TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/PAIRED-REVIEW.txt Original file SHA-256: f51a4d21315fac64cf42e2edceea20073b78f86b74e5e04e5bf66a5236280867 Inclusive original lines: 1-200 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 | INDEPENDENT PROOF REVIEW: TRUE DYSON EVOLUTION AND FULL GRAPH PAIRING 000002 | 2 November 2026. New application theorem on the given original source. 000003 | 000004 | Verdict 000005 | 000006 | The proposed positive paired-source theorem is mathematically correct with 000007 | the domain, prefix-endpoint and centering details written below. The true 000008 | one-particle Dyson evolution differs from identity by S2. Its actual full 000009 | two-face graph admits a canonical strictly positive paired Fredholm kernel. 000010 | The positive answer is not obtained by fitting a scalar counterphase. 000011 | 000012 | The polar extension is also correct: for an actual closed injective dense- 000013 | range transfer K the companion on the second face is V_K W V_K^dagger, 000014 | not a second literal W in an unrelated face chart. V_K is fixed by K. 000015 | This is uniqueness relative to that specified polar transport rule, not a 000016 | claim that every positive physical operation has this form. 000017 | 000018 | These are genuine full-graph statements with crossings retained. They do 000019 | not assert that the unpaired Dyson vacuum amplitude, physical OS time, 000020 | absolute Z_rel[0], or the full primitive S/Z field map has been derived. 000021 | 000022 | 1. Given one-particle source and the true time-ordered unitary 000023 | 000024 | Use the same D-family hypotheses independently reviewed in 000025 | work/source_continuation/VARIABLE-REVIEW.txt: 000026 | h=gamma D0+V, D0=-i partial_theta+alpha, alpha1/4 or3/4, 000027 | gamma=gamma*, gamma^2=I, V=V*, ||V||<=M, 000028 | Dom(h)=H1 on the original compact circle and finite internal space. 000029 | F(t,theta)=F*=sum f_a Ta is the actual smooth odd insertion, 000030 | {gamma,F}=0, compactly supported in time, with the existing compatible 000031 | twist/domain. V is time independent, possibly spatially variable. 000032 | No new geometry, coupling, state or physical time is selected. 000033 | 000034 | Set V_I(t)=exp(ith)F(t)exp(-ith), W(a)=I, and 000035 | W'(t)=-i V_I(t) W(t). 000036 | This is the actual Dyson unitary. It is not replaced by exp(-i A_F). 000037 | For t in[a,b] put A_(a,b)=int_a^b V_I(t)W(t)dt. Then exactly 000038 | W(b)-I=-i A_(a,b). (1) 000039 | 000040 | 2. Domain invariant Dyson prefixes and the bounded hA identity 000041 | 000042 | Multiplication by smooth F(t) maps H1 boundedly into itself. The graph 000043 | norms of h and D0 are equivalent by the bounded V perturbation. exp(ith) 000044 | preserves Dom(h) and is isometric in its h graph norm. Consequently V_I(t) 000045 | is a bounded operator on the graph-norm Banach space H1, strongly continuous 000046 | there with an integrable norm on every compact time interval. The Dyson 000047 | construction on H1 therefore proves W(t)H1 subset H1, with its inverse 000048 | preserving the same domain. This supplies the domain needed when h passes 000049 | through W(t) in the following weak-form calculation. 000050 | 000051 | Reuse the exact identity 000052 | {h,V_I(t)}=B_I(t), 000053 | B(t)=-i gamma partial_theta F(t)+{V,F(t)}, 000054 | B_I(t)=exp(ith)B(t)exp(-ith). 000055 | Also V_I'(t)=i[h,V_I(t)]+dotF_I(t) as forms on H1. Therefore 000056 | [V_I(t)W(t)]_a^b 000057 | =i int_a^b [h,V_I(t)]W(t)dt 000058 | +int_a^b dotF_I(t)W(t)dt-i int_a^b V_I(t)^2 W(t)dt. 000059 | Adding the integrated anticommutator gives the correct PREFIX identity 000060 | 2h A_(a,b) 000061 | =int_a^b [B_I+i dotF_I+V_I^2]W dt 000062 | -i [V_I(t)W(t)]_a^b. (2) 000063 | For a full compact pulse F(a)=F(b)=0 the last term vanishes, giving exactly 000064 | the identity proposed by root. For a prefix ending inside the pulse it 000065 | must be retained. It is bounded and does not obstruct the proof. 000066 | 000067 | First interpret(2) between H1 vectors. The right side is bounded in the 000068 | Hilbert norm. Selfadjointness of h gives A_(a,b)H1 subset Dom(h), and 000069 | closedness of h extends the range inclusion to ALL Hilbert vectors. 000070 | Thus(2) is a bounded operator identity, not merely a formal manipulation 000071 | of two undefined unbounded products. 000072 | 000073 | Let f0=int||F||dt, ftheta=int||partial_theta F||dt, 000074 | ft=int||partial_t F||dt, f2=int||F||^2dt and 000075 | e_ab=||F(a)||+||F(b)||. Unitarity gives ||A||<=f0, and 000076 | ||hA||<=(ftheta+2M f0+ft+f2+e_ab)/2, 000077 | ||D0 A||<= (ftheta+ft+f2+e_ab)/2+2M f0. 000078 | Hence, with the SAME Hilbert-Schmidt circle resolvent r_alpha, 000079 | ||A||_2<=r_alpha[(ftheta+ft+f2+e_ab)/2+(2M+1)f0], 000080 | r_alpha=||(D0+i)^-1||_2W strongly. Define the embedded W_N as identity 000091 | off P_N. Then A_N=i(W_N-I) is NORMAL. It need not be selfadjoint. 000092 | 000093 | The finite analogue of(2) has possible Galerkin boundary defects relative 000094 | to P_N B P_N but satisfies the SAME uniform bound(3). Therefore 000095 | ||(I-P_K)A_N||_2 000096 | <=||(I-P_K)(D0+i)^-1||_2 sup_N||(D0+i)A_N|| ->0 000097 | uniformly in N. Normality gives the exact equality 000098 | ||A_N(I-P_K)||_2=||(I-P_K)A_N||_2, 000099 | because A_N A_N^dagger=A_N^dagger A_N. This is the correct replacement 000100 | for the selfadjointness argument used for a linear smeared insertion. 000101 | Strong convergence on fixed finite corners and both tails prove 000102 | W_N-I -> W-I in S2. (4) 000103 | 000104 | 4. Full positive graph, not independent thermal faces 000105 | 000106 | Let K=exp(-ell h), ell>0 the GIVEN parameter. This positive selfadjoint 000107 | operator is densely defined, closed, injective and has dense range; it 000108 | need not be bounded or boundedly invertible. Let P be the orthogonal 000109 | projection onto graph(K) in H direct_sum H. Its bounded functional-calculus 000110 | form is the original full crossing-retaining graph, with top block 000111 | C=(I+K^2)^-1 and offdiagonal K(I+K^2)^-1. No face is discarded. 000112 | 000113 | Set U=diag(W,W), and, as an operator on Ran(P), 000114 | T=P U P|Ran(P), B=(I-P)U P. 000115 | Since U-I belongs to S2, B belongs to S2 and T-I_Ran(P) is compact. 000116 | Thus T is Fredholm of index0. The following elementary positive-graph 000117 | argument proves its kernel is0 WITHOUT assuming W preserves Dom(K). 000118 | 000119 | If T(x,Kx)=0, then U(x,Kx) belongs to graph(K)^perp. There exists 000120 | y in Dom(K) such that 000121 | Wx=-Ky, WKx=y. 000122 | Since x,y are both in Dom(K), 000123 | ===-. 000124 | The left side is strictly positive for x!=0, while the right side is 000125 | nonpositive. Therefore x=0, and the vector is0. Index0 now makes T 000126 | bijective with bounded inverse. In particular 000127 | T^dagger T=I_Ran(P)-B^dagger B is boundedly positive, 000128 | ||B||<1, and 000129 | det_F(I-B^dagger B)>0. (5) 000130 | This proves STRICT positivity, not just nonnegativity of a possibly 000131 | vanishing determinant. No artificial spectral lower bound on K was used. 000132 | 000133 | 5. Exact finite formula, centering and canonical infinite kernel 000134 | 000135 | In finite dimension write D=K^-1=exp(ell h). An isometry onto graph(K) 000136 | is Z=[I;K](I+K^2)^-1/2. Direct determinant algebra gives 000137 | det(Z^dagger U Z) 000138 | =det(W) det[I+W^dagger K W K]/det[I+K^2]. 000139 | The positive ratio equals 000140 | chi(W)=det[I+W^dagger D W D]/det[I+D^2]>0. (6) 000141 | The equality of the K and D ratios follows by inversion/cyclic determinant 000142 | identities. Positivity follows because (W^dagger D W)D is similar to the 000143 | strictly positive operator D^1/2 W^dagger D W D^1/2. 000144 | 000145 | Thus the RAW full Slater overlap contains det(W). The PAIRED CENTERED 000146 | overlap is chi(W), with that exact finite trace centering removed. 000147 | Equivalently its finite-Fock expression is 000148 | Tr[rho^1/2 Gamma(W)^dagger rho^1/2 Gamma(W)], 000149 | rho=Gamma(D^2)/det(I+D^2). 000150 | This is the adjoint-paired coefficient on the given full source. It is 000151 | not a product of two independent face expectation values. For the actual 000152 | odd finite source Tr(F(t))=0, det(W)=exp[-i int Tr(F)] is already1. 000153 | For a more general source centering must be displayed, not silently omitted. 000154 | 000155 | Since U is unitary, 000156 | |det(Z^dagger U Z)|^2=det(I-B^dagger B), 000157 | and |detW|=1. Combining with(6) gives exactly 000158 | chi(W)^2=det(I-B^dagger B). (7) 000159 | For the actual infinite graph define 000160 | chi_P(W)=sqrt(det_F(I-B^dagger B))>0, (8) 000161 | with the positive square root. This fixes the phase of THIS paired kernel 000162 | canonically; there is no fitted scalar counterterm. 000163 | 000164 | For actual Galerkin graphs, the bounded graph projections obtained from 000165 | htilde_N converge strongly to P. Together with(4), this gives B_N->B in 000166 | S2 and B_N^dagger B_N->B^dagger B in S1. Fredholm determinant continuity, 000167 | the strictly positive finite formula and(5) imply chi_N->chi_P. 000168 | Every finite family chi_N(W_i^dagger W_j) is a Gram kernel of the displayed 000169 | adjoint-paired vectors. Its entrywise limit is positive semidefinite too. 000170 | This establishes kernel positivity. It is not physical Euclidean OS 000171 | reflection positivity, a positive physical time generator or absolute Z0. 000172 | 000173 | 6. Canonical polar completion for an actual nonsymmetric normal transfer 000174 | 000175 | More generally let the original complete transfer K:H0->H1 be densely 000176 | defined, closed, injective and have dense range. Its polar decomposition 000177 | K=V_K |K| 000178 | has UNITARY V_K:H0->H1, and Dom|K|=DomK. The bounded unitary 000179 | T_K=diag(I,V_K^dagger):H0 direct_sum H1 -> H0 direct_sum H0 000180 | sends graph(K) exactly to graph(|K|). Therefore the source operation 000181 | U_K(W)=diag(W,V_K W V_K^dagger) (9) 000182 | satisfies 000183 | T_K U_K(W) T_K^dagger=diag(W,W). 000184 | All statements(5)-(8) transport unitarily to the actual graph(K), with its 000185 | full crossing blocks. U_K-I remains S2 because W-I is S2. Finite original 000186 | invertible transfers satisfy these hypotheses immediately. 000187 | 000188 | This genuinely corrects the WORD 'same W on both faces' when K is not 000189 | selfadjoint. The second operation is determined by the existing full K; 000190 | choosing an unrelated second W or deleting the polar factor would test a 000191 | different source word. The rule(9) is unique after requiring this specified 000192 | polar identification and the first-face W. It is not a uniqueness theorem 000193 | over every possible positive physical sewing or instrument. 000194 | 000195 | The operation may be spatially nonlocal and field dependent. That does not 000196 | affect the graph theorem and does not make it an already local physical 000197 | gate. If a continuum K is only a formal transfer, its closedness, dense 000198 | domain/range and the identification of its actual graph must be supplied 000199 | by that original transfer contract. The theorem does not infer them from 000200 | a finite numerical matrix. Convergence of a separate K_N construction