TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/PAIRED-LOGIC-REVIEW.txt Original file SHA-256: e374b53fa0f6b7f2a0f91b2fa4d12a74d083f5215322a9a80e3bc8c3e93d66cb 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 REVIEW: WHOLE TWO-FACE DYSON / POLAR-GRAPH SOURCE 000002 | 2 November2026. relational_logic. Analytical review of the parent's proposed 000003 | paired construction. No canonical write, new state, metric or counterphase. 000004 | 000005 | VERDICT 000006 | 000007 | The paired construction is mathematically correct under its stated original 000008 | operator/graph hypotheses. It completes the whole smooth Dyson reader on a 000009 | fixed positive two-face graph, retains all crossing blocks, gives a positive 000010 | canonical Fredholm amplitude and a positive-definite kernel on histories. 000011 | The polar transport extends this paired reader to the existing closed, 000012 | injective dense-range normal-profile graph class. Its polar unitary is 000013 | uniquely derived from that same K. 000014 | 000015 | This is a paired natural-cone / adjoint source, not the one-face complex 000016 | Dyson amplitude. The mathematical extension does not by itself prove that 000017 | the original full S_adm insertion uses the polar-transported paired operation. 000018 | In particular locality of V_K W V_K*, field variations of V_K and the original 000019 | absolute Z0/eta scalar require their actual source dictionary. These limits 000020 | are part of the statement, not a new universal TFPT requirement. 000021 | 000022 | 1. SMOOTH DYSON DIFFERENCE IS HILBERT-SCHMIDT 000023 | 000024 | Use the same original fixed circle h=gamma D0+V on periodic H1, with constant 000025 | gamma, bounded Hermitian V, and the original Hermitian odd test F(t,theta). 000026 | Assume the displayed integrable spatial/time derivatives and compact time 000027 | support. In the original interaction picture write 000028 | F_I(t)=exp(ith)F(t)exp(-ith), 000029 | W'(t)=-iF_I(t)W(t), W(-infinity)=I, 000030 | A=int F_I(t)W(t)dt, so W-I=-iA. 000031 | No replacement of this W by exp(-i int F_I) is made. 000032 | 000033 | Spatial W1,infty regularity makes F(t) preserve H1. The anticommutator 000034 | {h,F_I}=B_I, B=-i gamma partial_thetaF+{V,F} 000035 | gives an integrable graph-norm bound. The Dyson equation therefore preserves 000036 | Dom(h)=H1; it can first be solved on that Banach graph space and identified 000037 | with the bounded Hilbert-space unitary solution. Strong continuity and uniform 000038 | graph bounds suffice; operator-norm continuity of F_I is not assumed. 000039 | 000040 | Differentiating F_I W as a weak Dom(h) form gives 000041 | d_t(F_IW)=[2i hF_I-iB_I+(partial_tF)_I-iF_I²]W. 000042 | The boundary term vanishes. Consequently 000043 | 2hA=int [B_I+i(partial_tF)_I+F_I²]W dt. (1) 000044 | The sign of both i partial_tF and +F_I² is correct. Selfadjointness and 000045 | closedness of h turn the weak form identity into A(H) subset Dom(h), exactly 000046 | as in the independent review of D. The right side is bounded. 000047 | 000048 | With a0=int||F||, a1=int||partial_thetaF||, at=int||partial_tF||, 000049 | a2=int||F||² and M=||V||, 000050 | ||A||<=a0, 000051 | ||hA||<=(a1+2M a0+at+a2)/2, 000052 | ||A||2<=||(D0+i)^-1||2[(1+2M)a0+(a1+at+a2)/2]. (2) 000053 | Thus W-I is S2. This is a genuine smooth-Dyson step beyond D's linear 000054 | smeared exponential. No positive spectral gap or momentum factorization is 000055 | used. Zero modes are harmless. The resolvent argument is dimension-specific 000056 | to the given circle; it is not an automatic3+1 estimate. 000057 | 000058 | A is NOT generally selfadjoint. It is normal, because A=i(W-I) and W is 000059 | unitary. This is the necessary replacement for Hermiticity when transferring 000060 | the finite-cutoff proof: normality makes its left and right HS tail norms 000061 | equal. The same finite Galerkin anticommutator estimate, strong Dyson 000062 | convergence and resolvent tails therefore yield W_N-I -> W-I in S2. 000063 | The finite projection boundary defect must be bounded as in D, not set to0. 000064 | 000065 | 2. THE ACTUAL FINITE PAIRED AMPLITUDE 000066 | 000067 | Let K=exp(-ell h)>0 and P=P_Graph(K). At finite compatible regulator, for a 000068 | unitary W set U=diag(W,W). The full graph Slater expectation is det(S), 000069 | S=PUP restricted to Ran(P). 000070 | The reference-centered lift is 000071 | pi(W)=det(W)^-1 Gamma(diag(W,W)). 000072 | It is a genuine unitary representation: the scalar is an actual determinant 000073 | character, not a separately selected phase at each W. 000074 | 000075 | In an orthonormal graph frame, 000076 | det(S)=det(W+K W K)/det(I+K²). 000077 | Therefore 000078 | chi(W)=det(W)^-1 det(S) 000079 | =det(I+W* K W K)/det(I+K²) 000080 | =det(I+W* D W D)/det(I+D²), D=K^-1. (3) 000081 | Both equivalent finite formulas are correct. In particular the parent's 000082 | D=exp(+ell h) formula has the right sign and order. 000083 | 000084 | The numerator is positive: W*KW and K are positive, and their product is 000085 | similar to the positive operator K^(1/2)W*KW K^(1/2). Equivalently this is 000086 | the finite adjoint/natural-cone overlap from the same trace pairing. 000087 | No infinite determinant of unbounded D is assigned by(3); the full limit 000088 | is instead the bounded graph formula below. 000089 | 000090 | For the ACTUAL odd Higgs Dyson source, Tr F_N(t)=0, because the cutoff 000091 | commutes with gamma and F_N is odd. Hence det W_N(t)=1 by its Dyson ODE. 000092 | The centering character in(3) is then already identically1. This strengthens 000093 | the provenance: no new finite scalar is added to that actual paired source. 000094 | For more general W the determinant character remains explicit reference 000095 | normal ordering. 000096 | 000097 | 3. FREDHOLM FORMULA, NONVANISHING AND UNBOUNDED K DOMAIN 000098 | 000099 | Put B=(I-P)UP. Since U-I is S2 and (I-P)P=0, B is S2. On Ran(P), 000100 | S*S=I-B*B. (4) 000101 | Moreover S=I+P(U-I)P is a compact perturbation of identity and has index0. 000102 | 000103 | Its injectivity holds for an arbitrary positive selfadjoint closed K, even 000104 | unbounded. If (x,Kx) in Graph(K) lies in ker(PUP), then U(x,Kx) belongs to 000105 | the orthogonal complement Graph(K)^perp. Thus there is y in Dom(K) such that 000106 | Wx=-Ky, y=WKx. 000107 | Taking the scalar product with y gives 000108 | +=0. 000109 | Both terms are nonnegative. Therefore Kx=0, then y=0 and Wx=0, hence x=0. 000110 | This uses only the domains forced by graph orthogonality. W need not preserve 000111 | Dom(K) in advance. Compact-Fredholm index0 now gives invertibility of S. 000112 | Consequently ||B||<1 and 000113 | chi(W)^2=det_F(I-B*B), 000114 | chi(W)=the strictly positive square root. (5) 000115 | The determinant is nonzero: B*B is trace class, its norm is below1 and its 000116 | eigenvalue sum is finite. Positivity of K, not a false boundedness of K, 000117 | justifies the canonical sign. 000118 | 000119 | 4. POSITIVE-DEFINITE HISTORY KERNEL 000120 | 000121 | Pointwise positivity of(5) would not alone prove positive definiteness. 000122 | The finite representation pi(W) supplies the missing exact argument: 000123 | chi_N(W_i*W_j)=. 000124 | These matrices are positive semidefinite for every finite list. 000125 | The limit therefore preserves a positive-definite kernel. 000126 | 000127 | For actual source histories F_i the kernel is 000128 | K_ij=chi(W[F_i]* W[F_j]), 000129 | with the exact original orders. It is NOT generally chi(W[F_j-F_i]), since 000130 | the Dyson map is not additive. This convention is essential for retaining 000131 | the entire cross-history information. 000132 | 000133 | All face-diagonal and crossing terms remain in P and B. Replacing P by its 000134 | two diagonal marginal contractions would change(4) and the kernel. 000135 | 000136 | 5. CUTOFF CONVERGENCE AND MIXED CAR WORDS 000137 | 000138 | Let the finite compatible graph projections P_N converge strongly to P and 000139 | U_N-I converge to U-I in S2. Finite regulators are embedded into the same 000140 | one-particle space, with U_N identity outside the cutoff. A compatible 000141 | outside reference extension of the finite graph does not affect its finite 000142 | amplitude or the resulting strong limit. Then 000143 | B_N=(I-P_N)(U_N-I)P_N -> B in S2, 000144 | B_N*B_N -> B*B in S1. 000145 | Fredholm determinant continuity and the positive finite sign give exactly 000146 | chi_N->chi. This requires no independent phase limit for det W_N or det S_N. 000147 | Only their actual centered whole expression is taken to the limit. 000148 | 000149 | The construction can also carry mixed bounded CAR insertions, rather than 000150 | only an isolated group character. Since ||B||<1 and B_N->B in norm, the 000151 | compressed S_N are uniformly invertible on Ran(P_N). Extend them by identity 000152 | on its orthogonal complement. The extensions converge strongly to S and 000153 | their inverses converge strongly as well. 000154 | 000155 | The finite Slater transition Wick formula uses the transition density 000156 | Q_N=U_N P_N S_N^-1 P_N 000157 | for the convention with CAR insertions to the left of Gamma(U_N). Its 000158 | matrix entries converge strongly. All finite inserted determinants/minors 000159 | therefore converge together with the scalar chi_N. Other insertion orders 000160 | are transported by the same CAR automorphism, with the actual adjoints. 000161 | The full finite mixed-word Gram kernels retain positivity in the limit and 000162 | restrict at U=I to the original CAR covariance P. This is the relevant 000163 | whole-source kernel statement. Merely taking a square root in(5) would 000164 | not establish these mixed words. 000165 | 000166 | The standard implementability criterion is available because [P,U] is HS. 000167 | The transition formula supplies its phase with positive overlap. The finite 000168 | representation and full mixed kernels preserve the group action, rather 000169 | than assigning unrelated projective implementers to individual histories. 000170 | Any stronger statement about unbounded bosonic source moments still uses 000171 | their actual original integrability; it is not implied by bounded CAR words. 000172 | 000173 | 6. DERIVED POLAR COMPLETION OF THE EXISTING NORMAL GRAPH 000174 | 000175 | Let K be a closed densely defined injective operator with dense range from 000176 | the existing normal-profile transfer/Cauchy graph. Its polar decomposition 000177 | K=V_K |K| 000178 | has a UNIQUE unitary V_K, because initial and final polar spaces are both 000179 | whole one-particle spaces. The derived unitary 000180 | M=diag(I,V_K*) 000181 | sends Graph(K) exactly to Graph(|K|), on the same Dom(K). 000182 | Transport the positive-graph operation back: 000183 | U_K=M*diag(W,W)M=diag(W,V_K W V_K*). (6) 000184 | This retains the whole original K and every crossing block. It does not 000185 | replace K by an averaged exponential or choose a new metric/state. 000186 | The Fredholm kernel and positivity follow by this unitary conjugacy. 000187 | W-I in S2 implies U_K-I in S2 with the same ideal control. 000188 | 000189 | There is also no separate polar cutoff-continuity assumption in this graph 000190 | class. If P_Graph(K_N)->P_Graph(K) strongly, the bounded off-diagonal blocks 000191 | T_N=K_N(I+K_N*K_N)^-1 -> T=K(I+K*K)^-1 000192 | converge strongly together with adjoints. Thus |T_N|->|T| strongly. 000193 | Polar(T_N)=V_N and Polar(T)=V_K. The identity 000194 | V_N |T_N|=T_N 000195 | shows V_N |T|psi -> V_K |T|psi. Ran(|T|) is dense because K is injective 000196 | with dense range. Boundedness of the unitaries gives V_N->V_K strongly; 000197 | the unitary limit also gives strong adjoint convergence. Consequently 000198 | V_N(W_N-I)V_N* -> V_K(W-I)V_K* in S2. 000199 | This justifies the corresponding actual finite polar operation limit. 000200 |