TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-source-character-search-20261015/source_reconstruction/PROOF.txt
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000130 | 5. Constructive reconstruction without initially inserting a circle
000131 | 
000132 | Assume all of the following about the ACTUAL source pair on one Hilbert space:
000133 | (i) C orthogonal, U unitary and (1-C) U C=0;
000134 | (ii) S=U|CH is Fredholm with index -1;
000135 | (iii) S is pure: intersection_n S^n CH={0};
000136 | (iv) U is its minimal unitary dilation, span{U^n CH:n in Z} dense in H.
000137 | 
000138 | Set W=CH orthogonally minus U CH. By (i),(ii), dim W=1. Wold decomposition gives
000139 | CH=orthogonal_sum_(n>=0) U^n W. Minimality supplies the negative powers,
000140 | so H=orthogonal_sum_(n in Z) U^n W. Choosing a unit vector w in W gives
000141 | e_n=U^n w. Only a common scalar phase depends on this choice. Therefore
000142 | (C,U) is unitarily equivalent to (Hardy projection, bilateral shift).
000143 | This derivation uses Wold, not an assumed equality of dimensions.
000144 | 
000145 | Now require the ACTUAL selfadjoint B on the common invariant finite-span
000146 | core to satisfy U* B U=B+1 and B w=alpha w. These domain statements are
000147 | part of the premise, not inferred from bounded matrices. Iteration in both
000148 | directions gives B e_n=(n+alpha)e_n. The diagonal operator is essentially
000149 | selfadjoint on this core, so its selfadjoint closure is fixed. Thus
000150 | 
000151 |  B=N+alpha, C=1_[0,infinity)(B), exp(2pi i B)=exp(2pi i alpha) I.
000152 | 
000153 | Lean certifies the integer-index recurrence at the scalar level. The
000154 | infinite-domain and spectral statements are analytic parts of this proof.
000155 | The positive spectral projection follows for 0<alpha<1; it is not a
000156 | separate choice after these hypotheses.
000157 | 
000158 | For the diagnostic b_a family the bilateral basis is k_a b_a^n, n in Z.
000159 | It is complete and orthonormal: change variables by the circle diffeomorphism
000160 | b_a, with |k_a|^2 equal to its positive phase derivative. Its nonnegative
000161 | span is H2, so the resulting unitary F_a commutes with C. The operator
000162 | B_a=F_a(N+alpha)F_a* has the required covariance and quarter transition.
000163 | For a!=0 it is not the supplied B_alpha in section 4. Replacing B_alpha
000164 | by B_a proves a conditional reconstruction, not an original-source match.
000165 | 
000166 | 6. Actual conditional field, sea-state and time action
000167 | 
000168 | If the physical field algebra is CAR(H), unitarity of the derived map
000169 | F:delta_n -> U^n w from the reference orthonormal basis induces its CAR
000170 | isomorphism: products, adjoints and all
000171 | anticommutators are preserved, not just spectra. CAR is an additional
000172 | field-statistics premise; neither P1 reflection positivity nor Wold alone
000173 | implies it. With negative sea occupation Q_occ=1-C, the pure gauge-invariant
000174 | quasifree state is fixed by omega(c_m* c_n)=delta_mn 1_(n<0) and Wick's rule.
000175 | The physical identification with this quasifree state is also a premise.
000176 | Do not use the symbol Q_occ for the relative charge operator below.
000177 | 
000178 | On the algebraic finite-excitation sea core define
000179 | 
000180 |  Q = sum_n :c_n* c_n:,
000181 |  H_alpha = sum_n (n+alpha) :c_n* c_n:.
000182 | 
000183 | Each particle at n>=0 costs n+alpha; each hole at n<0 costs -(n+alpha).
000184 | Thus H_alpha is positive with a unique zero-energy sea vector. Its closure
000185 | is the positive free-Fock excitation Hamiltonian. The field time satisfies
000186 | exp(i t H_alpha)c_n exp(-i t H_alpha)=exp(-i t(n+alpha)) c_n.
000187 | 
000188 | The unit winding action is beta_U(c_n)=c_(n+1), not a four-periodic mode
000189 | permutation. Its change of polarization has rank one. In the charged sea
000190 | Fock representation it is implementable, up to scalar phase, by a unitary R,
000191 | with convention beta_U(A)=R A R*: R c_n R*=c_(n+1), equivalently
000192 | R* c_n R=c_(n-1).
000193 | Equivalently R shifts each occupation configuration, including the whole sea;
000194 | the CAR ordering fixes relative fermionic signs. Hence on the common core
000195 | 
000196 |  R* Q R=Q+1,       R* H_alpha R=H_alpha+Q+alpha.
000197 | 
000198 | The additive constant is the newly occupied n=0 sea mode. In particular
000199 | R Omega is a charge-one state of energy alpha; R^(-1) Omega has charge -1
000200 | and energy 1-alpha. Generally the charge-k shifted sea has energy
000201 | k(k-1)/2+alpha k, for every integer k. These formulas follow by summing
000202 | the finite occupation differences and hold beyond any numerical cutoff.
000203 | 
000204 | On every finite-excitation vector H_alpha-alpha Q is an integer. Therefore
000205 | exp(2pi i H_alpha)=exp(2pi i alpha Q): the quarter phase survives on charged
000206 | fields and states. The charged winding R, the representation equivalence F,
000207 | and the quarter transition are three distinct operators. R changes the sea
000208 | charge; requiring it to preserve the vacuum would contradict this unit flow.
000209 | No event rate, stochastic instrument or preferred measurement is inferred
000210 | from the existence of R.
000211 | 
000212 | The rank-one wandering space counts a primitive charged shift channel.
000213 | It is not the five-slot P2 interface, a spacetime dimension, or a D5
000214 | spinor. Mapping this reconstructed channel to the actual P2 carrier and
000215 | the earlier E8 extension remains a separate physical identification.