=0, gamma_F pairs each
000175 | nonzero eigenvalue m with -m. Therefore signature(Phi)=0. Equation (5) then
000176 | has zero net index in THIS infinite, one-dimensional, real scalar wall
000177 | ansatz. The finite double has no exact zero mode by section3; assigning
000178 | exact chiralities separately to its two coupled walls is not justified.
000179 | 
000180 | 5. The local deck relation is not a smooth global wall prescription
000181 | 
000182 | The earlier exact deck calculation with S R_r does change the displayed
000183 | Yukawa coefficient to -Phi on the reflected coordinate chart, under its
000184 | stated common-domain and coefficient-transport assumptions. A physical
000185 | global mass field additionally requires the actual spinor trivialization,
000186 | clutch and internal lift. A sign in two local descriptions alone does not
000187 | prove a gauge-independent domain wall in a globally identified bundle.
000188 | 
000189 | If one DOES choose a single trivialization and an ordinary scalar odd
000190 | periodic sigma, smoothness requires sigma to vanish at reflection-fixed
000191 | seams. Extending a nonzero constant mass with the opposite sign produces a
000192 | jump unless the coefficient is changed near the seams. The tanh(sin)
000193 | profile supplies precisely such a new smooth interpolation. It is not
000194 | derived by the local matrix S alone.
000195 | 
000196 | For a specified smooth profile and a fixed closed invertible domain the
000197 | resolvent identity delta(D^-1)=-D^-1(delta D)D^-1 is valid. It does not imply
000198 | that finite-wall eigenvalues stay exactly at |p|. Their normal splittings
000199 | in (3) can respond to the mass profile. Nor can the homogeneous KMS formula
000200 | of section2 be applied to an arbitrary normally varying coefficient: the
000201 | existing normal_dependent_transport result carries its actual ordered
000202 | transfer and full crossing blocks without making that substitution.
000203 | 
000204 | 6. Preserve the older positive normal-index result and the target scope
000205 | 
000206 | The existing spin normal geometry is not classified by (5). On the stated
000207 | Spin S2=CP1, K^(1/2)=O(-1), and the actual twist L2=O(1) gives
000208 | 
000209 |   ker D_(N,L2)^+ =H0(O(-1) tensor O(1))=H0(O)=C,
000210 |   ker D_(N,L2)^- =H1(O)=0.
000211 | 
000212 | This operator has index one. That topological index is not the signature
000213 | of a finite Yukawa matrix. The old result remains a positive conditional
000214 | TFPT normal attachment under its explicit operator and family premises.
000215 | 
000216 | Consequently the proposed inference that unit winding generally requires
000217 | a gamma_F-even mass of nonzero signature does not follow from the real
000218 | one-dimensional wall calculation. Neither the finite-wall correction nor
000219 | the odd-Phi signature identity is a no-go theorem for full P1, physical
000220 | chirality, or other already present normal geometries.
000221 | 
000222 | The useful retained result is precise: a given homogeneous antiperiodic
000223 | closed source reproduces the known graph blocks, and a chosen normally
000224 | varying source is already carried by the full fixed Clifford chart. The
000225 | new asserted origin and exact finite-wall modes are not established by
000226 | those facts. The finite normal-zero calculation above directly disproves
000227 | the latter assertion in its stated antiperiodic script class.