TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/QP-WHOLE-SOURCE-PULLBACK-REVIEW.txt
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000001 | q TO p: THE WHOLE LINEAR SOURCE PULLBACK AND ITS TARGET DATA
000002 | Independent bounded original-operator review, 2026-10-30.
000003 | No new source class, metric, cover, Atlas write or pinned-source mutation.
000004 | 
000005 | VERDICT
000006 | 
000007 | The proposed whole linear covariance is CORRECT with the transported
000008 | connection, domain and spin map. In a constant finite frame it sends
000009 | 
000010 |  h(A_q,Phi_q) -> h(A_p,-bar(Phi_q) composed with inversion),
000011 | 
000012 | where A_p is the orientation-reversed pullback of A_q. For the actual
000013 | v252 symmetric Majorana input this means Y_p=-bar(Y_q), MR_p=-bar(MR_q).
000014 | It holds on all96 coefficients and has an explicit whole F5 lift preserving
000015 | the given complex structure and C0. Its time covariance has PLUS time,
000016 | because this charged map is linear. It is distinct from the earlier
000017 | antiunitary E0 family transport, whose time covariance has minus time.
000018 | 
000019 | The original q+p results already provide the p sign/spin packet and its
000020 | pulled flat connection; those are reused. The checked sources do not also
000021 | identify the complete coupled96 p-background with the displayed negative
000022 | conjugate controls. Thus the whole operator arrow is proved to the
000023 | EXPLICIT TRANSPORTED BACKGROUND, and becomes an arrow to an independently
000024 | specified original p operator precisely when that operator has these data.
000025 | Calling the target p does not itself derive that final identification.
000026 | 
000027 | 1. THE ACTUAL FINITE MATRIX, INCLUDING MAJORANA
000028 | 
000029 | The original verification/v252_full_finite_triple.py:86-132 gives
000030 | 
000031 |  Phi(Y,MR)=[[S_Y,T_MR],[T_MR^dagger,bar(S_Y)]],
000032 |  gamma=diag(g_particle,-g_particle),
000033 |  J_F=K_F kappa, K_F=[[0,I48],[I48,0]],
000034 | 
000035 | where kappa is the actual component conjugation. In the signed wedge
000036 | dictionary it becomes kappa_split. Therefore K_F=J_F kappa is LINEAR;
000037 | it is the actual particle/antiparticle swap, not a presumed scalar
000038 | identification of the two half-spinors. Directly
000039 | 
000040 |  K_F^2=I, K_F gamma K_F=-gamma.
000041 | 
000042 | For MR^t=MR, the actual KO6 condition gives
000043 | 
000044 |  K_F Phi K_F=bar(Phi).
000045 | 
000046 | This is not valid for an arbitrary nonsymmetric matrix inserted into the
000047 | MR argument: the top-right block after the swap is T_MR^dagger, whereas
000048 | the corresponding block of bar(Phi) is bar(T_MR). Symmetry is precisely
000049 | what makes them equal. It is the original Majorana premise, not an
000050 | additional fit. Linearity of the constructor then gives
000051 | 
000052 |  -bar(Phi(Y,MR))=Phi(-bar(Y),-bar(MR)).
000053 | 
000054 | The minus sign applies also to MR. It cannot be inferred by treating that
000055 | block as another Yukawa block with the same two endpoint types; the
000056 | displayed actual constructor is what proves it here.
000057 | 
000058 | 2. CONSTANT FRAME: THE CHARGED OPERATOR AND TIME
000059 | 
000060 | Let Rx f(x)=f(-x). Assume the actual q and p connections satisfy
000061 | 
000062 |  Rx A_q Rx^-1=-A_p
000063 | 
000064 | in the compared constant frame. Put T=K_F Rx, an ordinary linear unitary.
000065 | For the already given charged Hamiltonian
000066 | 
000067 |  h_q=A_q gamma-i gamma Phi_q(x),
000068 | 
000069 | direct substitution gives
000070 | 
000071 |  T h_q T^-1=A_p gamma-i gamma Phi_p(x),
000072 |  Phi_p(x)=-bar(Phi_q(-x)).
000073 | 
000074 | Thus T exp(i t h_q) T^-1=exp(i t h_p). The actual supplied circle
000075 | evolution transports with the same parameter t; no new time is defined.
000076 | This is a source-family relation. It is not a fixed-background symmetry
000077 | unless the compared backgrounds are independently identified.
000078 | 
000079 | The same calculation includes variable smooth Phi and an even connection.
000080 | No [A,Phi]=0 is needed. If an internal matrix connection is present, its
000081 | K_F conjugation must be included in A_p; it cannot be commuted past K_F
000082 | without checking it.
000083 | 
000084 | 3. THE DOMAIN AND THE QUARTER SIGN ARE ESSENTIAL
000085 | 
000086 | For the periodic presentation A_alpha=-i partial_x+alpha,
000087 | 
000088 |  Rx A_alpha Rx^-1=+i partial_x+alpha=-A_(-alpha).
000089 | 
000090 | Consequently the target is A_(-alpha), not the SAME A_alpha. At alpha=1/4
000091 | the class is 3/4 modulo integral gauge. Choosing that representative
000092 | requires its actual integer gauge/sea/reference transport. Omitting the
000093 | connection transformation leaves the unwanted term -2alpha gamma relative
000094 | to the claimed same-A h(-bar(Phi)).
000095 | 
000096 | Equivalently in the twisted presentation A=-i partial_x,
000097 | 
000098 |  Dom A_q={f in H1: f(2pi)=M_q f(0)},
000099 | 
000100 | Rx maps the domain to the inverse twist. With a genuine local unitary
000101 | spin/source frame map H(x), put T=H(x)K_F Rx. Then exactly
000102 | 
000103 |  M_p=H(2pi)K_F M_q^-1 K_F^-1 H(0)^-1.
000104 | 
000105 | This is an H1 trace identity. The endpoints and the original spin phase
000106 | must be retained. Similarly a scalar clutch u(x)=exp(ix) becomes u^-1;
000107 | general gluing is conjugated and pulled back by the same endpoint maps.
000108 | The unitary image domain is the actual selfadjoint domain of the transported
000109 | operator, not a new arbitrary extension.
000110 | 
000111 | 4. THE GLOBAL q/p SPIN MAP, WITHOUT DISCARDING ITS PHASES
000112 | 
000113 | The old six-line completion and source-selected S0 already give the
000114 | holomorphic map R=sc^2, covering z->1/z and exchanging q,p. On the seam
000115 | x->-x. Its actual normalized coefficient lift and spin lift are available.
000116 | The spin lift r has r^6=-I and R^2=c_geo. A constant Rx chart must NOT
000117 | be used to conclude that the full global source lift has square one.
000118 | Its local H(x), deck, frame transition and spin factors carry the missing
000119 | iterations and endpoint phases.
000120 | 
000121 | For a local unitary source map H commuting with the displayed finite grading,
000122 | the complete covariant formulas are
000123 | 
000124 |  A_p=-T A_q T^-1,
000125 |  Phi_p=-T Phi_q T^-1,
000126 |  T=H(x)K_F Rx.
000127 | 
000128 | If A_q=-i partial_x+a_q(x), these read
000129 | 
000130 |  a_p=i H' H^-1-H K_F a_q(-x)K_F^-1 H^-1,
000131 |  Phi_p=-H K_F Phi_q(-x)K_F^-1 H^-1.
000132 | 
000133 | They follow from the differential operator, not from declaring a scalar
000134 | phase harmless. The derivative term is required in a nonconstant frame.
000135 | Unitarity here refers to the existing compared source metrics, or to
000136 | their actual pullback under the source isomorphism. An arbitrary GL frame
000137 | change alone would not justify the unitary spectral-calculus statement.
000138 | When H includes the actual family or determinant actions, extracting
000139 | Y_p and MR_p from this full matrix equation retains their different
000140 | endpoint/congruence types. The simple negative-conjugate control formula
000141 | belongs to the constant compared frame of section 2.
000142 | 
000143 | The existing global q/p spin and sign maps prove transport of those
000144 | bundles and their pulled flat structures. They do not on their own prove
000145 | that the independently supplied normal NS/quarter connection and all96
000146 | charged controls are this same associated global source. The whole96
000147 | wedge dictionary supplies the finite representation comparison; its
000148 | global determinant powers and source-connection map must also be used.
000149 | This is the exact location of the source identification, not a prohibition
000150 | on the already proved q/p open transport.
000151 | 
000152 | 5. WHOLE F5 LIFT, GIVEN GRAPH AND REFLECTION DATA
000153 | 
000154 | Retain the actual normal matrices
000155 | 
000156 |  Z=diag(1,-1), N=[[0,-1],[1,0]], S=NZ,
000157 |  Gamma=Z tensor gamma, Ical=i Gamma, C0=i N tensor I96,
000158 |  B_q=Z tensor A_q-S tensor Phi_q,
000159 |  E v=(P_gamma,+ v,i P_gamma,- v).
000160 | 
000161 | The correct whole constant-frame operator is
000162 | 
000163 |  V=(i N) tensor K_F Rx.
000164 | 
000165 | It satisfies
000166 | 
000167 |  V E=E T, V Gamma V^-1=Gamma,
000168 |  V Ical V^-1=Ical, V C0 V^-1=C0,
000169 |  V B_q V^-1=B_p.
000170 | 
000171 | Indeed iN reverses Z and S, while Rx reverses A and K_F reverses gamma.
000172 | Both grading signs are needed. K_F alone would not be the claimed map
000173 | of the positive total-grade charged chart. With H it is the corresponding
000174 | source-associated extension, retaining the same pointwise relations.
000175 | 
000176 | Unitary spectral calculus then transports the GIVEN full two-face graph
000177 | G(B_q) to G(B_p), and the charged-plus-dual chart transports with it.
000178 | The supplied quasifree/CAR state, if this is its original graph functor,
000179 | is pushed forward rather than selected anew. The supplied time group
000180 | likewise transports; reflection, face orientations, reference and sewing