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000001 | FULL REFERENCE TRANSPORT THROUGH THE EXISTING GEOMETRIC CARRIER CHART
000002 | L20261102G. Independent joint_origin review, 2 November 2026.
000003 | Entry Atlas a30058cba8bd21f1e5fd77b9, shared plan84. Canonical unchanged.
000004 | 
000005 | RESULT
000006 | 
000007 | The old bound D_ref is transportable through the SAME existing full E
000008 | chart. No new reference, P_F, grading rotation or state is required for
000009 | this operator transport. For the two actual R032 lifts its coefficient is
000010 | 
000011 |  Phi_ref,S=L(m),              Phi_ref,T=L(p)+D_M,
000012 |  p=(H+H_tau)/2,               m=(H-H_tau)/2.
000013 | 
000014 | The full chart U=(E,NE), with its full transported original domain, gives
000015 | 
000016 |  U* D U = mathfrakD_h,        U* D_ref U = mathfrakD_href,
000017 |  mathfrakD_h = [[0,-partial_r+h],[partial_r+h,0]],
000018 |  h=A gamma-i gamma Phi,       h_ref=A gamma-i gamma Phi_ref.       (A)
000019 | 
000020 | It follows exactly that F(D_ref)=(F(D))_ref when F denotes this full
000021 | unitary transport and the original tau is transported with it. This is
000022 | not a commutation of a logarithm with an arbitrary compression.
000023 | 
000024 | The ALREADY EXISTING complete v258 reader then gives, in its actual
000025 | normal-independent smooth product class,
000026 | 
000027 |  R_ref=mu_geo Pi_odd^gamma log((I-C_ref)C_ref^-1)
000028 |       =2ell mu_geo U_gamma Phi_ref U_gamma*,
000029 |  U_gamma=exp(-i pi gamma/4).                                    (B)
000030 | 
000031 | This applies the existing R003 theorem to the reference coefficient;
000032 | it does not posit raw D_mod=Phi. Retaining the given A, ell and mu_geo
000033 | recovers h_ref, the full graph, and (A). Thus the reference OPERATOR is
000034 | determined in this existing source route. A second independent choice of
000035 | that operator would be spurious. The full unnormalized reference source
000036 | still uses its actual spectral/Berezin/bosonic caller: those different
000037 | functionals are transported, not declared identical by their common input.
000038 | 
000039 | 1. THE SAME FULL CHART COMMUTES WITH THE ACTUAL REFERENCE OPERATION
000040 | 
000041 | Use precisely the supplied old-complex collar data
000042 |  N=[[0,-1],[1,0]], Z=diag(1,-1), S=NZ,
000043 |  B=Z A-S Phi, D=N(partial_r+B), Gamma=Z tensor gamma,
000044 |  gamma=gamma*=gamma^-1, [A,gamma]=0, Phi*=Phi, {gamma,Phi}=0,
000045 |  E v=(Pplus v,i Pminus v), Pplus/minus=(I+/-gamma)/2.
000046 | 
000047 | E is fixed independently of H, theta and r. Its range is the entire
000048 | positive TOTAL grade, including every original finite carrier entry.
000049 | Put F=NE and U=(E,F). Since E*F=0 and EE*+FF*=I, U is old-complex
000050 | unitary on the whole original normal-double space. No additional species
000051 | is introduced by writing its two summands. Directly,
000052 | 
000053 |  B E=E h, B F=-F h,
000054 |  D E=F(partial_r+h), D F=E(-partial_r+h).
000055 | 
000056 | These give (A), with Dom(mathfrakD_h)=U*Dom(D). The fixed U retains all
000057 | derivative terms when A or Phi vary; no [A,Phi]=0 is used. The same proof
000058 | holds for r-dependent smooth coefficients, provided the supplied original
000059 | closed operator and domain exist. It does not infer a bounded semigroup
000060 | on all L2 from the two-sided spectrum of h.
000061 | 
000062 | The reference is the original even operation on the common stable domain:
000063 |  D_ref=(D+tau D tau^-1)/2.
000064 | Set tau_U=U* tau U. Then the exact whole-class identity is
000065 | 
000066 |  U*D_ref U=(U*DU+tau_U(U*DU)tau_U^-1)/2.                       (1)
000067 | 
000068 | For the ACTUAL R032 same-symbol lifts, the full base pullback R_r is kept:
000069 | 
000070 |  tauS=S R_r,         tauS,U=[[0,gamma],[gamma,0]] R_r,
000071 |  tauT=S gamma R_r,   tauT,U=[[0,I],[I,0]] R_r.                (2)
000072 | 
000073 | The U chart therefore does not erase reflection or make it preserve one
000074 | grade. In both cases it exchanges the two full normal slots. Inserting
000075 | the original intertwining of H_tau, Y and M_R gives the already proved
000076 | 
000077 |  D_ref,S=D0+Z L(m),
000078 |  D_ref,T=D0+Z[L(p)+D_M], D0=N partial_r+SA.                  (3)
000079 | 
000080 | Equations (A)-(3) are an explicit commuting diagram, including tau, not
000081 | an identification by equal dimensions. Neither S nor T is selected anew
000082 | as the primitive lift. A different lift still carries its actual symbol;
000083 | the shape in (A) is asserted for the verified same-symbol class, whereas
000084 | the abstract full-unitary identity (1) holds for every supplied tau.
000085 | 
000086 | If the ACTUAL original tau depends on the field, (1) still holds pointwise.
000087 | For its differentiable family set Q_delta=(delta tau)tau^-1. Then
000088 | 
000089 |  delta D_ref=1/2[delta D+tau(delta D)tau^-1
000090 |                          +[Q_delta,tau D tau^-1]].        (3a)
000091 | 
000092 | The same U carries every term of (3a), since delta U=0 for these H jets.
000093 | Thus there is no tacit fixed-tau hypothesis in the full transport theorem.
000094 | The simplified R032 L(delta p/m) expressions use its actual fixed supplied
000095 | lift; if its base/bundle transport moves, delta H_tau and the displayed
000096 | commutator must move too. Smooth moving domains are transported as domains,
000097 | not differentiated by pretending their graph is fixed.
000098 | 
000099 | 2. ACTUAL GRADING AND REAL STRUCTURE ARE TRANSPORTED AS WELL
000100 | 
000101 | The exact matrices in this old-complex chart are
000102 | 
000103 |  U* Gamma U=diag(I,-I),
000104 |  U* N U=[[0,-I],[I,0]],
000105 |  U* S U=[[0,gamma],[gamma,0]].                              (4)
000106 | 
000107 | For the unchanged v252 antilinear J_F, the originals give
000108 |  J_F^2=1, J_F gamma=-gamma J_F, J_F Phi=Phi J_F.
000109 | Let Jhat=I_normal tensor J_F and use the existing local candidate
000110 |  Jloc=S Jhat, c=-i J_F.
000111 | Because Jloc E=E c and Jloc N=-N Jloc, its exact full transport is
000112 | 
000113 |  U* Jloc U=diag(c,-c), c^2=1.                              (5)
000114 | 
000115 | This is an antilinear block formula. Its coefficient matrix is transformed
000116 | with conjugate(U), not with U, on the right. The native check below uses
000117 | that rule. Formula (5) does not choose Jloc as a new physical total J;
000118 | any actually supplied total J is transported as U*JU instead.
000119 | 
000120 | For the actual charge-odd connection a^J=-a, c commutes with h and
000121 | Jloc anticommutes with D on a Jloc-stable full domain. A scalar quarter
000122 | connection is the different original case: c maps h_alpha to h_-alpha,
000123 | and the supplied winding e^-i theta maps this to h_(1-alpha). Thus
000124 | alpha=1/4 and 3/4 are exchanged, including Fourier k -> -k-1.
000125 | This chart does not replace that map by same-quarter charge invariance.
000126 | 
000127 | U_gamma in (B) already commutes with the actual J_F (antilinearity and
000128 | J_F gamma J_F^-1=-gamma cancel its two signs), with grading, and with the
000129 | represented even algebra. It is the existing dictionary comparison,
000130 | not a newly chosen physical gauge transformation.
000131 | 
000132 | 3. GREEN, CAUCHY/GRAPH, BOUNDARY LEGS AND BEREZIN COMMUTE
000133 | 
000134 | For each actual closed bulk realization and z in its resolvent set,
000135 | 
000136 |  U*(D-z)^-1 U=(mathfrakD_h-z)^-1,                         (6)
000137 | 
000138 | and identically for D_ref. The boundary trace is carried by U_boundary;
000139 | its Green boundary form carries N to the matrix in (4). Consequently
000140 | the actual Cauchy-data subspace, its projector, and every source/test leg
000141 | are carried by the SAME boundary unitary. Closed APS or sewn domains
000142 | that couple endpoint slots retain those couplings. Independent endpoint
000143 | conditions are not imposed after this transport.
000144 | 
000145 | For the supplied normal propagation of (partial_r+h)v=0, the actual
000146 | K and K_ref are obtained with h and h_ref respectively. In an autonomous
000147 | product source they are exp(-ell h) and exp(-ell h_ref) on their correct
000148 | closed graph domains. For nonautonomous backgrounds they remain the
000149 | actual ordered propagation, not a pointwise replacement exp(-ell h(r)).
000150 | 
000151 | When K is a closed injective dense-range transfer, the second graph face
000152 | is C2=K(I+K*K)^-1 K*=KK*(I+KK*)^-1. Therefore
000153 | 
000154 |  log((I-C2)C2^-1)=-log(KK*)                               (7)
000155 | 
000156 | on its spectral domain. The same is true for K_ref. The full crossing
000157 | blocks retain the polar part of K; C2 alone does not. This preserves the
000158 | normal-dependent source already constructed in E. It does not extrapolate
000159 | the autonomous local reader (B) to every normal-dependent profile.
000160 | 
000161 | At a compatible finite regulator a complex Berezin substitution
000162 |  psi=U chi, bar(psi)=bar(chi) U* has unit paired Jacobian. Thus its original
000163 | full bilinear matrix, determinant, covariance and all sourced moments are
000164 | identical after transport. If the original caller is instead a legal
000165 | Majorana/Pfaffian pairing, its skew matrix transforms by congruence
000166 | M -> U^T M U and its Pfaffian factor det(U) is compensated by the
000167 | transported Grassmann measure/orientation. The full paired reference uses
000168 | the same fixed U. No uncancelled reference phase or state rescaling is
000169 | created by this change of frame.
000170 | 
000171 | This is preservation of the actual original pairing. On a single E grade
000172 | the existing <Jloc x,Dy> pullback is zero; the present full U retains both
000173 | grades. It never replaces the full kinetic Pfaffian by the symmetric
000174 | one-slot expression <cv,hw>.
000175 | 
000176 | 4. THE ALREADY PROVED COMPLETE READER CLOSES THE REFERENCE OPERATOR
000177 | 
000178 | The exact R003 source (listed below) assumes A=-i partial_theta+a(theta)
000179 | selfadjoint on H1, smooth bounded Hermitian even a, and smooth bounded
000180 | Hermitian odd Phi. Both Phi_ref in (3) satisfy the SAME class whenever
000181 | the original reflected coefficient and pullback do. They keep all original
000182 | families/colors/Y and the appropriate original M_R term.
000183 | 
000184 | For the actual labeled second face, followed by its reducing E chart,
000185 | 
000186 |  C_ref=(I+exp(2ell h_ref))^-1.
000187 | 
000188 | Spectral calculus gives logodds(C_ref)=2ell h_ref on H1. The original
000189 | Pi_odd^gamma deletes A gamma and retains -i gamma Phi_ref. Hence (B)
000190 | follows without commuting a logarithm through the nonreducing full face
000191 | projection. Its inverse and reconstruction are
000192 | 
000193 |  Phi_ref=U_gamma* R_ref U_gamma/(2ell mu_geo),
000194 |  h_ref=A gamma+R_ref/(2ell mu_geo).                        (8)
000195 | 
000196 | No value mu_geo=1/(2ell) is chosen. In particular, for normal-independent
000197 | even H the actual S reference has Phi_ref=0, while T has Phi_ref=Phi;
000198 | the reader returns exactly those cases, not a fresh common reference.
000199 | 
000200 | R003 explicitly distinguishes its fiberwise full E implementation from