TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/TRANSFER-FIELD-SOURCE-PROOF.txt Original file SHA-256: f9f8954dba19fea962d3ef343ef7d664eda1b0e8f7f89a8c9ab3e4074c85b1d8 Inclusive original lines: 1-158 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 | R075. ORIGINAL TRANSFER, MARKED FIELD INSERTIONS AND UNIFORMITY 000002 | 1 November 2026. Scoped source audit, not physical F1 completion. 000003 | 000004 | 1. Decisive question and stop condition 000005 | 000006 | Can the original QFT004 transfer paragraph supply the missing marked field 000007 | source from Z_adm/S_prim, without adding a tensor, measure or geometry? 000008 | Test its first identifications, rather than scan new candidate couplings. 000009 | Success is an actual original decorated transfer, or a precise obstruction 000010 | to the proposed shortcut. Stop the diagnostic class when decided. The 000011 | counterexamples below are implication tests, not proposed TFPT dynamics. 000012 | 000013 | QFT004:137-155 describes T_V as obtained from finite C6 Green kernels, 000014 | diagonal admissibility weights and Padm. Lines302-353 then identify 000015 | Alg(P_V,D_V)=M_n with an injective MPS tensor and a volume-independent 000016 | parent gap. The finite Burnside conclusion itself is valid: a commuting 000017 | diagonal family separating all finitely many basis words generates every 000018 | coordinate projector. A strictly entrywise positive P has every matrix 000019 | unit E_ii P E_jj nonzero. Hence the generated algebra is M_n. 000020 | 000021 | The next identification requires the ACTUAL source-labelled local tensor. 000022 | An ordinary matrix on retained words, a CP map on matrices, and a local 000023 | tensor with a physical field leg are different objects. The paragraph 000024 | does not display their factorization or its supported field dictionary. 000025 | This is a gap in this cited source route, not proof that all original 000026 | sources lack a rule or that their implicit dependencies vanish. 000027 | 000028 | 2. Even a complete undecorated CP transfer does not fix marked responses 000029 | 000030 | Use convention E_O(X)=sum_ij O_ij A_j X A_i*, E=E_I. 000031 | Given the ACTUAL A_i and field matrix O, every finite ordered response is 000032 | obtained by composing these E_O with all intervening E powers and the 000033 | actual boundary/Perron data. No Wick assumption is required. 000034 | 000035 | For an exact implication countertest let the auxiliary space be C2 and 000036 | A_0=I/sqrt(2), A_j=sigma_j/sqrt(6), j=x,y,z. 000037 | Then E(I)=I, E(sigma_j)=sigma_j/3 and the unique stationary density is 000038 | l=I/2. The tensor already spans M2 at length1 and is primitive. 000039 | Set B_0=(A_0+A_x)/sqrt(2), B_x=(A_0-A_x)/sqrt(2), 000040 | B_y=A_y, B_z=A_z. For EVERY X, sum B_i X B_i*=E(X). 000041 | Thus the full transfer, Perron data, normalization and mixing eigenvalues 000042 | are identical. Yet for the SAME marked physical O=|0><0|, 000043 | tr(l E_O^A(I))=1/2, tr(l E_O^B(I))=1/3. 000044 | The difference occurs before higher-word reconstruction is attempted. 000045 | 000046 | The Kraus rotation is a physical-basis rotation. If O and every mark are 000047 | transported with it, the sources are equivalent. Holding O fixed tests 000048 | exactly the missing marked correspondence; it is not a new unmarked 000049 | theory or a pair satisfying the entire original TFPT master data. 000050 | No claim of physical locality, OS positivity or TFPT embedding is made 000051 | for this auxiliary countertest. Its sole conclusion is that E_I alone 000052 | does not determine E_O at fixed physical labels. 000053 | 000054 | 3. Finite positivity and separation do not give uniform transfer mixing 000055 | 000056 | For 0(a<0|+b<1|), A_1=|1>(b<0|+a<1|). 000070 | For 00. 000072 | E is unital and its nonzero eigenvalues are 1 and lambda=1-2e. 000073 | Its stationary density l=[[1/2,ab],[ab,1/2]] is faithful. 000074 | 000075 | On a periodic ring of L sufficiently large, the MPS amplitudes are 000076 | psi(i_1,...,i_L)=product_j sqrt(P_e[i_j,i_(j+1)]). 000077 | The squared norm is tr(P_e^L)=1+lambda^L. Let h be the ORTHOGONAL 000078 | projector onto the complement of the length3 MPS image, and 000079 | H_L(e)=sum_j h_(j,j+1,j+2). This is the canonical range3 parent, 000080 | with local terms norm1, independent of L at each fixed e. Injectivity 000081 | at length2 gives its unique periodic ground line for L>=4: the cited 000082 | Theorem10 requires ring size>=2L0 and interaction length>L0, here L0=2. 000083 | 000084 | All expectations below are divided by ||psi||^2. For Z=diag(1,-1), 000085 | M=sum_j Z_j, the ground state has =0 and 000086 | =(lambda^r+lambda^(L-r))/(1+lambda^L), 000087 | =L (1+lambda)/(1-lambda) (1-lambda^L)/(1+lambda^L). 000088 | These follow by the full periodic Markov weight, not an independent 000089 | site approximation. The nonzero trial vector M psi is orthogonal to 000090 | the unique ground line. Each h_j annihilates psi, and commutes with 000091 | Z outside its three sites. Therefore 000092 | =sum_j ||h_j (Z_j+Z_(j+1)+Z_(j+2)) psi||^2 <=9L. 000093 | The variational principle gives the DIRECT PARENT bound 000094 | gap H_L(e) <= 9 (1-lambda)/(1+lambda) 000095 | * (1+lambda^L)/(1-lambda^L). 000096 | 000097 | At e=e_L=1/(4L), lambda^L tends to exp(-1/2), so this bound tends 000098 | to0 as O(1/L). All bond/physical dimensions, blocking length, local 000099 | interaction range and projector norms stay fixed. Every member is 000100 | injective and has positive classical transfer and separated basis. 000101 | The tensor changes with L. Thus finite-volume injectivity alone does 000102 | not imply a UNIFORM gap over changing tensors, even for canonical 000103 | normalized parents. This does not contradict the fixed-tensor FNW 000104 | gap theorem, and is not a counterexample to a specified TFPT tensor. 000105 | 000106 | 5. Correct original continuation and retained results 000107 | 000108 | A fixed original C6 kernel may contribute a uniform local bound. To 000109 | use it one must derive how the original weights, carrier, selector and 000110 | blocking produce the same local tensor (or controlled varying family) 000111 | on the exhaustion. A bound on one Green matrix alone is not that map. 000112 | The correct sufficient route is the original local tensor plus its 000113 | source-labelled E_O, actual boundary state, support/reflection/time, 000114 | and fixed-tensor or uniform quantitative control. A different original 000115 | constructive field rule could also suffice; MPS is not a new axiom. 000116 | 000117 | This supplies a concrete next source calculation: derive the original 000118 | decorated block T_V[J] from C6/word/carrier weights, then evaluate its 000119 | first derivatives at0 and the same ordered two/four/higher insertions. 000120 | Retain whole neutral words with charged intermediates. Full moments 000121 | come from Z[J]/Z[0], not directly from log Z derivatives. That latter 000122 | repair was already established on25October and is REUSED, not new. 000123 | 000124 | The existing free Green/HS K_F functional remains a complete comparison 000125 | reader. It is not identified with the primitive physical source by 000126 | matching spectra, a Perron line or a positive projection. Neither a 000127 | universal whole-Fock intertwiner nor preservation of old Cauchy G is 000128 | imposed. H is not the signed one-particle B; QTheta is not declared 000129 | CAR parity; Pcenter is not enlarged to a full SU3 singlet projector. 000130 | Preserve original Ref/Eta/H0/Z0, R035/R037/R043, the whole higher source, 000131 | memory, State/Time/RP, E8/clocks and existing Yukawa/flavor/alpha. 000132 | The old ORIGINAL-BLOCKED-TV-WEIGHT-PRODUCER-REVIEW already established 000133 | that penalty-only functions are scalar on their joint kernel and that 000134 | the original paragraph names additional retained-word data. That result, 000135 | and its v330/v175/origin_transfer audit, are reused rather than re-counted. 000136 | Current code-graph readback of v330.symmetric_transfer and run confirms 000137 | its supplied three-state T and diagonal D are another concrete reader, 000138 | not a displayed source-labelled factorization of this QFT004 T_V. 000139 | 000140 | 6. Primary source comparison 000141 | 000142 | Perez-Garcia, Verstraete, Wolf, Cirac, Matrix Product State 000143 | Representations, quant-ph/0608197v2: Sec2.1 Eq3 gives decorated E_S; 000144 | Sec2.2 distinguishes the CP extension with a physical output leg; 000145 | Sec4.1 gives fixed-tensor parent uniqueness; Sec4.2 gap estimates 000146 | depend on the second transfer eigenvalue and blocking constants. 000147 | https://arxiv.org/html/quant-ph/0608197v2 000148 | Fannes, Nachtergaele, Werner, CMP144(1992)443-490: 000149 | https://doi.org/10.1007/BF02099178 000150 | The publisher abstract was read; the full1992 proof was not accessed. 000151 | The explicit counterfamilies and variational inequality are derived 000152 | above, not attributed as numerical results of either reference. 000153 | 000154 | Verdict: the named transfer shortcut does not derive the marked source 000155 | or a uniform parent gap from its stated algebraic criterion alone. 000156 | No actual TFPT gap collapse, new physical field law, 3+1 derivation, 000157 | universal spin2 source, Lean completion or ledger promotion follows. 000158 | complete_TFPT_solution=false; ledger_changed=false.