TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/E0-TRANSPORTED-QUARTER-ORIGINAL-SEED-REVIEW.txt Original file SHA-256: f51a35b26e1e9569115f81ce6fbf795448f35957872dbbfd3f21e34febd9fce4 Inclusive original lines: 1-187 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 | E0, TRANSPORTED QUARTER AND THE ACTUAL ORIGINAL SEED REQUIREMENT 000002 | Independent bounded original-source and operator review, 2026-10-29. 000003 | No new source class, gauge calibration, Atlas or old-source mutation. 000004 | 000005 | RESULT 000006 | The original operational-seed definition does NOT separately fix the v117 000007 | coefficient quarter. Therefore failure to centralize that matrix is not a 000008 | universal exclusion of a valid Seed or local P1 image. The checked original 000009 | collar/seam construction supplies no explicit map reconstructing this 000010 | particular flavor quarter from [u_Sigma] or D_coll. Such a map must not be 000011 | introduced as a new physical-time requirement. 000012 | 000013 | For a claim identifying the FULL originally marked flavor system, however, 000014 | the issue is concrete: left composition F_q E0 leaves a vertical sign in 000015 | the original E0/Q relation. It does not repair that fixed relation. A local 000016 | image can ignore this sign only when its adjoint action is trivial on that 000017 | whole image. Ordinary even intergeneration CAR bilinears already detect it. 000018 | Simultaneous transport of every source structure is a different operation 000019 | and preserves their relations by conjugation. 000020 | 000021 | 1. WHAT Seedop ACTUALLY CONTAINS 000022 | 000023 | Direct original read: 000024 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000025 | sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex: 000026 | 103-143. Its object is 000027 | 000028 | S=(A_loc,tau_t,Theta,omega,[u_Sigma],D_coll). 000029 | 000030 | Conditions111-117 require the local observable net and positive-time 000031 | reflection, reflection positivity and clustering, finite relative action 000032 | against the DECLARED reference, a nontrivial seam class, and a reflection- 000033 | regular first-order elliptic collar. Line119 makes morphisms unitary 000034 | intertwiners preserving the displayed structures. Q_v117 is not a separate 000035 | entry. A chosen matrix basis is not frozen by this categorical definition. 000036 | 000037 | Lines139-143 identify the reconstructed boundary operator B_Sigma from 000038 | D_coll, and the seam spectral-flow class from [u_Sigma]. They do not give 000039 | a formula for the particular v117 coefficient matrix U or A_q. 000040 | 000041 | Direct original source01 section001:42-80 likewise starts with the actual 000042 | one-sided data and general selfadjoint B_Sigma in 000043 | D_+=gamma_n(partial_n+B_Sigma). It derives the double, seam, relative/ 000044 | reference split and primitive seam response from that data. No displayed 000045 | identification with the rank3 v117 flat coefficient system occurs there. 000046 | 000047 | This is a scoped source conclusion, not a claim that no future/full 000048 | construction could reconstruct a quarter from a particular D_coll. 000049 | 000050 | 2. THE CONCRETE FLAVOR MARK IS REAL DATA OF ANOTHER ORIGINAL CONSTRUCTION 000051 | 000052 | The September28 original section004:203-289 fixes the actual Fuchs 000053 | system, U=diag(1,i,-i), Psi(0)=I, the marked triangle pair and its 000054 | genus2 coefficient cover. In particular255-259 fixes its positive 000055 | quarter marking; it is not just the number/order4. Later in that same 000056 | section the original explicitly warns that the point intertwiner is 000057 | neither a field intertwiner nor a derivation of the original seam projector. 000058 | 000059 | The old fullv21 section001:225-247 instead specifies the scalar seam 000060 | U_Gamma=e^(i theta) and a unit spectral-flow crossing. These are genuine 000061 | original data, but an integer winding and a rank3 flavor lift are different 000062 | types. No displayed source map identifies them in the P1 local cyclic image. 000063 | 000064 | The existing common K construction gives adapted native kappa,sigma0,c0; 000065 | F_q preserves those sections exactly. This removes that particular K-clock 000066 | obstruction. It does not mean that the Seed definition reconstructs those 000067 | clocks instead of Q, or that any of them has been identified with tau_t. 000068 | The appropriate original field/domain map remains the relevant comparison. 000069 | 000070 | 3. TEST THE COMPLETE COMPOSITION, NOT F_q IN ISOLATION 000071 | 000072 | Use the actual G coefficient order (L0,L2,L1), with 000073 | 000074 | K=[[0,1,0],[-1,0,0],[0,0,1]], S=diag(1,-1,1), 000075 | J=D_E C, D_E=omega^2 S, W=omega S. 000076 | 000077 | Here C is coefficient conjugation, while the FULL J additionally covers 000078 | E0=Q tau0 on the curve. Its base relation is 000079 | 000080 | E0 Q E0=Q^-1. 000081 | 000082 | These matrices use the existing section-pullback/right-frame convention. 000083 | The corresponding forward fiber quarter has inverse coefficient K^-1. 000084 | Replacing K by K^-1 consistently leaves all inversion tests and the 000085 | residual vertical D_Q=K^2 below unchanged; the base map must still be 000086 | included, rather than inferred from the matrix inverse alone. 000087 | 000088 | Since K is real and S K S=K^-1, the original coefficient relation 000089 | J K J^-1=K^-1 agrees with that full base relation. 000090 | 000091 | The SAME left composition used to repair real finite Phi is instead 000092 | 000093 | Jprime=W J=C 000094 | 000095 | on these horizontal family coefficients, with the base STILL E0. 000096 | It is an involution, but its coefficient conjugation now fixes K. Thus 000097 | as FULL bundle maps over their actual base maps, 000098 | 000099 | Jprime Qlift Jprime^-1=D_Q Qlift^-1, 000100 | D_Q=diag(-1,-1,+1) on the G triplet. 000101 | 000102 | D_Q is a parallel VERTICAL automorphism with identity base. It is not the 000103 | full Qlift^2, whose base is the nontrivial hyperelliptic map Q^2. In the 000104 | singlet-first family order (O,L0,L2,L1), D_Q=diag(1,-1,-1,1). 000105 | This is the exact residual sign, not a guessed inversion of native kappa. 000106 | Extra existing gamma/Ical factors in the whole F5 antiunitary commute 000107 | with this family matrix test and do not erase the residual sign. 000108 | 000109 | 4. GENUINE SIMULTANEOUS SOURCE TRANSPORT IS DIFFERENT 000110 | 000111 | Let Qnew=f Qlift f^-1 for the genuine vertical K extension f of W. 000112 | It has base Q and inverse coefficient K^-1. Its FULL inverse has base 000113 | Q^-1; this base distinction is retained in all equations above. 000114 | The left-composed Jprime interchanges the two inverse marks: 000115 | 000116 | Jprime Qlift Jprime^-1=Qnew^-1, 000117 | Jprime Qnew Jprime^-1=Qlift^-1. 000118 | 000119 | It therefore transports a marking orbit rather than preserving either 000120 | one as the original fixed reflected quarter by itself. 000121 | 000122 | In contrast, a genuine common equivalence transports EVERY displayed 000123 | structure by one unitary f, including 000124 | Qnew=f Qlift f^-1 and Jnew=f J f^-1, as well as the algebra, state, 000125 | time, collar, seam and actual reference/domain prescription. Then 000126 | Jnew Qnew Jnew^-1=Qnew^-1 follows immediately by conjugation. 000127 | This is not the formula Jprime=f J and must not be substituted for it. 000128 | 000129 | The Seedop definition permits such intertwiners when these conditions 000130 | actually hold. It does not decree every parallel K bundle automorphism a 000131 | physical gauge equivalence of a fixed Phi and reference. The previous 000132 | same-Phi identity for real controls and the simultaneous transport theorem 000133 | have different hypotheses; neither alone supplies the missing original map. 000134 | 000135 | 5. LOCAL CYCLIC IMAGES NEED LESS THAN GLOBAL EQUIVALENCE 000136 | 000137 | Original source01 section001:319-342 defines the physical local net by 000138 | restriction to an invariant admissible subspace, requiring invariance 000139 | under each image operator AND its adjoint. Hence a smaller local cyclic 000140 | image is explicitly allowed. A global onto equivalence of every ambient 000141 | charged field is stronger than the local mechanism requires. 000142 | 000143 | The source-reviewed PAPER-COLLAR-REFERENCE.txt, read through Atlas sources 000144 | research-source:a0a02c8724ef5d21a8a66f63 and 000145 | research-source:17eb50b8cbc171310fc470fd, spells out the actual sufficient 000146 | local test: an injective local star map preserving state, time, reflection 000147 | and the specified seam on its cyclic image. Complete Seed admissibility 000148 | additionally needs the actual restricted elliptic collar and finite relative 000149 | action with its declared reference and eta/domain data. These restrictions 000150 | are not replaced by compression, a determinant character or a bundle map. 000151 | 000152 | Consequently a Q-intertwining condition is required on that image IF the 000153 | claimed source identification includes this Q action or its actual 000154 | reconstruction from the displayed Seed structures. It is not an extra 000155 | universal condition for every local P1 mechanism. The checked originals 000156 | do not establish the needed flavor-Q-to-Seed identification. 000157 | 000158 | 6. SMALLEST CONDITIONAL IMAGE TEST 000159 | 000160 | If the full marked E0/Q relation is claimed on an image, the residual 000161 | Ad(D_Q) must be the identity there. This can hold on a smaller family- 000162 | invariant algebra, and cannot be ruled out merely by a global commutator. 000163 | It fails on the whole even three-generation CAR algebra already by a 000164 | bounded local word. Choose the same carrier/smearing in generations0,2: 000165 | 000166 | A=a0^* a2+a2^* a0, A not zero, A even, 000167 | Ad(D_Q)(A)=-A. 000168 | 000169 | Thus a faithful local image containing these intergeneration bilinears 000170 | cannot forget the residual sign. An image generated only by a different 000171 | neutral/family-invariant composite must be checked on its own generating 000172 | words. It need not inherit this exclusion. This calculation uses the 000173 | already given bounded CAR algebra, not arbitrary left multiplication on 000174 | an OS quotient. 000175 | 000176 | 7. PRECISE NEXT ORIGINAL STEP 000177 | 000178 | There is no source-backed universal fixed-Q veto. The positive K/E8 000179 | compensator and its preserved adapted native clocks remain available. 000180 | For the requested P1 join, exhibit the actual local generating image and 000181 | its original reflection/seam/collar/reference action. If that image is 000182 | claimed to carry the v117 marked quarter, apply the explicit Ad(D_Q) 000183 | test above; otherwise do not add flavor Q as a new physical-time axiom. 000184 | 000185 | This locates the first real decision at the source map already required by 000186 | the originals. It introduces neither a new source family nor a demand for 000187 | an additional state, clock, physical time or independent Berezin measure.