TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/PAIRED-DYSON-PROOF.txt Original file SHA-256: 556735a9132f328707be6c83315eb068e102ae4c1bf1b68461181cc3d03cdb57 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 | L20261102E. Nonlinear time-ordered completion of the given full graph source 000002 | 2 November2026. Conditional original-source theorem; no physical TOE promotion. 000003 | 000004 | 1. Outcome and actual original objects 000005 | 000006 | L20261102D proves the original local odd insertion A_F is Hilbert-Schmidt 000007 | for bounded spatially variable Higgs backgrounds on the given circle. 000008 | The true time-ordered perturbation is W'=-iF_I(t)W, not exp(-iA_F). 000009 | We now construct its whole paired source with the ORIGINAL coherent two-face 000010 | graph, including its crossing block. No independent face copies or freely 000011 | chosen counterphase are inserted. The general nonselfadjoint normal-profile 000012 | graph has a precise polar-transport extension, stated below. 000013 | 000014 | Keep K0=L2(S1,C^d), D0=-i partialtheta+alpha, alpha1/4 or3/4, 000015 | h=gammaD0+V onH1, gamma Hermitian involution and V bounded Hermitian. 000016 | F(t,theta)=sum f_a(t,theta)Ta is the original Hermitian gamma-odd 000017 | pointwise Higgs oneform. Use smooth compact time support and spatialW1,infty. 000018 | The internal reader is the original reducing quark36, not96 CAR copies. 000019 | All couplings, the given ell and the original h-time are retained. 000020 | 000021 | R080's two faces are a=(1+exp(-2ellh))^-1 and c=1-a. 000022 | a=S_AP(0+) is not automatically the occupation; the original field 000023 | leg/charge chart must determine that order. logodds(a)=-2ellh, 000024 | logodds(c)=+2ellh. D's estimates hold for either covariance, but their 000025 | physical source identities may not silently exchange them. 000026 | 000027 | 2. The D identity extends to the actual Dyson evolution 000028 | 000029 | Put V_I(t)=e^(ith)F(t)e^(-ith) and solve W'=-iV_IW,W(a)=I. 000030 | It is the unique unitary bounded-operator propagator. Each V_I is bounded 000031 | on the graph norm of Domh because F preservesH1 and h-time preservesDomh. 000032 | The same Dyson construction in that Banach norm first proves W(t)Domh 000033 | subsetDomh. No unproved W(Domh) manipulation is used below. 000034 | 000035 | Let A=int_a^b V_I(t)W(t)dt, so W(b)-I=-iA. On Domh, integrate weakly 000036 | d_t(V_IW)=i[h,V_I]W+(partialtF)_I W-iV_I^2 W. 000037 | Together with {h,V_I}=(-i gamma partialthetaF+{V,F})_I=B_I, this gives 000038 | 000039 | 2hA=int_a^b [B_I+i(partialtF)_I+V_I^2]W dt 000040 | -i[V_I(t)W(t)]_a^b. (1) 000041 | 000042 | All terms are bounded. For the complete pulse the boundary term vanishes. 000043 | The same selfadjointness/closedness argument asD proves A(K0)subsetDomh 000044 | and equality(1) as a bounded operator, not merely a formal integration. 000045 | Writing a0=int||F||, a1=int||partialthetaF||, at=int||partialtF||, 000046 | a2=int||F||^2, M=||V||, one obtains for the whole pulse 000047 | 000048 | ||W-I||2=||A||2 000049 | <=||(D0+i)^-1||2 [(1+2M)a0+(a1+at+a2)/2]. (2) 000050 | 000051 | Thus the ACTUAL nonlinear time-ordered evolution belongs to I+S2. 000052 | Prefixes have the displayed finite endpoint correction and the same property. 000053 | No commuting Magnus series, mass gap or independent momentum factorization 000054 | is needed. General scalar-current R066 was already solved by central 000055 | commutators; that result is not re-proved or misapplied to this Higgs tuple. 000056 | 000057 | Actual finite Galerkin propagators converge strongly by the bounded 000058 | Duhamel equation, with their uniformly bounded principal-symbol estimates. 000059 | Equation(1) supplies uniform LEFT S2 tails through the fixed D0 resolvent. 000060 | Now A is not Hermitian, but A=i(W-I) is NORMAL. Hence its right S2 tails 000061 | equal its left S2 tails. Finite corners plus both tails give convergence 000062 | inS2, exactly as required for the source below. Time-mesh integration errors 000063 | in the native diagnostic are kept separate from this spatial limit theorem. 000064 | 000065 | 3. True coherent graph and its positive paired source 000066 | 000067 | First use the stationary original normal transfer K=exp(-ellh), a positive 000068 | closed injective densely defined operator with dense range. Its whole 000069 | orthogonal graph projection P has blocks[a,X;X,c],X=sqrt(a c). 000070 | Let U=diag(W,W) on the two faces, Q=I-P, B=QUP. Then B isS2 because 000071 | QP=0 and U-I isS2. The compression T=PUP|RanP satisfies 000072 | 000073 | T* T=I_RanP-B*B. (3) 000074 | 000075 | At finite cutoff the centered full Slater source is 000076 | chi_N(W)=det(W)^-1 det(E* diag(W,W)E) 000077 | =det(I+W* D W D)/det(I+D^2)>0, D=exp(ellh_N), 000078 | where E=[sqrt(a);sqrt(c)]. Positivity follows because W*DW is positive 000079 | and its product with positiveD is similar to a positive matrix. 000080 | The crossingX is inside E; it is not omitted or replaced by a product of 000081 | two independent mixed-face characteristic functions. 000082 | 000083 | For the actual odd Higgs pulse TrF_N(t)=0. Therefore detW_N=1 identically; 000084 | the displayed centering character is already1. No new phase is subtracted. 000085 | For general finiteW the det character is the explicit reference normal 000086 | ordering, not an unstated convention. 000087 | 000088 | The continuum definition is therefore the positive Fredholm root 000089 | 000090 | chi_P(W)=sqrt(det_F(I_RanP-B*B)). (4) 000091 | 000092 | It is strictly positive in this graph class. Indeed PUP is I+compact on 000093 | RanP, so is Fredholm index0. If U(x,Kx) belongs to graphK perpendicular, 000094 | there is y inDomK with Wx=-Ky and WKx=y. Consequently 000095 | ==-. Positivity and injectivity force x=y=0. 000096 | The compression is injective, hence invertible. Thus ||B||<1 and (4)>0. 000097 | This proof needs no separate W(DomK) assumption: domain membership of y 000098 | is supplied by orthogonality to the closed graph. 000099 | 000100 | S2 convergence of U_N-I and strong convergence of the original P_N give 000101 | B_N->B inS2 and B_N*B_N->B*B inS1. The Fredholm determinant and its positive 000102 | root converge. For stationary h_N, strong graph convergence follows from 000103 | the bounded continuous a(h),c(h),X(h) functional calculus. 000104 | 000105 | Kernel positivity is not inferred from a positive square root alone. 000106 | At finite cutoff pi_N(W)=det(W)^-1 Gamma(diag(W,W)) is one genuine unitary 000107 | group representation. Thus chi_N(W_i*W_j) is a Gram matrix in the SAME 000108 | full-graph Slater state. Its limit is positive definite. It supplies a 000109 | canonical cyclic representation and every finite ordered paired source word. 000110 | This is a nonlinear quantum source, not Gaussian moment closure. 000111 | 000112 | This also extends to finite mixed CAR insertions, rather than an isolated 000113 | group characteristic. The finite Slater transition rule has matrix 000114 | Q_N=U_NP_N (P_NU_NP_N|RanP_N)^-1 P_N. The compressions are uniformly 000115 | invertible since (3) and ||B||<1 bound their smallest singular values. 000116 | After adjoiningI-P_N their inverses converge strongly; hence Q_N converges 000117 | strongly. Every finite mixed CAR determinant/minor converges with chi_N. 000118 | The finite exterior-algebra generating determinant proves this transition 000119 | rule without an additional state assumption. Positivity of the whole mixed 000120 | word kernel passes to the limit. The original operator ordering determines 000121 | the displayed transition convention; other placements use its adjoints. 000122 | In particular this is a representation of 000123 | the same graph CAR source, not just a formal overlap of group labels. 000124 | 000125 | If this conditional graph reader is already identified by the original 000126 | joint source, integration over its EXISTING bosonic weight preserves all 000127 | bounded unitary/CAR word limits by domination with the word norm. A uniform 000128 | bound on the entire unbounded set of H amplitudes is then unnecessary for 000129 | those bounded words. This does not construct that bosonic weight, remove 000130 | extra original vertices, or prove integrability of unbounded bosonic fields. 000131 | 000132 | 4. General normal-profile graphs and the required geometric transport 000133 | 000134 | The previous normal-dependent chart already supplies the ordered graphP_K 000135 | with its original mixed boson/CAR insertions. No normal-independence gap 000136 | is reopened. For closed densely defined injective K with dense range, 000137 | write its unique polar decomposition K=V_K|K|; V_K is unitary and 000138 | Dom|K|=DomK. The fixed unitary M=diag(I,V_K*) maps graphK to graph|K|. 000139 | Hence the transported paired operation is 000140 | 000141 | U_K(W)=diag(W,V_K W V_K*). (5) 000142 | 000143 | It is conjugate by M to diag(W,W). Formula(4), positivity and S2 continuity 000144 | therefore apply to this GIVEN whole graph, retaining all crossings. 000145 | No new state, metric, or compensating scalar is selected. The polar factor 000146 | is determined by K. This is the specified polar paired map, not a claim 000147 | that it is the only operation producing a positive number. 000148 | 000149 | Equal UNtransported W on both faces need not be the same paired source 000150 | when K is not selfadjoint. The native original two-step normal-profile test 000151 | shows a complex answer there and a positive answer after(5). This is an 000152 | operator-word distinction, not a physical no-go or new counterterm demand. 000153 | 000154 | V_K need not preserve point support. No local physical gate is inferred. 000155 | For differentiable K families, source variation must retain 000156 | d(V_KWV_K*)=V_K(dW)V_K*+[(dV_K)V_K*,V_KWV_K*]. 000157 | The differentiability/domain of that family must belong to the actual 000158 | original caller. The theorem does not erase this connection term. 000159 | For its actual cutoff limit no independent polar-convergence axiom is 000160 | required: strong graph projections give strong T_N=(P_KN)21 and adjoints, 000161 | T_N=V_N|T_N|. On denseRan|T|, V_N|T|psi->V|T|psi, hence V_N->V strongly; 000162 | unitary limits also give adjoint convergence. This preserves S2 convergence 000163 | of V_N(W_N-I)V_N* when the whole graph cutoff already converges. 000164 | 000165 | 5. What has actually been closed, and the common-source comparison 000166 | 000167 | This closes a full time-ordered PAIRED reader of the existing circle graph, 000168 | extending D from exponentials of linear insertions to nonlinear Dyson 000169 | evolution. For the larger normal-profile graph it supplies the exact polar 000170 | paired map in the stated closed-operator class. It does not identify every 000171 | original S_adm source argument with(5), nor turn the simultaneous face time 000172 | into the physical positive OS Hamiltonian. A thermal HS/GNS Liouvillian 000173 | and a positive excitation Hamiltonian are different data. 000174 | 000175 | The unpaired Dyson amplitude retains its own scalar. For C+ its raw linear 000176 | centering has second logarithmic coefficient -i sigma^2||f||2^2 logN, 000177 | sigma^2=3Tr(Yu*Yu+Yd*Yd). QUENCH-SOURCE-PROOF.txt constructs its actual 000178 | relative echo from the existing ground-zero excitation rule. Paired 000179 | positivity is not used to declare that unpaired phase or absoluteZ0 zero. 000180 | 000181 | The original whole S/Z/Gamma path remains controlling. Its normalized 000182 | paired restriction, unpaired determinant and geometric vacuum factor must 000183 | be associated with their actual field legs, reference and eta. If the 000184 | original caller uses this paired graph, the nonlinear reader is now closed; 000185 | no additional freely chosen source/phase is needed for that reader. 000186 | This theorem supplies neither a new Higgs law nor a complete primitive 000187 | metric-sector measure. It does not derive all TFPT physics fromD. 000188 | 000189 | 6. Observed controls and provenance 000190 | 000191 | check_paired_dyson.py imports originalv252, retains all36 quark/color modes, 000192 | uses both original quarters and spatially varying original H/F profiles. 000193 | The temporal pulse is C1 compact sin^2, with an independent finer time step. 000194 | All46 checks pass under -B -OO: truegraph positivity, Fredholm identity, 000195 | unitary Dyson evolution, retained crossings and actual polar normal-profile 000196 | transport. N1/2/4 are diagnostics, not a numerical continuum certificate. 000197 | An independent sixteen-state exterior-algebra calculation additionally 000198 | checks all16 two-field and256 four-field transition words on an actual 000199 | original singular component, without using the transition Wick formula 000200 | as its evaluator. All272 words agree to roundoff.