TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/graded_boundary_functor/GRADED-HS-REFLECTION-AND-TIME.txt
Original file SHA-256: 0d9727a9e167c64ef4a8413736c1555b90a7c5da5392eeea3a3455d0bb037986
Inclusive original lines: 1-200
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.

000001 | THE EXISTING GRADED HS SOURCE IMPLEMENTS THE PRODUCT REFLECTION
000002 | 29 October 2026. Analytic continuation, no new model, state fit or test run.
000003 | 
000004 | RESULT AND TYPE
000005 | The actual product reflection in EVEN-REFLECTION-POSITIVITY.txt has an exact
000006 | implementation in the already used graded thermal Hilbert-Schmidt source.
000007 | Use the natural CREATION chart of the complex-linear selfdual CAR field.
000008 | It is implemented by J_R=Q_Gamma^* J_HS, Q_Gamma=P_even-i P_odd. Conjugation by this
000009 | antiunitary is ANTILINEAR AND ORDER-PRESERVING. On the whole bounded EVEN
000010 | face algebra its reflection form is a positive HS Gram form. This identifies
000011 | the geometric reflection with a specific graded standard-form partner of
000012 | the GIVEN graph source, including its circle group with the SAME parameter.
000013 | It does not choose that quantization or physical time from P1.
000014 | Euclidean Grassmann word reversal is a separate source/reconstruction map.
000015 | 
000016 | 1. Actual h_Phi and its thermal source
000017 | 
000018 | Use exactly FINITE-COUPLED.txt:
000019 |   g=gamma_F, h=A g-i0 g Phi,
000020 |   A=-i0 partial_theta+a(theta), [A,g]=0,
000021 |   Phi=Phi^*, {g,Phi}=0.
000022 | For the geometric graph of collar length ell in EVEN-REFLECTION-POSITIVITY.txt,
000023 | fix beta=2ell. Thus the X below is exactly m_ell(h), not a different thermal
000024 | covariance at an independently varied temperature.
000025 | The background is smooth, bounded, classical and normal-independent here.
000026 | No commutation [A,Phi]=0 is needed. In a finite regulator h is the Hermitian
000027 | matrix built from the same actual finite coefficient, including the original
000028 | v252 Yukawa/Majorana block and grading. No replacement coefficient is used.
000029 | 
000030 | In the continuum h is selfadjoint on periodic H1. It is a bounded zeroth-order
000031 | perturbation of a first-order elliptic circle operator with finite internal
000032 | rank. Hence its eigenvalues lambda_j have N_|h|(E)=O(E), and
000033 |   Tr exp(-beta |h|)<infinity for every beta>0.
000034 | The source Fock presentation is relative to the negative-energy sea of this
000035 | given h. In its excitation basis define the honest positive operator
000036 |   H=sum_j |lambda_j| n_j^exc >=0.
000037 | Finite zero modes cause only finite multiplicity. In infinite dimension one
000038 | does NOT subtract two divergent expressions dGamma(h)-Tr(P_- h).
000039 | The positive particle/hole excitation formula defines H on its usual Fock
000040 | domain. The signed charged annihilators satisfy [H,a_j]=-lambda_j a_j:
000041 | positive lambda labels are particle annihilators; negative labels are hole
000042 | creators. Consequently the given free Gibbs source has
000043 |   rho=exp(-beta H)/Z_beta,
000044 |   Z_beta=product_j(1+exp(-beta |lambda_j|))<infinity,
000045 |   C=(1+exp(beta h))^-1,
000046 |   X=sqrt(C(1-C))=(2cosh(beta h/2))^-1.
000047 | 
000048 | This is a presentation of the conditional free graph source already assumed
000049 | in EVEN-REFLECTION-POSITIVITY.txt, not an independently chosen interacting
000050 | vacuum. The particle/antiparticle meaning of the original 96 labels is not
000051 | redefined by these excitation labels. A claim that the actual P1 source uses
000052 | this h and covariance remains a physical identification to prove.
000053 | 
000054 | The reused object is the GENERAL old HS construction. The density rho(h_Phi)
000055 | is determined by THIS graph, not asserted to be literally the old scalar NS
000056 | or QWZ density. Already at Phi=0, h=A gamma_F differs from A tensor I when
000057 | gamma_F has both signs. Equivalence to that old physical source requires
000058 | its actual polarization, field and time intertwiner. None is inferred here
000059 | by applying the general HS representation theorem to a different h.
000060 | 
000061 | 2. Reuse the existing HS double, with its full parity
000062 | 
000063 | The existing HALF-PERIOD-GNS-SEWING.txt, section4, uses
000064 |   HS(K), Omega=sqrt(rho), L_A Y=A Y, R_A Y=Y A,
000065 |   J_HS Y=Y^*, Gamma Y=Pi Y Pi.
000066 | Pi is the genuine excitation parity. Gamma Omega=Omega and J_HS Gamma=
000067 | Gamma J_HS. Right multiplication reverses composition, R_A R_B=R_(B A).
000068 | As operator conjugation, J_HS L_A J_HS=R_(A^*) is nevertheless antilinear
000069 | and MULTIPLICATIVE in L_A. For odd fields set, without a spectator-parity
000070 | replacement,
000071 |   a_R,j=L_(a_j), a_L,j=-Gamma R_(a_j), a_L,j^*=Gamma R_(a_j^*).
000072 | These are the same two anticommuting CAR families as in that existing result.
000073 | Their occupations are C and 1-C; their actual cross Gram is X. Ordinary
000074 | smeared annihilators a(w) are ANTI-linear in w and creators a^*(v) are linear
000075 | in v. With the inner product linear in the second variable, the intrinsic
000076 | source convention is
000077 |   omega(a^*(v)a(w))=<w,Cv>,
000078 |   omega(a(v)a^*(w))=<v,(1-C)w>.
000079 | This states the operator-C convention without a hidden transpose. The
000080 | mode formulas below can be read in any fixed orthonormal h eigenbasis.
000081 | The complex-linear geometric CAR field therefore has the natural map
000082 | 
000083 |   iota[c(T_faces(v0,v1,bar(w0),bar(w1)))]
000084 |     =a_R^*(v0)-a_L^*(v1)+a_R(w0)-a_L(w1).                (1)
000085 | 
000086 | Thus its positive-grade slots are p0(v)=a_R^*(v) and p1(v)=-a_L^*(v).
000087 | The conjugate slots are their actual adjoints. No spectral conjugation
000088 | kappa_h, redefined linear annihilator or extra field half is used.
000089 | For mode labels, ordinary CAR gives
000090 |   <p0^* p0>=<a_R a_R^*>=1-C,
000091 |   <p1^* p1>=<a_L a_L^*>=C,
000092 |   <p0^* p1>=-<a_R a_L^*>=<a_L^* a_R>=X.
000093 | The other cross block is the adjoint X. Hence the whole geometric covariance is
000094 |   P_h=[[1-C,X],[X,C]].
000095 | The existing GNS intertwiner ARGUMENT, applied to this P_h and rho(h),
000096 | identifies this graph state with its HS vector state on all CAR words.
000097 | It is not a previously proved unitary to the old scalar physical source.
000098 | In infinite dimension it uses bounded smeared
000099 | CAR fields and cyclic closure, not an everywhere bounded inverse of rho.
000100 | 
000101 | 3. Exact geometric R, including both odd phases
000102 | 
000103 | The existing geometric matrices are
000104 |   F=S tensor g, R_geo=face_swap tensor F,
000105 |   C0=i0 N, E v=(P_+v,i0 P_-v).
000106 | They obey F E=C0 E(-i0) and F C0=-C0 F. Under the oriented chart
000107 |   T_faces(v0,v1,bar(w0),bar(w1))
000108 |      =(Ev0+C0Ew0, Ev1-C0Ew1),
000109 | the full action is
000110 |   (v0,v1,bar(w0),bar(w1))
000111 |      ->(-i0 w1,i0 w0,overline(-i0 v1),overline(i0 v0)).
000112 | With the natural creation map (1), this is precisely
000113 |   Theta_geo(p0(v))=-i p1(v)^*=+i a_L(v),
000114 |   Theta_geo(p1(v))=+i p0(v)^*=+i a_R(v).
000115 | These are antilinear in v on both sides, as required. Equivalently
000116 |   Theta_geo(a_R^*(v))=+i a_L(v),
000117 |   Theta_geo(a_L^*(v))=-i a_R(v).
000118 | Both adjoint formulas follow by the star operation. The map on the algebra
000119 | is explicitly order-preserving, as in EVEN-REFLECTION-POSITIVITY.txt:59-62.
000120 | 
000121 | Put
000122 |   Q_Gamma=(1+Gamma)/2-i(1-Gamma)/2
000123 |          =exp(-i pi/4) exp(i pi Gamma/4),
000124 |   J_R=Q_Gamma^* J_HS.
000125 | Since J_HS is antiunitary and commutes with Gamma, J_R^2=1. Also
000126 | J_R Omega=Omega. For any odd left or right multiplication operator O,
000127 |   Q_Gamma^* O Q_Gamma=-i Gamma O.
000128 | Therefore direct operator calculation gives
000129 |   J_R a_R^*(v) J_R=-i Gamma R_(a(v))=+i a_L(v),
000130 |   J_R[-a_L^*(v)]J_R=+i a_R(v).
000131 | This is the exact geometric formula above, not just a matching cross norm.
000132 | The graph-to-HS GNS unitary for THIS h intertwines J_R with Theta_geo's cyclic
000133 | antiunitary, because it intertwines every bounded field and the cyclic vector.
000134 | Q_Gamma^* is the fixed grading phase forced by those two displayed odd phases.
000135 | It is not the isolated original finite charge J_F, nor is J_R identified
000136 | with a physical OS word reflection by this calculation alone.
000137 | 
000138 | 4. Entire even RP form and comparison with the independent Wick proof
000139 | 
000140 | For a bounded EVEN source operator A, Q_Gamma commutes with L_A and R_A.
000141 | Thus
000142 |   J_R L_A J_R=R_(A^*).
000143 | In particular J_R n_R J_R=1-n_L, not n_L. For any bounded even A_i,A_j,
000144 |   <Omega,(J_R L_Ai J_R)L_Aj Omega>
000145 |      =Tr(sqrt(rho) A_i^* sqrt(rho) A_j)
000146 |      =<rho^(1/4) A_i rho^(1/4),rho^(1/4) A_j rho^(1/4)>_HS.
000147 | For i=j this is nonnegative. The Schatten inequality makes the displayed
000148 | vectors HS for bounded A, since rho^(1/4) belongs to Schatten4. This proves
000149 | positivity on the entire even source von Neumann algebra, hence on its
000150 | bounded even CAR subalgebra. The corresponding formula for an even right
000151 | operator gives positivity on the other face as well.
000152 | 
000153 | This is the same even reflection and state as in
000154 | EVEN-REFLECTION-POSITIVITY.txt:105-131. There the cross contraction on real
000155 | first-face fields is -i M, and the cross-pairing sign cancels its phase at
000156 | every even Wick length. Here the same statement is the HS Gram identity.
000157 | The real-field -i phase agrees with the two odd transformations in section3.
000158 | Neither proof replaces the actual same-face contractions or the crossing
000159 | blocks. The Wick proof needs no normal traceclass presentation; the present
000160 | HS identification uses precisely the stated normal thermal source class.
000161 | Unbounded currents and point fields retain their existing domain/limit tasks.
000162 | 
000163 | 5. The normalized sech factor does not obstruct positive physical heat
000164 | 
000165 | For one already quantized fermion let H=E n, E>0. Its positive heat operators
000166 | exp(-sH) exist for every s>=0, at every finite cylinder length. Nevertheless
000167 |   Tr(exp(-ell H) a exp(-ell H) a^*) / Tr exp(-2ell H)
000168 |      =exp(-ell E)/(1+exp(-2ell E))
000169 |      =1/(2cosh(ell E)).
000170 | Its second derivative at ell=0 is -E^2/2. There is no contradiction:
000171 | varying ell changes both the Gibbs normalization and the thermal state.
000172 | This normalized, parameter-dependent midpoint correlation is not asserted
000173 | to be a fixed positive spectral measure's Laplace transform. The numerator
000174 | already has the ordinary positive-energy heat representation. Consequently
000175 | sech''(0)<0 cannot show that positive heat or a source Hilbert space appears
000176 | only in a half-infinite-cylinder limit.
000177 | 
000178 | The same physical heat factors exist for the stated H built from h_Phi.
000179 | For bounded even insertions at 0<t1<...<tn<ell, their reflected Euclidean
000180 | trace functional, WHEN defined with the actual adjoint and word order, has
000181 | the standard product
000182 |   K_F=exp(-(ell-tn)H) A_n exp(-(tn-t_(n-1))H) ...
000183 |                          A_1 exp(-t1 H).
000184 | Its reflected pairing is Tr(K_F^* K_G)/Z_(2ell), a positive HS Gram form.
000185 | The heat intervals sum to ell, and Schatten Holder gives
000186 |   ||K_F||_2 <= sqrt(Z_(2ell)) product_j ||A_j||.
000187 | This does not assume that every imaginary-time conjugate exp(tH)Aexp(-tH)
000188 | is an everywhere bounded operator. The Euclidean word/adjoint map must be
000189 | the actual source map; the fiber matrix R_geo alone does not define it.
000190 | 
000191 | 6. Same-parameter circle transport; distinct physical and GNS generators
000192 | 
000193 | The real-time source automorphism has canonical HS implementation
000194 |   U_t Y=exp(i t H)Y exp(-i t H),
000195 |   L_HS=L_H-R_H, U_t=exp(i t L_HS), U_t Omega=Omega.
000196 | For the ordinary CAR convention, one has on BOTH HS faces
000197 |   U_t a_R^*(v) U_t^*=a_R^*(exp(i t h)v),
000198 |   U_t a_L^*(v) U_t^*=a_L^*(exp(i t h)v),
000199 | and the adjoint formulas
000200 |   U_t a_R(w) U_t^*=a_R(exp(i t h)w),