TFPT scientific source excerpt Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/PROOF.txt Original file SHA-256: 3537a2b080df0bb8b5d30c73da7dd83dc8e1ae5a6cfea4bb48a8b99f0080dfce Inclusive original lines: 34-128 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000034 | 2. The original source data and what they do not assert 000035 | 000036 | The known product collar gives (partial_r+B)psi=0, R_l=exp(-l B) and its full 000037 | two-face orthogonal Cauchy graph projection G_l. The previous contract proves 000038 | that R_l and signed B are recoverable from the labeled full graph and known l. 000039 | In infinite dimension this includes the closure of the spectral quotient. 000040 | These are one-particle/classical boundary data. 000041 | 000042 | Original source references: 000043 | 01_boundary_kernel_source.tex:302-316: S_prim is a relative spectral trace 000044 | for a DECLARED test profile f, plus the eta phase. It is not an explicitly 000045 | supplied quadratic Grassmann action on the whole physical field algebra. 000046 | 01_boundary_kernel_source.tex:319-343: local insertions are introduced as a 000047 | separate source algebra. Lines480-520: the operational seed tuple already 000048 | includes its algebra, time, reflection and state. 000049 | 02_carrier_source.tex:8-39: S+=Lambda_even(E) is an INTERNAL half-spinor 000050 | carrier, not automatically the physical many-particle Fock representation. 000051 | 04_qft_source.tex:2270-2301: higher derivatives S_adm^(n), n>=3, are retained 000052 | as interaction vertices. Therefore free dynamics cannot be imposed as a 000053 | general consequence of the original action. 000054 | 000055 | GATE.METRIC.17/v161 and QFT.NETRP.01/v175 already supply second-quantization 000056 | functoriality under an actual Bogoliubov field identification. The construction 000057 | below reuses that mathematics; it does not rediscover a physical selection rule. 000058 | 000059 | 3. A projection CAR covariance forces all state moments 000060 | 000061 | Let c(f) obey CAR on a finite-dimensional one-particle space, and suppose the 000062 | ACTUAL state has covariance C_ij=omega(c_j^* c_i)=G_ij with G=G*=G^2. 000063 | For f in Ker G, positivity gives ||c(f)Omega||^2=0. For g in Ran G, CAR gives 000064 | ||c^*(g)Omega||^2=0. These annihilation conditions and CAR determine all moments 000065 | by normal ordering. In finite Fock space their common solution space is one 000066 | dimension, the occupied Slater state for Ran G. Thus an independent assumption 000067 | that this SAME state is quasifree is redundant. 000068 | 000069 | state_from_projection implements these linear annihilation constraints and 000070 | checks every entry of the supplied covariance. It does not start with a Wick 000071 | formula or choose a vacuum occupation by minimizing a function. 000072 | 000073 | IMPORTANT: the classical Cauchy projector being G does not prove that the 000074 | physical state's CAR covariance is G. That identification remains a substantive 000075 | source/field premise. A mixed subface covariance alone also does not satisfy 000076 | this projection argument. 000077 | 000078 | 4. Oriented boundary-state pairing determines the full operator ray 000079 | 000080 | On two labeled m-dimensional faces (input first, output second), let V=[I;R] and 000081 | G=V(V*V)^(-1)V*. Its forced boundary state ray is 000082 | wedge_j (e_in,j + sum_i R_ij e_out,i). 000083 | Use the fixed ordered exterior input dual: an input subset J-complement is 000084 | paired with input index J, with sign (-1)^inv(J), where inv(J) counts pairs 000085 | j in J, k not in J, k>j. Expanding the wedge gives 000086 | Q_(I,J) / Q_(empty,empty) = det R_(I,J) if |I|=|J|, 000087 | and zero otherwise. Therefore Q is proportional to direct_sum_n Lambda^n R. 000088 | boundary_amplitude reconstructs the state from G, performs this oriented 000089 | unvectorization, and independently agrees with the creator-covariance recurrence. 000090 | Complex nondiagonal controls verify the signs and avoid an unnoticed transpose 000091 | or conjugation. Independent frame changes U_in,V_out give exactly 000092 | Q'=Gamma(V_out) Q Gamma(U_in^*). An internal quarter iI gives phases i^n on 000093 | n-field products. Projection of G alone does not imply positivity of Q: 000094 | R=-1 gives Q=diag(1,-1). A positive amplitude additionally uses the actual 000095 | selfadjoint collar transfer R=exp(-lB)>0. 000096 | 000097 | The oriented operator-state identification is part of this specified fermionic 000098 | quantization. It is not deduced just by observing that a matrix has matching 000099 | dimension. This construction does not equate the internal CAR rotation with 000100 | the physical chiral quarter domain, which remains separately labeled. 000101 | 000102 | Only an OPERATOR RAY is determined by normalized G. Its overall complex factor, 000103 | vacuum determinant and unnormalized source partition function are not recovered. 000104 | The chart Q_(empty,empty)=1 is a normalization of that ray, not a physical 000105 | derivation of the scalar, its seam phase, or a gravitational area coefficient. 000106 | 000107 | 5. Complete multi-field covariance and chronological sewing 000108 | 000109 | Equivalently suppose Q Omega=Omega and the FULL operator relations hold: 000110 | Q c_j^* = sum_i R_ij c_i^* Q. 000111 | Pushing Q past a word of creators determines its value on every Fock basis 000112 | vector. These vectors span, so the result is unique and equals the exterior 000113 | lift above. It follows for all words, not only vacuum matrix elements. For two 000114 | successive source maps, 000115 | Q(R2 R1)=Q(R2)Q(R1). 000116 | This is verified for noncommuting R1,R2 so chronological order is actually tested. 000117 | For a source semigroup R_(s+t)=R_s R_t this gives the normalized free multi-field 000118 | sewing rule at every particle number; amplitudes are minors, not fitted factors. 000119 | 000120 | For positive Euclidean R, this is NOT a CAR *-automorphism. The adjoint relation 000121 | is c_j Q = Q sum_i conjugate(R_ij)c_i, not Q c_j=sum_i conjugate(R_ij)c_i Q. 000122 | For unitary real-time source maps the exterior lift is unitary and conjugation 000123 | does define the corresponding CAR *-automorphism. 000124 | 000125 | An interaction can have the same reference and one-particle sector while 000126 | violating this full covariance. For instance a different two-particle diagonal 000127 | entry in diag(1,r,r,b) passes the vacuum creator relation but fails the full 000128 | relation unless b=r^2. This is the adversarial test, not a proposed new TFPT model.