TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-full-collar-reconstruction-20261015/PROOF.txt
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Inclusive original lines: 96-143
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000096 | 4. A common state and time from that same datum, with explicit field premise
000097 | 
000098 | Now additionally assume that the actual field algebra is charged CAR(H direct_sum H)
000099 | and that its state is the gauge-invariant quasifree state with covariance G_ell.
000100 | These are field/statistical premises, not consequences of P1 alone. A projection
000101 | covariance defines a pure quasifree state. Restriction to the second labeled face
000102 | has occupation n_ell=I-A=(I+exp(2ell B))^(-1), with 0<n_ell<I spectrally.
000103 | For the free CAR dynamics a(f)->a(exp(itB)f), this is the quasifree KMS state
000104 | at inverse temperature beta=2ell (using the convention where occupation is
000105 | (I+exp(beta B))^(-1)). Its one-particle modular logarithm is
000106 | log((I-n_ell)/n_ell)=2ell B.
000107 | Thus the covariance and time generator are read from the SAME two-face datum;
000108 | one does not fit a second Hamiltonian to a prescribed occupation list. In infinite
000109 | dimension this is a state on the CAR algebra, not automatically a trace-class
000110 | Gibbs density matrix in an arbitrarily chosen Fock representation.
000111 | 
000112 | This is the known faithful-covariance inverse from the earlier Atlas contract
000113 | seam-state-dynamics-injectivity-20260918, now linked to the exact finite-collar
000114 | boundary equation. Neither the logarithm nor quasifree purification is new general
000115 | mathematics. The constructive contribution here is the corrected TFPT data path.
000116 | As ell->infinity, n_ell tends strongly to 1_(-infinity,0)(B)+1/2*1_{0}(B).
000117 | The sign-only limit loses the energy magnitudes that the finite two-face data keep.
000118 | On that limit one must not continue using a faithful-state logarithm.
000119 | 
000120 | Normal distance and physical time are distinct. Analytic continuation of the
000121 | positive transfer constructs a unitary one-particle group exp(itB). Its physical
000122 | identification, time unit and event interpretation require the original field
000123 | realization and scale. Neither a finite matrix test nor KMS uniqueness proves them.
000124 | 
000125 | 5. Quarter phase, markings and actual field action
000126 | 
000127 | All spectral functions above act on the given domain of B. They retain an existing
000128 | quarter transition; they do not select alpha=1/4 from unrestricted holonomies.
000129 | In the explicit witness the negative branch has eigenvalues
000130 | -(n+1/4)=(-n-1)+3/4. This is a spectral relabeling. The matrix J_n acts linearly
000131 | between graded components of the SAME domain. An anti-linear charge conjugation
000132 | of a literal scalar twisted section instead sends H_alpha to H_(1-alpha).
000133 | These statements must not be merged into an untyped scalar identity between domains.
000134 | 
000135 | For a supplied unitary source transport W, B'=WBW*, G'=diag(W,W)Gdiag(W*,W*)
000136 | and n'=WnW*. A supplied marking M must transform M'=WMW*, including
000137 | [B',M']=W[B,M]W*. On the CAR algebra the associated field map is a(f)->a(Wf).
000138 | The checks transport a noncommuting marking and test the operator action, not
000139 | only equal eigenvalues. They do NOT derive the original M, the P2 split, a local
000140 | geometric W, or an implementer in every infinite-dimensional Fock representation.
000141 | In particular a function of B alone commutes with B. A noncommuting physical
000142 | marking cannot be manufactured from spectral data of B alone; local algebra,
000143 | winding or another source structure has to supply its actual operator action.