TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/native-arc-charge-source-interface-20261026/PROJECTOR-STATE-RIGIDITY.txt
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000030 | These are source/type distinctions, not an assertion that every possible
000031 | fermion term is absent from the entire archive.
000032 | 
000033 | 2. Exact CAR uniqueness theorem; no Gaussian premise
000034 | Let H be a complex Hilbert space with its charged CAR C* algebra and let
000035 | omega be ANY normalized positive state. Define its occupation covariance C
000036 | by omega(a*(f)a(g)) with the consistent inner-product convention. Positivity
000037 | and CAR imply 0<=C<=I. Suppose C=P=P*=P^2 on the WHOLE specified H.
000038 | In the GNS representation, for f in kerP,
000039 |  ||a(f)Omega||^2=omega(a*(f)a(f))=0;
000040 | for f in ranP,
000041 |  ||a*(f)Omega||^2=omega(a(f)a*(f))=||f||^2-<f,Pf>=0.    (1)
000042 | Thus the empty-particle annihilators and filled-hole annihilators annihilate
000043 | the cyclic vector. Split every smeared field using H=ranP direct_sum kerP.
000044 | The corresponding particle/hole operators obey CAR. Anticommuting their
000045 | annihilators to the right, and creators to the left, leaves only the scalar
000046 | CAR contractions in every finite word. All remaining terms have expectation0
000047 | by(1) and its adjoint. This determines every polynomial expectation uniquely.
000048 | Polynomials are norm dense in the CAR algebra, so the whole state is unique.
000049 | It is precisely the pure gauge-invariant quasifree/Fock state omega_P.
000050 | Odd and anomalous pair expectations vanish as CONSEQUENCES, not assumptions.
000051 | Every grouped neutral word has its Wick determinant with covariance P.
000052 | Araki1970, Lemma4.3, proves this arbitrary-state Fock uniqueness in self-dual
000053 | CAR form. The argument above is the charged CAR version, not a new general
000054 | CAR theorem. On infinite H it is uniqueness as a CAR state and its GNS
000055 | representation; no implementability in a separately fixed old Fock space
000056 | or global density matrix is asserted.
000057 | 
000058 | 3. Apply to the ORIGINAL complete chiral annulus graph
000059 | For the actual NS eigenmode and r_n=exp(-W lambda_n)>0, its full projector is
000060 |  P_n=1/(1+r_n^2) [[1,r_n],[r_n,r_n^2]].                  (2)
000061 | On the complete two-face module its bounded functional calculus form is
000062 |  P_A=[[I-C_A,X_A],[X_A,C_A]],
000063 |  C_A=(I+exp(2W B_Sigma))^-1,
000064 |  X_A=(2cosh(W B_Sigma))^-1=sqrt(C_A(I-C_A)).             (3)
000065 | All functions commute and X_A^2=C_A(I-C_A), hence P_A^2=P_A=P_A*.
000066 | These expressions are bounded even though exp(-W B_Sigma) is unbounded.
000067 | The graph is understood as its closed Cauchy space, not an everywhere
000068 | defined bounded transfer operator. W=1/2 is the existing width, no fit.
000069 | The labelled second-face C_A is faithful with0<c_n<1 and is NOT a projector.
000070 | Under the previous quarter map c_r=(1+exp(4pi(r+alpha)))^-1. No state change
000071 | or physical temperature/time is inferred from these generator normalizations.
000072 | 
000073 | Consequently the minimal sufficient physical-state test is the following:
000074 |  A. Supply an actual normalized positive source on this same JOINT charged
000075 |     CAR algebra, with its full fields, marks, domains and boundary pairing.
000076 |  B. Derive its actual complete covariance to be(3), including X_A.
000077 | Then Section2 supplies all its CAR words. No separate Gaussian measure
000078 | postulate or independent four-point matching is required for that algebra.
000079 | Its neutral current and native composites inherit the preceding source
000080 | functional when their existing smearing/subtraction/domain limits apply.
000081 | Conversely, a non-Wick word in a positive state on this very algebra rules
000082 | out EXACT full covariance P_A under this dictionary, not the whole theory.
000083 | 
000084 | The missing cross-face source equation is therefore explicit:
000085 |  omega_P1(a_L*(f)a_R(g))=<f,(2cosh(W B_Sigma))^-1 g>,     (4)
000086 | together with the two diagonal blocks and the actual joint CAR embedding.
000087 | It is NOT enough to call a geometric Calderon projector a covariance, to
000088 | purify a marginal by choice, or to identify the old Hardy state with C_A.
000089 | The former would insert the physical-state step that A/B must derive.
000090 | The map in(4) refers to the already given chiral scalar two-face source.
000091 | It is not an identification with the complete two-component Dirac vector
000092 | source, its normal Clifford action, OS admissible projector or physical time.
000093 | Existing full-Dirac vector-current cancellation results remain unchanged.
000094 | 
000095 | 4. Exact positive countercontrols: why the missing coherence matters
000096 | For one actual mode put p=c_n, x=sqrt(p(1-p)). In the occupation basis
000097 | |00>,|10>,|01>,|11>, the geometric state is
000098 |  |Psi_p>=sqrt(1-p)|10>+sqrt(p)|01>.
000099 | Its full covariance is [[1-p,x],[x,p]]=P_n, with p=r_n²/(1+r_n²).
000100 | Compare two other positive states: