TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-partner-original-action-20261015/PROOF.txt
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Inclusive original lines: 154-203
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000154 | 5. A REAL SOURCE PARTNER WITH CHARGE AND TIME, IN ITS STATED DOMAIN
000155 | 
000156 | There is also a direct construction on the already existing one-channel chiral
000157 | source, without adding a half-charge sector. It is not a selection of that
000158 | representative from P1. Use the fold's positive phase convention:
000159 | 
000160 |     f_(n,alpha)(theta)=exp(i(n+alpha)theta),
000161 |     D_alpha=-i partial_theta, f(theta+2pi)=exp(2pi i alpha)f(theta).
000162 | 
000163 | Complex conjugation C maps the ACTUAL operator domains as
000164 | 
000165 |     C f_(n,alpha)=f_(-n-1,1-alpha),
000166 |     C D_alpha C^-1=-D_(1-alpha).
000167 | 
000168 | For alpha=1/4 and 3/4 it exchanges the two quarter holonomies; it does not
000169 | replace either by a periodic phase. Functional calculus gives
000170 | C |D_alpha| C^-1=|D_(1-alpha)|. No mode is zero and the excitation gap remains
000171 | 1/4 in original cover units. If P_alpha is the filled negative-energy projector,
000172 | 
000173 |     C P_alpha C^-1=I-P_(1-alpha).
000174 | 
000175 | The resulting particle-hole map identifies the two CAR representations with
000176 | their respective sea states. In explicit finite-excitation language, a state
000177 | with occupied positive modes P and holes H maps to holes {-n-1:n in P} and
000178 | particles {-n-1:n in H} in the other domain. The positive excitation energy
000179 | sum_(n in P union H)|n+alpha| in the displayed original-cover normalization is
000180 | exactly preserved and normal-ordered charge
000181 | |P|-|H| reverses. The empty excitation state maps to the unique empty excitation
000182 | state of the other domain. More explicitly, the mode CAR map sends a_n to
000183 | a'*(-n-1). It preserves the CAR and annihilates the corresponding sea vacuum
000184 | on both sides after polarization. Apply this map to ordered excitation words
000185 | and set U Omega_alpha=Omega_(1-alpha). Equal CAR inner products make U a unitary
000186 | intertwiner between these two polarized representations. This is not an inner
000187 | implementer in a single fixed representation. Its reciprocal phase convention
000188 | and square are not an additional physical selection result. The checker tests
000189 | the mode formulas and an actual finite four-mode polarized Fock intertwiner,
000190 | including fermionic ordering signs, charge and energy. The infinite extension
000191 | follows from the finite-word inner-product argument, not a matrix truncation.
000192 | 
000193 | This supplies an actual state/charge/time conjugation for the specified source.
000194 | It yields neither an additional zero vector in one fixed-alpha representation
000195 | nor the affine r0 current. It also must not be substituted for the unanswered
000196 | comparison of opposite charges WITHIN a single fixed-alpha domain. That latter
000197 | question is the distinct discriminator in native-charged-conjugation-20261015.
000198 | 
000199 | Together with section 4 this is the positive direction worth retaining: carrier
000200 | conjugation and source particle-hole conjugation have explicit opposite-charge
000201 | actions, preserve the appropriate positive excitation energies and exchange the
000202 | two sheet/domain labels. No original seam-to-carrier intertwiner connecting
000203 | these two constructions is supplied merely by their parallel form.