TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-source-character-search-20261015/source_reconstruction/PROOF.txt
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000217 | 6a. A same-source variational selection, without changing B
000218 | 
000219 | Nonzero leakage does NOT make a seam transformation physically invalid.
000220 | It prevents the pure-isometric Wold route just examined. In the given CAR
000221 | source any unitary multiplication U=M_u still defines the quasifree state
000222 | with occupation U Q_occ U*. For smooth u its off-diagonal polarization
000223 | blocks are Hilbert-Schmidt, so it is also implementable in the sea Fock
000224 | representation. Thus non-inner candidates must not be discarded as impossible
000225 | physics merely because they fail the isometry test.
000226 | 
000227 | There is a stronger constructive comparison using the SAME B_alpha and
000228 | the same original sea representation and reference vector Omega. The
000229 | transformed state is R_u Omega with covariance U Q_occ U*, not Omega.
000230 | Let u(theta)=exp(i phi(theta)) have winding w and
000231 | square-integrable phase derivative p=phi'. Write u_k for its Fourier
000232 | coefficients. The transformed diagonal occupation is
000233 | 
000234 |  occupation_u(n)=sum_(m<0) |u_(n-m)|^2.
000235 | 
000236 | Subtract the original sea occupation. A single Fourier shift k has charge
000237 | k and relative energy E_k=k(k-1)/2+alpha k by section 6. Therefore
000238 | 
000239 |  <Q>_u=sum_k k |u_k|^2=w,
000240 |  <H_alpha>_u=sum_k [k(k-1)/2+alpha k] |u_k|^2
000241 |             =1/2 <p^2> + (alpha-1/2) w,
000242 | 
000243 | where <.> denotes normalized integration over the circle. These identities
000244 | follow directly from the CAR covariance, not from an inserted cost function.
000245 | The charge sum is absolutely convergent when sum k^2|u_k|^2 is finite.
000246 | For energy, all E_k are nonnegative for 0<alpha<1, allowing the positive
000247 | particle/hole sums to be interchanged; the same H1 condition gives finiteness.
000248 | Parseval and u* (-i partial_theta)u=p give the displayed integral expressions.
000249 | For H1 u the implementability condition is also visible explicitly:
000250 | ||[C,U]||_HS^2=sum_k |k| |u_k|^2 < infinity.
000251 | 
000252 | At w=1 this reduces to the exact nonnegative remainder
000253 | 
000254 |  <H_alpha>_u = alpha + 1/2 <(p-1)^2>.
000255 | 
000256 | Equality holds precisely when p=1 almost everywhere, hence
000257 | u=e^(i chi) z. Thus the original time and sea select a UNIQUE lowest-energy
000258 | representative of winding one, up to scalar phase, among all smooth scalar
000259 | symbols (and the stated H1 class). This uniqueness is within scalar
000260 | multiplication operators, not the full restricted unitary group: additional
000261 | block-diagonal vacuum symmetries can produce the same minimal state energy.
000262 | This is stronger than selecting a winner
000263 | among the finite diagnostic grid and does not change B to fit U.
000264 | 
000265 | For real Blaschke parameters, <P_a P_b>=(1+ab)/(1-ab), from the absolutely
000266 | convergent Fourier series P_a=sum_(k in Z) a^|k| exp(i k theta).
000267 | Hence the program computes <p^2> exactly for every signed factor list.
000268 | Both b_a and z^2/b_a have energy excess a^2/(1-a^2). At a=1/4 it is 1/15,
000269 | and their alpha=1/4 energy is 19/60, compared with 1/4 for z. They differ
000270 | in leakage despite having the same energy. A noncentered Blaschke isometry
000271 | creates a charge-one wavepacket, not the lowest original-time charge-one
000272 | state; choosing its adapted B_a would hide this excess.
000273 | 
000274 | The variational THEOREM is established in the given source. The SELECTION
000275 | RULE that the actual P1 seam must minimize this charge-one excitation energy
000276 | is not stated or derived in the original postulates. Moreover the theorem
000277 | does not select the original B, field algebra or state used to define that
000278 | energy. Jointly conjugating (U,B,C) preserves the comparison; minimizing
000279 | inside one chosen source is not a proof that it is the physical source.