TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/graded_boundary_functor/normal_dependent_transport/PROOF.txt Original file SHA-256: 1ae1ff8cf27210381ba0c37fcbc7feeec9a10ee2781e2265fc761dbe3c878ae3 Inclusive original lines: 1-182 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 | NORMAL-DEPENDENT TRANSPORT IN THE SAME GRADED BOUNDARY CHART 000002 | 29 October 2026 000003 | 000004 | Purpose and scope 000005 | The existing coupled dictionary need not require a normal-independent 000006 | classical background. The same constant chart transports the entire 000007 | normal-dependent Cauchy graph, including its crossing blocks. This removes 000008 | the restriction in FINITE-COUPLED.txt:7-8 for the stated product class. 000009 | It does not construct a physical bosonic measure, identify OS time, or prove 000010 | that P1 chooses this Clifford quantization. No pinned source is changed. 000011 | 000012 | 1. Fixed geometric data, varying actual coefficients 000013 | 000014 | Retain the product collar and its fixed normal matrices 000015 | 000016 | Z=diag(1,-1), N=[[0,-1],[1,0]], S=NZ, 000017 | Gamma_tot=Z tensor gamma_F, Ical=i0 Gamma_tot, C0=i0 N tensor I. 000018 | 000019 | Let gamma_F be fixed. At each r let A(r)=A(r)^* be even and let 000020 | Phi(r)=Phi(r)^* be odd: [A(r),gamma_F]=0, {Phi(r),gamma_F}=0. 000021 | Allow smooth dependence on r and theta, with the same original finite 000022 | Dirac/Higgs coefficient type. Any additional original J_F-even condition on 000023 | Phi is retained; it is not needed for the following transport identity. 000024 | Put 000025 | 000026 | B(r)=Z tensor A(r)-S tensor Phi(r), 000027 | E v=(P_F,+ v, i0 P_F,- v), 000028 | h(r)=A(r) gamma_F-i0 gamma_F Phi(r). 000029 | 000030 | Then E is independent of r AND of the background, E^*E=I, and pointwise 000031 | 000032 | [B(r),Gamma_tot]=0, C0 B(r) C0=-B(r), B(r)E=E h(r). 000033 | 000034 | C0 is complex-antilinear for Ical; it is linear for the old i0 structure. 000035 | The fixed one-face unitary chart is 000036 | 000037 | T(v,bar(w))=Ev+C0Ew, 000038 | T^-1 B(r) T=h(r) directsum(-bar(h(r))). (1) 000039 | 000040 | No derivative of E occurs. A moving grading, moving normal frame, or moving 000041 | metric trivialisation would require the corresponding connection terms. 000042 | They are outside this fixed product assertion. 000043 | 000044 | 2. Exact finite regulated transfer, without a commutativity premise 000045 | 000046 | Work first in any finite regulator preserving the displayed constant maps 000047 | and adjoints. B(r) is then a smooth selfadjoint matrix for the new Hermitian 000048 | metric. Let K(0)=I and 000049 | 000050 | K'(r)=-B(r)K(r). 000051 | 000052 | Finite evolution is invertible. Set M(r)=C0 K(r) C0. Since r is real, 000053 | 000054 | M'=B(r)M, M(0)=I. 000055 | 000056 | Also (K^*)'=-K^*B(r). Consequently 000057 | 000058 | ((K^*)^-1)'=B(r)(K^*)^-1. 000059 | 000060 | Uniqueness for this finite ordinary differential equation proves 000061 | 000062 | C0 K(r) C0=(K(r)^*)^-1. (2) 000063 | 000064 | No commutation of B(r) with B(s) is assumed. In general the right side is 000065 | NOT K(r)^-1. This is the transport version of the oriented Green pairing. 000066 | 000067 | 3. Entire two-face projector in the charged chart 000068 | 000069 | Write L_K=Graph(K) in H_boundary directsum H_boundary. Its orthogonal 000070 | projection is 000071 | 000072 | P_K=[I;K](I+K^*K)^-1[I,K^*]. 000073 | 000074 | For C_faces=C0 directsum(-C0), equation (2) gives 000075 | 000076 | C_faces(v,Kv)=(C0v,-C0Kv) in Graph(K)^perp. 000077 | 000078 | Indeed set w=-C0Kv. Then -K^*w=C0v, so the displayed vector is (-K^*w,w), 000079 | the general form of a vector in Graph(K)^perp. Dimensions, or invertibility, 000080 | give equality of the two subspaces. Thus 000081 | 000082 | C_faces P_K C_faces=I-P_K. (3) 000083 | 000084 | Let K_+'=-h(r)K_+, K_+(0)=I. Equation (1) and the constant E imply 000085 | 000086 | T^-1 K T=K_+ directsum overline((K_+^*)^-1). 000087 | 000088 | Use the same oriented two-face chart as before: 000089 | 000090 | T_faces(v0,v1,bar(w0),bar(w1)) 000091 | =(Ev0+C0Ew0, Ev1-C0Ew1). 000092 | 000093 | The first charged sector satisfies v1=K_+v0. The second satisfies 000094 | w1=-(K_+^*)^-1 w0. Its conjugate graph is exactly the orthogonal complement 000095 | of Graph(bar(K_+)). Therefore 000096 | 000097 | T_faces^-1 P_K T_faces 000098 | =P_K+ directsum(I-bar(P_K+)). (4) 000099 | 000100 | All face-diagonal and crossing correlations are retained. Constant B is 000101 | the earlier exponential special case, not a prerequisite for (4). 000102 | Composition K_20=K_21 K_10 intertwines through the same E. This is an exact 000103 | composition statement, not a new determinant-line or physical sewing theorem. 000104 | 000105 | 4. Small exact ordering check in the same coefficient type 000106 | 000107 | For a purely algebraic finite check take gamma_F=diag(1,-1) on one even/odd 000108 | pair and X=[[0,1],[1,0]]. Set 000109 | 000110 | B1=Z tensor I, B2=-S tensor X. 000111 | 000112 | They have precisely the displayed even-A/odd-Phi form, B_j^2=I, and 000113 | C0 B_j C0=-B_j. Choose ordered disjoint smooth normal bumps f1,f2, each 000114 | with integral t>0, and B(r)=f1(r)B1+f2(r)B2. The exact transfer is 000115 | 000116 | K=exp(-t B2)exp(-t B1). 000117 | 000118 | Then 000119 | 000120 | C0KC0-K^-1 000121 | =exp(t B2)exp(t B1)-exp(t B1)exp(t B2) 000122 | =sinh(t)^2 [B2,B1] 000123 | =-2 sinh(t)^2 N tensor X != 0. 000124 | 000125 | This is an ordering counterexample to the wrong inverse formula, not a new 000126 | physical source or a replacement for the actual 96-dimensional coefficient. 000127 | It uses smooth normal dependence; no suite or numerical evolution is needed. 000128 | 000129 | 5. Existing joint bosonic insertions can pass through the fixed chart 000130 | 000131 | Suppose a common bosonic measure mu(db), its original insertions q(b), and 000132 | the conditional fermionic functionals omega_b are already given, with the 000133 | required integrability. At each background b use precisely its actual B_b 000134 | and graph, not a fitted covariance. If these functionals use the Clifford 000135 | graph as quantum covariance, (4) transports every conditional CAR word 000136 | through the SAME background-independent algebra map alpha. Hence 000137 | 000138 | integral q(b) omega_b(F) mu(db) 000139 | = integral q(b) omega'_b(alpha(F)) mu(db). (5) 000140 | 000141 | The same holds for arbitrary existing integrable bosonic functionals q and 000142 | finite linear combinations of joint words, including distinct insertion 000143 | points. Bounded CAR words allow the usual dominated integration when q is 000144 | integrable and the conditional states are normalized. Any determinant or 000145 | Pfaffian factor already included in the original common measure must be 000146 | retained on both sides. This identity does not supply that factor or measure. 000147 | It concerns conditional classical-background integration, not an automatic 000148 | identification of an arbitrary noncommuting bosonic operator algebra. 000149 | 000150 | For a positive normalized mu and conditional states, the integrated fermion 000151 | state is generally a mixture. With a complex eta or Pfaffian density, (5) 000152 | remains an integrable algebraic identity and asserts no positive mixture. 000153 | The averaged two-point function need not be a projector, and higher moments are integrals 000154 | of the actual conditional moments. They are not obtained by applying Wick 000155 | to the averaged two-point function. No projector-rigidity or Gaussian 000156 | assumption for the integrated source is made here. Likewise no bosonic 000157 | reflection positivity follows just from this intertwiner. 000158 | 000159 | 6. Unbounded operators and provenance 000160 | 000161 | The finite proof asserts no well-posed arbitrary Hilbert-space Cauchy problem 000162 | for an unbounded elliptic generator. In the continuum, use the actual closed 000163 | elliptic Cauchy data space with its Green form and compatible domain. If that 000164 | space has the standard Lagrangian property and is invariant under the fixed 000165 | grading, the same constant chart gives (4) by splitting its two grades and 000166 | using C_faces L=L^perp. No everywhere bounded K is required in that argument. 000167 | Additional APS, admissibility, Pin, or gluing restrictions must preserve the 000168 | specific maps before this continuum statement applies to them. 000169 | 000170 | The general geometric Lagrangian input was already present in the primary 000171 | construction: Ludewig and Roos, The Chiral Anomaly of the Free Fermion in 000172 | Functorial Field Theory (2020), section 3.3, equations 36-38 and Theorem 3.12, 000173 | as cited in ../PROOF.txt:43-48. The present extra step is the explicit fixed 000174 | E transport of the actual odd finite coefficient through a normal-dependent 000175 | family, beyond ../FINITE-COUPLED.txt:7-8. It does not create a new quantization 000176 | principle from the paper's geometric anomaly-twist construction. 000177 | 000178 | Original fermion action and source distinctions are recorded in 000179 | ../charge_transport/ORIGINAL-SOURCE-AUDIT.txt. In particular the already given 000180 | full operational Seed needs no newly chosen Berezin measure for its own 000181 | reconstruction. Identifying its actual physical insertions with this Clifford 000182 | functor, its charge J and physical time remains the stated origin task.