TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-local-source-variable-background-20261102/OS-VACUUM.txt Original file SHA-256: 469b77e3a452f777ad90abbc363a12eca7d84193b6acded6443c79fe83026218 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 | ORIGINAL FULL SOURCE TO ITS CANONICAL VACUUM 000002 | joint_origin, 3 November2026. J continuation; Atlas entry457bcebe946f1d5ac3b6f403, 000003 | plan90. Canonical repo and Atlas unchanged. Earlier G/H/I scopes retained. 000004 | 000005 | CENTRAL RESULT 000006 | 000007 | The vacuum vector is not an independent fit after the full normalized 000008 | source has been given. It is the class of the empty word. In the already 000009 | constructed ORIGINAL Hamiltonian class, there is now a complete direct 000010 | proof: every bounded Euclidean ground word, its entire local source and 000011 | RP kernel converge on the SAME state subnet, and their OS space is the 000012 | same ground GNS representation with Omega=[1] and H>=0. See the complete 000013 | new analytic proof in GROUND-SOURCE.txt, especially(1)-(8). Its Gibbs 000014 | version(9)-(11) uses the whole physical ground projection and retains 000015 | all original absolute factors. No new mass gap, oscillator or temperature 000016 | fit was used. 000017 | 000018 | For a FULL Euclidean covariant original source the relativistic vacuum 000019 | also follows canonically. Euclidean covariance and reflection positivity 000020 | are the actual input; an additional independently chosen Lorentz-invariant 000021 | vacuum vector is then redundant. Clustering gives vacuum uniqueness or 000022 | ergodicity, rather than constructing or selecting that vector. The first 000023 | extra premise in a proposed shortcut is identifying principal-symbol or 000024 | background-family covariance with invariance of the SAME normalized 000025 | full source at its actual fixed flat background/state. The original 000026 | bounded operator route avoids an unbounded Wightman moment requirement. 000027 | 000028 | 1. THE FULL MOMENT PARENT ALREADY EXISTS IN THE ORIGINAL PAPERS 000029 | 000030 | The original regulated functional is exp W_k[J], not just its Hessian 000031 | or connected derivatives. With all the original graded sources retained, 000032 | 000033 | q[J]=Z_rel[J]/Z_rel[0]=Z_adm[J]/Z_adm[0] 000034 | =exp(W_rel[J]-W_rel[0]), q[0]=1. (1) 000035 | M_n=delta^n q|J=0. (2) 000036 | 000037 | The source-independent Z_ref[0] cancels only in this NORMALIZED word 000038 | functional. This does not remove that factor, eta data or original 000039 | zero-source normalization from the unnormalized original S/Z caller. 000040 | Bare S_adm, the relative spectral seed and Gamma_ren remain distinct. 000041 | 000042 | The old45 QFT005:144-184 defines S_n^T=delta^n log Z and then puts that 000043 | truncated family into its OS theorem. The already source-reviewed 000044 | 25October repair supplies(1)-(2), with the full graded moment/cumulant 000045 | expansion and the correct complete reflected word form. The older42 000046 | FRG original itself already supplies exp W, so repairing this slot does 000047 | not invent a second generating functional. That repair is reused here; 000048 | the old Gaussian counterexample was not repeated. 000049 | 000050 | If the original integral is only a formal expression, (1)-(2) still 000051 | supplies its FORMAL full moments. A positive Hilbert completion requires 000052 | the actual positive full functional, not order-by-order positivity of 000053 | selected coefficients. H's concrete bounded Hamiltonian source supplies 000054 | such a full functional in its original class; G carries the original 000055 | full source and every external normalization under coordinate change. 000056 | This is not an absence claim about a new general TFPT measure. 000057 | 000058 | 2. CANONICAL OS VACUUM AND THE EXACT ROLE OF CLUSTERING 000059 | 000060 | Let S be the actual normalized full reflected-word functional. On the 000061 | correct actual half-word algebra use 000062 | (F,G)_OS=S(Theta F G), N={F:(F,F)_OS=0}. 000063 | RP gives the quotient completion H_OS. The empty word gives 000064 | 000065 | Omega=[1], ||Omega||^2=S(1)=1. (3) 000066 | 000067 | Actual Euclidean time shifts preserve1. If their induced operators are 000068 | well-defined strongly continuous SYMMETRIC contractions, then 000069 | T(t)=exp(-tH), H selfadjoint>=0, T(t)Omega=Omega, 000070 | H Omega=0. (4) 000071 | Symmetry, not the generic Hille-Yosida theorem alone, is needed for 000072 | selfadjointness. The accepted25October repair states this explicitly. 000073 | GROUND-SOURCE(8) constructs these operators directly from the existing 000074 | H_0, with the exact reflected order, adjoint and half-line convention. 000075 | 000076 | For regular full Euclidean-covariant sources, the corresponding 000077 | Poincare reconstruction has invariant Omega and translation spectrum 000078 | in the closed FORWARD light cone. A bounded-net version needs the actual 000079 | Euclidean net and regular invariant RP functional. Constructing its 000080 | local net through the standard time-zero route additionally uses a 000081 | nontrivial actual time-zero algebra in that Euclidean net. The direct 000082 | Hamiltonian ground reconstruction above already has its original 000083 | equal-time algebra and does not need this alternative theorem. Neither 000084 | route proves full Euclidean rotations for a fixed lattice merely from 000085 | the existence of equal-time operators. For unbounded Wightman fields use the corrected 000086 | OS1975 full-family growth/distribution hypothesis instead of merely 000087 | separate temperedness of every S_n. These are two applicable routes, 000088 | not cumulative gates imposed on every original observable source. 000089 | 000090 | Clustering of the SAME full reconstructed moments gives a rank-one 000091 | translation-invariant projection. On a dense word domain, 000092 | <[F],U(a)[G]> -> <[F],Omega> 000093 | for large spacelike separation implies that the translation-invariant 000094 | vectors are exactly C Omega. With the forward-cone spectrum, a zero-energy 000095 | vector is translation invariant. This supplies vacuum uniqueness. 000096 | A mass gap is a sufficient stronger route to exponential time decay 000097 | within its actual source sector; it is not needed for(3)-(4) or for the 000098 | existence of an invariant relativistic vacuum from full Euclidean data. 000099 | 000100 | 3. THE SOURCE TRANSPORT ALSO TRANSPORTS THE VACUUM 000101 | 000102 | Reuse G's ACTUAL unital whole-source operation map iota, with actual 000103 | grading, adjoints, reflection and time operations transported along it. 000104 | If 000105 | S_target(Theta_target iota(F) iota(G)) 000106 | =S_original(Theta_original F G) (5) 000107 | for the complete original word functional, then 000108 | [F] -> [iota(F)] (6) 000109 | is an isometry, onto the corresponding generated target sector. It 000110 | intertwines the actual semigroups and sends Omega_original=[1] to 000111 | Omega_target=[1]. Its Hamiltonians are related by the same unitary map. 000112 | This is the already proved complete OS transport theorem applied to 000113 | G's actual full source. No new vacuum rule follows from changing chart, 000114 | and none is required. A lossy scalar current reader alone is not(5). 000115 | 000116 | All operations in(5), including reflection and time, must be the original 000117 | transported ones. An unrelated thermal circle reflection, modular boost, 000118 | carrier clock or abstract same-dimensional matrix cannot replace them. 000119 | This preserves G's whole-source conclusion without adding a new global 000120 | P_F/CAR clock condition. 000121 | 000122 | 4. THE POSITIVE OLD GAP PROOF, ACTUALLY READ WITH ITS STATED SCOPE 000123 | 000124 | Root's two cited sources contain the SAME160-line section, the snapshot 000125 | with line offset3305. They are not two independent proofs. The actual 000126 | finite C6 transfer has spectrum {1,(2/3)^6,(1/3)^6}, with simple Perron 000127 | line and H=-log T spectrum {0,6log(3/2),6log3}. Given the actual positive 000128 | selfadjoint transfer, this produces the full chain semigroup words, 000129 | unique Omega, positive H and explicit exponential clustering. The source 000130 | itself expressly calls the result a finite gapped transfer system, 000131 | 1D constructive QFT/gapped quantum mechanics, rather than a4D theorem. 000132 | Those positive conclusions are retained. Eigenvalues by themselves would 000133 | not establish selfadjointness or the physical insertion dictionary; here 000134 | the actual given transfer and its full words carry those structures. 000135 | 000136 | The actual old v330 symmetric_transfer constructor was also read1-136 000137 | through source:f15dfc8c547c01da89f1, SHA256 000138 | 21cbeb2b96d9ee581da3a9c45be400d77d232730ba37bbb19cb4cec07fb5e5be. 000139 | It supplies T=v1v1*+lambda2 u2u2*+lambda3 u3u3* with the explicit 000140 | orthonormal vectors and v1=(1,1,1)/sqrt3. Thus T is symmetric, positive, 000141 | entrywise positive and T1=1. The actual cylinder probabilities 000142 | mu(i0,...,in)=(1/3) PRODUCT_j T_(i_j,i_(j+1)) 000143 | are positive and EXACTLY consistent by summing the last row. Their 000144 | stationary path measure exists directly; no susceptibility-to-tightness 000145 | inference is required for this actual finite alphabet. This is a source 000146 | read and analytic consequence of the original T, not a new simulation. 000147 | 000148 | Source45 QFT004:111-353 was also reread in full. It proves finite 000149 | primitivity, a positive finite gap, and a Cauchy thermodynamic state 000150 | conditional on the stated uniform Perron contraction in its boundary 000151 | decoupling lemma. Its later Burnside step proves the full finite matrix 000152 | algebra; the parent-gap inference applies when these matrices belong 000153 | to the SAME fixed actual marked MPS tensor and canonical local parent. 000154 | That fixed-tensor positive theorem remains valid. The existing R075 000155 | variable-tensor counterfamily excludes only inferring uniformity from 000156 | separate volume-wise injectivity; it was not rerun or misapplied. 000157 | 000158 | QFT005:1-122 has a valid parent-domination implication when each added 000159 | positive term annihilates the same Perron line, and uses the preceding 000160 | uniform gap and boundary-decoupling assumptions for the thermodynamic 000161 | cluster conclusion. v1013 later supplies the actual common Hamiltonian 000162 | family and ground-state existence independently of that fixed-MPS gap 000163 | claim. Its own boundary does not assert phase uniqueness or a uniform 000164 | thermodynamic gap. J's direct ground source therefore uses this stronger 000165 | available existence route without requiring that separate gap proof. 000166 | 000167 | The finite C6 proof cannot by its stated scope discharge full spatial 000168 | Euclidean rotations of all coupled4D source insertions. This does not 000169 | undo any of its genuine full chain-word, gap or clustering conclusions. 000170 | 000171 | 5. QFT008 PREDECESSOR CHAIN AND ITS PRECISE APPLICATION 000172 | 000173 | QFT008:179-254 was read in full together with its preceding RP argument, 000174 | QFT005:1-184, QFT004:1-381 and the decisive geometric/primitive originals. 000175 | 000176 | 179-196: separate truncated temperedness invokes finite-cutoff elliptic 000177 | selfadjointness plus the constructive geometric measure. The geometric 000178 | original Metrology005:137-178 explicitly calls its blocked D_Vn and 000179 | nu_Vn compatible under exhaustion. The previous R080/H exact effective 000180 | blocking and full bounded source results are retained; no new claim that 000181 | the bare spectral ansatz must survive marginalization is introduced. 000182 | These elliptic and block premises alone are not the corrected OS1975 000183 | full-family Wightman growth condition. For the actual bounded operator 000184 | source, GROUND-SOURCE(4)-(7) instead provides the needed full-word norm 000185 | control and strong time continuity constructively. 000186 | 000187 | 199-213: graded symmetry follows from the original graded source 000188 | derivatives. Its covariance proof refers to the Dirac principal symbol 000189 | and low-curvature reduction on U_scat. QFT005:126-142 defines that 000190 | region by ||g-eta||C2 <<1 and small background gradients. Approximate 000191 | flatness and common principal symbol alone do not prove EXACT Euclidean 000192 | invariance of the fixed full quantum source. The actual full source 000193 | Ward/covariance identity must be used when invoking exact relativistic 000194 | OS reconstruction. This is precisely I's existing source hypothesis, 000195 | not a new requirement to reconstruct the source once again. 000196 | 000197 | 215-254: tailored OS/CAR theorem has the correct conditional architecture: 000198 | full RP source plus regular covariance/symmetry and clustering reconstructs 000199 | its admissible Hilbert theory. Use full moments(2), a proper symmetric 000200 | semigroup as in(4), and an applicable regularity theorem. With these